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SINGLE- AND CROSS-GENERATION NATURAL HEDGING OF LONGEVITY AND FINANCIAL RISK.

INTRODUCTION

Longevity risk, that is, the risk of unexpected changes in survivorship, is now perceived as an important threat to the safety of insurance companies and pension funds. Most actors in the financial market are long longevity risk. This has stimulated the transformation of contracts subject to longevity risk into an asset class, as originally suggested by Blake and Burrows (2001). The creation of q-forwards, s-forwards, longevity bonds, and swaps represents a step in this direction, but this asset class is still in its infancy. In the meanwhile, insurance companies can benefit from natural hedging, that is, from natural offsetting between the longevity risk exposures of death benefits and life contracts, such as annuities. The importance of exploiting this natural offsetting extends beyond theory. Cox and Lin (2007) find empirical evidence that insurers whose liability portfolios benefit from natural hedging have a competitive advantage and charge lower premiums. Despite being safe, sound, and comparatively cheap, natural hedging is not trivial in the presence of longevity risk because the latter is difficult to capture per se in a parsimonious and manageable way and even more difficult to couple with a satisfactory model of financial risk, such as interest rate risk. However, the interactions between longevity and financial risk cannot be avoided from the perspective of immunization in the form of liability management, as the value of the reserves is subject to interest-rate risk, and a fortiori, from the perspective of asset and liability management (ALM).

Natural hedging of longevity risk without financial risk is recently addressed by Cox and Lin (2007), Wang et al. (2010), and Gatzert and Wesker (2012, 2013). Cox and Lin (2007), motivated by the empirical evidence mentioned above, propose the use of mortality swaps between annuity providers and life insurance writers. Wang et al. (2010) propose an immunization strategy that matches the duration and convexity of life insurance and annuity benefits. They demonstrate that this strategy is effective in reducing longevity risk by calibrating it to U.S. mortality data. However, they consider only liabilities, while we consider both assets and liabilities as well as financial risk. Gatzert and Wesker (2012) use simulations to select portfolios of policies that immunize the insurer's solvency against changes in mortality. Gatzert and Wesker (2013) consider the interactions among systematic, unsystematic, basis risk, and adverse selection in determining the effectiveness of natural hedging.

Natural hedging with financial risk is studied by Stevens et al. (2011). They show that financial risk has a clear impact on the overall initial riskiness of the annuity--life insurance mix. The effect of natural hedging may be overestimated when financial risk is ignored, affecting hedging possibilities. In their case, financial risk occurs only from potential losses from assets, while in our case, it affects both the assets and the fair value of liabilities.

We extend the existing literature in four directions. First, we model longevity and financial risk at the same time, and we assess their impact on the fair value of the insurer's net liabilities or reserves. We aim at hedging changes in reserves, at the first- and second-order approximations (delta and gamma hedging, respectively). The risk factors to hedge against are the differences between the mortality and interest-rate intensities forecasted today and their actual realizations in the future. Second, to hedge liabilities, we let the insurer use new sales of insurance contracts, reinsurance, and bonds, so we extend previous research by using both assets and liabilities for immunization. Third, we exploit hedging within a single generation and across generations (or across genders) to capture the fact that some products may be not marketed. For instance, death benefits for older generations may not be marketed. Thus, we develop a cohort-based mortality model, and we split the longevity risk factor of each generation into common and idiosyncratic parts. Fourth, we provide all delta--gamma hedges in closed form. This enhances the computation and comprehension of the hedge drivers. Moreover, optimal hedges solve linear systems. Consequently, assessing whether reserves can be perfectly hedged (up to any chosen level of accuracy) and whether the mix of assets and/or liabilities that achieves the hedge is unique is a trivial matter. The framework can accommodate some important practical aspects, such as self-financing constraints, sales constraints, and limited availability or absent products.

The UK-calibrated application that concludes this article computes the hedging portfolios of an annuity. First, this example permits the comparison of the magnitudes of first- and second-order effects, that is, the deltas and gammas, within a single type of risk, across risks, and across generations. Second, the application illustrates the straightforward computation of sensitivities and hedges, given closed-form solutions and linear systems. Third, it demonstrates the effectiveness of delta and gamma hedging, even when the hedge is adjusted discretely in time, as occurs in practice, rather than renewed continuously. The application is extended to two generations to demonstrate the effectiveness of delta--gamma intra- and cross-generation hedging strategies, even in the presence of constraints on the products that can be sold or reinsured.

