# On the economics of postassessments in insurance guaranty funds: a stakeholders' perspective.

ABSTRACT

This article proposes a model that suggests there are contagion effects among members of an insurance guaranty fund when postassessments are charged to all other insurers upon the failure of a member company. Indeed, these extraordinary payments are shown to increase the default rate of other firms in the industry, ultimately lowering the value of corporate claims as well as government tax claims. The model is also used to examine the efficiency of different recoupment mechanisms (both existing and new) used by regulators and insurers to potentially reduce these contagion effects. Analysis allows us to stipulate the conditions under which a "tax carryforward" provision could be more efficient than the usual recoupment mechanisms known as "premium rate surcharge" and "premium tax credit."

INTRODUCTION

In most countries, it is common to find a guaranty fund (GF) system that covers part of the claims obligations of insolvent insurers. For example, in the United States, each state has two guaranty associations, one for life and the other for property-liability insurance. Each GF is a mandatory association of all licensed insurers doing business in that state (Klein and Wang, 2009).

This GF system is interdependent and represents a cooperative effort among regulators and insurers in the states where the insolvent insurer operated. If the home-state insurance commissioner considers that the financially troubled insurer is not likely to be able to solve its problems, then he can petition a state court for an order of liquidation. Insurance regulators in the other states where the insurer operated will do the same. The order of liquidation in each state generally triggers the operation of that state's GF. Once the insolvent's liabilities and assets are valued and a shortfall estimated, then postassessments are requested from each participating insurer in proportion to their premium volume in that state. In property-liability insurance, assessments are usually first imposed on the particular coverage that occasioned the insolvency and are limited to 1 percent to 2 percent annually of each insurer's premium volume in that line in that state. If these assessments prove insufficient, then other property-liability lines may be assessed as well (Cobb, 2003).

The existence of GFs has been shown to potentially affect the risk-taking incentives of insurance firms. Empirical evidence seems to support the claim of greater risk-taking on the part of insurers following the adoption of GFs in the United States (Lee, Mayers, and Smith, 1997; Lee and Smith, 1999). This kind of moral hazard, similar to the one in banking due to deposit insurance, has led researchers to suggest risk-based rather than flat assessment of premiums for GFs as well (Cummins, 1988). More recently, Ligon and Thistle (2007) have also shown that non-risk-based GF premiums do create an economic incentive for the fleet form of organization, which is quite important in the U.S. property-liability industry.

The design of GFs has also been analyzed from an agency cost perspective (Han, Lai, and Witt, 1997). Such an analysis allows for either ex ante or ex post assessment of GF premiums. "Ex ante premiums give shareholders less incentive for risk-taking behavior than the actual ex post assessment scheme observed most frequently in practice. In the case of the latter, only insurers who survive pay the cost of the scheme, so the share of the cost paid by safer insurers is more highly disproportionate than in an ex ante assessment" (Ligon and Thistle, 2007, p. 858).

In practice however, "most states allow assessed insurers to include the assessments in their rates, though some states allow insurers to surcharge policyholders or offset the assessments against their premium taxes" (Cobb, 2003, p. 4). Hence, through premium rate surcharges (PRSs) and/or premium tax offsets or credits (PTCs), surviving insurers can expect to recoup most their assessment costs. (1)

However, this may lead to a timing issue in that insurers may be compelled to stretch out their recoupments over a number of years. For example, in the case of a PTC provision, it is very common for states to allow insurers to offset 20 percent of their assessment from their premium taxes each year for a maximum of 5 years. Hence, such staggering of premium tax offsets (or credits) does not achieve the sought objective of balance sheet neutrality for surviving insurers. Likewise, recouping GF assessments through rate surcharges contains significant lags, perhaps as long as 12 months if they can be implemented immediately and possibly much more if the assessment base is in a rate regulated line that requires supervisory approval.

Ligon and Thistle (2007) argue that these recoupment mechanisms may further exacerbate the incentives for risk-increasing behavior on the part of insurers. For regulators, these mechanisms are deployed to mitigate the strain potentially imposed by postassessments on surviving insurers following the liquidation of a member company. In their view, such recoupment mechanisms may have a role to play in preventing contagion effects due to postassessments that might weaken other insurers. (2) Said differently, whether or not these mechanisms create ex ante incentives for risk taking is actually viewed as less critical compared to the desired ex post resolution of financial distress issues they are supposed to achieve.

This is what we hope to investigate in our article that is organized as follows. First, we develop a realistic cash flow model of financial distress of a property-liability insurance company in which the insurer's financial capacity is modeled respecting the specificities of the industry in terms of main sources of both revenues and risks. Then, we rely upon our basic model in order to investigate whether or not the presence of a GF charging postassessments to other insurers following the failure of a member company can be a source of contagion in the industry. Here, we define contagion as being a deterioration of the financial situation of surviving insurers as indicated by their level of leverage that impacts on their probability of default. We show that this type of contagion is indeed possible in such a GF system. Then, by extending our model, we also examine the optimality of the different recoupment mechanisms (both existing and new) designed to reduce these contagion effects from the point of view of different stakeholders, namely, shareholders, policyholders, and the tax authority. Our basic model and its many variants are implemented through Monte Carlo simulation analyses. Hence, we are also able to examine and compare the efficiency of the existing recoupment mechanisms (PTCs and PRSs) versus a newly proposed provision calling instead for a tax carryforward in order to deal with contagion effects in such a context. In the final section, we summarize our findings and we conclude.

THE BASIC MODEL OF THE INSURANCE FIRM

We describe here a model of the business and financial risks of a property-liability insurance firm. We consider an economic framework where the time line is divided into equally distanced dates [t.sub.0], [t.sub.1], [t.sub.2], ..., T. For instance, the length of each time increment [DELTA]t = ([t.sub.i+1] - [t.sub.i]) could be viewed as a financial quarter.

Insurance Activity and Net Underwriting Income

The business activity of the insurance firm consists of selling insurance protection (insurance policies) to customers against predetermined sources of pure risk. This activity permits the firm to generate a periodic underwriting revenue but also implies the payment of a periodic and random amount of claims as well as operational costs.

We assume that the insurance firm holds a portfolio of a fixed number N of business lines/clients over time, which could viewed as a portfolio of the same N lines the company serves as an insurer. The assumption of flat volume of insurance activity is not required for our model, but it permits us to avoid making supplementary assumptions about business or market share growth not relevant for our purpose.

On one side, the portfolio of N insurance lines allows the company to earn a time-periodic amount of underwriting revenue, [P.sub.t] (t = [t.sub.0], [t.sub.1], ...), representing the periodic amount charged to clients against the supplied insurance. On the other side, the company suffers a time-periodic aggregate loss due to claims payments, which we denote by [L.sub.t]. Without loss of generality, we assume that each business line or client i = 1, ..., N pays a periodic insurance premium pit over time, but may at any time trigger a claim payment [l.sub.it]. We assume that the number of claims payments made over a given time period for each particular client or line i is unconstrained (not capped) as long as the client keeps on paying the same periodic amount of insurance premium [p.sub.it]. This assumption could easily be justified under our time-unchanging mix of clients/lines, since the insurance firm will be fairly remunerated over the whole portfolio independent of individual claims records. Under this simple model of a stationary portfolio of insurance policies, the net underwriting income is given by

[NUI.sub.t] = [P.sub.t] - [L.sub.t], (1)

[P.sub.t] = [N.summation over (i=1)] [p.sub.it], [L.sub.t] = [N.summation over (i=1)] [l.sub.it]. (2)

Now, we describe the risk model of aggregate loss [L.sub.t]. We assume that the claim payment [l.sub.it] due to the business line/client i (i = 1, ..., N) is randomly distributed as follows:

[l.sub.it] = [q.sub.it][s.sub.it] for any t > [t.sub.0], (3)

where [s.sub.it] is the time t claim size and [q.sub.it] is the time t claim intensity or frequency. Both claim size and claim frequency are randomly distributed and mutually independent. Namely, we assume a homogenous and time-stationary portfolio of claims, so that at each time t > [t.sub.0] and for any i = 1, ..., N, we have

Pr[[q.sub.it] = 1] = [lambda], Pr[[q.sub.it] = 0] = 1 - [lambda] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

with

[([z.sub.it]).sub.i=1, ..., N] ~ N([0.bar], [summation]), (6)

[[summation].sub.ij] = E[[z.sub.it][z.sub.jt]] = [rho][[sigma].sup.2], (7)

[[summation].sub.ii] = E[[z.sup.2.sub.it]] = [[sigma].sup.2]. (8)

This means that claim sizes are correlated and exhibit a shifted joint lognormal distribution generated from the shift parameter [??] and the time-constant covariance matrix [SIGMA] of the correlated Gaussian shocks [([z.sub.it]).sub.i=1, ..., N]. Relaxing the assumption of zero serial-correlation between realizations of claim size is easy to do, but we decided to keep the model as simple as described earlier. It then follows that line i's expected claim size is constant over time,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

We introduce here two random variables describing the distribution of the aggregate loss [L.sub.t]. First, let [n.sub.t] ([n.sub.t] [member of] {0,1, 2, ..., N}) be the random frequency of claims experienced at any time instant t over the whole portfolio of business lines. According to the individual claims severities [([q.sub.it]).sub.i=1, ..., N], we have

Q(k) [equivalent to] Pr[[n.sub.t] = k] = [N!/k![(N - k).sup.!]][[lambda].sup.k][(1 - [lambda]).sup.N-k] with k = 0, 1, 2, ..., N. (10)

Then, let us define [S.sub.t] as the time t random sum,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

representing the aggregate claim severity conditional on the claims frequency [n.sub.t]. Therefore, the probability density function (p.d.f.) of the aggregate loss [L.sub.t] reads as follows:

Pr[[L.sub.t] [member of] da] = [N.summation over (k=0)] Q(k)Pr[[S.sub.t] [member of] da | [n.sub.t] = k], for any t > [t.sub.0]. (12)

Since claim sizes are correlated lognormals, we cannot derive an analytic formula for the conditional p.d.f. Pr[[S.sub.t] [member of] da | [n.sub.t] = k] for any arbitrarily high integer k. To obtain the numerical values of both the mean and the variance of the aggregate loss, which we make use in our subsequent analysis, we employ the Fast Fourier Transforms method.

Now, assuming that insurance firms charge risk-adjusted premiums to policyholders, it follows that the total periodic product of collected premiums (i.e., the underwriting revenue) is given by the expected aggregate loss [bar.L] loaded by a proportional margin rate m > 0; that is, for any t [greater than or equal to] [t.sub.0],

[P.sub.t] = (1 + m)[bar.L], (13)

which in turn implies that

[NUI.sub.t] = (1 + m)[bar.L] - [L.sub.t]. (14)

We can expect that the insurance firm will charge the price of aggregate loss risk (the loss variance, which already includes claims correlation risk, could be a good estimator of this risk) through the markup m. More precisely, we shall assume that,

(1 + m)[bar.L] = [bar.L] + [eta][[sigma].sup.2.sub.L], (15)

or,

m = [eta] [[sigma].sup.2.sub.L]/[bar.L] (16)

where [eta] > 0 is a constant parameter.

Real Assets and Franchise Value

Real assets are illiquid assets held by the insurance firm other than pure cash reserves and investment assets. These assets are essentially needed to sustain underwriting activity and they are valued at their opportunity cost, so that their acquisition value over the secondary market equalizes the discounted value of the future rents they allow to be earned. In this article, by real assets we refer to what is known as "economic rents" in the economic literature and the "franchise value" in the insurance literature. Babbel and Merrill (1999, pp. 245-278) define this component of the insurance firm as follows. It is the present value of these rents that an insurer is expected to garner because it has scarce resources, scarce capital, charter value, licenses, a distribution network, personnel, reputation, and so forth. It includes renewal business. In order to use a term that contrasts well with pure cash reserves and marketable securities, we shall adopt in this article the term "real assets" to refer to this present value of economic rents known as the franchise value.

More precisely, we assume that the real assets or franchise value of the firm (the economic value of an equivalent unlevered firm) satisfies

[A.sub.t] = [E.sub.t] [[T.summation over (u=t+1)] [e.sup.-(u-t)[phi]][NUI.sub.u][DELTA]t], [t.sub.0] [less than or equal to] t [less than or equal to] T-1, (17)

= [E.sub.t][[T.summation over (u=t+1)] [e.sup.-(u-t)[phi]][(1 + m)[bar.L] - [L.sub.u]][DELTA]t, (18)

= m[bar.L][DELTA]t [[T.summation over (u=t+1)] [e.sup.-(u-t)[phi]], (19)

where [E.sub.t][.] is the time t risk-neutral expectation conditional on the information set [F.sub.t], while the quantity [e.sup.-(u-t)[phi]] captures a time-increasing opportunity cost (with a constant speed of [phi]) of business assets in place over the secondary market. Note that there exists a constant [bar.[phi]] reflecting the average return required by insurers on investments in franchise value such that the expression of real assets reduces to

[A.sub.t] = m[bar.L]/[bar.[phi]]. (20)

The process by which the value of the firm's real assets is determined over the secondary market is important for our purpose, since it is a determinant of the cash flows that will go to the firm's claimants upon bankruptcy, and hence, it determines the loss-given failure as well as bankruptcy costs.

Investment Activity

The main economic function of insurance firms is risk reduction through pooling (diversification) of policies, which allows them to cover the insured risk, while being able to generate economic surplus thanks to the investment of underwriting revenue into fixed-income-generating assets. Consistent with this asset-risk transformation function, the insurance firm in our model holds an investment portfolio essentially composed of fixed income securities. For simplicity, we assume that the instantaneous rate of return realized by this investment portfolio could be replicated by a linear combination of the rate of return of the instantaneous short rate [r.sub.t] (the interest rate earned on T-bills) and a discount bond with a relatively long maturity [T.sub.r]. This assumption reflects the fact that return realized over fixed-income portfolios are essentially explained by both interest income flows and gains/losses due to positions rebalancing. Let [R.sub.t] be the instantaneous total rate of return of the investment portfolio and [alpha] (0, 1) be the fraction replicating the weight of gains/losses realized from bond trading in this total return, we have

[R.sub.t]dt = [alpha] [d B(t, [T.sub.r], [r.sub.t])/B(t, [T.sub.r], [r.sub.t])] + (1 - [alpha])[r.sub.t]dt, (21)

where B(t, u, [r.sub.t]) represents the time t market value of a discount bond paying one unit of face value at time u, given [r.sub.t]. Since the investment portfolio is composed of very liquid assets and marketable securities, it is viewed as the cash reserves of the insurance firm. As we will see later, these cash reserves play a key role in our model, since the default risk of insurance firms is tightly dependent upon their repayment capacity captured through the amount of available cash reserves.

Defining [v.sub.t] as the market value of the investment portfolio, which is equal to the amount of cash reserves, the net interest income generated by the investment activity over the time interval [t - [DELTA]t, t] will be given by

[NII.sub.t] = [[integral].sup.t.sub.t-[DELTA]t] [R.sub.s][v.sub.s]ds. (22)

We suppose here that the dynamics of the term structure are described by the simple mean-reverting short rate model of Vasicek (1977),

[dr.sub.t] = [[theta].sub.r]([bar.r] - [r.sub.t])dt + [[sigma].sub.r]d[W.sup.r.sub.t], (23)

where [bar.r] is the equilibrium interest rate level, [[sigma].sub.r] is the volatility of the interest rate, [[theta].sub.r] is the mean reversion speed of the short rate process, and ([W.sup.y.sub.t]) is a standard Brownian motion. Note that the market value of the discount bond B(t, [T.sub.r], [r.sub.t]) is given by the well-known Vasicek formula,

B(t, [T.sub.r], [r.sub.t]) = exp([[phi].sub.1](t, [T.sub.r]) - [r.sub.t][[phi].sub.2](t, [T.sub.r])),

where [[phi].sub.1](.) and [[phi].sub.2](.) are defined deterministic functions.

Financial Leverage, EBIT, and Net Income

To finance her business activity, the insurance firm periodically issues debt securities over the bond market. At each time, the firm also repays a fraction of principal on its outstanding debt. In particular, as in Leland and Toft (1996), we consider a stationary debt structure where the firm, as long as it remains solvent, periodically issues a constant principal amount b of debt with a given term [T.sub.b] and simultaneously repays the same amount on the previously issued debt reaching their maturity. Hence, at any time t, the maturities of total principal amount B of outstanding debt are uniformly distributed over the interval (t, t + [T.sub.b]), where B = b[T.sub.b]. Each single debt pays a periodic coupon flow of (C P x [DELTA]t)/[T.sub.b], which means that the total interest fees the firm pays on outstanding debt is C P = (wac x B) per year or C P x [DELTA]t per each time period of length [DELTA]t, where wa c is the weighted average coupon rate. Under this model setup, the earnings before interests and taxes, EBIT, realized over the time interval [t - [DELTA]t, t] are given by

[EBIT.sub.t] = [NUI.sub.t] + [NII.sub.t], (24)

while the net income of the insurance firm will be given by

[[pi].sub.t] = [EBIT.sub.t] - C P - [g.sub.t], (25)

where

[g.sub.t] = [[tau].sub.I] max[0, [EBIT.sub.t] - C P] (26)

is the amount of income taxes paid out to the government given a marginal corporate income tax rate of [[tau].sub.I].

Cash Reserves Dynamics and Bankruptcy

The cash reserves represent what the firm holds in the form of pure cash and very liquid assets. These reserves constitute the repayment capacity of the firm and thus are fundamental for its solvency. These cash reserves evolve over time as follows:

[v.sub.t] = ([v.sub.t-[DELTA]t] + [[pi].sub.t][[DELTA].sub.t])(1 - [delta]), [t.sub.1] [less than or equal to] t [less than or equal to] T-1, (27)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [delta] is the dividend payout rate. Notice here that because we are dealing with a stationary debt structure, new financial debt issues offset the repayment of maturing debt claims, so that there is no impact on the outstanding cash reserves.

