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Valuation of early exercisable interest rate guarantees.

INTRODUCTION

This article focuses on the valuation of financial contracts that entail an explicit interest rate guarantee. In particular, we concentrate on interest rate guarantees with the possibility of premature or early exercise. The contract in question is arranged between two agents. An investor makes a deposit with a fund manager, who invests in a definite reference portfolio and guarantees a certain minimum return on the investment. The guarantee applies until the maturity date of the contract, but the investor may decide to invoke the guarantee prematurely if doing so is advantageous.

An arrangement of this sort is an explicit or implicit characteristic of many products in the financial markets as well as in the insurance industry. Because of the early exercise feature of the guarantee element, the financial economist will quickly recognize the similarity between the above contract and American options, which can be exercised at any time until maturity - contrary to European options, which can be exercised at maturity only.

For the actuary, the early exercise feature brings to mind the surrender feature or sell-back option of some life insurance products. In particular, the contract has characteristics in common with unit-linked or equity-linked life insurance policies (ELLIPs) with interest rate guarantees included.

This article examines the intersection between the financial and the actuarial sciences and attempts to apply some recent option theoretical results in order to obtain a market-based valuation of contracts linked to some reference portfolio (the unit) and with an American type of interest rate guarantee added.

Previous literature in this area reflects the fact that the option to exercise early - the surrender feature - has not been a common element of unit-linked life insurance products. For example, Brennan and Schwartz (1976) obtain equilibrium prices for the most common ELLIPs by an application of European option pricing theory.(1) The analogy between European options and guaranteed ELLIPs is further developed in subsequent articles by Boyle and Schwartz (1977), Brennan and Schwartz (1979), Collins (1982), Nielsen and Sandmann (1994), Ekern and Persson (1995), and Aase and Persson (1997).

However, ELLIPs with an American type of interest rate guarantee do in fact exist. In Norway, for example, a variety of these types of policies are offered by major insurance companies, and, in Denmark, banks have recently been offering their clients participation in investment programs with contract characteristics that are very similar to the contracts that we study here.(2)

Furthermore, since the option to exercise early is an embedded characteristic in many traditional life insurance contracts, our model may prove helpful in the valuation of other contracts - or elements of contracts - which, to a reasonable degree of accuracy, can be approximated as a pure American type of ELLIP.

Our motivation for the present article was further nourished by the observation that, in the last twenty years, there has been a dramatic drop in the safety margin between the earning power of the insurance companies and the issued interest rate guarantees leading to potential solvency problems. In response to this development, many companies have lowered the level of their guaranteed interest rates. Other companies have responded to the potential solvency problems by changing the type of option implicit in the interest rate guarantee from American to European. As a result, it is now quite common that the insurance companies have policyholders with different types of interest rate guarantees in the same fund. Ignorance of the fair valuation of these guarantees will naturally lead to an inequitable treatment of different classes of policyholders. We provide some numerical results which illustrate that the redistribution of wealth among the various classes of policyholders stemming from incorrect pricing of the guarantees can be substantial.

As in many related studies on ELLIPs, we utilize the powerful framework of continuous time contingent claims valuation. Our contribution to the existing literature consists in taking explicitly into account the possibility of early exercise and, in particular, doing this by applying and manipulating concepts and results from American option pricing theory.

The next section further discusses the American option element of the interest rate guarantee. Then the formal model is introduced and the equilibrium value of the interest rate guarantee is derived. We discuss some alternative schemes for compensating the issuer of interest rate guarantees and provide some numerical examples.

THE INTEREST RATE GUARANTEE

The interest rate guarantee contract analyzed in this article is arranged between two agents. In the case where ELLIPs are analyzed, it is natural to refer to these agents as the insurance company and the insured individual, respectively. However, in other situations where interest rate guarantees occur as part of more general investment programs, the agents could be called the fund manager and the investor. We maintain the latter terminology for the remainder of the article.

The terms of the basic contract are as follows: At the time of initiation of the contract, t = 0, the agents enter into a contract of nominal value G. The equilibrium value of the contract - that is, the amount the investor must pay the fund manager in exchange for the contract - is denoted by [[Gamma].sub.0]. There are no further payments from the investor to the fund manager during the life of the contract. Hence, the term single premium contract. The fund manager immediately invests the amount G in a well-defined reference portfolio (e.g., a market index portfolio) and keeps track of the continuous development of the market value of this investment on behalf of the investor.(3) Any dividends from the reference portfolio are immediately reinvested.

The investor has the right to terminate the contract without prior notification at any time, t, during the interval [0,T], where T denotes the maturity date of the contract. At this time, and hence at any time, the reference portfolio investment is insured in the sense that the fund manager guarantees a rate of return of at least [r.sub.G] up to the time where the investor chooses to terminate the contract. In other words, at the termination date, the investor is entitled to receive the greater of either the market value of the reference fund investment or an amount equal to the initial investment, G, compounded continuously up to the termination date with the guaranteed interest rate, [r.sub.G]. Thus, we can refer to G [e.sup.[r.sub.G]t] as the guaranteed amount at time t.

For the moment we leave aside the issue of credit risk - that is, default on the side of the fund manager - but some ideas for a possible extension of the model in this respect are sketched at the end of the article.