The article is structured as follows. The "Longevity and Interest-Rate Risks" section reviews the mortality and interest-rate model and examines the pricing and hedging of annuities and life insurance policies. The "Natural Hedging Within a Single Generation" section focuses on single-generation natural hedging. The "Natural Hedging Across Generations" section presents hedging on a multiple-generation portfolio. The "UK-Calibrated Application" section presents a hedge example calibrated to UK data. Assessment of hedge effectiveness is also provided. The "Conclusions" section concludes.

LONGEVITY AND INTEREST-RATE RISKS

We place ourselves in a continuous-time framework, as suggested by Cairns et al. (2006, 2008). In this framework, known as a "stochastic mortality" approach, we use a parsimonious, continuous-time model for cohort-specific mortality intensity that extends the classic Gompertz law and a benchmark model for interest-rate risk, the Hull and White model. This section introduces the models for longevity and interest-rate risk and delta--gamma hedging for a pure endowment, as obtained in Luciano et al. (2012).

Model for Longevity Risk

We consider the time of death of an individual as the first jump time of a Poisson process with stochastic intensity, that is, a Cox process. Let us introduce a probability space ([OMEGA], F, P) equipped with a filtration {[F.sub.t]: 0 [less than or equal to] t [less than or equal to] T} that satisfies the usual properties of right-continuity and completeness. (1) Our approach is generation based. We use x to indicate both the group (generation, cohort, or gender) and its age at time 0. The spot mortality intensity at calendar time t of a head belonging to generation x is denoted by [[lambda].sub.x](t). We assume that under the historical P measure, [[lambda].sub.x](t) follows an Ornstein--Uhlenbeck process without mean reversion (OU):

d[[lambda].sub.x](t) = [a.sub.x][[lambda].sub.x](t)dt + [[sigma].sub.x]d[W.sub.x](t), (1)

where [a.sub.x] > 0, [[sigma].sub.x] [greater than or equal to] 0, and [W.sub.x] is a standard one-dimensional Brownian motion. Our choice of the OU process is motivated by its parsimony--very few parameters must be calibrated--and its appropriateness to fit cohort life tables because of its lack of mean reversion. It is an affine process for which we can find closed-form expressions for the survival probability. Moreover, it is a natural, stochastic extension of the Gompertz model for the force of mortality and is easy to interpret in light of traditional actuarial practice. Its major drawback is the positive probability that [[lambda].sub.x] will become negative. However, in each application of our model, we verify that this probability is negligible and that the survival probability is decreasing over the duration of human life. (2) Together with the spot intensity, we consider the forward instantaneous intensity, denoted [f.sub.x](t,T). This is the best forecast at time t of the spot intensity at T because it converges to it when the horizon of the forecast goes to zero, or T [right arrow] t:[f.sub.x](t,t) = [[lambda].sub.x](t). Standard properties of affine processes allow us to represent the survival probability from time t to T as follows:

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

As due in a stochastic mortality environment, survival probabilities in the future [S.sub.x](t,*), t > 0 are random variables at time 0. Following a hint in Jarrow and Turnbull (1994), we can write the survival probability as follows:

[mathematical expression not reproducible]

The term I(t), the difference between the instantaneous mortality intensity at t and its forecast at time 0, is the longevity risk factor, that is, the error in the forecast that exposes insurance companies and pension funds to longevity risk. At time 0, this is the only random quantity in future survival probabilities [S.sub.x](t,T). Because [X.sub.x](t,T) > 0 and I(t) = 0 provide the "base" survival probability curve [S.sub.x](t,T) = [S.sub.x](0.T)/[S.sub.x](0,t) exp [-[Y.sub.x](t,T)], a positive I(t) implies a survival curve lower than the base curve, and vice versa. Indeed, I(t) is greater than 0 if the realized mortality intensity is higher than its forecast, and consequently, the survival probability for every duration T > t is lower than its forecast at time 0. Our hedging technique exploits the crucial feature that I(*) depends only on t. The same factor affects all survivals at each horizon T in the future.

Model for interest-rate Risk

While in the longevity domain we model spot intensities, in the financial domain we adopt the standard Heath, Jarrow, and Morton (1992) framework and directly model the instantaneous forward rate F(t,T), which is the rate that applies at instant T,as agreed upon at t < T.

Additionally, we assume arbitrages do not exist, and we start modeling directly under the risk-neutral measure equivalent to P, which we call Q. (3) We assume that the process for the forward interest rate F(t,T), defined on the probability space ([OMEGA], F, Q), is the well-known Hull and White (1990) model with constant parameters:

dF(t,T) = -gF(t,T)dt + [SIGMA][e.sup.-g(T-t)]d[W.sub.F] (t), (5)

where g > 0, [SIGMA] > 0, and [W.sub.F] is a univariate Brownian motion. The limit of the forward rate when T [right arrow] t is the short rate that applies instantaneously at t, r(t): F(t,t) = r(t). The price of a zero-coupon bond issued at time t expiring at time T is as follows:

[mathematical expression not reproducible].