We assume that the payout policy of the firm consists of paying shareholders a payout rate [delta], such that the dividend payment at time t is given by

[div.sub.t] = [delta]([v.sub.t-[DELTA]t] + [[pi].sub.t][[DELTA].sub.t]).

This means that when net income goes negative, both the debt repayment and dividends will not necessarily be cut, thanks to the cumulated cash reserves. This captures in a realistic way the well-observed facts that: (1) firms facing business losses do not enter into financial distress immediately, and (2) dividends are not strongly correlated with profits realized over the same periods. Under this realistic model, the firm enters into bankruptcy once her cash reserves falls below zero. The random bankruptcy time is then defined as follows:

[theta] = inf{t [member of] ([t.sub.1], ..., T-1): [v.sub.t] < 0}. (28)

When the firm is still solvent by time [t.sub.i] (i.e., {[v.sub.u] > 0, u = [t.sub.1, ..., [t.sub.i]}), we denote this event by {[theta] > [t.sub.i]}. However, the event {[theta] [less than or equal to] [t.sub.i]} means that bankruptcy has occurred at any prior time t [member of] ([t.sub.1], ..., [t.sub.i]). Finally, to designate the event that the firm has not defaulted by time (T-1) (i.e., {[v.sub.u] > 0, u = [t.sub.1], ..., T-1}) and that [v.sub.T] > 0, we shall use the notation {[theta] > T}.

At time T, economic surplus vanishes, business activity ceases and the firm's real assets are worth nothing. At that time, the firm (if default has not occurred in the past) is supposed to be liquidated. The liquidation process takes place as follows: the firm stops issuing new debt, and shareholders gets the residual claim:

max[0; [v.sub.T] - B],

with

[v.sub.T] = [v.sub.T-1] + [[pi].sub.T][DELTA]t. (29)

EQUITY AND GOVERNMENT CLAIMS

Under the developed model aforementioned, the value of equity claim at time to is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

where D(t, u) denotes the stochastic discount factor used to set the price at time t, given [r.sub.t], of a riskless security paying one dollar at time u > t.

The first term represents the present value of expected dividend payoffs, while the firm is still solvent. The second term constitutes the present value of the expected residual payoff, after the payment of outstanding debt, which will go to shareholders upon bankruptcy. Once the firm becomes bankrupt, cash reserves are zero, and shareholders receive nothing in the form of cash. They receive only the residual liquidation value of assets, net of bankruptcy costs evaluated at [omega][A.sub.[theta]] (with 0 [less than or equal to] [omega] < 1), after repayment of outstanding debt. The last term is the present value of the expected residual claim of shareholders upon the liquidation of the firm at date T.

On the other side, the government claim is given by the expected present value of both income tax and premium tax flows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

where [[tau].sub.P][P.sub.t] is the amount of premium tax flows to be received at time t by the tax authority over the realized sales of insurance policies, computed based on a premium tax rate of [[tau].sub.P] (generally, we have [[tau].sub.P] < [[tau].sub.I]). Of course, in contrast to income taxes paid out by the firm on realized profits, premium taxes are sort of built-in fees charged over the insurance policy premium, and they are supported by policyholders. Summarizing, the total tax payoff could be viewed as the sum of an asymmetric payoff representing the income taxes, considered a call option on the firm's EBIT, and a linear premium tax flow.

We see that the government receives taxes while the firm remains solvent, since once the firm becomes bankrupt the tax claim vanishes. It is important to note that contrary to shareholders who will not receive dividends at bankruptcy, the government may receive taxes at bankruptcy time [theta], and this because [[tau].sub.P][P.sub.[theta]] > 0 and [g.sub.[theta]] = [[tau].sub.I] max [0, [EBIT.sub.[theta]] - CP] is not always zero. Recall that the firm is assumed to fail once her cash reserves after-tax repayment are below zero, such that she is no longer able to meet her financial obligations. This explains why we have used the event {[theta] [greater than or equal to] [t.sub.i]} rather than {[theta] > [t.sub.i]}, as in the expression of equity claim, to indicate that taxes at bankruptcy are not necessarily zero.

Observe that the government claim could be decomposed into two subcomponents: an income tax claim, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and a premium tax claim, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]P, as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Before moving on, one can see that the values of both equity and government claims (as well as the economic value of the firm's assets) depends on the length of the time horizon, T. As the concept of duration reflects a time-based measure of interest rate risk in the context of fixed-income securities, the variable T in the model reflects a kind of time-based value of the attractiveness of the insurance industry in terms of future economic surplus.

Under this setup, at any time t, the economic (market) value of the firm, [V.sub.t], is given by the sum of equity, debt and government claims at time t, to which we add the time t present value of the fraction [omega] of the firm's assets lost at default, that is, bankruptcy costs. Note that because taxes are explicitly treated within the model, the economic value of the tax advantage of debt is already included in the value of equity claim. Recalling the definition of the economic value of the firm's real assets and based on the expressions of both equity and government claims, one can easily deduce that the economic (market) value of the firm will be simply given by the sum of cash reserves, investment portfolio, and the economic value of real assets. That is,

[V.sub.t] = [v.sub.t] + [A.sub.t], [t.sub.0] [less than or equal to] t [less than or equal to] T. (35)

MONTE CARLO IMPLEMENTATION OF THE MODEL

The model is implemented numerically using the Monte Carlo method. First, for each time date t = [t.sub.1], [t.sub.2], ..., T, we generate a random sample of M realizations of the sequence of correlated Gaussian variables [([z.sub.i]).sub.i=1, ..., N]. The T time scenarios of these Gaussian samples transformed into the samples of shifted-lognormal claims sizes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], each represents a Monte Carlo path, so we are able to project M paths of aggregate loss [L.sub.t] over the time horizon. Then, at each time date t, the aggregate loss [L.sub.t] is generated using the conditional p.d.f. Pr[[S.sub.t] [member of] da | [n.sub.t] = [bar.n]]. More formally, we proceed as follows. We generate at each time date t a sample of M realizations of the discrete variable [n.sub.t] representing M paths of the claims frequency, using a simple transformation of uniformly distributed variable and the discrete distribution Q(.). Thereafter, for each time t realization of the variable [n.sub.t] corresponding to one realized path, we recover the associated random sum St capturing the claim severity conditional on the realized value of [n.sub.t]. Thus, the M aggregate loss realizations are constructed from the M realizations of the claim frequency n and the sample of N claim size. The uniformity of the random generation of the discrete variable [n.sub.t], as well as the Gaussian numbers [([z.sub.i]).sub.i=1, ..., N] serving to project the correlated lognormal claim size, is enhanced using the stratified sampling technique. The moments matching technique is similarly employed to simulate the correlated Gaussian numbers [(z.sub.i]).sub.i=l, ..., N].

The interest rate process is simulated using the exact transition density technique. Simulation is performed at a very small time increment of length equal to 1/120, which ensures the convergence of our Monte Carlo paths to the true diffusion equation of the short rate process. The same variance reduction techniques of stratified sampling and moments matching are employed to simulate the transition of the short rate process. The stochastic discount facto paths are also constructed from these quasi-continuous paths of the short rate. Afterward, the paths of short rates, stochastic discount factors, and bond prices B(t, [T.sub.r], [r.sub.t]) are determined analytically from the Vasicek's bond price formula and recovered at the discrete time dates t = [t.sub.1], [t.sub.2], ...,T from the originally simulated quasi-continuous paths.

INTRODUCING GF ASSESSMENTS AND RECOUPMENT MECHANISMS

We extend here the basic model developed above in order to capture the financial risk borne by property-liability insurance companies that are members of a GF when failure events occur. We consider the same insurance firm as described earlier. But now the occurrence of a failure event at time [t.sub.0] makes the firm, as well as the other member companies of the GF, engaged in a sort of a loss-recovery program. The program implies that the firm will pay to the GF a periodic amount of h[DELTA]t during the time calendar [t.sub.1], [t.sub.2] ... [T.sub.h] < T. These funds collected by the GF will serve to meet the obligations of the failed company toward its clients. In return, the GF will be treated as a claimant of the failed company and will receive a fraction of the liquidation proceeds. This revenue from liquidation will be returned by the GF to its members in the form of dividends. We assume that this amount H of dividends will be received by our firm at date ([T.sub.h] + 1) and is equal to a fraction (1 - [psi]) [member of] (0, 1) of the sum of amounts h paid previously, that is, H = (1 - [psi])h[DELTA][T.sub.h]. The introduction of the loss fraction [psi] is justified by the fact that the contribution to the GF represents a very risky investment made by member firms to prevent systematic reputation risk generated by the failure events of insurance firms.

How the amount h is fixed for each company as well as the time schedule of the loss-recovery program, ([t.sub.1], [t.sub.2], ..., [T.sub.h]), is based on criteria resulting from an agreement between the GF's affiliated members. Later, we discuss this point in more detail when we describe the calibration of the model.

The main consequence of the occurrence of such failure event is that the financial strength of the firm will be impacted by the payment of the extraordinary obligation h, which makes the firm's cash reserves shifting below its normal level during time period [[t.sub.1], [T.sub.h]] as follows:

[v.sub.t] = ([v.sub.t-[DELTA]t] + [DELTA][v.sub.t])(1 - [delta]), [t.sub.1] [less than or equal to] t [less than or equal to] T-1, (36)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

Under this setup, the occurrence of a failure event can be viewed as a financial risk factor that suddenly pushes up the firm's leverage and thus increases its own failure risk in return. The implications of this "contagion effect" are that the values of both equity and government claims will decrease in response to the higher failure risk, making the probability distribution of cash reserves more skewed. Note that the equity and government claims' valuation formulas derived earlier remain the same under this basic model of contagion risk.

Considering the PTC Recoupment Mechanism

One of the most commonly used recoupment mechanism aiming to attenuating the potential contagion effects of GF assessments seen earlier is the PTC. Under the PTC, firms exposed upon failure events to GF assessments are allowed by the tax authority to keep during the period [[t.sub.1], [T.sub.h]] of GF assessments a fraction [phi] [member of] (0, 1) of premium tax flows collected from policyholders. This amount of tax flows is directly injected into the cash reserves of the firm, which contributes to enhance the solvency profile of the insurer. Formally, under the PTC, the incremental change of cash reserves of the insurer is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

The retained amount of premium taxes is considered as a government subsidy for insurers, in the sense that it is not refundable once the regime of GF assessments is over. The expression of equity claim under the PTC is the same than under the basic model, except that cash reserves dynamics are, first, perturbed by GF assessments and, second, smoothed by the premium tax subsidy. In contrast, the expression of the government claim under the PTC changes to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

This means that during the period [[t.sub.1], [T.sub.h]], the tax authority is supposed to be receiving the income tax in full but only a fraction of premium taxes collected from policyholders. Once the regime of GF assessments ends, tax revenues are established back to their regular level. We see that the PTC has no direct incidence on the expression of equity claim, but it is assumed to affect positively the value of the equity claim by enhancing the solvency probability of the firm through an immediate increase of available cash reserves. For the government, however, allowing for the PTC does not lead to an obvious expected effect on the value of the tax claim. On the one hand, foregoing the fraction of premium tax flows without any subsequent complete or partial refunding from the insurer is a pure loss for the government. On the other hand, allowing for this tax credit will improve the chances of the firm to survive over the stressing environment of GF assessments and, hence, increase the expected value of tax revenues to be generated by the firm beyond time [T.sub.h]. In sum, these two competing effects when regrouped together make the tax authority facing a nontrivial trade-off. We shall examine later the determinants of the solution of this trade-off.

Remark also that the decomposition of the government claim into income tax claim and a premium tax claim is still valid here, with the exception that the premium tax claim is cut by the rate [phi] of tax credit during the program period [[t.sub.1], [T.sub.h]].

Considering the PRS Recoupment Mechanism

Another alternative to the PTC is the PRS provision. Under the PRS, insurers are allowed after supervisory approval to increase premiums charged to policyholders during the period [[t.sub.1], [T.sub.h]] in order to cope with the cash flow pressure caused by GF assessments. Note that the approval of supervisory authorities is only required in the case of rate regulated lines. In contrast to the PTC, the PRS does not imply any transfer of tax revenues from the government to the insurance firms. Rather, the incremental revenues generated by the surcharged premiums are supposed to maintain the cash reserves of the firm away from the bankruptcy floor. Under our model, the PRS is modeled by shifting the net underwriting income upward during the period [[t.sub.1], [T.sub.h]] through some surcharge rate [zeta] > 0 as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

One can easily note that under the PRS, the valuation expressions of both equity and government claims as well as the equation describing the dynamics of the firm's cash reserves remain the same than under the basic model of GF assessments. This results from the fact that there is no transfer of tax revenues from the tax authority to the insurer. As we will discuss later, this property constitutes a main particularity of the PRS in comparison to tax-based mitigating mechanisms.

Considering the Tax Carryforward as a New Recoupment Mechanism

Here, we introduce a new mitigation mechanism and study its properties. To the best of our knowledge, this provision has not been used before in the insurance industry. As we will see later in the article when we implement the model, this new mechanism is designed to yield different and attractive features that are not generally obtained by the usual recoupment provisions described earlier. We call it a tax carryforward provision (TCF), and it is an income tax-based mechanism operating as follows. During the period [[t.sub.1], [T.sub.h]] of GF assessments, the tax authority will receive a fraction [beta] [member of] (0, 1) of the regular income tax and then recuperate the amount,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

at time ([T.sub.h] + 1). This will permit the company to reduce its failure risk caused by the GF extraordinary obligation. Of course, as a consequence of adverse business conditions and/or inadequately aggressive leverage or payout policies, the company may fail before the repayment at date ([T.sub.h] + 1) of residual income taxes. This scenario makes the TCF risky for the government. But, on the other hand, business may turn bad for some periods during the time calendar [[t.sub.1], [T.sub.h]] due to rather normal economic shocks. Over those time periods, the firm will face business losses (i.e., EBIT < 0), and the regular income tax payoff g that would have been paid out to the government in the absence of the TCF is zero.

As a consequence of the TCF, the value of the government claim at time to will be given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (41)

where for any [t.sub.] [less than or equal to] t [less than or equal to] [T.sub.h],

[[pi].sub.t] = [[EBIT.sub.t] - C P] - [BETA][g.sub.t], (42)

[DELTA][v.sub.t] = ([[pi].sub.t] - h)[DELTA]t, (43)

[v.sub.t] = ([v.sub.t-[DELTA]t] + [DELTA][v.sub.t])(1 - [delta]), (44)

with

[[bar.X].sub.t] = [t-1.summation over ([t.sub.i]=[t.sub.1])](1 - [beta])[g.sub.t], [DELTA]t, for [t.sub.1] < t [less than or equal to] [T.sub.h] + 1, (45)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (46)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (47)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (48)

For any [T.sub.h] + 2 [less than or equal to] t [less than or equal to] T, the expressions of the net income [[pi].sub.t] and cash reserves [v.sub.t] are the same as under the basic model. The first term in the expression of the government claim represents the present value of the expected income tax payoffs the government will receive under the TCF during the time period [[t.sub.1], [T.sub.h]] while the firm remains solvent. The second term captures the present value of the expected payoff the government receives once the firm becomes bankrupt at any time during the tax carryforward period [[t.sub.1], [T.sub.h]]. As we can see, once the firm becomes bankrupt, the government will get the full amount of income taxes (without TCF) due on the current EBIT, that is, [g.sub.[theta]], plus the minimum between the liquidation value of the firm's real assets and the amount of residual income taxes not paid by the firm during previous periods owing to the TCF. Since the amount of taxes due on the generated EBIT is collected in full at time of bankruptcy, we have used the event {[theta] > [t.sub.i]} rather than {[theta] [greater than or equal to] [t.sub.i]} in the first term in order to avoid double counting of the same amount of income taxes. The third term can be defined in the same way as the second term. The exception is that the payoff [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represents the terminal payoff the government will receive, in priority over bondholders, when the TCF ends. This payoff is scheduled even if the firm has not defaulted at time ([T.sub.h] + 1). Indeed, as we can see, because the TCF could be viewed as indirect debt financing, the government has a prior claim similar to that of the holders of a debt claim. Finally, the last term describes the income tax payoff the government receives in normal circumstances once the TCF has ended, exactly the same as under the basic model.

The same remark on the decomposition of the government claim into income tax claim and a premium tax claim applies here. Under the TCF, the premium tax claim is unchanged (i.e., the same under the basic model), but the income tax claim gets largely affected by the exchange of cash flows between the insurer and the tax authority during the TCF period [[t.sub.1], [T.sub.h]].

Furthermore, the equity claim at time to under the TCF is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

The first term represents the expected value of dividends to be paid out during the time period [[t.sub.1], [T.sub.h]] while the firm is still solvent. The second term consists of the expected residual claim of shareholders at bankruptcy if the firm defaults at any time during the tax carryforward period [[t.sub.1], [T.sub.h]]. The third term is the expected residual claim of shareholders upon default if the bankruptcy time [theta] coincides with ([T.sub.h] + 1), the date at which the firm is supposed to repay the government the amount of differed income taxes. Again here, once the firm becomes bankrupt, its cash reserves are zero, and shareholders receive nothing in the form of cash. They only receive the residual liquidation value of assets after repayment of government and debt claims.

The remaining terms are associated with the payoffs that will go to shareholders once the TCF as well as the GF's extraordinary obligation have ended and thus can be interpreted similarly as under the basic model. As one can note, the sequence of expected dividends once the TCF has ended starts at date ([T.sub.h] + 1). This is explained by the fact that if the firm stays solvent at date ([T.sub.h] + 1) and pays out the amount [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of differed income taxes to the government from its cash reserves, it will be able to pay out dividends to shareholders. The firm will be liquidated at time ([T.sub.h] + 1) only if cash reserves do not cover the repayment of debt principal and the amount [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of differed income taxes (which also explains the use of the event {[theta] = [T.sub.h] + 1} rather than {[theta] [greater than or equal to] [T.sub.h] + 1} as in the expression of the government claim).