The financial contract described above clearly has many characteristics in common with those of a derivative security. In particular, the possibility of early exercise brings to mind the American type of option. We attempt below to describe the characteristics of the interest rate guarantee in more detail, and we address the problem of determining the relation between G and [[Gamma].sub.0] in a fairly constructed contract. Clearly, the difference between the initial investment, [[Gamma].sub.0], and the guaranteed amount, G, should serve to compensate the fund manager for the issued guarantee. In the basic form of the contract, this necessary compensation takes the form of an initial payment - an entrance fee, if you like - but other constructions are possible. For instance, we later analyze the situation where the investor pays the compensation in the form of a proportional fee on exit.

The following sections first determine the equilibrium value of the basic contract described above and, second, discuss the various forms a compensation to the fund manager can take and determine accompanying analytic formulas for the exact determination of such premiums.

THE MODEL

Initially, we consider a simple financial market in which all activity occurs on a filtered probability space ([Omega],T, {[T.sub.t]}, Q) supporting Brownian motion with a finite horizon date T.(4) We suppose that a risky and a riskless asset trade in continuous time, and we refer to these assets as a reference portfolio of risky assets and a default free bond, respectively. The risk-free rate of interest, or, alternatively, the rate of return on the bond, r, is assumed constant, while it is assumed that the value of the risky asset, [Mathematical Expression Omitted], evolves through time according to the stochastic differential equation

[Mathematical Expression Omitted]. (1)

In accordance with the assumption in Black and Scholes' (1973) pioneering article, the drift and volatility parameters, [Mu] and [Sigma], of the above process are assumed constant, and equation (1) thus defines the well-known geometric Brownian motion.(5) In addition, we maintain the standard assumptions about continuous time perfect markets; that is, we assume that assets trade continuously and that there are no transactions costs or taxes of any kind.

The Value of the Contract

Denoting by v(t) the market value at time t, t [element of] [0,T], of the fund manager's investment in the reference portfolio on behalf of the client, we have

dV(t): [Mu] v(t) dt+ [Sigma] V(t) dz(t), V(0) = G, (2)

where [Mu] and [Sigma] are as in equation (1).

Now, according to the contract conditions described above, the investor can claim the amount

C(t) [equivalent to] max {G [e.sup.[r.sub.G]t], V(t)}, (3)

at any time t [equivalent to] [0,T]. We restrict ourselves below to considering the economically meaningful contracts with [r.sub.G] [equivalent to] [0,r].

Clearly the payoff function C(t) defined in equation (3) can be evaluated at any given date (it is measurable with respect to the information available at time t) and, hence, given an exercise date and the contemporaneous value of the reference portfolio investment, the investor will know whether to invoke the interest rate guarantee or to simply claim the reference portfolio market value. The problem is, however, that at time zero the exercise date is itself a stochastic variable, and we must take proper account of this fact in our attempt to determine the equilibrium value of the contract. As a consequence, this valuation exercise must include the identification of a rule for choosing the optimal date for exercising the contract. Alternatively, in the terminology of stochastic control theory, equation (3) is a (stochastic) reward function, and the task of determining the present value of this stochastic reward involves the identification of an optimal stopping time. So in essence the problem is an optimal stopping-stochastic horizon problem since it is not possible to predict at time zero the investor's choice of termination date. Whether the investor will find it optimal to terminate the contract at a given date will depend on the investor's rational assessment of current gains and future possibilities - precisely at that date.

The solution of problems of optimal stopping is often quite difficult, if not impossible, to characterize by analytic formulas, but we demonstrate below that in this particular case a simple reformulation of the reward function (3), allows us to decompose the valuation problem into two minor problems. The first of these subproblems is rather trivial, while the other is shown to be a variation of the American put option pricing problem. This latter problem is a "classic" in financial theory, and a quasi-analytic solution has been identified in a series of independent articles by Kim (1990), Jamshidian (1992), Myneni (1992), and Carr, Jarrow, and Myneni (1992).

We proceed by first rewriting equation (3) to a form that demonstrates more clearly the presence of an American option payoff element in the interest rate guarantee contract. Simple manipulation of equation (3) yields

C(t) [equivalent to] max{G [e.sup.[r.sub.G]t], V(t)},

= max{G [e.sup.[r.sub.G]t] - V(t), 0} + V(t), (4)

or, alternatively,

C(t) = [e.sup.[r.sub.G]t] max{G - [e.sup.[-r.sub.G]t] V(t), 0} + V(t).(5)

From equation (4), it is evident that the contract essentially is a claim on the reference fund plus something that is similar to an American put option, however with an exercise price that increases in time at the rate [r.sub.G].(6)

We now demonstrate that an analytic expression for the value at tune zero of the interest rate guarantee contract can be obtained by the help of the risk-neutral pricing methodology of Harrison and Kreps (1979) and some recent results on American option pricing theory.