As with survival probabilities, bond prices at time t are random variables at time 0. They may be written as follows (see Jarrow and Turnbull, 1994):

[mathematical expression not reproducible].

As in the longevity case, the financial risk factor K(t), which is the difference between the time t spot and forward rates, is the only source of randomness that affects bonds. It is the same across all bond maturities T.

ALM and Greeks

In the presence of both longevity and interest-rate risks, the fairly priced, future reserves of every insurance product become stochastic because survival probabilities and bond prices are stochastic. This generates the need for liability hedging and opens the way to ALM. If we compute the fair value of the reserves and assume that insurance companies hedge this value, we implement liability hedging. If we assume that insurance companies set up a hedged portfolio under a self-financing constraint (which means that the premiums received for death or life benefits are used to build the hedge) and may include bonds in the menu of available assets, we perform asset and liability management.

To compute the fair value of an insurance liability, a change of probability measure is still needed. We assume that there exists a measure Q that allows the mortality intensity to remain affine. (4) This standard choice is equivalent to fixing a risk premium (5) [[theta].sub.x](t) = q[[lambda].sub.x](t)/[[sigma].sub.x], q [member of] R, q > -[a.sub.x]. We assume independence between longevity and financial risks after the change of measure. Thereby, we can provide expressions for the fair value of insurance liabilities as products of survival or death probabilities and discount factors.

Consider first a pure endowment contract starting at time 0 and paying one unit of account if the individual aged x is alive at time T. The fair value of such an insurance policy at time t [greater than or equal to] 0is [Z.sub.Ex](t,T). Assuming a single premium paid at the policy issue, [Z.sub.Ex] is also the time t reserve for the policy and the value that must be hedged by the life office. We have the following:

[mathematical expression not reproducible] (6)

where the parameter [a.sub.x] in [X.sub.x], [Y.sub.x] has become [a'.sub.x] = [a.sub.x] + q > 0 to account for the measure change. For the sake of simplicity, we suppress the subscript x until the "Natural Hedging Across Generations" section, which introduces multiple cohorts. Using Ito's lemma, for a given t, we obtain the dynamics of the reserve [Z.sub.E] as a function of the changes in the risk factors:

[mathematical expression not reproducible],

where

[mathematical expression not reproducible].

To simplify the notation, we define the Greeks as follows:

[[DELTA].sub.E.sup.M](t,T) = B(t,T)[[DELTA].sup.M](t,T) < 0,

[[GAMMA].sub.E.sup.M](t,T) = B(t,T)[[GAMMA].sup.M](t,T) > 0, (7)

[[DELTA].sub.E.sup.F](t,T) = S(t,T)[[DELTA].sup.F](t,T) < 0,

[[GAMMA].sub.E.sup.F](t,T) = S(t,T)[[GAMMA].sup.F](t,T) > 0. (8)

Using those definitions, the following is the change in the pure endowment:

d[Z.sub.E] = [[DELTA].sub.E.sup.M][DELTA]I + 1/2[[GAMMA].sub.E.sup.M][DELTA][I.sup.2] + [[DELTA].sub.E.sup.F]K + 1/2[[GAMMA].sub.E.sup.F][DELTA][K.sup.2].

Let us consider an annuity--with annual installments R--issued at time 0 to an individual belonging to a certain generation. Assuming the payment of a single premium at policy inception, we obtain the prospective reserve [Z.sub.A] from t > 0 to T: (6)

[Z.sub.A](t,T) = R[T-t.summation over (u=1)][B.sub.t,u][S.sub.t,u],

where we use the short notation [B.sub.t,u] for B(t,t + u). We use the same shortcut for S and the Greeks below. The change in the reserve for given t is straightforward to compute:

d[Z.sub.A] = R [[[DELTA].sub.A.sup.M][DELTA]I + 1/2[[GAMMA].sub.A.sup.M][DELTA][I.sup.2] + [[DELTA].sub.A.sup.F][DELTA]K + 1/2[[GAMMA].sub.A.sup.F][DELTA][K.sup.2]],

where

[mathematical expression not reproducible].

As expected, deltas are negative and gammas are positive for both risks.

Let us consider now a life insurance issued at time 0. It has maturity T and sum assured C, paid at the end of the year of death. If a single premium is paid at policy inception, the following is the prospective reserve [Z.sub.D] from t > 0 to horizon T:

[Z.sub.D](t,T) = C[T-t.summation over (u=1)][B.sub.t,u]([S.sub.t,u-1] - [S.sub.t,u]).