Optimality Criteria for the Different Recoupment Mechanisms

We provide here the criteria upon which the different recoupment mechanisms will be evaluated. We begin by defining the decision rule for the tax authority as the decision of fixing the tax deferral rate [beta] in the context of the TCF or the tax credit rate [phi] under the PTC, based on the maximization of the expected utility of (discounted) future income tax revenues projected over the time horizon ([t.sub.1, [t.sub.2], ..., T). We shall assume in our subsequent numerical analysis that the tax authority exhibits preferences toward risk described by the simple power utility function,

U(x) = [[gamma].sup.-1][x.sup.[gamma]], (50)

where [gamma] (0 < [gamma] [less than or equal to] 1) is a parameter capturing the risk tolerance/aversion of the tax authority (i.e., higher [gamma] implies lower risk aversion, with the extreme case [gamma] = 1 indicating risk-neutrality).

Now, let's define ([E.sup.Basic], [G.sup.Basic]), ([E.sup.GF], [G.sup.GF]) and ([E.sup.PRG], [G.sup.PRG]) as the equity and government claims under the basic model, the GF contagion model and with different recoupment mechanisms (with PRG = PTC, PRS, and TCF), respectively. Therefore, a recoupment program PRG is said sustainable if it verifies the incentive compatibility conditions:

[G.sup.PRG] > [G.sup.GF], (51)

[E.sup.PRG] > [E.sup.GF], (52)

where both equity and government claims, ([E.sup.PRG], [G.sup.PRG]), are valued after optimizing the program's intervention rate [beta] (under the TCF) or [phi] (under the PTC). Moreover, a recoupment program is said economically optimal if it permits a reduction in the level of the cumulative probability of default/insolvency of the firm by comparison to the case of contagion risk where such a provision is not allowed.

An important fact to pointing out here is the particularity of the PRS. In contrast to the programs PTC and TCF implying a kind of government subsidy either via premium taxes or income taxes, the PRS does not imply any intervention from the tax authority. Rather, the PRS requires only the allowance of insurers to raise their premium rates, which means that policyholders are the direct payers for the program instead of the tax authority. This makes the PRS particular in the sense that the government authorities are not explicitly involved in the contagion mitigation process. In the context of our analysis, this creates a challenging issue, since the optimality of the PRS cannot be defined or measured appropriately through the formal criteria described above in the context of PTC and TCF. One approach would consist of introducing a cost or disutility function for the government measuring the collective decrease in welfare caused by the surcharge of premium rates for policyholders. But this avenue would turn out to be subjective, since one cannot ensure a fair comparison between this disutility function and the utility function of tax revenues in the context of the PTC and TCF. Moreover, allowing the insurance firm or even the tax authority to optimize the surcharge rate will inevitably result in an infinitely high solution, since corporate claims are increasing with the net underwriting income. One would impose in that case a trade-off between premium rates and premiums volume to capture the elasticity of clients to charged rates, but this may also turn out to be subjective for comparative purposes. For these reasons, we shall limit our analysis of the PRS to the situation where an exogenous rate of surcharge is applied. We shall also be careful in comparing the outcomes resulting from this setup to those implied by the PTC and TCF provisions.

NUMERICAL RESULTS DISCUSSION

Sustainability and Optimality of Recoupment Mechanisms

Table i reports the equity and government claims for different dividend payout rates ranging from 0.25 percent to 2.50 percent under the different models: the basic model, the contagion model of GF assessments, and the GF assessments coupled with the different recoupment programs (i.e., PTC, PRS, and TCF). For any dividend payout considered, the values of corporate claims under the PTC and TCF are those obtained at the optimal tax credit rate [[phi].sup.*] and the optimal tax deferral rate [[beta].sup.*], respectively, both maximizing the government' utility function. In the context of the PRS, the surcharge rate [zeta] is exogenously fixed at 5 percent, given the lack of government intervention under this program. The table contains three panels, each corresponding to a given level of initial cash reserves. The numerical results are drawn from reasonable basic case parameters (see Table A1 in the Appendix) and assuming a GF assessment amount h of $1. As we can see in Panels B and C, the move from the basic environment to the GF model will lower both the equity and government claims for most dividend payouts considered. This is a direct consequence of the contagion effect. Paying the extraordinary GF assessment amount will increase the failure rate and thus lower the values of the firm's claims. However, this contagion effect is considerably reduced for relatively high cash reserves, given the inverse relationship between default intensity and initial cash reserves. We also observe that as expected, equity claim is increasing with dividend payout, while the government claim is decreasing with dividends. This dividend effect holds no matter the environment under consideration. This is because dividends are a payout for shareholders while at the same time they shrink available cash reserves and thus increase the probability of losing tax revenues upon bankruptcy for tax authority. Furthermore, note that although corporate claims under the PRS are affected in the same way, one can eliminate the contagion effect completely (or even outperform the basic environment claims where there is no contagion risk at all) by assuming a higher premium surcharge rate.

From the perspective of shareholders, we see that for the lowest value of cash reserves (Panel A), the equity claim is lower under the recoupment provisions than under the GF case. This particularly holds for low dividend payouts and could be explained by the option-to-default effect. Indeed, when the intrinsic risk of default due to low level of cash reserves is high, shareholders are better off liquidating the company immediately and suffering the bankruptcy loss rather than maintaining the firm alive thanks to the mitigating recoupment programs and thus reducing the liquidation payoff it receives upon delayed default. When cash reserves are higher (Panels B and C), the firm is worthy and the option-to-default effect is offset. In that situation, indeed, prospective dividend payouts are significant, thus making shareholders more inclined to agree to a recoupment provision rather than resorting to an immediate liquidation of the firm.

With regard to recoupment mechanisms, numerical results are clear: shareholders are better off with the PTC and PRS rather than the TCE This preference holds independently from the level of cash reserves and is particularly pronounced for high dividend payouts. However, increasing cash reserves will considerably reduce the relative preference for the PTC against the TCE The idea here is simple: both the PRS (depending on the applied surcharge rate) and the PTC allow shareholders to cope with contagion risk without bearing any cost for the enhancement of their solvency profile. In contrast, the TCF is designed to transform the mitigation of contagion risk into a time-delayed debt the firm will reimburse once the turmoil period of GF assessments has passed away.

From the position of the tax authority, preferences toward the recoupment mechanisms are quite different. When cash reserves are limited (Panel A) and the risk of insolvency is high, the government claim is increased thanks to the recoupment provisions with respect to the GF case. In that situation, we also note that the TCF yields a higher tax claim than the PTC, while the PRS promises the highest claim value, given the zero-cost the government bears under that program. The relative preference toward the TCF against the PTC is reflected in the selected optimal credit rates. We see that the PTC requires almost a full credit allowance, that is, [[phi].sup.*] near 100 percent, while the TCF achieves a better return on claim with only a tax deferral rate [[beta].sup.*] of 20 percent on average. However, whenever cash reserves are increased and the insolvency risk is reduced (Panels B and C), the tax authority will exhibit a pronounced preference toward the TCF rather than the alternative tax-based program of PTC. In fact, the PTC in that situation is so poor that the government is better off letting the firm go eventually to bankruptcy (the GF case) rather than supplying the premium tax subsidy. The optimal tax deferral rate [[beta].sup.*] in these cases is increased to be around 60 percent on average in Panel B and 90 percent on average in Panel C, since the debt supplied by the government in the form of refundable tax credit is less risky than in the previous case. The optimal tax credit rate [[phi].sup.*], however, reflects exactly the poor performance of the PTC: it is considerably cut by more than a half on average in Panel B with comparison to the previous case (i.e., Panel A) and is considerably reduced in Panel C. Because of the associated zero cost for the government, the PRS (again, conditional on the assumed surcharge rate) continues in that case to outperform the tax-based programs including the TCF.

When confronting the impact of the recoupment programs on both the government and equity claims together, we conclude that these programs (even including the PRS) are not sustainable all the time. When cash reserves are very low (Panel A) and scheduled dividend payouts are low, the option-to-default effect makes shareholders better off with proceeding to an immediate liquidation of the firm, in spite of the tax authority incentive in that case to supply credit. When cash reserves are relatively high and insolvency risk is not alarming (Panel C), the PTC will be optimally rejected by the government (since the GF case of no intervention yields a higher claim value in that case), while both the TCF and, of course, the costless PRS will be retained by the two parties. In the case of medium level of cash reserves (Panel B), the PTC is optimally rejected by the tax authority most of the time (depending on the dividend payout), while both the TCF and the costless PRS are sustainable, with a strict preference for the PRS from the two parties as expected (again, based on the assumed premium surcharge rate).

Figure 1 depicts the default rates (cumulated default probability) over a time horizon of 20 years associated with the different environments considered in Table 1. In each of the three panels of Table 1, we have selected the default rates corresponding to the basic case value of the dividend payout of 1.25 percent to illustrate in Figure 1. As one can see, the contagion effect is well illustrated through the default rates. We observe that the credit curve jumps upward once we move from the basic model to the contagion model GF. The impact of contagion risk caused by GF assessments is propagated over a long time horizon, highlighting the potential significant deterioration of insurers' solvency that would be caused by failure/ruin events.

We also observe how default rates shift downward from the GF case after introducing the recoupment mechanisms. An important result is that the positive effect of these programs in terms of lower default rates takes place independent of the level of initial cash reserves. Numerical results not reported here also show that this effect holds independent of the dividend payout considered. Overall, this means that the recoupment mechanisms will always generate the desired control over contagion risk (lower risk of default), and this independent of the incentive compatibility considerations discussed above. Moreover, we observe that the reduction of the contagion effect one would measure based on the downward displacement of the credit curve from the GF case to the mitigant programs cases, and thus the extent to which these programs add economic value by lowering expected bankruptcy costs, essentially depends on the cash reserves capturing the intrinsic repayment capacity of the firm. The higher these cash reserves, the higher is the intrinsic solvency of the firm and the lower will be the relative gain in term of lower default rates.

When comparing the impact of the TCF on default rates with respect to the PTC, we note that the latter generates a higher impact, no matter the intrinsic solvency of the firm captured through the initial level of cash reserves. However, both of them yields the desired control over default events. Taking into account the sustainability outcomes discussed above and this impact on default rates, one can conclude that the TCF offers a larger scope for economic optimality than the alternative tax-based PTC. Because of its costless nature, a similar comparison against the PRS cannot be directly assessed unless the introduction of additional and arbitrary assumptions about the economic cost of increasing premium rates.

[FIGURE 1 OMITTED]

By keeping the values of cash reserves and dividend payout rate fixed at $7.50 and 1.25 percent, respectively, both Table 2 and Figure 2 report the same numerical results as those reported in Table I and Figure 1. Here, however, the amount h of GF assessment ranges from $0.20 to $2.00 (the credit curves depicted in Figure 2 are those associated with h = $1.00). Three cases are considered, where each one corresponds to a different value of the principal amount of outstanding debt, B. Similar to our earlier results, we note that the introduction of the GF assessment amount generates a contagion effect illustrated through the downward shifts of the values of the firm's claims and the upward shift of the credit curve resulting after the move from the basic model toward the contagion model. We also note that the introduction of the TCF and the PRS (based on the assumed premium surcharge rate) produces a systematic appreciation of the government claim value over the GF case. However, in the case of the PTC, the government claim appreciation is not ensured all the time. Similar to our previous results reported in Table 1, this appreciation is tightly dependent on the solvency risk of the firm measured here through financial leverage. Indeed, the tax authority is found in most of cases better off waiting for the payoffs under contagion pressure as shown in the GF case rather than supplying the tax credit to the firm stipulated by the PTC.

From the view point of shareholders, the homogenous impact of the dividend payout reported in our previous results is no longer valid everywhere now. Independently from the level of financial leverage or the environment considered (including the GF and the recoupment provisions cases, except the flat equity value case of the basic model), the equity claim is found a nonmonotonic function of the assessment amount h. This is a manifestation of the option-to-default effect. When the assessment amount h is small shareholders are better off waiting. Increasing progressively the assessment amount over that continuation region will lower the value of equity. However, once the assessment amount reaches a critical barrier, shareholders exchange their continuation strategy for an immediate liquidation of the firm. The higher the amount h over that liquidation region, the sooner will be the liquidation decision and the higher will be the liquidation proceedings thus contributing to a higher equity value. In sum, this tension between continuation and immediate liquidation generates a U-shape for the equity claim as a function of the assessment amount h. The lower the financial leverage of the firm, the higher the potential solvency of the firm, and thus the more severe this continuation/immediate liquidation trade-off for shareholders. This mechanism explains the striking result according to which shareholders are by far better off with the PTC and PRS rather than the TCF, and this in more important proportions in comparison to the previous results obtained as a function of dividend payouts. Indeed, in the context of this trade-off between continuation and immediate liquidation, the introduction of the TCF implies an additional but delayed leverage for the firm. As a consequence, this makes the TCF equivalent to the GF case over the continuation region where the assessment amount h is small but worse than the GF outcome over the liquidation region where the assessment amount h is high. Of course, all of these mechanisms at play take place as conditional on the optimal tax deferral rate [[beta].sup.*] set by the tax authority.

[FIGURE 2 OMITTED]

By confronting the incentive compatibility conditions for the tax authority and shareholders, one comes to the conclusion that the PTC is only sustainable in the case where financial leverage (or insolvency risk) is high (i.e., Panel A). In contrast, the TCF offers a wider scope for sustainability, but this sustainability is conditional. We see that the TCF's sustainability does not depend on the financial leverage, but rather it is limited to the continuation region. This means that the two parties will always agree on the implementation of the TCF no matter financial leverage, given that the prospects of an immediate liquidation are not so high to make shareholders inclined to accept the government offer. In the case of the PRS, the sustainability scope is maximized with respect to the tax-based recoupment programs, but in general, it is tightly dependent on the assumed surcharge rate [zeta]. Since the mitigation impact on default rates is preserved no matter the financial leverage considered, as illustrated in Figure 2, we are forced to conclude the same way with respect to the economic optimality of these recoupment programs than those concluded with respect to their sustainability.

AGENCY CONFLICTS AND BANKRUPTCY COSTS: COMPARATIVE STATICS

Here we want to explore the same outcomes discussed earlier but after imposing exogenous credit tax rate and tax deferral rate for the tax-based programs, PTC and TCF, as we have already done in the context of the PRS. The results will be very useful, since it is hard to expect that tax authorities will adjust their intervention plan in consideration of the specific solvency and balance-sheet profiles of the different insurance firms, members of the same GF, applying for the program. Our focus is on the tax-based programs, PTC and TCF, where for each scenario we shall assume the same mitigation rate for both programs. In doing so, we also stress the impact of dividend payouts and bankruptcy costs on the sensitivity of equity and government claims to recoupment provision rates. The choice of dividend payout as a variate for the government claim sensitivity is motivated by the potential agency conflicts mechanism. When shareholders apply a high dividend payout rate, available cash reserves of the firm are continuously reduced, which fragilizes the solvency of the firm. In the context of tax-based recoupment programs, this represents an agency cost for the tax authority, since both PTC and income tax report are not adequately compensated by a prudent cash management in that case. With respect to bankruptcy costs, we have seen in our earlier results that the option to default may provide shareholders with the incentive to proceeding to an immediate liquidation of the firm. If the liquidation option is more attractive than the continuation option, we have observed that none of the recoupment programs could be economically sustainable from the perspective of shareholders. Therefore, by stressing bankruptcy costs, we are also able to highlight the impact of the option to default.

Our numerical results are illustrated in Figures 3 and 4. In Figure 3, we consider a highly stressed firm, where the contagion risk caused by the GF assessments is very high. Namely, we assume in that case initial cash reserves of $5 and a GF assessment amount h of $2 (which implies h/[v.sub.0] = 40 percent). Figure 4 depicts the opposite scenario, where the solvency of the firm is high and contagion risk is low. More precisely, we assume that cash reserves are equal to $10 and consider a low GF assessment amount h of $0.40 (which implies h/[v.sub.0] = 4 percent). We report the sensitivity of government and equity claims to the recoupment program rate. Both the PTC and TCF are implemented using the same credit rate; that is, we impose the parity [beta] = [phi]. In addition to the mitigant programs cases, two horizontal lines are also illustrated corresponding to the claims values under the basic environment and the GF case (without mitigant program), respectively. In each figure, Panel A reports the government claim sensitivity to the mitigant program's rate, by varying the dividend payout rate. In Panel A-l, the dividend payout is set equal to the base case parameter [delta] = 1.25 percent, while it is increased to [delta] = 2.50 percent in Panel A-2. In both cases, the rest of model parameters (including bankruptcy costs fraction) follow from the base case parameters. Panel B reports the equity claim sensitivity to the mitigant program's rate by changing the fraction [omega] representing bankruptcy costs to be suffered upon the liquidation of the firm. In Panel B-1, bankruptcy costs are set equal to the base case parameter [omega] = 0.25, while they are increased to [omega] = 0.75 in Panel B-2. In both cases, the rest of model parameters (including the dividend payout rate) follow from the base case parameters.

[FIGURE 3 OMITTED]

In the first case described by Figure 3, where the contagion risk is significant, we see that the tax authority most of the time is better off implementing the TCF rather than the PTC, although both programs achieve the desired appreciation of the tax claim in comparison to the GF case. In fact, this holds independently from the dividend payout rate (Panel A-1 vs. A-2). The relative preference of the tax authority for the TCF against the PTC occurs for credit rates approximately below 70 percent, while the inverse case happens when moving beyond that rate. This is mainly due to the fact that for excessive credit rates (higher than 70 percent), the supply of debt in the form of income tax report becomes a very risky investment for the government, thus making the alternative solution of foregoing premium tax revenues more attractive in that case. In the opposite case of reasonable credit rates (below 70 percent), the TCF offers an enhancement of the firm solvency (lower default rates), which translates into a higher probability of getting reported income taxes paid as well as protecting subsequent premium and income tax revenues (those occurring after Th) from bankruptcy events, while the PTC does not provide the same enhancement. Interestingly, we see that while the same mechanisms take place in both Panels A-1 and A-2, the level of the government claim is shifting downward after increasing the dividend payout when moving from Panels A-1 to A-2. This highlights the agency cost the tax authority bears when supplying tax credit advantages. This means that the only way insurers will maximize the chance of getting the tax credit advantages or of improving the mitigant program conditions (i.e., higher credit rates) is by taking control over capital expenditures, mostly dividend payments. Most importantly, the relative preference of the government to the TCF versus the PTC is unchanged when varying dividend payouts. This is because no matter the dividend payout and its negative impact on the solvency of the firm, the attributes of the TCF compared to those of the PTC remain unchanged. This signals the robustness of the existing differentiation between the two tax-based programs.