The initial value of the contract, [[Gamma].sub.0], can be represented (cf., e.g., Karatzas, 1988, 1989) as follows:

[Mathematical Expression Omitted], (6)

where [S.sub.0,T] denotes the class of [T.sub.t]-stopping times taking values in [0,T], and where expectation is taken with respect to the risk-neutral probability measure, [Mathematical Expression Omitted].(7)

The representation in equation (6) of the time zero contract value as the supremum (over all permissible stopping times) of the risk-neutral expectation of the discounted reward is quite logical and follows from a definition of equilibrium prices as the minimum initial wealth required to initiate a replicating strategy for the contract in an arbitrage free market (see Karatzas, 1988, 1989).

Since under the risk-neutral probability measure [e.sup.-r t] V(t) is a [Mathematical Expression Omitted]-martingale (hence the alternative term martingale measure), the expectation of the second component in equation (6) can be dealt with separately by the help of the optional sampling theorem (see, e.g., Karatzas and Shreve, 1988, p. 19). Optional sampling tells us that

[Mathematical Expression Omitted], (7)

which means that, for purposes of determining the optimal stopping time in relation to equation (6), we may restrict our attention to finding the supremum of the expectation of the first component inside the expectation. Summing up, we have shown that we may write

[Mathematical Expression Omitted]. (8)

Next, substituting [Mathematical Expression Omitted]for V([Tau]) in equation (8), we obtain

[Mathematical Expression Omitted], (9)

where we have defined [r.sup.*] [equivalent to] r - [r.sub.G]. Hence, [r.sup.*] denotes the spread between the risk-free rate of interest and the guaranteed interest rate.

At this point, the supremum remaining in equation (9) is a version of a well-known problem from the theory of optimal stopping of diffusions in general and from recent research on American put option pricing in particular. A comparison with the problem studied by Myneni (1992) reveals that the stopping problem (the "supremum part" of equation (9)) corresponds to Myneni's optimal stopping time formulation of the equilibrium price of a T-period at-the-money normalized American put option in a Black-Scholes economy with a risk-free interest rate equal to [r.sup.*] and a volatility parameter, [Sigma].(8) This is the key observation of our article. Therefore, denoting by P(S,K,T,r,[Sigma]) the value of the more general American put option with time to maturity T and with exercise price K in a Black-Scholes type of model with risk-free interest rate r, current stock price S, and a volatility of [Sigma], we have shown that the interest rate guarantee is correctly priced when

[[Gamma].sub.0] = GP (1, 1, T, [r.sup.*], [Sigma]) + 1). (10)

Analytic Representation of the Guarantee Value

We have argued above that

[Mathematical Expression Omitted], (11)

with [Mathematical Expression Omitted]. In this section we demonstrate, by an application of the main idea in Myneni (1992), how to obtain an analytic formula for P(1,1,T,[r.sup.*],[Sigma]).

The optimal stopping time, [Mathematical Expression Omitted], of a problem of the type (11) is known to be characterized as follows (see, e.g., Krylov, 1980):

[Mathematical Expression Omitted]. (12)

The intuition behind equation (12) is that the process is optimally stopped the first time it hits a certain (time varying) level known as the optimal exercise boundary. This boundary is given by the function [x.sup.*](t), [for every]t [equivalent to] [0,T]. At the outset of the analysis, the location of this boundary is, of course, unknown, but it is determinable as part of the overall solution, as discussed, for example, in Kim (1990).

The main result of Myneni (1992) is that the left-hand side of equation (11) is decomposable as follows:

[Mathematical Expression Omitted], (13)

where [I.sub.A] denotes the indicator function on the set A.

The first part of the decomposition is obviously the equilibrium value of a normalized European put option, which is otherwise identical to the American option we are concentrating on here. Let us denote the value of this European put by p(1, 1, T, [r.sup.*], [Sigma]). Applying the famous result of Black and Scholes (1973) we have

p(1, 1, T, [r.sup.*], [Sigma]) = [e.sup.[-r.sup.*]T][Phi] (- ([r.sup.*] - 1/2 [[Sigma].sup.2]) [square root of T]/[Sigma]) - [Phi] (-([r.sup.*] + 1/2 [[Sigma].sup.2]) [square root of T]/[Sigma]), (14)

where [Phi]([center dot]) denotes the standard normal distribution function;

[Phi](v) = [integral of 1/[square root of 2 [Pi] [e.sup.-1/2][y.sup.2]]] dy between limits v and -[infinity].

The second part of the decomposition in equation (13) is known as the potential or the early exercise premium and can be manipulated further. This was neglected by Myneni (1992), but an analysis similar to what follows can be found in Jamshidian (1992) or Kim (1990). Elaborating on the potential we get

[Mathematical Expression Omitted], (15)

where we have defined

d_ ([x.sup.*] (u), u) [equivalent to] ln (1/[x.sup.*](u)) + ([r.sup.*] - 1/2 [[Sigma].sup.2] u/[Sigma][square root of u]). (16)

The analysis can now be summed up by stating that the value of the basic contract is given by

[[Gamma].sub.0] = G(P(1, 1, T,[r.sup.*], [Sigma]) + 1)

= G(p(1, 1, T, [r.sup.*], [Sigma]) + e(1, 1, T, [r.sup.*], [Sigma]) + 1)

= G([e.sup.-[r.sup.*]T] [Phi](-d_(1, T)) - [Phi](-d+(1, T))