The change in the reserve [Z.sub.D] at time t is as follows:

d[Z.sub.D] = C [[[DELTA].sub.D.sup.M][DELTA]I + 1/2[[GAMMA].sub.D.sup.M][DELTA][I.sup.2] + [[DELTA].sub.D.sup.F][DELTA]K + 1/2[[GAMMA].sub.D.sup.F][DELTA][K.sup.2]],

where

[mathematical expression not reproducible],

[mathematical expression not reproducible], (9)

[mathematical expression not reproducible],

[mathematical expression not reproducible]. (10)

As intuition would suggest, the Greeks for the longevity risks of life insurance and annuities have the opposite signs. If the actual mortality intensity is higher than forecasted ([DELTA]I > 0), life insurance contracts increase in value while endowments decrease, up to the first order. Second-order derivatives also have opposite values because they mitigate the approximation given by the first derivative. The Greeks of life insurance and annuities with respect to financial risk have the same sign because they are all present values.

The Greeks with respect to the risk factors provide the exposure to mortality and interest-rate forecast errors in closed form. Actually, they do more: they create the possibility of hedging. If the sole aim is mortality-risk hedging, one can create a hedged portfolio taking only short positions (i.e., issued policies). This is the case analyzed by Wang et al. (2010). If instead the aim is mortality- and financial-risk hedging using solely insurance products, a mix of short and long positions (i.e., issued policies and reinsurance purchased) is needed to neutralize the risk exposure. If the insurer cannot or does not want to buy reinsurance, natural hedging for mortality risk can be used; after creating a portfolio of issued policies, long and short positions can be taken in bonds to hedge financial risk. In this case, one should first hedge the mortality risk and then the interest-rate risk because the insurance products used to hedge mortality risk produce additional interest-rate risk.

NATURAL HEDGING WITHIN A SINGLE GENERATION

To implement natural hedging, it is possible to compute the number of offsetting contracts in closed form. Consider an insurer who has issued m annuities with rate R and maturity [T.sub.1] as well as n life insurance policies with face value C and maturity [T.sub.2] on the same generation. (7) The change in the value of its liabilities [Z.sub.H](t) for t < min{[T.sub.1], [T.sub.2]}, can be written as follows:

[mathematical expression not reproducible] (11)

where the coefficients of the changes in the risk factors are simply the weighted sums of the sensitivities of each liability. This follows from the fact that the risk factors depend only on t. What differs by product is the first- and second-order exposure--that is, the Greeks.

Natural hedging can be achieved because it is possible to select m and n such that the coefficients of [DELTA]I and [DELTA][I.sup.2] are both equal to 0:

[mathematical expression not reproducible].

This is a linear system of two equations with two unknowns m and n. Negative solutions for m or n indicate that the insurer must take a short position or sell the contract. Positive solutions indicate long positions, that is, reinsurance needs. Thus, longevity risk is hedged according to standard risk management techniques, up to first- and second-order approximations. As usual, because the risk factors are modeled in continuous time, a perfect hedge would require continuous adjustments of the positions. We show in the "Conclusions" section that delta--gamma strategies are quite robust to discrete implementation under realistic calibrations. The hedge described thus far entails only liabilities. We could extend it to assets by including bonds, which we do so in the application. We also consider the case of a self-financing hedge.

NATURAL HEDGING ACROSS GENERATIONS

Suppose that we must hedge the liability of a pure endowment written on generation x with that written on another generation y. All generations are subject to the same financial risk factor because the term structure of interest rates is the same for all generations. We assume that each generation has its own intensity and, therefore, its own longevity-risk factor, but the intensities of two generations are instantaneously correlated. This captures the idea that the drivers of longevity risk, while not exactly the same across generations, are all affected by progress in medicine and improved welfare on the one hand and by possible pandemics on the other. Using the notation from the "Longevity and Interest-Rate Risks" section, we have the following:

d[[lambda].sub.x] = [a'.sub.x][[lambda].sub.x]dt + [[sigma].sub.x]d[W.sub.x](t), (12)

d[[lambda].sub.y] = [a'.sub.y][[lambda].sub.y]dt + [[sigma].sub.y]d[W.sub.y](t), (13)

where the two Brownians have correlation coefficient [rho]. Appendix A shows that by reparametrizing the two sources of risk through independent Wieners, one can isolate the mortality risk that affects generations x and y (the common risk) from the risk that affects only y (the idiosyncratic risk). We identify the common risk factor as I(t) and denote the idiosyncratic risk factor describing the y-specific variations as I'(t). Obviously, I'(t) is instantaneously uncorrelated with I(t), and the survival probability of generation y is affected by both I and I'. With this parametrization of the two factors, the Greeks for pure endowments, annuities, and life insurance policies written on generation x, which we derived in the previous sections, still apply. We now provide the analogous Greeks for generation y. Appendix A shows that:

[mathematical expression not reproducible], (14)

where

[mathematical expression not reproducible], (15)

[mathematical expression not reproducible], (16)

[mathematical expression not reproducible], (17)

[mathematical expression not reproducible]. (18)

Obviously, the Greeks of generation y with respect to the common risk factor x,[[DELTA].sub.y.sup.M,x], and [[GAMMA].sub.y.sup.M,x], depend on [rho]. The sign of the first derivative with respect to the common risk [[DELTA].sub.y.sup.M,x] is opposite as the sign of the correlation coefficient [rho], while the derivative with respect to idiosyncratic risk [[DELTA].sub.y.sup.M,y] is negative, as expected. Both gamma coefficients are non-negative, as usual. The gamma with respect to the common risk [[GAMMA].sub.y.sup.M,x] is positive (and equals zero whenever [rho] equals zero), while the gamma with respect to idiosyncratic risk [[GAMMA].sub.y.sup.M,y] is strictly positive.

Using notation similar to (7) and (8), the change in the reserve for a pure endowment on y is as follows:

d[Z.sub.Ey] = [[DELTA].sup.M,x.sub.Ey][DELTA]I + 1/2[[GAMMA].sup.M,x.sub.Ey][DELTA][I.sup.2] + [[DELTA].sup.M,y.sub.Ey][DELTA]I' + 1/2[[DELTA].sup.F.sub.Ey][DELTA][I'.sup.2] + [[DELTA].sup.F.sub.Ey][DELTA]K + 1/2[[GAMMA].sup.F.sub.Ey][DELTA][K.sup.2].

The change in the reserve of an annuity on y is as follows:

[mathematical expression not reproducible],

with j = x,y. While the sign of [[DELTA].sup.M,y.sub.Ay] is negative as usual, the sign of [[DELTA].sup.M,x.sub.Ay] is opposite the sign of [rho]. The gammas are strictly positive, with the exception of [[GAMMA].sub.Ay.sup.M,x] (t,T), which is null when [rho] = 0.

The change in the reserve for life insurance is as follows:

[mathematical expression not reproducible]

with j = x, y. For positive p, the comments on the sign of [[DELTA].sup.M,x.sub.Dy] (t, T) are the same as in (9). The opposite comments apply for negative correlations. Provided that [rho] [not equal to] 0, the comments on the sign of [[GAMMA].sup.M,x.sub.Dy] (t, T) are the same as in (10). The same comments as in (9) and (10) hold for the delta and gamma of generation y with respect to its factors [[DELTA].sup.M,x.sub.Dy] (t, T) and [[GAMMA].sup.M,x.sub.Dy] (t, T).

Hedging

If the insurer wants to hedge mortality risk on generation x but does not have enough insurance products on that generation, products on generation y can be used to offset exposure to the common risk factor. However, coverage of the common risk factor I implies exposing the portfolio to the idiosyncratic risk I' of generation y unless generations are perfectly correlated. This risk can be evaluated and can be either traded away by the insurer in a market or reinsured. Imagine an insurer who has issued [n.sub.H] products on generation x with fair value [Z.sub.H]. He can delta--gamma hedge his liability by assuming positions in [n.sub.i] units of some other N instruments with fair value [Z.sub.i], i = 1, ***, N. The products available for hedging the longevity risk factor of each cohort can be written either on the same cohort as [Z.sub.H] or on different ones. To simplify the notation, the index i of each product denotes both the type of product (i.e., E, A, D) and its maturity. As in the single-generation case, we interpret negative positions [n.sub.i] < 0 as short positions on the corresponding product and positive solutions [n.sub.i] > 0 as reinsurance purchases. (8) Hedging portfolios are obtained by equating to zero [[DELTA].sup.M,j.sub.[PI]], [[DELTA].sup.F.sub.[PI]], [[GAMMA].sup.M,j.sub.[PI]] and [[GAMMA].sup.F.sub.[PI]], where the subscript [PI] refers to the portfolio itself; the superscript M,j refers to the jth longevity risk factor; j = 1, ***, J is the cohort on which the product is written; (9) and F refers to the financial risk factor, which is unique across cohorts.