[FIGURE 4 OMITTED]

Now, from the perspective of shareholders, and still in the case of high contagion risk (Figure 3), obtained results suggest a completely different trade-off. First, and not surprisingly, Panel B-1 describes the typical situation where immediate liquidation of the firm is more worthy solution for shareholders than the continuation option. This option-to-default effect is due to the relatively low assumed bankruptcy costs and implies that the equity claim under the GF case for which liquidation is imminent is higher than the claim value coming from the continuation option under the mitigant programs, and even under the basic environment (remember here that in addition to the high contagion risk, we have also assume a very low level of cash reserves). When moving to Panel B-2, bankruptcy costs are significantly increased, and thus the payoffs of immediate liquidation of the firm become now less attractive. As a consequence, this makes the mitigant programs worthy for shareholders and transforms the GF case (basic environment) the worst (best) among the all scenarios. Interestingly, when only focusing on the situation where recoupment programs are both more worthy than the GF case (i.e., Panel B-2), we observe that shareholders exhibit the same relative preference for these programs than the tax authority. We see the TCF more attractive than the PTC when a credit rate below 60 percent approximately is applied, while the inverse is true beyond that level. This alignment of incentives between the two parties comes from the threat of loss caused by higher bankruptcy costs. Indeed, when the liquidation of the firm is less attractive than the continuation option because of higher bankruptcy costs, shareholders are more inclined to implement the TCF for its most effective enhancement of solvency (since applying the same tax deferral rate than PTC has a higher impact on cash reserves), and this even after considering the provision of reimbursement of reported taxes at the end of the program. But this preference is completely switched in favor of the PTC when a high credit rate is applied, since in that case the "free lunch" representing the tax subsidy of the PTC is sufficient enough to cope with contagion risk, while the TCF--even always doing good in terms of solvency enhancement--starts to be costly in terms of delayed reimbursement of reported taxes.

In Figure 4 different patterns occur where contagion risk is negligible and the intrinsic solvency of the insurer is important. By looking at the different four cases depicted by Panels A-1 to B-2, we see a situation where the mitigant programs are not worthy for the tax authority, while by symmetry effect, they totally represent a sort of "free lunch" for shareholders. Indeed, the contagion risk issue in those cases is so minor that the government claim is almost the same between the basic environment and the GF case. This, as a result, inevitably makes the mitigant programs strictly unattractive in comparison to the alternative solution of not supplying any tax advantages. When comparing the two mitigant programs in that case, we note without any surprise a pronounced preference toward the TCF, since it allows the tax authority to minimize the dead loss caused by an eventual intervention. Again, by the same symmetry effect, shareholders will exhibit an opportunistic preference for the PTC for its costless nature. Overall, the situation described here alerts us that not any government intervention to cope with contagion risk is desired in the sense of collective welfare. Indeed, when the contagion issue is not significant, the tax authority is better off cutting intervention plans rather than providing assistance plans in a systematic way.

FURTHER CONSIDERATIONS" BUSINESS CYCLE AND DEFAULT CORRELATION

GF assessments are usually procyclical in the sense that most of property-casualty insurance failures occur either near the bottom of the insurance business cycle, where the return on the equity-to-loss ratio is at its lowest level or either just as the recovery begins. This means that the need for assessments generally amplifies the natural financial fragility of insurers caused by the regular business cycle. In our previous numerical simulations, we have shown that weaker initial financial conditions for an insurer, captured either through low cash reserves or a high level of outstanding debt obligations, increase considerably the stress caused by GF assessments. To evaluate the impact of the timing of these assessments as they relate with the insurance business cycle, one simply needs to stress the model's variables even further. These include the insurer's profitability due to its insurance operations reflecting the magnitude of its aggregated losses. Based on additional numerical simulations (not reported here), we can easily conclude that the business cycle effect is equivalent to our scenario of weaker financial conditions that we examined and discussed earlier.

Furthermore, in all of our modeling exercise, the tax authority is put in a position where it is concerned with the effect of assessments on the financial situation of only one surviving insurer. However, in real-world situations, the tax authority faces a pool of surviving firms that will consider taking advantage of these recoupment measures. We can intuitively postulate that the incentives for the government to allow these provisions will be boosted by the threat of a default correlation among surviving firms. Indeed, when most surviving insurance firms are subject to strain caused by GF assessments, then there is a significant risk for the government to lose most of its future tax revenues upon the realization of more failures. Clearly, this portfolio-wide risk would encourage the tax authority to implement such tax-based mechanisms even much more quickly than when the number of distressed insurers is small.

CONCLUSIONS

In this article, we have examined the impact of assessments charged by a GF following the failure of an insurance company on the financial safety of surviving insurers. After building a realistic contingent claim model of the property-casualty insurance firm, GF assessments are integrated into the model to highlight the implications in terms of default rates and corporate claims values for the fund's surviving members. Our analysis reveals that although GF assessments do not necessarily cause an immediate failure of surviving members in the usual sense associated with the notion of contagion, the long-run financial safety of these members is shown to be significantly perturbed by these ex post charges. Indeed, our results show the existence of a latent type of contagion whose impacts would be exacerbated whenever the financial conditions of insurers---captured in our model through available cash reserves and outstanding financial obligations--are weak.

The main issue we investigate in the article, after highlighting the contagion effect of GF assessments, is to what extent recoupment mechanisms would be sustainable and optimal from an economic perspective. Our criteria here are twofold: (1) the enhancement of default rates prospects for sustainability outcomes and (2) the appreciation of corporate claims for optimality outcomes. Both the PTC and PRS provisions, largely used in the U.S. property-casualty industry, are examined. We also introduce a new tax-based recoupment program, the TCF provision, which aims to yield different attributes from those of existing mechanisms. Our analysis unambiguously reveals that while all recoupment mechanisms are sustainable in the sense they reduce default probabilities of insurers, tax-based recoupment programs fail most of the time to achieve a simultaneous improvement of both equity and tax claims. This is mainly due to the option to default and dividend payout mechanisms that create a sort of divergence between the relative preferences of equityholders and tax authority to one particular recoupment program. Our results also suggest that the main attractiveness of the PRS provision is limited to the simple fact the bill of this recoupment program is passed through to customers. From an economic welfare perspective this provision does certainly not lead to a superior outcome when compared to the other tax-based programs.

Finally, we believe that further research would be highly constructive if a new focus is made in order to propose GFs that could better mitigate the contagion risk problem. The allowance for ex post recoupment provisions that could be implemented upon failure events to enhance the solvency profile of surviving insurers simultaneously with ex ante risk-adjusted assessments, analogous to deposit insurance in the banking industry, represents an alternative to the existing practice that should attract the interest of both insurers and regulators.

DOI: 10.1111/j.1539-6975.2010.01367.x

APPENDIX: BASIC CASE PARAMETERS

REFERENCES

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Babbel, D. E, and C. B. Merrill, 1999, Toward a Unified Valuation Model for Life Insurers, in: J. D. Cummins and A. M. Santomero, eds., Changes in the Life Insurance Industry: Efficiency, Technology and Risk Management (Boston: Kluwer Academic Publications).

Chen, Y., 1999, Banking Panics: The Role of the First-Come First-Served Rule and Information Externalities, Journal of Political Economy, 107: 946-968.

Cobb, C., 2003, Insurance Guaranty Funds, CRS Report for Congress, The Library of Congress, Order Code RL 32175.

Cummins, J. D., 1988, Risk-Based Premiums for Insurance Guaranty Funds, Journal of Finance, 43: 823-839.

De Bondt, O., 1995, Competition Among Financial Intermediaries and the Risk of Contagious Failures, Banque de France, Notes d'etudes et de recherche, 30: 45.

Garber, P. M., and V. Grilli, 1989, Bank Runs in Open Economics and the International Transmission of Panics, Journal of International Economics, 27: 165-175.

Han, L.-M., G. C. Lai, and R. C. Witt, 1997, A Financial-Economic Evaluation of Insurance Guaranty Fund System: An Agency Cost Perspective, Journal of Banking and Finance, 21: 1107-1129.

Klein, R. W., and S. Wang, 2009, Catastrophe Risk Financing in the US and the EU: A Comparative Analysis of Alternative Regulatory Approaches, Journal of Risk and Insurance, 76: 607-637.

Lee, S.-J., D. Mayers, and C. W. Smith, Jr., 1997, Guaranty Funds and Risk-Taking: Evidence From the Insurance Industry, Journal of Financial Economics, 44: 3-24.

Lee, S.-J., and M. L. Smith, Jr., 1999, Property-Casualty Insurance Guaranty Funds and Insurer Vulnerability to Misfortune, Journal of Banking and Finance, 23: 1437-1456.

Leitner, Y., 2005, Financial Networks: Contagion, Commitment and Private Sector Bailout, Journal of Finance, 60: 2925-2953.

Leland, H., and K. Toft, 1996, Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads, Journal of Finance, 51: 987-1019.

Ligon, J. A., and P. D. Thistle, 2007, The Organization Structure of Insurance Companies: The Role of Heterogeneous Risks and Guaranty Funds, Journal of Risk and Insurance, 74: 851-861.

Miller, R. K., and J. Polonchek, 1999, Contagion Effects in the Insurance Industry, Journal of Risk and Insurance, 66: 459-475.

Schoenmaker, D., 1996, Contagion Risk in Banking, LES Financial Markets Group Discussion Paper 329, London School of Economics.

Shiller, R. J., 1989, Market Volatility (Cambridge, MA: MIT Press).

Vasicek, O., 1977, An Equilibrium Characterization of the Term Structure, Journal of Financial Economics, 5: 177-188.

Gilles Bernier is at the Department of Finance and Insurance and the Industrial-Alliance Insurance Chair, Laval University, Pavilion Palasis-Prince, Quebec City, Quebec, Canada GIK 7P4. Ridha M. Mahfoudhi is at Desjardins, Montreal, Quebec, Canada; Affiliated Researcher, Department of Finance and Insurance, Laval University. The authors can be contacted via e-mail: gilles.bernier@fsa.ulaval.ca and ridha.mahfoudhi@fsa.ulaval.ca, respectively. We acknowledge the financial support of the Property and Casualty Insurance Compensation Corporation (PACICC) for this research. Our special thanks to David Babbel and Darrell Leadbetter for their detailed comments and very helpful suggestions, which have helped us improve the quality of this article. We would also like to thank two anonymous referees of this journal for the relevance of their comments that largely contributed to improve the content of the article. We are grateful to Norma Nielson for her helpful discussion of the paper at the 2007 meeting of the American Risk and Insurance Association, as well as Jean-Francois Larocque and Jean Cote for their comments and suggestions. All remaining errors are ours. This article is dedicated to our deceased colleague, Dr. Klaus Fischer, who was a starting member of this research project.

(1) Indeed, the recoupment provision in property-casualty GF laws in the various states is described as follows: "This section details the method provided for in the guaranty association act which allows member insurers to recover the cost of assessments; the usual methods being 'rates and premiums' or 'premium tax offset.' The rates and premiums method allows member insurers to recover the cost of assessments by permitting member insurers to include the cost of assessments as a factor in determining rates and premiums that the insurer is allowed to charge for its policies. The premium tax offset method allows member insurers to recover the cost of assessments by permitting a reduction in premium taxes payable by the member insurer." See http://beta.ncigf.org/guaranty/definitions.asp.

(2) Contagion risk typically refers to the spillover effects of shocks from one or more firms to other firms. The vast majority of research carried out on contagion risk is related to the banking sector. Contagion in banking has been studied at the domestic level by Shiller (1989), De Bondt (1995), Schoenmaker (1996), Chen, (1999), and, at the international level, by Garber and Grilli (1989). These authors observed different transmission mechanisms of contagion, such as panics, revision of expectations, rumors on liquidity shocks, and transmission of the loss in value of assets. Likewise, Leitner (2005) analyzed private bailout mechanisms designed to cushion contagion effects. Although more subtle than a run on bank deposits, there also appears to exist contagion risk in the insurance industry. For example, Angbazo and Narayanan (1996) and Miller and Polonchek (1999) show the existence of contagion effects in the insurance industry due to things such as asset quality problems and/or catastrophic events.

This article proposes a model that suggests there are contagion effects among members of an insurance guaranty fund when postassessments are charged to all other insurers upon the failure of a member company. Indeed, these extraordinary payments are shown to increase the default rate of other firms in the industry, ultimately lowering the value of corporate claims as well as government tax claims. The model is also used to examine the efficiency of different recoupment mechanisms (both existing and new) used by regulators and insurers to potentially reduce these contagion effects. Analysis allows us to stipulate the conditions under which a "tax carryforward" provision could be more efficient than the usual recoupment mechanisms known as "premium rate surcharge" and "premium tax credit."

INTRODUCTION

In most countries, it is common to find a guaranty fund (GF) system that covers part of the claims obligations of insolvent insurers. For example, in the United States, each state has two guaranty associations, one for life and the other for property-liability insurance. Each GF is a mandatory association of all licensed insurers doing business in that state (Klein and Wang, 2009).

This GF system is interdependent and represents a cooperative effort among regulators and insurers in the states where the insolvent insurer operated. If the home-state insurance commissioner considers that the financially troubled insurer is not likely to be able to solve its problems, then he can petition a state court for an order of liquidation. Insurance regulators in the other states where the insurer operated will do the same. The order of liquidation in each state generally triggers the operation of that state's GF. Once the insolvent's liabilities and assets are valued and a shortfall estimated, then postassessments are requested from each participating insurer in proportion to their premium volume in that state. In property-liability insurance, assessments are usually first imposed on the particular coverage that occasioned the insolvency and are limited to 1 percent to 2 percent annually of each insurer's premium volume in that line in that state. If these assessments prove insufficient, then other property-liability lines may be assessed as well (Cobb, 2003).

The existence of GFs has been shown to potentially affect the risk-taking incentives of insurance firms. Empirical evidence seems to support the claim of greater risk-taking on the part of insurers following the adoption of GFs in the United States (Lee, Mayers, and Smith, 1997; Lee and Smith, 1999). This kind of moral hazard, similar to the one in banking due to deposit insurance, has led researchers to suggest risk-based rather than flat assessment of premiums for GFs as well (Cummins, 1988). More recently, Ligon and Thistle (2007) have also shown that non-risk-based GF premiums do create an economic incentive for the fleet form of organization, which is quite important in the U.S. property-liability industry.

The design of GFs has also been analyzed from an agency cost perspective (Han, Lai, and Witt, 1997). Such an analysis allows for either ex ante or ex post assessment of GF premiums. "Ex ante premiums give shareholders less incentive for risk-taking behavior than the actual ex post assessment scheme observed most frequently in practice. In the case of the latter, only insurers who survive pay the cost of the scheme, so the share of the cost paid by safer insurers is more highly disproportionate than in an ex ante assessment" (Ligon and Thistle, 2007, p. 858).

In practice however, "most states allow assessed insurers to include the assessments in their rates, though some states allow insurers to surcharge policyholders or offset the assessments against their premium taxes" (Cobb, 2003, p. 4). Hence, through premium rate surcharges (PRSs) and/or premium tax offsets or credits (PTCs), surviving insurers can expect to recoup most their assessment costs. (1)

However, this may lead to a timing issue in that insurers may be compelled to stretch out their recoupments over a number of years. For example, in the case of a PTC provision, it is very common for states to allow insurers to offset 20 percent of their assessment from their premium taxes each year for a maximum of 5 years. Hence, such staggering of premium tax offsets (or credits) does not achieve the sought objective of balance sheet neutrality for surviving insurers. Likewise, recouping GF assessments through rate surcharges contains significant lags, perhaps as long as 12 months if they can be implemented immediately and possibly much more if the assessment base is in a rate regulated line that requires supervisory approval.

Ligon and Thistle (2007) argue that these recoupment mechanisms may further exacerbate the incentives for risk-increasing behavior on the part of insurers. For regulators, these mechanisms are deployed to mitigate the strain potentially imposed by postassessments on surviving insurers following the liquidation of a member company. In their view, such recoupment mechanisms may have a role to play in preventing contagion effects due to postassessments that might weaken other insurers. (2) Said differently, whether or not these mechanisms create ex ante incentives for risk taking is actually viewed as less critical compared to the desired ex post resolution of financial distress issues they are supposed to achieve.

This is what we hope to investigate in our article that is organized as follows. First, we develop a realistic cash flow model of financial distress of a property-liability insurance company in which the insurer's financial capacity is modeled respecting the specificities of the industry in terms of main sources of both revenues and risks. Then, we rely upon our basic model in order to investigate whether or not the presence of a GF charging postassessments to other insurers following the failure of a member company can be a source of contagion in the industry. Here, we define contagion as being a deterioration of the financial situation of surviving insurers as indicated by their level of leverage that impacts on their probability of default. We show that this type of contagion is indeed possible in such a GF system. Then, by extending our model, we also examine the optimality of the different recoupment mechanisms (both existing and new) designed to reduce these contagion effects from the point of view of different stakeholders, namely, shareholders, policyholders, and the tax authority. Our basic model and its many variants are implemented through Monte Carlo simulation analyses. Hence, we are also able to examine and compare the efficiency of the existing recoupment mechanisms (PTCs and PRSs) versus a newly proposed provision calling instead for a tax carryforward in order to deal with contagion effects in such a context. In the final section, we summarize our findings and we conclude.