+ [integral of [e.sup.-[r.sup.*]u]] [r.sup.*] [Phi](-d_ ([x.sup.*] (u), u)) du + between limits T and 0 1), (17)

where

[Mathematical Expression Omitted]. (18)

ALTERNATIVE COMPENSATION SCHEMES

From equation (10) (or (17) for that matter), it follows that [[Gamma].sub.0] [greater than or equal to] G (option prices are never negative) and that therefore the fund manager would never issue interest guarantees of this type based on an initial reference fund investment of value G without any form of compensation in excess of G. In the basic contract, the compensation takes the form of an additional upfront payment of size [[Gamma].sub.0] - G from the investor to the fund manager when the contract is signed. In the life insurance-related version of our model, this could be interpreted as an upfront payment for membership of the insurance fund. But other compensation schemes are possible as well. These include the periodic or continuous payment of management fees or the imposition of an exit fee to be paid by the investor if or when (two separate possibilities) the investor decides to terminate the contract. Variations of these compensation schemes - and combinations of them - all appear in practice as part of many actual marketed contracts, and below we discuss some of these alternatives in more detail. We begin with the basic contract and determine the equilibrium price of membership given the guaranteed rate of interest and vice versa.

Payment for Membership

The situation where the compensation to the fund manager for the issue of the interest guarantee takes the form of an upfront payment at the initiation time of the contract is very easy to analyze. Recall that formula (10) provides the value of the total claim that the investor has on the fund manager at time zero. By definition this value is decomposable into two parts; the guaranteed amount, G, and the value of the guarantee, which we henceforth denote [Pi]. We are therefore in a position to conclude that

[Pi] = G P(1, 1, T, [r.sup.*], [Sigma]). (19)

In the life insurance interpretation of our model, [Pi] is the amount that the fund manager must demand as payment for membership of the investment/pension fund - the entrance fee - unless the admittance of new members is to be subsidized by present (or future) members.

An alternative way of stating equation (19) is the following:

[Pi]/G = P(1, 1, T, [r.sup.*], [Sigma]), (20)

which shows that P(1, 1, T, [r.sup.*], [Sigma]) is merely the per-dollar cost of the interest rate guarantee.

Here we provide some numerical results in relation to the basic contract. Generally, the values for the tables and figures have been obtained by solving for the American put option premium by numerical integration of the integrals in equation (17) (see, e.g., Kim, 1990, for details). However, as explained below, closed-form formulas are obtained for T [right arrow] [infinity] and for [r.sub.G] = r, so in these cases the results are obtained without the use of numerical algorithms.

We consider an interest rate guarantee on an investment with an initial value equal to 100 units of account, that is, G = 100. Further, suppose the risk-free rate of interest, r, is equal to 10 percent. Figure 1 supports the intuition about the exercise problem at hand. Figure 1 depicts two simulated price paths for the reference portfolio from time zero to time T and with parameters r = 0.10 and [Sigma] = 0.30. Also shown are the evolution of the guaranteed amount for [r.sub.G] = 0.05 and G = 100 as well as the optimal exercise boundary relating to this particular problem. These two curves intersect at the maturity date of the contract.

In accordance with intuition, for t [less than] T the optimal exercise boundary lies everywhere below the curve depicting the guaranteed amount as a function of time. The reason for this is that the value of the reference portfolio must drop significantly below the guaranteed amount before premature exercise is triggered. Path 1 symbolizes one such situation; in the other situation (path 2), exercise does not occur.

In Tables 1 and 2, in which the volatility parameter, [Sigma], has been set to equal 10 percent and 30 percent, respectively, we have tabulated the value of the interest rate guarantee, [Pi], for some representative combinations of the time to maturity, T, and the guaranteed rate of interest, [r.sub.G]. The values of the corresponding interest rate guarantees which are not early exercisable (i.e, of European type) accompany the American guarantee values in parentheses immediately below.
Table 1

The Basic Contract
Value of the American Type Interest Rate Guarantee, [Pi]
(Values of European Type Guarantees in Parentheses)
G = 100, r = 10 Percent, and [Sigma] = 10 Percent

 [r.sub.G] (%)
T (Years) 0.00 2.00 4.00 6.00 8.00 10.00

0.25 1.23 1.34 1.47 1.62 1.79 1.99
 (0.98) (1.14) (1.32) (1.53) (1.75) (1.99)
0.50 1.45 1.63 1.85 2.11 2.42 2.82
 (0.97) (1.24) (1.55) (1.91) (2.34) (2.82)
1.00 1.63 1.90 2.24 2.67 3.22 3.99
 (0.79) (1.15) (1.64) (2.26) (3.04) (3.99)
2.00 1.74 2.09 2.57 3.24 4.19 5.64
 (0.45) (0.84) (1.47) (2.42) (3.79) (5.64)
5.00 1.79 2.21 2.85 3.88 5.63 8.90
 (0.08) (0.27) (0.80) (2.04) (4.55) (8.90)
10.00 1.79 2.23 2.93 4.18 6.68 12.56
 (0.00) (0.04) (0.26) (1.26) (4.55) (12.56)
20.00 1.79 2.23 2.94 4.30 7.52 17.69
 (0.00) (0.00) (0.03) (0.43) (3.66) (17.69)
30.00 1.79 2.23 2.94 4.32 7.84 21.58
 (0.00) (0.00) (0.00) (0.15) (2.74) (21.58)
40.00 1.79 2.23 2.94 4.33 8.00 24.82
 (0.00) (0.00) 0.00) (0.05) (2.01) (24.82)
[infinity] 1.79 2.23 2.94 4.33 8.19 100.00
 (0.00) (0.00) (0.00) (0.00) (0.00) (100.00)