The quantities [n.sub.i], i = 1, ***, N therefore solve the following system (for a given [n.sub.H]):

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

The expressions for the delta and gamma coefficients take the forms we derived in the previous sections, depending on the ith product type (pure endowment, annuity, life insurance). When we simultaneously solve:

* the J Equations (19), we delta hedge longevity risk;

* the J + 1 Equations (19) and (20), we delta hedge both risks;

* the 2J Equations (19) and (21), we delta-gamma hedge longevity risk;

* all 2 + 2J Equations (19),(20), (21), and (22), we delta-gamma hedge both risks.

In all cases, we perform liability hedging. ALM occurs when we either require the portfolio to be self-financing or add a bond. In the former case, a further equation must be added to the system. Because we have assumed that a single fair premium is paid at policy issuance, self-financing strategies are characterized by the self-financing constraint:

[n.sub.H][Z.sub.H] + [N.summation over (i=1)] [n.sub.i][Z.sub.i] = 0. (23)

The self-financing constraint equates the inflows from sales and the outflows for asset purchases; it means that the premiums are used to buy hedging instruments. To include a zero-coupon bond, (10) it is sufficient to expand our notation to include a contract [Z.sub.N+1] whose value is the discount factor (or to imagine a fake generation that has a constant survival probability equal to one). Because bonds are unaffected by mortality risk, their Greeks with respect to mortality are null, and bonds can be used as hedging instruments for financial risk.

We obtain a unique solution to the system of equations that solves the delta or delta-gamma hedging problem if the matrix of the [n.sub.i] coefficients is full rank and the number of hedging instruments equals this rank. This imposes a restriction on how many life insurance liabilities (and bonds, when they are admitted) are used for coverage:

* N = J instruments for delta hedging longevity risk;

* N = J + 1 instruments for delta hedging both risks;

* N = 2J instruments for delta-gamma hedging longevity risk;

* N = 2 + 2J instruments for delta-gamma hedging both risks.

If one accepts multiple solutions, this restriction can be relaxed. A further instrument is required if the portfolio must be self-financing.

As anticipated in the "Longevity and Interest-Rate Risks" section, one can also implement a two-step procedure that requires solving two systems of equations sequentially. First, neutralize the Greeks with respect to mortality using insurance products. Then, given the insurance portfolio, use bonds to cover the resulting financial risk. In the special case in which the term structure of interest rates is flat and interest-rate risk is absent, one can obviously omit the second step.

Practical Issues

The delta--gamma hedging procedure proposed in the previous sections appears relatively simple, but its implementation must cope with some important practical issues. It might be difficult for a life office to sell the exact number of policies required for hedging purposes and/or to buy the exact amount of reinsurance. If the desired number of policies does not match the availability of potential customers in the market, the system of Equations (19)-(23) should be accompanied by some additional constraints. The additional restrictions may imply that the system has no longer a solution. In this case, it is necessary to include additional hedging instruments to obtain at least one solution to the system.

If reinsurance is not available, the delta--gamma hedging procedure should be performed in two steps, as described in the "Longevity and Interest-Rate Risks" section. The two-step procedure provides more freedom to choose insurance products and may avoid reinsurance. The following simple example illustrates this.

Suppose that a life office sells one annuity written on generation x ([n.sub.H] = -1) and wants to hedge it with the appropriate number ([n.sub.1], [n.sub.2]) of life insurance policies with different maturities issued on the same generation. The numbers that achieve delta-gamma hedging solve the following system:

[mathematical expression not reproducible] (24)

Suppose that we want to avoid reinsurance; that is we want both [n.sub.1] and [n.sub.2] to be negative. Assuming that [[DELTA].sub.1]/[[GAMMA].sub.1] [not equal to] [[DELTA].sub.2]/[[GAMMA].sub.2], it can be proved that the goal is achieved if and only if the following inequality holds:

min {[[DELTA].sub.1]/[[GAMMA].sub.1], [[DELTA].sub.2]/[[GAMMA].sub.2]} < [[DELTA].sub.H]/[[GAMMA].sub.H] < max {[[DELTA].sub.1]/[[GAMMA].sub.1], [[DELTA].sub.2]/[[GAMMA].sub.2]} (25)

This means that the ratio between delta and gamma of the annuity lies between the corresponding ratios of the life insurance policies. In the "Intragenerational Hedging" section, we provide an example in which condition (25) is satisfied, and the quantities [n.sub.1] and [n.sub.2] are negative.

Finally, reinsurance is costly. We should separate two cases: non-self-financing and self-financing strategies. In the first case, reinsurance premiums do not enter the system; hence, they do not affect the solution of the system directly. The hedging strategy does not change, even though its total cost is higher. In the second case, reinsurance premiums can be introduced into the model by adding the corresponding loadings to the fair value [Z.sub.i] of the hedging instruments in Equation (23). In this case, the solution to the system is affected, but the method remains valid.