THE BASIC MODEL OF THE INSURANCE FIRM

We describe here a model of the business and financial risks of a property-liability insurance firm. We consider an economic framework where the time line is divided into equally distanced dates [t.sub.0], [t.sub.1], [t.sub.2], ..., T. For instance, the length of each time increment [DELTA]t = ([t.sub.i+1] - [t.sub.i]) could be viewed as a financial quarter.

Insurance Activity and Net Underwriting Income

The business activity of the insurance firm consists of selling insurance protection (insurance policies) to customers against predetermined sources of pure risk. This activity permits the firm to generate a periodic underwriting revenue but also implies the payment of a periodic and random amount of claims as well as operational costs.

We assume that the insurance firm holds a portfolio of a fixed number N of business lines/clients over time, which could viewed as a portfolio of the same N lines the company serves as an insurer. The assumption of flat volume of insurance activity is not required for our model, but it permits us to avoid making supplementary assumptions about business or market share growth not relevant for our purpose.

On one side, the portfolio of N insurance lines allows the company to earn a time-periodic amount of underwriting revenue, [P.sub.t] (t = [t.sub.0], [t.sub.1], ...), representing the periodic amount charged to clients against the supplied insurance. On the other side, the company suffers a time-periodic aggregate loss due to claims payments, which we denote by [L.sub.t]. Without loss of generality, we assume that each business line or client i = 1, ..., N pays a periodic insurance premium pit over time, but may at any time trigger a claim payment [l.sub.it]. We assume that the number of claims payments made over a given time period for each particular client or line i is unconstrained (not capped) as long as the client keeps on paying the same periodic amount of insurance premium [p.sub.it]. This assumption could easily be justified under our time-unchanging mix of clients/lines, since the insurance firm will be fairly remunerated over the whole portfolio independent of individual claims records. Under this simple model of a stationary portfolio of insurance policies, the net underwriting income is given by

[NUI.sub.t] = [P.sub.t] - [L.sub.t], (1)

[P.sub.t] = [N.summation over (i=1)] [p.sub.it], [L.sub.t] = [N.summation over (i=1)] [l.sub.it]. (2)

Now, we describe the risk model of aggregate loss [L.sub.t]. We assume that the claim payment [l.sub.it] due to the business line/client i (i = 1, ..., N) is randomly distributed as follows:

[l.sub.it] = [q.sub.it][s.sub.it] for any t > [t.sub.0], (3)

where [s.sub.it] is the time t claim size and [q.sub.it] is the time t claim intensity or frequency. Both claim size and claim frequency are randomly distributed and mutually independent. Namely, we assume a homogenous and time-stationary portfolio of claims, so that at each time t > [t.sub.0] and for any i = 1, ..., N, we have

Pr[[q.sub.it] = 1] = [lambda], Pr[[q.sub.it] = 0] = 1 - [lambda] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

with

[([z.sub.it]).sub.i=1, ..., N] ~ N([0.bar], [summation]), (6)

[[summation].sub.ij] = E[[z.sub.it][z.sub.jt]] = [rho][[sigma].sup.2], (7)

[[summation].sub.ii] = E[[z.sup.2.sub.it]] = [[sigma].sup.2]. (8)

This means that claim sizes are correlated and exhibit a shifted joint lognormal distribution generated from the shift parameter [??] and the time-constant covariance matrix [SIGMA] of the correlated Gaussian shocks [([z.sub.it]).sub.i=1, ..., N]. Relaxing the assumption of zero serial-correlation between realizations of claim size is easy to do, but we decided to keep the model as simple as described earlier. It then follows that line i's expected claim size is constant over time,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

We introduce here two random variables describing the distribution of the aggregate loss [L.sub.t]. First, let [n.sub.t] ([n.sub.t] [member of] {0,1, 2, ..., N}) be the random frequency of claims experienced at any time instant t over the whole portfolio of business lines. According to the individual claims severities [([q.sub.it]).sub.i=1, ..., N], we have

Q(k) [equivalent to] Pr[[n.sub.t] = k] = [N!/k![(N - k).sup.!]][[lambda].sup.k][(1 - [lambda]).sup.N-k] with k = 0, 1, 2, ..., N. (10)

Then, let us define [S.sub.t] as the time t random sum,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

representing the aggregate claim severity conditional on the claims frequency [n.sub.t]. Therefore, the probability density function (p.d.f.) of the aggregate loss [L.sub.t] reads as follows:

Pr[[L.sub.t] [member of] da] = [N.summation over (k=0)] Q(k)Pr[[S.sub.t] [member of] da | [n.sub.t] = k], for any t > [t.sub.0]. (12)

Since claim sizes are correlated lognormals, we cannot derive an analytic formula for the conditional p.d.f. Pr[[S.sub.t] [member of] da | [n.sub.t] = k] for any arbitrarily high integer k. To obtain the numerical values of both the mean and the variance of the aggregate loss, which we make use in our subsequent analysis, we employ the Fast Fourier Transforms method.

Now, assuming that insurance firms charge risk-adjusted premiums to policyholders, it follows that the total periodic product of collected premiums (i.e., the underwriting revenue) is given by the expected aggregate loss [bar.L] loaded by a proportional margin rate m > 0; that is, for any t [greater than or equal to] [t.sub.0],

[P.sub.t] = (1 + m)[bar.L], (13)

which in turn implies that

[NUI.sub.t] = (1 + m)[bar.L] - [L.sub.t]. (14)

We can expect that the insurance firm will charge the price of aggregate loss risk (the loss variance, which already includes claims correlation risk, could be a good estimator of this risk) through the markup m. More precisely, we shall assume that,

(1 + m)[bar.L] = [bar.L] + [eta][[sigma].sup.2.sub.L], (15)

or,

m = [eta] [[sigma].sup.2.sub.L]/[bar.L] (16)

where [eta] > 0 is a constant parameter.

Real Assets and Franchise Value

Real assets are illiquid assets held by the insurance firm other than pure cash reserves and investment assets. These assets are essentially needed to sustain underwriting activity and they are valued at their opportunity cost, so that their acquisition value over the secondary market equalizes the discounted value of the future rents they allow to be earned. In this article, by real assets we refer to what is known as "economic rents" in the economic literature and the "franchise value" in the insurance literature. Babbel and Merrill (1999, pp. 245-278) define this component of the insurance firm as follows. It is the present value of these rents that an insurer is expected to garner because it has scarce resources, scarce capital, charter value, licenses, a distribution network, personnel, reputation, and so forth. It includes renewal business. In order to use a term that contrasts well with pure cash reserves and marketable securities, we shall adopt in this article the term "real assets" to refer to this present value of economic rents known as the franchise value.

More precisely, we assume that the real assets or franchise value of the firm (the economic value of an equivalent unlevered firm) satisfies

[A.sub.t] = [E.sub.t] [[T.summation over (u=t+1)] [e.sup.-(u-t)[phi]][NUI.sub.u][DELTA]t], [t.sub.0] [less than or equal to] t [less than or equal to] T-1, (17)

= [E.sub.t][[T.summation over (u=t+1)] [e.sup.-(u-t)[phi]][(1 + m)[bar.L] - [L.sub.u]][DELTA]t, (18)

= m[bar.L][DELTA]t [[T.summation over (u=t+1)] [e.sup.-(u-t)[phi]], (19)

where [E.sub.t][.] is the time t risk-neutral expectation conditional on the information set [F.sub.t], while the quantity [e.sup.-(u-t)[phi]] captures a time-increasing opportunity cost (with a constant speed of [phi]) of business assets in place over the secondary market. Note that there exists a constant [bar.[phi]] reflecting the average return required by insurers on investments in franchise value such that the expression of real assets reduces to

[A.sub.t] = m[bar.L]/[bar.[phi]]. (20)

The process by which the value of the firm's real assets is determined over the secondary market is important for our purpose, since it is a determinant of the cash flows that will go to the firm's claimants upon bankruptcy, and hence, it determines the loss-given failure as well as bankruptcy costs.

Investment Activity

The main economic function of insurance firms is risk reduction through pooling (diversification) of policies, which allows them to cover the insured risk, while being able to generate economic surplus thanks to the investment of underwriting revenue into fixed-income-generating assets. Consistent with this asset-risk transformation function, the insurance firm in our model holds an investment portfolio essentially composed of fixed income securities. For simplicity, we assume that the instantaneous rate of return realized by this investment portfolio could be replicated by a linear combination of the rate of return of the instantaneous short rate [r.sub.t] (the interest rate earned on T-bills) and a discount bond with a relatively long maturity [T.sub.r]. This assumption reflects the fact that return realized over fixed-income portfolios are essentially explained by both interest income flows and gains/losses due to positions rebalancing. Let [R.sub.t] be the instantaneous total rate of return of the investment portfolio and [alpha] (0, 1) be the fraction replicating the weight of gains/losses realized from bond trading in this total return, we have

[R.sub.t]dt = [alpha] [d B(t, [T.sub.r], [r.sub.t])/B(t, [T.sub.r], [r.sub.t])] + (1 - [alpha])[r.sub.t]dt, (21)

where B(t, u, [r.sub.t]) represents the time t market value of a discount bond paying one unit of face value at time u, given [r.sub.t]. Since the investment portfolio is composed of very liquid assets and marketable securities, it is viewed as the cash reserves of the insurance firm. As we will see later, these cash reserves play a key role in our model, since the default risk of insurance firms is tightly dependent upon their repayment capacity captured through the amount of available cash reserves.

Defining [v.sub.t] as the market value of the investment portfolio, which is equal to the amount of cash reserves, the net interest income generated by the investment activity over the time interval [t - [DELTA]t, t] will be given by

[NII.sub.t] = [[integral].sup.t.sub.t-[DELTA]t] [R.sub.s][v.sub.s]ds. (22)

We suppose here that the dynamics of the term structure are described by the simple mean-reverting short rate model of Vasicek (1977),

[dr.sub.t] = [[theta].sub.r]([bar.r] - [r.sub.t])dt + [[sigma].sub.r]d[W.sup.r.sub.t], (23)

where [bar.r] is the equilibrium interest rate level, [[sigma].sub.r] is the volatility of the interest rate, [[theta].sub.r] is the mean reversion speed of the short rate process, and ([W.sup.y.sub.t]) is a standard Brownian motion. Note that the market value of the discount bond B(t, [T.sub.r], [r.sub.t]) is given by the well-known Vasicek formula,

B(t, [T.sub.r], [r.sub.t]) = exp([[phi].sub.1](t, [T.sub.r]) - [r.sub.t][[phi].sub.2](t, [T.sub.r])),

where [[phi].sub.1](.) and [[phi].sub.2](.) are defined deterministic functions.

Financial Leverage, EBIT, and Net Income

To finance her business activity, the insurance firm periodically issues debt securities over the bond market. At each time, the firm also repays a fraction of principal on its outstanding debt. In particular, as in Leland and Toft (1996), we consider a stationary debt structure where the firm, as long as it remains solvent, periodically issues a constant principal amount b of debt with a given term [T.sub.b] and simultaneously repays the same amount on the previously issued debt reaching their maturity. Hence, at any time t, the maturities of total principal amount B of outstanding debt are uniformly distributed over the interval (t, t + [T.sub.b]), where B = b[T.sub.b]. Each single debt pays a periodic coupon flow of (C P x [DELTA]t)/[T.sub.b], which means that the total interest fees the firm pays on outstanding debt is C P = (wac x B) per year or C P x [DELTA]t per each time period of length [DELTA]t, where wa c is the weighted average coupon rate. Under this model setup, the earnings before interests and taxes, EBIT, realized over the time interval [t - [DELTA]t, t] are given by

[EBIT.sub.t] = [NUI.sub.t] + [NII.sub.t], (24)

while the net income of the insurance firm will be given by

[[pi].sub.t] = [EBIT.sub.t] - C P - [g.sub.t], (25)

where

[g.sub.t] = [[tau].sub.I] max[0, [EBIT.sub.t] - C P] (26)

is the amount of income taxes paid out to the government given a marginal corporate income tax rate of [[tau].sub.I].

Cash Reserves Dynamics and Bankruptcy

The cash reserves represent what the firm holds in the form of pure cash and very liquid assets. These reserves constitute the repayment capacity of the firm and thus are fundamental for its solvency. These cash reserves evolve over time as follows:

[v.sub.t] = ([v.sub.t-[DELTA]t] + [[pi].sub.t][[DELTA].sub.t])(1 - [delta]), [t.sub.1] [less than or equal to] t [less than or equal to] T-1, (27)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [delta] is the dividend payout rate. Notice here that because we are dealing with a stationary debt structure, new financial debt issues offset the repayment of maturing debt claims, so that there is no impact on the outstanding cash reserves.

We assume that the payout policy of the firm consists of paying shareholders a payout rate [delta], such that the dividend payment at time t is given by

[div.sub.t] = [delta]([v.sub.t-[DELTA]t] + [[pi].sub.t][[DELTA].sub.t]).

This means that when net income goes negative, both the debt repayment and dividends will not necessarily be cut, thanks to the cumulated cash reserves. This captures in a realistic way the well-observed facts that: (1) firms facing business losses do not enter into financial distress immediately, and (2) dividends are not strongly correlated with profits realized over the same periods. Under this realistic model, the firm enters into bankruptcy once her cash reserves falls below zero. The random bankruptcy time is then defined as follows:

[theta] = inf{t [member of] ([t.sub.1], ..., T-1): [v.sub.t] < 0}. (28)

When the firm is still solvent by time [t.sub.i] (i.e., {[v.sub.u] > 0, u = [t.sub.1, ..., [t.sub.i]}), we denote this event by {[theta] > [t.sub.i]}. However, the event {[theta] [less than or equal to] [t.sub.i]} means that bankruptcy has occurred at any prior time t [member of] ([t.sub.1], ..., [t.sub.i]). Finally, to designate the event that the firm has not defaulted by time (T-1) (i.e., {[v.sub.u] > 0, u = [t.sub.1], ..., T-1}) and that [v.sub.T] > 0, we shall use the notation {[theta] > T}.

At time T, economic surplus vanishes, business activity ceases and the firm's real assets are worth nothing. At that time, the firm (if default has not occurred in the past) is supposed to be liquidated. The liquidation process takes place as follows: the firm stops issuing new debt, and shareholders gets the residual claim:

max[0; [v.sub.T] - B],

with

[v.sub.T] = [v.sub.T-1] + [[pi].sub.T][DELTA]t. (29)

EQUITY AND GOVERNMENT CLAIMS

Under the developed model aforementioned, the value of equity claim at time to is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

where D(t, u) denotes the stochastic discount factor used to set the price at time t, given [r.sub.t], of a riskless security paying one dollar at time u > t.

The first term represents the present value of expected dividend payoffs, while the firm is still solvent. The second term constitutes the present value of the expected residual payoff, after the payment of outstanding debt, which will go to shareholders upon bankruptcy. Once the firm becomes bankrupt, cash reserves are zero, and shareholders receive nothing in the form of cash. They receive only the residual liquidation value of assets, net of bankruptcy costs evaluated at [omega][A.sub.[theta]] (with 0 [less than or equal to] [omega] < 1), after repayment of outstanding debt. The last term is the present value of the expected residual claim of shareholders upon the liquidation of the firm at date T.

On the other side, the government claim is given by the expected present value of both income tax and premium tax flows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

where [[tau].sub.P][P.sub.t] is the amount of premium tax flows to be received at time t by the tax authority over the realized sales of insurance policies, computed based on a premium tax rate of [[tau].sub.P] (generally, we have [[tau].sub.P] < [[tau].sub.I]). Of course, in contrast to income taxes paid out by the firm on realized profits, premium taxes are sort of built-in fees charged over the insurance policy premium, and they are supported by policyholders. Summarizing, the total tax payoff could be viewed as the sum of an asymmetric payoff representing the income taxes, considered a call option on the firm's EBIT, and a linear premium tax flow.

We see that the government receives taxes while the firm remains solvent, since once the firm becomes bankrupt the tax claim vanishes. It is important to note that contrary to shareholders who will not receive dividends at bankruptcy, the government may receive taxes at bankruptcy time [theta], and this because [[tau].sub.P][P.sub.[theta]] > 0 and [g.sub.[theta]] = [[tau].sub.I] max [0, [EBIT.sub.[theta]] - CP] is not always zero. Recall that the firm is assumed to fail once her cash reserves after-tax repayment are below zero, such that she is no longer able to meet her financial obligations. This explains why we have used the event {[theta] [greater than or equal to] [t.sub.i]} rather than {[theta] > [t.sub.i]}, as in the expression of equity claim, to indicate that taxes at bankruptcy are not necessarily zero.

Observe that the government claim could be decomposed into two subcomponents: an income tax claim, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and a premium tax claim, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]P, as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Before moving on, one can see that the values of both equity and government claims (as well as the economic value of the firm's assets) depends on the length of the time horizon, T. As the concept of duration reflects a time-based measure of interest rate risk in the context of fixed-income securities, the variable T in the model reflects a kind of time-based value of the attractiveness of the insurance industry in terms of future economic surplus.