[TABULAR DATA FOR TABLE 2 OMITTED]

The data in these tables reveal several interesting insights. First, it should be observed that the value of the guarantee is an increasing function of the guaranteed rate of interest. This is completely as expected. Note also that [Pi] is relatively modest for low levels of [r.sub.G] and that the sharpest increase in the value of the guarantee occurs as [r.sub.G] approaches the risk-free rate. This behavior is particularly pronounced for longer maturities. In this respect, observe that in the limiting case when [r.sub.G] = r - that is, when the guaranteed interest rate is at its maximum level - no premature exercise will occur. To understand this, recall that to establish the contract value in this case we price an American put with [r.sup.*] = 0. But it is a well-known result that, when interest rates are zero, the European and American put options are equivalent (see, e.g., Johnson, 1983).

In the present application, the intuition behind this result is that the vehicle driving early exercise is market interest rates that are higher than the guaranteed rate in combination with poor performance of the reference fund. If the market value of the reference fund investment is sufficiently low (the option is far in the money), the investor can expect to receive a return of only [r.sub.G] on the investment for the remaining life of the contract. Hence, the investor is likely to exercise prematurely, tempted by a higher risk-free market interest rate. In contrast, when [r.sub.G] = r, this incentive to exercise prematurely is removed. Note also that these arguments combined with simple manipulation of equation (5) show that, in the special case where [r.sub.G] = r, the contract is equivalent to a risk-free bond of value G plus a European option (call or put) on an investment of value G in the risky reference fund and with a stepped-up exercise price of G [e.sup.rT].(9) Evaluating the latter using the Black-Scholes formula yields the same result as using our valuation formula (17) in its degenerate version where [r.sup.*] = 0 and where the early exercise premium is zero. From formula (17), the value of the guarantee in this special case can easily be shown to degenerate to the expression G(1 - 2 [Phi](- [Sigma][square root of T]/2), which, using a first-order Taylor expansion of [Phi]([center dot]) about the origin, can be quickly approximated by G [Sigma] [square root of T]/[square root of 2[Pi]].

Also as expected, the value of the interest rate guarantee is an increasing function of time to maturity. This is a standard feature of American options since extending T for a given American option contract effectively amounts to adding further "options" of non-negative value to the existing contract. The value of the guarantee does not increase without bound, however. For [r.sub.G] [element of] [0; r[, [Pi] converges to an upper bound given by G (1/1 + [Gamma] [(1 + [Gamma]/[Gamma]).sup.-[Gamma]]), where [Gamma] = 2[r.sup.*]/[[Sigma].sup.2]. This is a simple transformation of a well-known result regarding perpetual options which is due to Merton (1973).

Now let us compare the computed American type interest rate guarantee values with their European counterparts (numbers in parentheses).(10) The first and unsurprising fact to note is that the interest rate guarantees which are immediately exercisable (American type) are always worth at least as much as the otherwise identical European type guarantees. In fact, the difference in value between the American and European type of guarantees is precisely the value of the surrender option - that is, the option to exercise prematurely. The difference is modest for relatively short horizon contracts, but for longer maturities the difference in values of the European and American contracts becomes very large. In fact, whereas the value of the American interest rate guarantee increases in T as argued above, the value of the European interest rate guarantee converges to zero as T [right arrow] [infinity] (except for the case where [r.sub.G] = r). As an example, consider the column in Table 1 where [r.sub.G] = 8 percent. The value of the American guarantee increases monotonically in T toward 8.19 while the value of the European guarantee reaches its maximum of approximately 4.6 for T somewhere in [5; 10] and subsequently converges to zero as T [right arrow] [infinity].(11) This pattern is typical for European put option prices as a function of the time to maturity.

The data in Tables 1 and 2 can also be used to exemplify the potential redistribution of wealth that can take place among policyholders with different levels and/or types of interest rate guarantees in the same fund. For example, a 4 percent interest rate guarantee of the American type together with a 2 percent guarantee of the European type may give rise to substantial redistribution, provided that no fees or incorrect fees are charged for the issue of the guarantees. For a 30-year contract of nominal value 100, the difference in value of the above-mentioned contracts amounts to 2.94 (2.94-0.0) when the volatility is 10 percent and as much as 18.85 (20.10-1.25) when volatility is 30 percent.

A comparison of the pricing results in Tables 1 and 2 reveals that a higher volatility uniformly increases the value of the interest rate guarantees. The qualitative effects in the two tables are identical.