UK-Calibrated Application

In the "Calibration" and "Greeks" sections, we calibrate the model to UK data and present the corresponding hedge ratios. In the "Intragenerational Hedging" section, we compute the portfolio mix of annuities and life insurance contracts within each generation that immunizes the portfolio up to the first and second orders. We also assess the effectiveness of the hedging strategies using Monte Carlo simulation. In the "Cross-Generational Natural Hedging" section, we consider cross-generation immunization.

Calibration

In calibrating the model to UK data, we assume that the risk premium on longevity risk is null: q = 0, so that a' = a. (11) We calibrate the parameters of the mortality intensity processes using UK cohort tables taken from the Human Mortality Database. We consider contracts written on the lives of male individuals who were 35 (generation 1973, which we call x), 65 (generation 1943, which we call y), and 75 (generation 1933, which we call z) years old on December 31, 2008. We calibrate the OU model to the data using a standard least squares method. We denote the set of t observed survival probabilities for each generation j = x, y, z and different horizons [tau] = 1,2, ***, n as [tau][[??].sub.j]. We jointly estimate the parameters and instantaneous correlations of the three generations. The calibrated values of [a.sub.j], [[sigma].sub.j], and vector [rho] = [[[rho].sub.xy], [[rho].sub.xz]] minimize the error:

[summation over (j=x,y,z)] [n.summation over ([tau]=1)] [([tau][[??].sub.j] - [S.sub.j]([tau];[a.sub.j],[[sigma].sub.j],[rho])).sup.2],

where [S.sub.j]([tau];[a.sub.j],[[sigma].sub.j],[rho]) are the theoretical survival probabilities for [tau] years. We derive them following the framework described in Duffie et al. (2000), as explained in Appendix B. The value of [[lambda].sub.j](0) is equal to - ln [p.sub.j]. We select 20 years of observations (n = 20), using December 31, 1988 as the observation starting point. It follows that the initial ages for generations x, y, and z are 15, 45, and 55, respectively. Table 1 reports the result of our calibration. The overall calibration error is 7.72 * [10.sup.-5].

The calibrated parameters satisfy the sufficient condition for biological reasonableness for the OU model, as in Luciano and Vigna (2008), and the shape of the predicted survival curve is reasonable (decreasing up to the terminal age [omega] = 120). However, because the model is used to predict survival probabilities that are in the distant future, survivorship at older ages is forecasted with some degree of uncertainty. (12) Although the OU model does not guarantee positive mortality intensity, we find that the probabilities of negative [[lambda].sub.x], [[lambda].sub.y], [[lambda].sub.z] are negligible given the parameters in Table 1. At the 1-year horizon considered in our simulations, the probability of negative intensity is [10.sup.-37] for generation x, [10.sup.-53] for generation y, and [10.sup.-52] for generation z.

In the application that follows, we consider longevity risk alone as well as longevity and interest-rate risks together. To introduce interest-rate risk, we calibrate the constant parameter Hull--White model to UK government bonds on January 2, 2009. We use data on the nominal government spot yields provided by the Bank of England. We obtain the zero-coupon prices on that date and calibrate the parameters of the interest-rate dynamics using the least squares method, that is, minimizing the squared differences between observed and fitted bond prices. The results are presented in Table 1.

Greeks

We first analyze each generation separately. Consider whole life annuities with a unit benefit on generations y and z and life insurance policies with different maturities on all three generations. The life insurance contracts (LIs) have sum insured of C = 100 and maturities of 10, 12, 15, 20, and 25 years. Table 2 summarizes the reserves and Greeks of the policies.

The table shows that the prices of LIs written on the youngest generation x are approximately 5 and 10 times smaller than those on y and z, respectively, because the death probabilities of generation x are much smaller than the others. The annuity price of generation z is lower than y because that generation has a shorter life expectancy at policy inception.

It is evident from the table that within each generation, the deltas and gammas with respect to longevity are greater (in absolute value terms) than the Greeks for financial risk. This happens for every product and maturity and is consistent with the result in Luciano et al. (2012) for pure endowments. Despite the relative magnitude of the Greeks, a one-standard-deviation change in the financial-risk factor has a greater impact than a comparable change in the longevity-risk factor, especially over the short term because the standard deviation of the financial-risk factor is higher than that of the mortality-risk factor. (13)

If we consider a single insurance product and compare the Greeks across generations, we notice the following:

* the sensitivity to mortality risk decreases in absolute value when moving from younger to older generations;

* the sensitivity to financial risk increases in absolute value when moving from younger to older generations for LIs and decreases for annuities.