Under this setup, at any time t, the economic (market) value of the firm, [V.sub.t], is given by the sum of equity, debt and government claims at time t, to which we add the time t present value of the fraction [omega] of the firm's assets lost at default, that is, bankruptcy costs. Note that because taxes are explicitly treated within the model, the economic value of the tax advantage of debt is already included in the value of equity claim. Recalling the definition of the economic value of the firm's real assets and based on the expressions of both equity and government claims, one can easily deduce that the economic (market) value of the firm will be simply given by the sum of cash reserves, investment portfolio, and the economic value of real assets. That is,

[V.sub.t] = [v.sub.t] + [A.sub.t], [t.sub.0] [less than or equal to] t [less than or equal to] T. (35)

MONTE CARLO IMPLEMENTATION OF THE MODEL

The model is implemented numerically using the Monte Carlo method. First, for each time date t = [t.sub.1], [t.sub.2], ..., T, we generate a random sample of M realizations of the sequence of correlated Gaussian variables [([z.sub.i]).sub.i=1, ..., N]. The T time scenarios of these Gaussian samples transformed into the samples of shifted-lognormal claims sizes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], each represents a Monte Carlo path, so we are able to project M paths of aggregate loss [L.sub.t] over the time horizon. Then, at each time date t, the aggregate loss [L.sub.t] is generated using the conditional p.d.f. Pr[[S.sub.t] [member of] da | [n.sub.t] = [bar.n]]. More formally, we proceed as follows. We generate at each time date t a sample of M realizations of the discrete variable [n.sub.t] representing M paths of the claims frequency, using a simple transformation of uniformly distributed variable and the discrete distribution Q(.). Thereafter, for each time t realization of the variable [n.sub.t] corresponding to one realized path, we recover the associated random sum St capturing the claim severity conditional on the realized value of [n.sub.t]. Thus, the M aggregate loss realizations are constructed from the M realizations of the claim frequency n and the sample of N claim size. The uniformity of the random generation of the discrete variable [n.sub.t], as well as the Gaussian numbers [([z.sub.i]).sub.i=1, ..., N] serving to project the correlated lognormal claim size, is enhanced using the stratified sampling technique. The moments matching technique is similarly employed to simulate the correlated Gaussian numbers [(z.sub.i]).sub.i=l, ..., N].

The interest rate process is simulated using the exact transition density technique. Simulation is performed at a very small time increment of length equal to 1/120, which ensures the convergence of our Monte Carlo paths to the true diffusion equation of the short rate process. The same variance reduction techniques of stratified sampling and moments matching are employed to simulate the transition of the short rate process. The stochastic discount facto paths are also constructed from these quasi-continuous paths of the short rate. Afterward, the paths of short rates, stochastic discount factors, and bond prices B(t, [T.sub.r], [r.sub.t]) are determined analytically from the Vasicek's bond price formula and recovered at the discrete time dates t = [t.sub.1], [t.sub.2], ...,T from the originally simulated quasi-continuous paths.

INTRODUCING GF ASSESSMENTS AND RECOUPMENT MECHANISMS

We extend here the basic model developed above in order to capture the financial risk borne by property-liability insurance companies that are members of a GF when failure events occur. We consider the same insurance firm as described earlier. But now the occurrence of a failure event at time [t.sub.0] makes the firm, as well as the other member companies of the GF, engaged in a sort of a loss-recovery program. The program implies that the firm will pay to the GF a periodic amount of h[DELTA]t during the time calendar [t.sub.1], [t.sub.2] ... [T.sub.h] < T. These funds collected by the GF will serve to meet the obligations of the failed company toward its clients. In return, the GF will be treated as a claimant of the failed company and will receive a fraction of the liquidation proceeds. This revenue from liquidation will be returned by the GF to its members in the form of dividends. We assume that this amount H of dividends will be received by our firm at date ([T.sub.h] + 1) and is equal to a fraction (1 - [psi]) [member of] (0, 1) of the sum of amounts h paid previously, that is, H = (1 - [psi])h[DELTA][T.sub.h]. The introduction of the loss fraction [psi] is justified by the fact that the contribution to the GF represents a very risky investment made by member firms to prevent systematic reputation risk generated by the failure events of insurance firms.

How the amount h is fixed for each company as well as the time schedule of the loss-recovery program, ([t.sub.1], [t.sub.2], ..., [T.sub.h]), is based on criteria resulting from an agreement between the GF's affiliated members. Later, we discuss this point in more detail when we describe the calibration of the model.

The main consequence of the occurrence of such failure event is that the financial strength of the firm will be impacted by the payment of the extraordinary obligation h, which makes the firm's cash reserves shifting below its normal level during time period [[t.sub.1], [T.sub.h]] as follows:

[v.sub.t] = ([v.sub.t-[DELTA]t] + [DELTA][v.sub.t])(1 - [delta]), [t.sub.1] [less than or equal to] t [less than or equal to] T-1, (36)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

Under this setup, the occurrence of a failure event can be viewed as a financial risk factor that suddenly pushes up the firm's leverage and thus increases its own failure risk in return. The implications of this "contagion effect" are that the values of both equity and government claims will decrease in response to the higher failure risk, making the probability distribution of cash reserves more skewed. Note that the equity and government claims' valuation formulas derived earlier remain the same under this basic model of contagion risk.

Considering the PTC Recoupment Mechanism

One of the most commonly used recoupment mechanism aiming to attenuating the potential contagion effects of GF assessments seen earlier is the PTC. Under the PTC, firms exposed upon failure events to GF assessments are allowed by the tax authority to keep during the period [[t.sub.1], [T.sub.h]] of GF assessments a fraction [phi] [member of] (0, 1) of premium tax flows collected from policyholders. This amount of tax flows is directly injected into the cash reserves of the firm, which contributes to enhance the solvency profile of the insurer. Formally, under the PTC, the incremental change of cash reserves of the insurer is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

The retained amount of premium taxes is considered as a government subsidy for insurers, in the sense that it is not refundable once the regime of GF assessments is over. The expression of equity claim under the PTC is the same than under the basic model, except that cash reserves dynamics are, first, perturbed by GF assessments and, second, smoothed by the premium tax subsidy. In contrast, the expression of the government claim under the PTC changes to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

This means that during the period [[t.sub.1], [T.sub.h]], the tax authority is supposed to be receiving the income tax in full but only a fraction of premium taxes collected from policyholders. Once the regime of GF assessments ends, tax revenues are established back to their regular level. We see that the PTC has no direct incidence on the expression of equity claim, but it is assumed to affect positively the value of the equity claim by enhancing the solvency probability of the firm through an immediate increase of available cash reserves. For the government, however, allowing for the PTC does not lead to an obvious expected effect on the value of the tax claim. On the one hand, foregoing the fraction of premium tax flows without any subsequent complete or partial refunding from the insurer is a pure loss for the government. On the other hand, allowing for this tax credit will improve the chances of the firm to survive over the stressing environment of GF assessments and, hence, increase the expected value of tax revenues to be generated by the firm beyond time [T.sub.h]. In sum, these two competing effects when regrouped together make the tax authority facing a nontrivial trade-off. We shall examine later the determinants of the solution of this trade-off.

Remark also that the decomposition of the government claim into income tax claim and a premium tax claim is still valid here, with the exception that the premium tax claim is cut by the rate [phi] of tax credit during the program period [[t.sub.1], [T.sub.h]].

Considering the PRS Recoupment Mechanism

Another alternative to the PTC is the PRS provision. Under the PRS, insurers are allowed after supervisory approval to increase premiums charged to policyholders during the period [[t.sub.1], [T.sub.h]] in order to cope with the cash flow pressure caused by GF assessments. Note that the approval of supervisory authorities is only required in the case of rate regulated lines. In contrast to the PTC, the PRS does not imply any transfer of tax revenues from the government to the insurance firms. Rather, the incremental revenues generated by the surcharged premiums are supposed to maintain the cash reserves of the firm away from the bankruptcy floor. Under our model, the PRS is modeled by shifting the net underwriting income upward during the period [[t.sub.1], [T.sub.h]] through some surcharge rate [zeta] > 0 as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

One can easily note that under the PRS, the valuation expressions of both equity and government claims as well as the equation describing the dynamics of the firm's cash reserves remain the same than under the basic model of GF assessments. This results from the fact that there is no transfer of tax revenues from the tax authority to the insurer. As we will discuss later, this property constitutes a main particularity of the PRS in comparison to tax-based mitigating mechanisms.

Considering the Tax Carryforward as a New Recoupment Mechanism

Here, we introduce a new mitigation mechanism and study its properties. To the best of our knowledge, this provision has not been used before in the insurance industry. As we will see later in the article when we implement the model, this new mechanism is designed to yield different and attractive features that are not generally obtained by the usual recoupment provisions described earlier. We call it a tax carryforward provision (TCF), and it is an income tax-based mechanism operating as follows. During the period [[t.sub.1], [T.sub.h]] of GF assessments, the tax authority will receive a fraction [beta] [member of] (0, 1) of the regular income tax and then recuperate the amount,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

at time ([T.sub.h] + 1). This will permit the company to reduce its failure risk caused by the GF extraordinary obligation. Of course, as a consequence of adverse business conditions and/or inadequately aggressive leverage or payout policies, the company may fail before the repayment at date ([T.sub.h] + 1) of residual income taxes. This scenario makes the TCF risky for the government. But, on the other hand, business may turn bad for some periods during the time calendar [[t.sub.1], [T.sub.h]] due to rather normal economic shocks. Over those time periods, the firm will face business losses (i.e., EBIT < 0), and the regular income tax payoff g that would have been paid out to the government in the absence of the TCF is zero.

As a consequence of the TCF, the value of the government claim at time to will be given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (41)

where for any [t.sub.] [less than or equal to] t [less than or equal to] [T.sub.h],

[[pi].sub.t] = [[EBIT.sub.t] - C P] - [BETA][g.sub.t], (42)

[DELTA][v.sub.t] = ([[pi].sub.t] - h)[DELTA]t, (43)

[v.sub.t] = ([v.sub.t-[DELTA]t] + [DELTA][v.sub.t])(1 - [delta]), (44)

with

[[bar.X].sub.t] = [t-1.summation over ([t.sub.i]=[t.sub.1])](1 - [beta])[g.sub.t], [DELTA]t, for [t.sub.1] < t [less than or equal to] [T.sub.h] + 1, (45)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (46)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (47)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (48)

For any [T.sub.h] + 2 [less than or equal to] t [less than or equal to] T, the expressions of the net income [[pi].sub.t] and cash reserves [v.sub.t] are the same as under the basic model. The first term in the expression of the government claim represents the present value of the expected income tax payoffs the government will receive under the TCF during the time period [[t.sub.1], [T.sub.h]] while the firm remains solvent. The second term captures the present value of the expected payoff the government receives once the firm becomes bankrupt at any time during the tax carryforward period [[t.sub.1], [T.sub.h]]. As we can see, once the firm becomes bankrupt, the government will get the full amount of income taxes (without TCF) due on the current EBIT, that is, [g.sub.[theta]], plus the minimum between the liquidation value of the firm's real assets and the amount of residual income taxes not paid by the firm during previous periods owing to the TCF. Since the amount of taxes due on the generated EBIT is collected in full at time of bankruptcy, we have used the event {[theta] > [t.sub.i]} rather than {[theta] [greater than or equal to] [t.sub.i]} in the first term in order to avoid double counting of the same amount of income taxes. The third term can be defined in the same way as the second term. The exception is that the payoff [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represents the terminal payoff the government will receive, in priority over bondholders, when the TCF ends. This payoff is scheduled even if the firm has not defaulted at time ([T.sub.h] + 1). Indeed, as we can see, because the TCF could be viewed as indirect debt financing, the government has a prior claim similar to that of the holders of a debt claim. Finally, the last term describes the income tax payoff the government receives in normal circumstances once the TCF has ended, exactly the same as under the basic model.

The same remark on the decomposition of the government claim into income tax claim and a premium tax claim applies here. Under the TCF, the premium tax claim is unchanged (i.e., the same under the basic model), but the income tax claim gets largely affected by the exchange of cash flows between the insurer and the tax authority during the TCF period [[t.sub.1], [T.sub.h]].

Furthermore, the equity claim at time to under the TCF is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

The first term represents the expected value of dividends to be paid out during the time period [[t.sub.1], [T.sub.h]] while the firm is still solvent. The second term consists of the expected residual claim of shareholders at bankruptcy if the firm defaults at any time during the tax carryforward period [[t.sub.1], [T.sub.h]]. The third term is the expected residual claim of shareholders upon default if the bankruptcy time [theta] coincides with ([T.sub.h] + 1), the date at which the firm is supposed to repay the government the amount of differed income taxes. Again here, once the firm becomes bankrupt, its cash reserves are zero, and shareholders receive nothing in the form of cash. They only receive the residual liquidation value of assets after repayment of government and debt claims.

The remaining terms are associated with the payoffs that will go to shareholders once the TCF as well as the GF's extraordinary obligation have ended and thus can be interpreted similarly as under the basic model. As one can note, the sequence of expected dividends once the TCF has ended starts at date ([T.sub.h] + 1). This is explained by the fact that if the firm stays solvent at date ([T.sub.h] + 1) and pays out the amount [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of differed income taxes to the government from its cash reserves, it will be able to pay out dividends to shareholders. The firm will be liquidated at time ([T.sub.h] + 1) only if cash reserves do not cover the repayment of debt principal and the amount [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of differed income taxes (which also explains the use of the event {[theta] = [T.sub.h] + 1} rather than {[theta] [greater than or equal to] [T.sub.h] + 1} as in the expression of the government claim).

Optimality Criteria for the Different Recoupment Mechanisms

We provide here the criteria upon which the different recoupment mechanisms will be evaluated. We begin by defining the decision rule for the tax authority as the decision of fixing the tax deferral rate [beta] in the context of the TCF or the tax credit rate [phi] under the PTC, based on the maximization of the expected utility of (discounted) future income tax revenues projected over the time horizon ([t.sub.1, [t.sub.2], ..., T). We shall assume in our subsequent numerical analysis that the tax authority exhibits preferences toward risk described by the simple power utility function,

U(x) = [[gamma].sup.-1][x.sup.[gamma]], (50)

where [gamma] (0 < [gamma] [less than or equal to] 1) is a parameter capturing the risk tolerance/aversion of the tax authority (i.e., higher [gamma] implies lower risk aversion, with the extreme case [gamma] = 1 indicating risk-neutrality).

Now, let's define ([E.sup.Basic], [G.sup.Basic]), ([E.sup.GF], [G.sup.GF]) and ([E.sup.PRG], [G.sup.PRG]) as the equity and government claims under the basic model, the GF contagion model and with different recoupment mechanisms (with PRG = PTC, PRS, and TCF), respectively. Therefore, a recoupment program PRG is said sustainable if it verifies the incentive compatibility conditions:

[G.sup.PRG] > [G.sup.GF], (51)

[E.sup.PRG] > [E.sup.GF], (52)

where both equity and government claims, ([E.sup.PRG], [G.sup.PRG]), are valued after optimizing the program's intervention rate [beta] (under the TCF) or [phi] (under the PTC). Moreover, a recoupment program is said economically optimal if it permits a reduction in the level of the cumulative probability of default/insolvency of the firm by comparison to the case of contagion risk where such a provision is not allowed.

An important fact to pointing out here is the particularity of the PRS. In contrast to the programs PTC and TCF implying a kind of government subsidy either via premium taxes or income taxes, the PRS does not imply any intervention from the tax authority. Rather, the PRS requires only the allowance of insurers to raise their premium rates, which means that policyholders are the direct payers for the program instead of the tax authority. This makes the PRS particular in the sense that the government authorities are not explicitly involved in the contagion mitigation process. In the context of our analysis, this creates a challenging issue, since the optimality of the PRS cannot be defined or measured appropriately through the formal criteria described above in the context of PTC and TCF. One approach would consist of introducing a cost or disutility function for the government measuring the collective decrease in welfare caused by the surcharge of premium rates for policyholders. But this avenue would turn out to be subjective, since one cannot ensure a fair comparison between this disutility function and the utility function of tax revenues in the context of the PTC and TCF. Moreover, allowing the insurance firm or even the tax authority to optimize the surcharge rate will inevitably result in an infinitely high solution, since corporate claims are increasing with the net underwriting income. One would impose in that case a trade-off between premium rates and premiums volume to capture the elasticity of clients to charged rates, but this may also turn out to be subjective for comparative purposes. For these reasons, we shall limit our analysis of the PRS to the situation where an exogenous rate of surcharge is applied. We shall also be careful in comparing the outcomes resulting from this setup to those implied by the PTC and TCF provisions.

NUMERICAL RESULTS DISCUSSION

Sustainability and Optimality of Recoupment Mechanisms

Table i reports the equity and government claims for different dividend payout rates ranging from 0.25 percent to 2.50 percent under the different models: the basic model, the contagion model of GF assessments, and the GF assessments coupled with the different recoupment programs (i.e., PTC, PRS, and TCF). For any dividend payout considered, the values of corporate claims under the PTC and TCF are those obtained at the optimal tax credit rate [[phi].sup.*] and the optimal tax deferral rate [[beta].sup.*], respectively, both maximizing the government' utility function. In the context of the PRS, the surcharge rate [zeta] is exogenously fixed at 5 percent, given the lack of government intervention under this program. The table contains three panels, each corresponding to a given level of initial cash reserves. The numerical results are drawn from reasonable basic case parameters (see Table A1 in the Appendix) and assuming a GF assessment amount h of $1. As we can see in Panels B and C, the move from the basic environment to the GF model will lower both the equity and government claims for most dividend payouts considered. This is a direct consequence of the contagion effect. Paying the extraordinary GF assessment amount will increase the failure rate and thus lower the values of the firm's claims. However, this contagion effect is considerably reduced for relatively high cash reserves, given the inverse relationship between default intensity and initial cash reserves. We also observe that as expected, equity claim is increasing with dividend payout, while the government claim is decreasing with dividends. This dividend effect holds no matter the environment under consideration. This is because dividends are a payout for shareholders while at the same time they shrink available cash reserves and thus increase the probability of losing tax revenues upon bankruptcy for tax authority. Furthermore, note that although corporate claims under the PRS are affected in the same way, one can eliminate the contagion effect completely (or even outperform the basic environment claims where there is no contagion risk at all) by assuming a higher premium surcharge rate.

From the perspective of shareholders, we see that for the lowest value of cash reserves (Panel A), the equity claim is lower under the recoupment provisions than under the GF case. This particularly holds for low dividend payouts and could be explained by the option-to-default effect. Indeed, when the intrinsic risk of default due to low level of cash reserves is high, shareholders are better off liquidating the company immediately and suffering the bankruptcy loss rather than maintaining the firm alive thanks to the mitigating recoupment programs and thus reducing the liquidation payoff it receives upon delayed default. When cash reserves are higher (Panels B and C), the firm is worthy and the option-to-default effect is offset. In that situation, indeed, prospective dividend payouts are significant, thus making shareholders more inclined to agree to a recoupment provision rather than resorting to an immediate liquidation of the firm.