For a fixed time to maturity of 10 years and a risk-free interest rate of 10 percent, Figure 2 provides an alternative view on the relation between [Pi], [Sigma], and [r.sub.G] for both European and American types of guarantees. The difference between the American and the European guarantee values (the vertical distance between the surfaces in [ILLUSTRATION FOR FIGURE 2 OMITTED]) is depicted in Figure 3.

Exit Fees

It is also possible to construct a compensation to the fund manager in the form of an exit fee to be paid by the investor when he or she chooses to terminate the contract. It could be arranged in the following way: Suppose that, as before, the investor has the option to terminate the contract at any time at or before the maturity date and demand either the nominal contract value, G, compounded continuously with [r.sub.G] to the date in question or the market value of the reference portfolio investment. But suppose that an extra restriction now applies in that immediately upon exercise and regardless of the payout scheme chosen by the investor, [Delta] 100 percent of the total value must be deducted and left with the fund manager. This arrangement is similar to a deferred premium option - an instrument that has been available in the over-the-counter market in recent years. Properly constructed, this penalty could serve to fully compensate the fund manager for the issued guarantee in the sense that no initial fee would be required. In other words, provided the cutoff fee, [Delta], is specified correctly at initiation of the contract, we could obtain [[Gamma].sub.0] = G. We call this the par contract. Another possibility is to exogenously specify [Delta] and [r.sub.G] and subsequently derive the contract value. Only by coincidence would such a contract be valued at par and hence the investor would typically be allowed to enter the contract either at a premium or at a discount. We therefore label this type of instrument a nonpar contract.

In any case, with a proportional exit fee imposed, the investor's final payoff will be as follows:(12)

[Mathematical Expression Omitted]. (21)

This stochastic payoff can again be transformed to an equilibrium present value by the help of the risk-neutral pricing methodology. Using the results of our model along with the linear homogeneity of the option payoff function and denoting the contract value by [Mathematical Expression Omitted], we get

[Mathematical Expression Omitted]. (22)

In the case where an initial fee is to be avoided (the par contract), [Mathematical Expression Omitted] must equal the guaranteed amount at time zero, G. This condition leads to the determination of [Delta]:

[Mathematical Expression Omitted],

where [P.sub.am] serves as short notation for P(1, 1, T, [r.sup.*], [Sigma]). Note that, for fixed T, [Sigma], and r, the above equation defines a unique correspondence between [Delta] and [r.sub.G].

In the case where [Delta] and [r.sub.G] are exogenously specified (the nonpar contract), the contract value per guaranteed dollar is clearly

[Mathematical Expression Omitted], (23)

and the contract could trade below par, at par, or above par value. Again, it is illustrative to consider some numerical examples.

Numerical Examples: The Par and Nonpar Contracts

For the par contract, as in the previous example, suppose that the risk-free interest rate, r, equals 10 percent. Then using equation (17) along with the relation between [Delta] and [P.sub.am] derived above, the fair exit fee percentage, [Delta], can be calculated for combinations of the time to maturity, T, and the guaranteed rate of interest, [r.sub.G]. We do this in Tables 3 and 4, where [Sigma] has again been set at 10 percent and 30 percent, respectively.

Not surprisingly, the pattern for the exit fee percentage in Tables 3 and 4 is very similar to that of Tables 1 and 2. Since [Delta] represents the compensation to the fund manager for the issued guarantee, it should be an increasing function of [[Gamma].sub.G] as well as of T. When the guarantee is of European type, however, the exit fee percentage need not necessarily increase in T. Finally, the exit fee percentage should increase in the volatility. All these effects can be observed in Tables 3 and 4.

For the nonpar contract, Table 5 shows contract values for T = 10 and for some combinations of [r.sub.G] and [Delta]. Contract values are increasing in [r.sub.G] and decreasing in [Delta], and below par values are indeed possible.
Table 3

The Par Contract
The Exit Fee Percentage (EFP), [Delta]
(EFP for European Type Guarantee in Parentheses)
r = 10 Percent and [Sigma] = 10 Percent

 [r.sub.G](%)
T (Years) 0.00 2.00 4.00 6.00 8.00 10.00

0.25 1.21 1.32 1.45 1.59 1.76 1.96
 (0.97) (1.13) (1.31) (1.50) (1.72) (1.96)

0.50 1.43 1.61 1.82 2.06 2.37 2.74
 (0.96) (1.22) (1.52) (1.88) (2.28) (2.74)

1.00 1.61 1.86 2.19 2.60 3.12 3.83
 (0.79) (1.14) (1.61) (2.21) (2.95) (3.83)

2.00 1.71 2.05 2.51 3.14 4.02 5.34
 (0.45) (0.83) (1.45) (2.36) (3.65) (5.34)

5.00 1.76 2.16 2.77 3.74 5.33 8.17
 (0.08) (0.27) (0.79) (2.00) (4.35) (8.17)

10.00 1.76 2.18 2.84 4.01 6.27 11.16
 (0.00) (0.04) (0.26) (1.24) (4.35) (11.16)

20.00 1.76 2.18 2.86 4.13 6.99 15.03
 (0.00) (0.00) (0.03) (0.43) (3.53) (15.03)

30.00 1.76 2.18 2.86 4.15 7.27 17.75
 (0.00) (0.00) (0.00) (0.15) (2.67) (17.75)