With regard to recoupment mechanisms, numerical results are clear: shareholders are better off with the PTC and PRS rather than the TCE This preference holds independently from the level of cash reserves and is particularly pronounced for high dividend payouts. However, increasing cash reserves will considerably reduce the relative preference for the PTC against the TCE The idea here is simple: both the PRS (depending on the applied surcharge rate) and the PTC allow shareholders to cope with contagion risk without bearing any cost for the enhancement of their solvency profile. In contrast, the TCF is designed to transform the mitigation of contagion risk into a time-delayed debt the firm will reimburse once the turmoil period of GF assessments has passed away.

From the position of the tax authority, preferences toward the recoupment mechanisms are quite different. When cash reserves are limited (Panel A) and the risk of insolvency is high, the government claim is increased thanks to the recoupment provisions with respect to the GF case. In that situation, we also note that the TCF yields a higher tax claim than the PTC, while the PRS promises the highest claim value, given the zero-cost the government bears under that program. The relative preference toward the TCF against the PTC is reflected in the selected optimal credit rates. We see that the PTC requires almost a full credit allowance, that is, [[phi].sup.*] near 100 percent, while the TCF achieves a better return on claim with only a tax deferral rate [[beta].sup.*] of 20 percent on average. However, whenever cash reserves are increased and the insolvency risk is reduced (Panels B and C), the tax authority will exhibit a pronounced preference toward the TCF rather than the alternative tax-based program of PTC. In fact, the PTC in that situation is so poor that the government is better off letting the firm go eventually to bankruptcy (the GF case) rather than supplying the premium tax subsidy. The optimal tax deferral rate [[beta].sup.*] in these cases is increased to be around 60 percent on average in Panel B and 90 percent on average in Panel C, since the debt supplied by the government in the form of refundable tax credit is less risky than in the previous case. The optimal tax credit rate [[phi].sup.*], however, reflects exactly the poor performance of the PTC: it is considerably cut by more than a half on average in Panel B with comparison to the previous case (i.e., Panel A) and is considerably reduced in Panel C. Because of the associated zero cost for the government, the PRS (again, conditional on the assumed surcharge rate) continues in that case to outperform the tax-based programs including the TCF.

When confronting the impact of the recoupment programs on both the government and equity claims together, we conclude that these programs (even including the PRS) are not sustainable all the time. When cash reserves are very low (Panel A) and scheduled dividend payouts are low, the option-to-default effect makes shareholders better off with proceeding to an immediate liquidation of the firm, in spite of the tax authority incentive in that case to supply credit. When cash reserves are relatively high and insolvency risk is not alarming (Panel C), the PTC will be optimally rejected by the government (since the GF case of no intervention yields a higher claim value in that case), while both the TCF and, of course, the costless PRS will be retained by the two parties. In the case of medium level of cash reserves (Panel B), the PTC is optimally rejected by the tax authority most of the time (depending on the dividend payout), while both the TCF and the costless PRS are sustainable, with a strict preference for the PRS from the two parties as expected (again, based on the assumed premium surcharge rate).

Figure 1 depicts the default rates (cumulated default probability) over a time horizon of 20 years associated with the different environments considered in Table 1. In each of the three panels of Table 1, we have selected the default rates corresponding to the basic case value of the dividend payout of 1.25 percent to illustrate in Figure 1. As one can see, the contagion effect is well illustrated through the default rates. We observe that the credit curve jumps upward once we move from the basic model to the contagion model GF. The impact of contagion risk caused by GF assessments is propagated over a long time horizon, highlighting the potential significant deterioration of insurers' solvency that would be caused by failure/ruin events.

We also observe how default rates shift downward from the GF case after introducing the recoupment mechanisms. An important result is that the positive effect of these programs in terms of lower default rates takes place independent of the level of initial cash reserves. Numerical results not reported here also show that this effect holds independent of the dividend payout considered. Overall, this means that the recoupment mechanisms will always generate the desired control over contagion risk (lower risk of default), and this independent of the incentive compatibility considerations discussed above. Moreover, we observe that the reduction of the contagion effect one would measure based on the downward displacement of the credit curve from the GF case to the mitigant programs cases, and thus the extent to which these programs add economic value by lowering expected bankruptcy costs, essentially depends on the cash reserves capturing the intrinsic repayment capacity of the firm. The higher these cash reserves, the higher is the intrinsic solvency of the firm and the lower will be the relative gain in term of lower default rates.

When comparing the impact of the TCF on default rates with respect to the PTC, we note that the latter generates a higher impact, no matter the intrinsic solvency of the firm captured through the initial level of cash reserves. However, both of them yields the desired control over default events. Taking into account the sustainability outcomes discussed above and this impact on default rates, one can conclude that the TCF offers a larger scope for economic optimality than the alternative tax-based PTC. Because of its costless nature, a similar comparison against the PRS cannot be directly assessed unless the introduction of additional and arbitrary assumptions about the economic cost of increasing premium rates.

[FIGURE 1 OMITTED]

By keeping the values of cash reserves and dividend payout rate fixed at $7.50 and 1.25 percent, respectively, both Table 2 and Figure 2 report the same numerical results as those reported in Table I and Figure 1. Here, however, the amount h of GF assessment ranges from $0.20 to $2.00 (the credit curves depicted in Figure 2 are those associated with h = $1.00). Three cases are considered, where each one corresponds to a different value of the principal amount of outstanding debt, B. Similar to our earlier results, we note that the introduction of the GF assessment amount generates a contagion effect illustrated through the downward shifts of the values of the firm's claims and the upward shift of the credit curve resulting after the move from the basic model toward the contagion model. We also note that the introduction of the TCF and the PRS (based on the assumed premium surcharge rate) produces a systematic appreciation of the government claim value over the GF case. However, in the case of the PTC, the government claim appreciation is not ensured all the time. Similar to our previous results reported in Table 1, this appreciation is tightly dependent on the solvency risk of the firm measured here through financial leverage. Indeed, the tax authority is found in most of cases better off waiting for the payoffs under contagion pressure as shown in the GF case rather than supplying the tax credit to the firm stipulated by the PTC.

From the view point of shareholders, the homogenous impact of the dividend payout reported in our previous results is no longer valid everywhere now. Independently from the level of financial leverage or the environment considered (including the GF and the recoupment provisions cases, except the flat equity value case of the basic model), the equity claim is found a nonmonotonic function of the assessment amount h. This is a manifestation of the option-to-default effect. When the assessment amount h is small shareholders are better off waiting. Increasing progressively the assessment amount over that continuation region will lower the value of equity. However, once the assessment amount reaches a critical barrier, shareholders exchange their continuation strategy for an immediate liquidation of the firm. The higher the amount h over that liquidation region, the sooner will be the liquidation decision and the higher will be the liquidation proceedings thus contributing to a higher equity value. In sum, this tension between continuation and immediate liquidation generates a U-shape for the equity claim as a function of the assessment amount h. The lower the financial leverage of the firm, the higher the potential solvency of the firm, and thus the more severe this continuation/immediate liquidation trade-off for shareholders. This mechanism explains the striking result according to which shareholders are by far better off with the PTC and PRS rather than the TCF, and this in more important proportions in comparison to the previous results obtained as a function of dividend payouts. Indeed, in the context of this trade-off between continuation and immediate liquidation, the introduction of the TCF implies an additional but delayed leverage for the firm. As a consequence, this makes the TCF equivalent to the GF case over the continuation region where the assessment amount h is small but worse than the GF outcome over the liquidation region where the assessment amount h is high. Of course, all of these mechanisms at play take place as conditional on the optimal tax deferral rate [[beta].sup.*] set by the tax authority.

[FIGURE 2 OMITTED]

By confronting the incentive compatibility conditions for the tax authority and shareholders, one comes to the conclusion that the PTC is only sustainable in the case where financial leverage (or insolvency risk) is high (i.e., Panel A). In contrast, the TCF offers a wider scope for sustainability, but this sustainability is conditional. We see that the TCF's sustainability does not depend on the financial leverage, but rather it is limited to the continuation region. This means that the two parties will always agree on the implementation of the TCF no matter financial leverage, given that the prospects of an immediate liquidation are not so high to make shareholders inclined to accept the government offer. In the case of the PRS, the sustainability scope is maximized with respect to the tax-based recoupment programs, but in general, it is tightly dependent on the assumed surcharge rate [zeta]. Since the mitigation impact on default rates is preserved no matter the financial leverage considered, as illustrated in Figure 2, we are forced to conclude the same way with respect to the economic optimality of these recoupment programs than those concluded with respect to their sustainability.

AGENCY CONFLICTS AND BANKRUPTCY COSTS: COMPARATIVE STATICS

Here we want to explore the same outcomes discussed earlier but after imposing exogenous credit tax rate and tax deferral rate for the tax-based programs, PTC and TCF, as we have already done in the context of the PRS. The results will be very useful, since it is hard to expect that tax authorities will adjust their intervention plan in consideration of the specific solvency and balance-sheet profiles of the different insurance firms, members of the same GF, applying for the program. Our focus is on the tax-based programs, PTC and TCF, where for each scenario we shall assume the same mitigation rate for both programs. In doing so, we also stress the impact of dividend payouts and bankruptcy costs on the sensitivity of equity and government claims to recoupment provision rates. The choice of dividend payout as a variate for the government claim sensitivity is motivated by the potential agency conflicts mechanism. When shareholders apply a high dividend payout rate, available cash reserves of the firm are continuously reduced, which fragilizes the solvency of the firm. In the context of tax-based recoupment programs, this represents an agency cost for the tax authority, since both PTC and income tax report are not adequately compensated by a prudent cash management in that case. With respect to bankruptcy costs, we have seen in our earlier results that the option to default may provide shareholders with the incentive to proceeding to an immediate liquidation of the firm. If the liquidation option is more attractive than the continuation option, we have observed that none of the recoupment programs could be economically sustainable from the perspective of shareholders. Therefore, by stressing bankruptcy costs, we are also able to highlight the impact of the option to default.

Our numerical results are illustrated in Figures 3 and 4. In Figure 3, we consider a highly stressed firm, where the contagion risk caused by the GF assessments is very high. Namely, we assume in that case initial cash reserves of $5 and a GF assessment amount h of $2 (which implies h/[v.sub.0] = 40 percent). Figure 4 depicts the opposite scenario, where the solvency of the firm is high and contagion risk is low. More precisely, we assume that cash reserves are equal to $10 and consider a low GF assessment amount h of $0.40 (which implies h/[v.sub.0] = 4 percent). We report the sensitivity of government and equity claims to the recoupment program rate. Both the PTC and TCF are implemented using the same credit rate; that is, we impose the parity [beta] = [phi]. In addition to the mitigant programs cases, two horizontal lines are also illustrated corresponding to the claims values under the basic environment and the GF case (without mitigant program), respectively. In each figure, Panel A reports the government claim sensitivity to the mitigant program's rate, by varying the dividend payout rate. In Panel A-l, the dividend payout is set equal to the base case parameter [delta] = 1.25 percent, while it is increased to [delta] = 2.50 percent in Panel A-2. In both cases, the rest of model parameters (including bankruptcy costs fraction) follow from the base case parameters. Panel B reports the equity claim sensitivity to the mitigant program's rate by changing the fraction [omega] representing bankruptcy costs to be suffered upon the liquidation of the firm. In Panel B-1, bankruptcy costs are set equal to the base case parameter [omega] = 0.25, while they are increased to [omega] = 0.75 in Panel B-2. In both cases, the rest of model parameters (including the dividend payout rate) follow from the base case parameters.

[FIGURE 3 OMITTED]

In the first case described by Figure 3, where the contagion risk is significant, we see that the tax authority most of the time is better off implementing the TCF rather than the PTC, although both programs achieve the desired appreciation of the tax claim in comparison to the GF case. In fact, this holds independently from the dividend payout rate (Panel A-1 vs. A-2). The relative preference of the tax authority for the TCF against the PTC occurs for credit rates approximately below 70 percent, while the inverse case happens when moving beyond that rate. This is mainly due to the fact that for excessive credit rates (higher than 70 percent), the supply of debt in the form of income tax report becomes a very risky investment for the government, thus making the alternative solution of foregoing premium tax revenues more attractive in that case. In the opposite case of reasonable credit rates (below 70 percent), the TCF offers an enhancement of the firm solvency (lower default rates), which translates into a higher probability of getting reported income taxes paid as well as protecting subsequent premium and income tax revenues (those occurring after Th) from bankruptcy events, while the PTC does not provide the same enhancement. Interestingly, we see that while the same mechanisms take place in both Panels A-1 and A-2, the level of the government claim is shifting downward after increasing the dividend payout when moving from Panels A-1 to A-2. This highlights the agency cost the tax authority bears when supplying tax credit advantages. This means that the only way insurers will maximize the chance of getting the tax credit advantages or of improving the mitigant program conditions (i.e., higher credit rates) is by taking control over capital expenditures, mostly dividend payments. Most importantly, the relative preference of the government to the TCF versus the PTC is unchanged when varying dividend payouts. This is because no matter the dividend payout and its negative impact on the solvency of the firm, the attributes of the TCF compared to those of the PTC remain unchanged. This signals the robustness of the existing differentiation between the two tax-based programs.

[FIGURE 4 OMITTED]

Now, from the perspective of shareholders, and still in the case of high contagion risk (Figure 3), obtained results suggest a completely different trade-off. First, and not surprisingly, Panel B-1 describes the typical situation where immediate liquidation of the firm is more worthy solution for shareholders than the continuation option. This option-to-default effect is due to the relatively low assumed bankruptcy costs and implies that the equity claim under the GF case for which liquidation is imminent is higher than the claim value coming from the continuation option under the mitigant programs, and even under the basic environment (remember here that in addition to the high contagion risk, we have also assume a very low level of cash reserves). When moving to Panel B-2, bankruptcy costs are significantly increased, and thus the payoffs of immediate liquidation of the firm become now less attractive. As a consequence, this makes the mitigant programs worthy for shareholders and transforms the GF case (basic environment) the worst (best) among the all scenarios. Interestingly, when only focusing on the situation where recoupment programs are both more worthy than the GF case (i.e., Panel B-2), we observe that shareholders exhibit the same relative preference for these programs than the tax authority. We see the TCF more attractive than the PTC when a credit rate below 60 percent approximately is applied, while the inverse is true beyond that level. This alignment of incentives between the two parties comes from the threat of loss caused by higher bankruptcy costs. Indeed, when the liquidation of the firm is less attractive than the continuation option because of higher bankruptcy costs, shareholders are more inclined to implement the TCF for its most effective enhancement of solvency (since applying the same tax deferral rate than PTC has a higher impact on cash reserves), and this even after considering the provision of reimbursement of reported taxes at the end of the program. But this preference is completely switched in favor of the PTC when a high credit rate is applied, since in that case the "free lunch" representing the tax subsidy of the PTC is sufficient enough to cope with contagion risk, while the TCF--even always doing good in terms of solvency enhancement--starts to be costly in terms of delayed reimbursement of reported taxes.

In Figure 4 different patterns occur where contagion risk is negligible and the intrinsic solvency of the insurer is important. By looking at the different four cases depicted by Panels A-1 to B-2, we see a situation where the mitigant programs are not worthy for the tax authority, while by symmetry effect, they totally represent a sort of "free lunch" for shareholders. Indeed, the contagion risk issue in those cases is so minor that the government claim is almost the same between the basic environment and the GF case. This, as a result, inevitably makes the mitigant programs strictly unattractive in comparison to the alternative solution of not supplying any tax advantages. When comparing the two mitigant programs in that case, we note without any surprise a pronounced preference toward the TCF, since it allows the tax authority to minimize the dead loss caused by an eventual intervention. Again, by the same symmetry effect, shareholders will exhibit an opportunistic preference for the PTC for its costless nature. Overall, the situation described here alerts us that not any government intervention to cope with contagion risk is desired in the sense of collective welfare. Indeed, when the contagion issue is not significant, the tax authority is better off cutting intervention plans rather than providing assistance plans in a systematic way.

FURTHER CONSIDERATIONS" BUSINESS CYCLE AND DEFAULT CORRELATION

GF assessments are usually procyclical in the sense that most of property-casualty insurance failures occur either near the bottom of the insurance business cycle, where the return on the equity-to-loss ratio is at its lowest level or either just as the recovery begins. This means that the need for assessments generally amplifies the natural financial fragility of insurers caused by the regular business cycle. In our previous numerical simulations, we have shown that weaker initial financial conditions for an insurer, captured either through low cash reserves or a high level of outstanding debt obligations, increase considerably the stress caused by GF assessments. To evaluate the impact of the timing of these assessments as they relate with the insurance business cycle, one simply needs to stress the model's variables even further. These include the insurer's profitability due to its insurance operations reflecting the magnitude of its aggregated losses. Based on additional numerical simulations (not reported here), we can easily conclude that the business cycle effect is equivalent to our scenario of weaker financial conditions that we examined and discussed earlier.

Furthermore, in all of our modeling exercise, the tax authority is put in a position where it is concerned with the effect of assessments on the financial situation of only one surviving insurer. However, in real-world situations, the tax authority faces a pool of surviving firms that will consider taking advantage of these recoupment measures. We can intuitively postulate that the incentives for the government to allow these provisions will be boosted by the threat of a default correlation among surviving firms. Indeed, when most surviving insurance firms are subject to strain caused by GF assessments, then there is a significant risk for the government to lose most of its future tax revenues upon the realization of more failures. Clearly, this portfolio-wide risk would encourage the tax authority to implement such tax-based mechanisms even much more quickly than when the number of distressed insurers is small.

CONCLUSIONS

In this article, we have examined the impact of assessments charged by a GF following the failure of an insurance company on the financial safety of surviving insurers. After building a realistic contingent claim model of the property-casualty insurance firm, GF assessments are integrated into the model to highlight the implications in terms of default rates and corporate claims values for the fund's surviving members. Our analysis reveals that although GF assessments do not necessarily cause an immediate failure of surviving members in the usual sense associated with the notion of contagion, the long-run financial safety of these members is shown to be significantly perturbed by these ex post charges. Indeed, our results show the existence of a latent type of contagion whose impacts would be exacerbated whenever the financial conditions of insurers---captured in our model through available cash reserves and outstanding financial obligations--are weak.

The main issue we investigate in the article, after highlighting the contagion effect of GF assessments, is to what extent recoupment mechanisms would be sustainable and optimal from an economic perspective. Our criteria here are twofold: (1) the enhancement of default rates prospects for sustainability outcomes and (2) the appreciation of corporate claims for optimality outcomes. Both the PTC and PRS provisions, largely used in the U.S. property-casualty industry, are examined. We also introduce a new tax-based recoupment program, the TCF provision, which aims to yield different attributes from those of existing mechanisms. Our analysis unambiguously reveals that while all recoupment mechanisms are sustainable in the sense they reduce default probabilities of insurers, tax-based recoupment programs fail most of the time to achieve a simultaneous improvement of both equity and tax claims. This is mainly due to the option to default and dividend payout mechanisms that create a sort of divergence between the relative preferences of equityholders and tax authority to one particular recoupment program. Our results also suggest that the main attractiveness of the PRS provision is limited to the simple fact the bill of this recoupment program is passed through to customers. From an economic welfare perspective this provision does certainly not lead to a superior outcome when compared to the other tax-based programs.

Finally, we believe that further research would be highly constructive if a new focus is made in order to propose GFs that could better mitigate the contagion risk problem. The allowance for ex post recoupment provisions that could be implemented upon failure events to enhance the solvency profile of surviving insurers simultaneously with ex ante risk-adjusted assessments, analogous to deposit insurance in the banking industry, represents an alternative to the existing practice that should attract the interest of both insurers and regulators.

DOI: 10.1111/j.1539-6975.2010.01367.x

APPENDIX: BASIC CASE PARAMETERS

TABLE A1 Numerical Values Assigned to Model Parameters for Simulation Purposes Basic Case Parameters Net underwriting income Claims number N 100 Claim intensity [lambda] 5% Claim size--average level parameter [bar.s] $0.25 Claim size-volatility parameter [sigma] $0.10 Claim size-correlation parameter [rho] 0.25 Premium margin parameter [eta] 0.40 Interest rate dynamics Equilibrium level [bar.r] 4.00% Initial value [r.sub.w] 4.00% Volatility [sigma] r 1.00% Mean-reversion speed [theta] r 0.20 Cash reserves and investment activity Initial cash reserves [v.sub.0] $7.50 Investment portfolio return parameter [alpha] 0.50 Equivalent bond maturity T r 5 years Financial debt Total principal amount B $25 Maturity Tb 20 years Weighted average coupon wac 8.00% Other parameters Dividend payout rate [delta] 1.25% Corporate tax rate [tau] 35% Bankruptcy costs [omega] 25% Required return on franchise value [bar.[PHI]] 6.00% Loss fraction [PSI] 20% Guarantee fund obligation time horizon T h 5 years Government's risk aversion parameter [gamma] 0.50

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Gilles Bernier is at the Department of Finance and Insurance and the Industrial-Alliance Insurance Chair, Laval University, Pavilion Palasis-Prince, Quebec City, Quebec, Canada GIK 7P4. Ridha M. Mahfoudhi is at Desjardins, Montreal, Quebec, Canada; Affiliated Researcher, Department of Finance and Insurance, Laval University. The authors can be contacted via e-mail: gilles.bernier@fsa.ulaval.ca and ridha.mahfoudhi@fsa.ulaval.ca, respectively. We acknowledge the financial support of the Property and Casualty Insurance Compensation Corporation (PACICC) for this research. Our special thanks to David Babbel and Darrell Leadbetter for their detailed comments and very helpful suggestions, which have helped us improve the quality of this article. We would also like to thank two anonymous referees of this journal for the relevance of their comments that largely contributed to improve the content of the article. We are grateful to Norma Nielson for her helpful discussion of the paper at the 2007 meeting of the American Risk and Insurance Association, as well as Jean-Francois Larocque and Jean Cote for their comments and suggestions. All remaining errors are ours. This article is dedicated to our deceased colleague, Dr. Klaus Fischer, who was a starting member of this research project.

(1) Indeed, the recoupment provision in property-casualty GF laws in the various states is described as follows: "This section details the method provided for in the guaranty association act which allows member insurers to recover the cost of assessments; the usual methods being 'rates and premiums' or 'premium tax offset.' The rates and premiums method allows member insurers to recover the cost of assessments by permitting member insurers to include the cost of assessments as a factor in determining rates and premiums that the insurer is allowed to charge for its policies. The premium tax offset method allows member insurers to recover the cost of assessments by permitting a reduction in premium taxes payable by the member insurer." See http://beta.ncigf.org/guaranty/definitions.asp.

(2) Contagion risk typically refers to the spillover effects of shocks from one or more firms to other firms. The vast majority of research carried out on contagion risk is related to the banking sector. Contagion in banking has been studied at the domestic level by Shiller (1989), De Bondt (1995), Schoenmaker (1996), Chen, (1999), and, at the international level, by Garber and Grilli (1989). These authors observed different transmission mechanisms of contagion, such as panics, revision of expectations, rumors on liquidity shocks, and transmission of the loss in value of assets. Likewise, Leitner (2005) analyzed private bailout mechanisms designed to cushion contagion effects. Although more subtle than a run on bank deposits, there also appears to exist contagion risk in the insurance industry. For example, Angbazo and Narayanan (1996) and Miller and Polonchek (1999) show the existence of contagion effects in the insurance industry due to things such as asset quality problems and/or catastrophic events.

TABLE 1 Equity and Government Claims for Different Values of Cash Reserves and Dividend Payouts Basic Model GF Model Equity Government Equity Government Claim Claim Claim Claim Panel A: Initial Cash Reserves = $5.00 Dividend payout (%) 0.25 2.99 23.04 4.70 18.74 0.50 4.71 22.88 6.12 18.36 0.75 6.19 22.66 7.29 18.03 1.00 7.40 22.41 8.28 17.54 1.25 8.40 22.12 9.06 17.12 1.50 9.26 21.76 9.74 16.63 1.75 10.01 21.38 10.33 16.08 2.00 10.69 20.95 10.83 15.61 2.25 11.28 20.48 11.27 15.13 2.50 11.83 20.01 11.67 14.56 Panel B: Initial Cash Reserves = $7.50 Dividend payout (%) 0.25 3.36 23.49 3.43 22.61 0.50 5.33 23.40 5.27 22.41 0.75 7.01 23.29 6.83 22.17 1.00 8.37 23.15 8.08 21.90 1.25 9.50 22.95 9.12 21.56 1.50 10.45 22.70 9.99 21.18 1.75 11.28 22.37 10.76 20.72 2.00 12.03 22.00 11.43 20.25 2.25 12.70 21.59 12.03 19.74 2.50 13.32 21.10 12.57 19.15 Panel C: Initial Cash Reserves = $10.00 Dividend payout (%) 0.25 3.93 23.65 3.63 23.46 0.50 6.12 23.57 5.71 23.37 0.75 8.02 23.48 7.50 23.25 1.00 9.55 23.39 8.93 23.12 1.25 10.81 23.22 10.11 22.94 1.50 11.85 23.04 11.10 22.70 1.75 12.76 22.80 11.95 22.43 2.00 13.57 22.50 12.71 22.08 2.25 14.29 22.15 13.39 21.70 2.50 14.95 21.74 14.00 21.26 GF Mixed With PRS Equity Government Claim Claim Panel A: Initial Cash Reserves = $5.00 Dividend payout (%) 0.25 3.70 21.86 0.50 5.36 21.61 0.75 6.75 21.35 1.00 7.87 21.08 1.25 8.80 20.77 1.50 9.58 20.43 1.75 10.29 19.93 2.00 10.91 19.45 2.25 11.45 18.90 2.50 11.94 18.38 Panel B: Initial Cash Reserves = $7.50 Dividend payout (%) 0.25 3.42 23.73 0.50 5.38 23.61 0.75 7.07 23.49 1.00 8.42 23.30 1.25 9.53 23.11 1.50 10.46 22.89 1.75 11.27 22.56 2.00 12.00 22.20 2.25 12.63 21.83 2.50 13.22 21.34 Panel C: Initial Cash Reserves = $10.00 Dividend payout (%) 0.25 3.92 24.03 0.50 6.10 23.97 0.75 8.00 23.90 1.00 9.53 23.79 1.25 10.77 23.66 1.50 11.80 23.49 1.75 12.70 23.24 2.00 13.50 22.96 2.25 14.19 22.67 2.50 14.84 22.28 GF Mixed With PTC Tax Equity Government Credit Claim Claim Rate Panel A: Initial Cash Reserves = $5.00 Dividend payout (%) 0.25 3.54 20.21 88% 0.50 5.27 20.02 90% 0.75 6.74 19.80 91% 1.00 7.92 19.54 92% 1.25 8.91 19.20 93% 1.50 9.73 18.89 94% 1.75 10.44 18.52 95% 2.00 11.08 18.08 96% 2.25 11.64 17.68 97% 2.50 12.25 17.26 98% Panel B: Initial Cash Reserves = $7.50 Dividend payout (%) 0.25 3.38 22.40 37% 0.50 5.32 22.12 40% 0.75 7.00 21.85 47% 1.00 8.34 21.66 51% 1.25 9.52 21.34 56% 1.50 10.45 21.09 62% 1.75 11.46 20.61 70% 2.00 12.20 20.25 77% 2.25 12.73 19.94 83% 2.50 13.56 19.39 86% Panel C: Initial Cash Reserves = $10.00 Dividend payout (%) 0.25 3.67 23.29 6% 0.50 5.77 23.19 7% 0.75 7.58 23.08 9% 1.00 9.03 22.95 10% 1.25 10.21 22.79 12% 1.50 11.20 22.57 14% 1.75 12.06 22.30 18% 2.00 12.83 21.96 21% 2.25 13.76 21.38 26% 2.50 14.52 20.83 32% GF Mixed With TCF Tax Equity Government Deferral Claim Claim Rate Panel A: Initial Cash Reserves = $5.00 Dividend payout (%) 0.25 3.61 20.79 18% 0.50 5.11 20.64 19% 0.75 6.42 20.38 19% 1.00 7.50 20.05 19% 1.25 8.40 19.66 19% 1.50 9.14 19.29 19% 1.75 9.82 18.78 19% 2.00 10.39 18.36 20% 2.25 10.92 17.85 21% 2.50 11.39 17.27 22% Panel B: Initial Cash Reserves = $7.50 Dividend payout (%) 0.25 3.28 22.84 55% 0.50 5.16 22.66 57% 0.75 6.79 22.36 59% 1.00 8.02 22.22 63% 1.25 9.07 21.99 65% 1.50 9.95 21.68 66% 1.75 10.75 21.33 67% 2.00 11.45 20.89 64% 2.25 12.07 20.32 61% 2.50 12.65 19.89 58% Panel C: Initial Cash Reserves = $10.00 Dividend payout (%) 0.25 3.63 23.46 99% 0.50 5.73 23.37 98% 0.75 7.53 23.25 97% 1.00 8.97 23.12 96% 1.25 10.15 22.94 94% 1.50 11.15 22.71 93% 1.75 12.01 22.45 91% 2.00 12.79 22.10 90% 2.25 13.47 21.71 89% 2.50 14.10 21.27 88% Note: This table provides the values of both equity and government claims for different levels of initial cash reserves and dividend payouts under the following different models: the basic model, the model of GF assessments, and guaranty fund assessments with the recoupment programs PTC, PRS, and WE Except for the dividend payout, the model parameters are those of the basic case (see Table A1 in the Appendix). The GF obligation amount is supposed equal to $1. TABLE 2 Equity and Government Claims for Different Values of Total Debt and the Guaranty Fund Assessment Amount Basic Model GF Model Equity Government Equity Government Claim Claim Claim Claim Panel A: Total Debt = $30 GF assessment (h) 0.20 7.35 17.60 7.26 17.60 0.40 7.35 17.60 7.19 17.49 0.60 7.35 17.60 7.14 17.20 0.80 7.35 17.60 7.14 16.56 1.00 7.35 17.60 7.15 15.59 1.20 7.35 17.60 7.25 14.02 1.40 7.35 17.60 7.35 11.92 1.60 7.35 17.60 7.44 9.61 1.80 7.35 17.60 7.49 7.47 2.00 7.35 17.60 7.52 5.54 Panel A: Total Debt = $25 GF assessment (h) 0.20 9.46 22.82 9.37 22.79 0.40 9.46 22.82 9.30 22.72 0.60 9.46 22.82 9.27 22.54 0.80 9.46 22.82 9.29 22.18 1.00 9.46 22.82 9.37 21.45 1.20 9.46 22.82 9.56 20.25 1.40 9.46 22.82 9.86 18.44 1.60 9.46 22.82 10.25 16.03 1.80 9.46 22.82 10.71 13.00 2.00 9.46 22.82 11.12 10.18 Panel A: Total Debt = $20 GF assessment (h) 0.20 12.50 25.66 12.41 25.62 0.40 12.50 25.66 12.34 25.59 0.60 12.50 25.66 12.28 25.52 0.80 12.50 25.66 12.24 25.39 1.00 12.50 25.66 12.27 25.05 1.20 12.50 25.66 12.37 24.44 1.40 12.50 25.66 12.60 23.33 1.60 12.50 25.66 12.91 21.69 1.80 12.50 25.66 13.45 19.28 2.00 12.50 25.66 14.06 16.19 GF Mixed With PRS Equity Government Claim Claim Panel A: Total Debt = $30 GF assessment (h) 0.20 7.73 18.93 0.40 7.64 18.92 0.60 7.56 18.84 0.80 7.51 18.62 1.00 7.47 18.12 1.20 7.47 17.20 1.40 7.49 15.80 1.60 7.55 13.87 1.80 7.58 11.49 2.00 7.61 9.14 Panel A: Total Debt = $25 GF assessment (h) 0.20 10.00 23.54 0.40 9.92 23.53 0.60 9.84 23.48 0.80 9.79 23.35 1.00 9.78 23.08 1.20 9.81 22.56 1.40 9.94 21.57 1.60 10.11 20.09 1.80 10.44 17.87 2.00 10.78 15.09 Panel A: Total Debt = $20 GF assessment (h) 0.20 13.15 26.12 0.40 13.07 26.12 0.60 12.99 26.09 0.80 12.92 26.06 1.00 12.86 25.97 1.20 12.84 25.76 1.40 12.87 25.34 1.60 12.99 24.54 1.80 13.24 23.12 2.00 13.60 21.13 GF Mixed With PTC Tax Equity Government Credit Claim Claim Rate Panel A: Total Debt = $30 GF assessment (h) 0.20 7.78 17.08 68% 0.40 7.69 17.07 74% 0.60 7.70 16.93 77% 0.80 7.63 16.73 80% 1.00 7.67 16.31 84% 1.20 7.66 15.58 88% 1.40 7.64 14.51 92% 1.60 7.73 13.08 94% 1.80 7.75 11.10 96% 2.00 7.74 8.79 98% Panel A: Total Debt = $25 GF assessment (h) 0.20 9.47 22.62 10% 0.40 9.49 22.45 16% 0.60 9.43 22.34 22% 0.80 9.60 21.84 40% 1.00 9.69 21.39 54% 1.20 10.03 20.63 77% 1.40 10.09 19.90 88% 1.60 10.28 18.81 92% 1.80 10.47 17.26 95% 2.00 10.75 15.03 99% Panel A: Total Debt = $20 GF assessment (h) 0.20 12.53 25.42 5% 0.40 12.45 25.39 7% 0.60 12.39 25.33 11% 0.80 12.35 25.21 15% 1.00 12.45 24.80 22% 1.20 12.66 24.21 43% 1.40 12.89 23.52 61% 1.60 13.14 22.72 78% 1.80 13.41 21.75 94% 2.00 13.64 20.28 97% GF Mixed With TCF Tax Equity Government Deferral Claim Claim Rate Panel A: Total Debt = $30 GF assessment (h) 0.20 7.27 17.60 99% 0.40 7.21 17.49 97% 0.60 7.17 17.25 90% 0.80 7.14 16.90 60% 1.00 7.12 16.37 40% 1.20 7.06 15.84 20% 1.40 7.01 14.97 12% 1.60 6.96 13.50 11% 1.80 6.88 11.80 9% 2.00 6.78 9.89 9% Panel A: Total Debt = $25 GF assessment (h) 0.20 9.38 22.79 97% 0.40 9.32 22.72 95% 0.60 9.30 22.57 90% 0.80 9.30 22.30 80% 1.00 9.31 21.98 58% 1.20 9.36 21.47 40% 1.40 9.40 20.79 20% 1.60 9.46 19.88 14% 1.80 9.59 18.46 11% 2.00 9.77 16.41 10% Panel A: Total Debt = $20 GF assessment (h) 0.20 12.42 25.62 98% 0.40 12.36 25.59 97% 0.60 12.30 25.52 96% 0.80 12.28 25.39 92% 1.00 12.30 25.11 80% 1.20 12.33 24.80 64% 1.40 12.35 24.50 44% 1.60 12.43 23.90 30% 1.80 12.52 23.05 20% 2.00 12.63 22.02 10% Note: This table provides the values of both equity and government claims for different levels of principal amount of total debt and GF obligation amount under the following different models: the basic model, the model of GF assessments, and GF assessments with the recoupment programs PTC, PRS, and TCF. The model parameters, including the dividend payout, are those of the basic case (see Table Al in the Appendix). Initial cash reserves are supposed equal to $7.50.

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Author: | Bernier, Gilles; Mahfoudhi, Ridha M. |
---|---|

Publication: | Journal of Risk and Insurance |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Dec 1, 2010 |

Words: | 16050 |

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