40.00 1.76 2.18 2.86 4.15 7.40 19.88
 (0.00) (0.00) (0.00) (0.05) (1.97) (19.88)

[infinity] 1.76 2.18 2.86 4.15 7.57 50.00
 (0.00) (0.00) (0.00) (0.00) (0.00) (50.00)
Table 4

The Par Contract
The Exit Fee Percentage (EFP), [Delta]
(EFP for European Type Guarantee in Parentheses)
r = 10 Percent and [Sigma] = 30 Percent

 [r.sub.G](%)
T (Years) 0.00 2.00 4.00 6.00 8.00 10.00

0.25 4.75 4.91 5.07 5.25 5.44 5.64
 (4.54) (4.75) (4.96) (5.18) (5.41) (5.64)
0.50 6.14 6.43 6.73 7.05 7.40 7.79
 (5.69) (6.07) (6.48) (6.90) (7.34) (7.79)
1.00 7.70 8.18 8.70 9.28 9.92 10.65
 (6.73) (7.43) (8.17) (8.95) (9.78) (10.65)
2.00 9.25 10.03 10.90 11.89 13.03 14.38
 (7.28) (8.46) (9.76) (11.18) (12.73) (14.38)
5.00 10.92 12.22 13.76 15.62 17.89 20.80
 (6.27) (8.29) (10.76) (13.67) (17.03) (20.80)
10.00 11.65 13.33 15.46 18.19 21.76 26.73
 (3.86) (6.21) (9.56) (14.10) (19.86) (26.73)
20.00 11.92 13.86 16.46 20.06 25.21 33.23)
 (1.24) (2.87) (6.12) (11.93) (21.03) (33.23)
30.00 11.96 13.96 16.74 20.73 26.78 37.05
 (0.38) (1.23) (3.59) (9.19) (20.21) (37.05)
40.00 11.96 13.99 16.83 21.02 (27.63 39.66
 (0.11) (0.52) (2.03) (6.79) (18.68) (39.66)
[infinity] 11.96 14.00 16.89 21.32 29.08 50.00
 (0.00) (0.00) (0.00) (0.00) (0.00) (50.00)




[TABULAR DATA FOR TABLE 5 OMITTED]

CONCLUSION

This article studies the equilibrium valuation of interest rate guarantees incorporating the possibility of premature exercise. The analysis is performed within the framework of contingent claims valuation and in particular by applying concepts and results from American option pricing theory. The numerical examples provided show that the value of interest rate guarantees, in particular of the American type, can be quite significant. It is furthermore demonstrated that, for longer maturities, the difference in value of a European and an American type of interest rate guarantee will become very large. This latter observation is particularly interesting for insurance companies wishing to reduce potential future solvency problems. This can in part be obtained by changing the option element in future uncompensated issues of interest rate guarantees from the American to the European type. On the other hand, in the absence of correct valuation of these different types of options, such a step may aggravate the problems of fairly distributing the insurance fund wealth among the different classes of policyholders. This article derives explicit pricing formulas as well as various compensation schemes that may help to ensure the fair treatment of all policyholders.

A number of natural paths emerge for further research. First, an extension to periodic premium contracts would be desirable. Second, realism could be added to the model by considering mortality risk explicitly. Taken together, such extensions might also help to lead us from the model for single premium American type ELLIPs toward a more accurate and realistic model for the valuation of traditional life insurance contracts with profit. Third, an extension to the case of stochastic interest rates would be natural to pursue. Finally, another relevant extension of our work would be to incorporate credit risk. The possibility of default of the fund manager should tend to lower the contract value (and consequently the value of the guarantee) for a given guaranteed interest rate and possibly also affect the exercise strategy of the investor. In this context, it might even be possible to deduce an optimal guaranteed interest rate that would maximize the value of the contract. This could be so since, from the investor's point of view, "higher" would not necessarily mean "better" because of the embedded moral. hazard problem (fund managers offering high guaranteed interest rates are more likely to default). This is the subject of current research by the authors.(13)

The authors are grateful for constructive criticism by two anonymous referees. Peter Lochte Jorgensen acknowledges financial support from Aarhus University Research Foundation (Research Grants # E-1994-SAM-1-1-72 and E-1995-SAM-1-59).

1 We adopt the terminology of Brennan and Schwartz (and many others) and use the term "equilibrium price" synonymously with "arbitrage free prices" in the sense of Black and Scholes (1973). Throughout the article, we use "price" and "value" interchangeably.

2 Details are available from the authors upon request.

3 As pointed out by Brennan and Schwartz (1976), it is not essential that the fund manager actually makes an investment of G in the reference portfolio at time zero. What matters is merely that both pasties recognize that the investor's claim on the fund manager depends at all times on what would have been the current value of such an investment in the reference fund.

4 For a rigorous treatment of the theory of stochastic processes, see, e.g., Karatzas and Shreve (1988).

5 As pointed out by Black and Scholes, allowing the drift parameter to depend on time and/or the state has no effect on the value of derivative assets priced by no arbitrage arguments in this economy. The assumption of a constant [Mu]t can thus be regarded merely as a simplification of notation and implies no loss of generality.

6 During the course of his analysis of the pricing of an option to exchange one asset for another, Margrabe (1978) establishes some results regarding options with exercise prices rising at the risk-free interest rate. Johnson (1983) applies these results to approximate the value of American put options.

7 See Karatzas and Shreve (1988) for an introduction to the concept of stopping times and their properties. By Girsanov's theorem (see, e.g., Karatzas and Shreve, 1988), the equivalent risk-neutral probability measure, [Mathematical Expression Omitted], is defined by the Radon-Nikodym derivative

[Mathematical Expression Omitted].

Note that the risk-neutral valuation principle of calculating present values as discounted expectations of future values is parallel to the well-established actuarial principle of equivalence. However, as noted by Ekern and Persson (1995), the principle of equivalence does not adjust for risk. In risk-neutral valuation, the risk adjustment takes place via the probability measure. Hence, one could also refer to [Mathematical Expression Omitted] as a risk-adjusted probability measure.

8 By "normalized" we mean with exercise price and current stock price both equal to one; that is, S = K = 1 in Myneni's notation.

9 Under the stated terms, the European call and put values are equal by put-call parity.

10 The value of the otherwise equivalent European guarantees have been obtained by using the version of Black-Scholes formula provided in equation (14).

11 The maximum value of the guarantee is 4.64 and occurs at T = 7.15.

12 With this specification, the investor pays the proportional fee regardless of whether the guarantee comes into effect, that is, whether the option is exercised. A variation of this scheme would be an arrangement where the penalty is paid only if the option is exercised. Options of this kind are often labeled contingent premium options. It is straightforward to extend our analysis to such a contract. Note, however, that in this case an initial fee would have to accompany the exit fee in order for the contract to be fair to both sides. In other words, as in the basic case, it would not be possible to construct a contract such that G = [[Gamma].sub.0].

13 We thank an anonymous referee for pointing out this interesting research subject.

REFERENCES

Aase, K. and S.-A. Persson, 1997, Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Products, Journal of Risk and Insurance, forthcoming.

Albizzati, M. O. and H. Geman, 1994, Interest Rate Risk Management and Valuation of the Surrender Option in Life Insurance Policies, Journal of Risk and Insurance, 62: 616-637.

Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81: 637-654.

Boyle, P. P. and E. S. Schwartz, 1977, Equilibrium Prices of Guarantees under Equity-Linked Contracts, Journal of Risk and Insurance, 44: 639-660.

Brennan, M. and E. S. Schwartz, 1976, The Pricing of Equity-Linked Life Insurance Policies with an Asset Value Guarantee, Journal of Financial Economics, 3: 195-213.

Brennan, M. and E. S. Schwartz, 1979, Alternative Investment Strategies for the Issuers of Equity-Linked Life Insurance Policies with an Asset Value Guarantee, Journal of Business, 52: 63-93.

Carr, P., R. Jarrow, and R. Myneni, 1992, Alternative Characterizations of American Put Options, Mathematical Finance, 2:87-106.

Collins, T. P., 1982, An Exploration of the Immunization Approach to Provisions for Unit-Linked Policies with Guarantees, Journal of the Institute of Actuaries, 109:241-284.

Ekern, S. and S.-A. Persson, 1995, Exotic Unit-Linked Life Insurance Contracts, Working Paper, Norwegian School of Economics and Business Administration, Bergen.

Harrison, M. and D. Kreps, 1979, Martingales and Arbitrage in Multiperiod Securities Markets, Journal of Economic Theory, 20:381-408.

Jamshidian, F., 1992, An Analysis of American Options, Review of Futures Markets, 11: 72-80.

Johnson, H. E., 1983, An Analytic Approximation for the American Put Price, Journal of Financial and Quantitative Analysis, 18:141-148.

Karatzas, I., 1988, On the Pricing of the American Option, Applied Mathematics and Optimization, 17: 37-60.

Karatzas, I., 1989, Optimization Problems in the Theory of Continuous Trading, SIAM Journal of Control and Optimization, 27:1221-1259.

Karatzas, I. and S. E. Shreve, 1988, Brownian Motion and Stochastic Calculus (New York: Springer Verlag).

Kim, I. J., 1990, The Analytic Valuation of American Options, Review of Financial Studies, 3: 547-572.

Krylov, N. V., 1980, Controlled Diffusion Processes (New York: Springer Verlag).

Margrabe, W., 1978, The Value of an Option to Exchange One Asset for Another, Journal of Finance, 33: 177-186.

Merton, R., 1973, The Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, 4: 141-183.

Myneni, R., 1992, The Pricing of the American Option, Annals of Applied Probability, 2: 1-23.

Nielsen, J. and K. Sandmann, 1995, Equity-Linked Life Insurance - A Model with Stochastic Interest Rates, Insurance: Mathematics and Economics, 16: 225-253.

Anders Grosen and Peter Lochte Jorgensen are Associate Professors of Finance at the Aarhus School of Business and the University of Aarhus, Denmark, respectively.
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Author:Grosen, Anders; Jorgensen, Peter Lochte
Publication:Journal of Risk and Insurance
Date:Sep 1, 1997
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