The Greeks with respect to the longevity risks of annuities and LIs have opposite signs, while the sensitivities with respect to financial risk match in sign. Thus, there exists no portfolio mix of annuities and life insurance contracts without long positions that is able to neutralize the exposure against both longevity and interest-rate risks--either reinsurance or bonds are needed.

Intragenerational Hedging

Consider an insurer who has issued a whole life annuity with a unit benefit on an individual belonging to generation y, that is, [n.sub.H] = -1. Using insurance contracts and bonds, he aims to achieve instantaneous neutrality to longevity and interest-rate shocks. We compute the hedging coefficients, that is, the positions the insurer must hold when he uses LIs on the same cohort y as the annuitant and different hedging objectives (listed in the first column of Table 3). We assume the existence of enough instruments to provide a unique solution to the hedging system. The objectives are specified in the first column of Table 3 and include coverage of the first or second order of different risks, with or without self-financing. Empty cells refer to redundant hedging instruments, which are not used in the hedging strategy. The rest of Table 3 reports the quantities [n.sub.i] needed for hedging. The last column contains the initial value of the hedged portfolio. This value is the sum of the premiums collected from the sale of policies net of reinsurance costs, where the value of self-financing strategies is zero.

Given the set of LIs with maturities ranging from 10 to 25 years, the first row of the table shows the only case in which a portfolio of policies issued by the insurer is naturally hedged (without resorting to reinsurance or bonds), which is delta hedging of longevity risk only, as in Wang et al. (2010). Delta hedging of the mortality risk of the annuity is accomplished by issuing 0.27 10-year LIs. When we create the other hedging strategies, that is, we want to neutralize exposure to both sources of risk or establish self-financing strategies, we must combine short positions with at least one long position on a contract. This is needed to make the portfolio self-financing or to neutralize exposure with respect to financial risk, which increases when adding the LI policies issued (together with the annuity) to the portfolio. For instance, row 2 of Table 3 reports the delta hedging portfolio for longevity and financial risks. Reinsurance on 6.43 12-year LIs is needed to reach the objective. If we consider a strategy involving 10- and 30-year LIs, it is possible to design a delta--gamma hedging strategy for longevity risk without resorting to reinsurance, as described in the "Practical Issues" section. Indeed, in this case, condition (25) is satisfied as follows:

[[DELTA].sub.1]/[[GAMMA].sub.1] = -0.0141 < [[DELTA].sub.H]/[[GAMMA].sub.H] = -0.00294 < [[DELTA].sub.2]/[[GAMMA].sub.2] = -0.00268.

The delta--gamma non-self-financing hedging strategy (not reported in Table 3) consists in issuing 0.03 10-year LIs and 0.15 30-year LIs.

As explained in the "ALM and Greeks" and "Hedging" sections, it is also possible to substitute some hedging instruments with bonds. The use of bonds in this strategy has the advantage of not requiring reinsurance on LIs. For instance, delta hedging of both longevity and financial risks can be achieved by using one LI and one bond from the interest-rate market rather than two LIs. Such a hedging portfolio (not reported in Table 3) is made by issuing 0.27 10-year LIs (as in longevity only delta hedging) and buying 20.60 10-year zero-coupon bonds. The value of this portfolio, 2.63, is smaller than that of natural hedging without bonds, 13.42 (row 2 of Table 3).

Hedge Effectiveness. This section shows the effectiveness of some of the hedging portfolios reported in Table 3 in reducing longevity and financial risks. We compare nonhedged, delta-hedged, and delta--gamma-hedged portfolios. For illustrative purposes, we consider the following three portfolios:

1. Portfolio 1 (NH): nonhedged, self-financing portfolio containing 1 (m =-1) issued annuity (not reported in Table 3).

2. Portfolio 2 (DH): self-financing longevity and financial delta-hedged portfolio containing the annuity and the LIs with the maturities and positions as in row 6 of Table 3.

3. Portfolio 3 (DGH): self-financing longevity and financial delta--gamma-hedged portfolio containing the annuity and the LIs as in row 8 of Table 3.

After the hedge has been established, the approximate value of the change in the overall reserve (portfolio value) is given by the following:

[mathematical expression not reproducible]

where N is the number of hedging instruments involved in the strategy, which is 0 in the NH, 3 in the DH, and 5 in the DGH portfolios.

In the DH portfolio, only the coefficients multiplying [DELTA]I(t) and [DELTA]K(t) are nullified, while in the DGH portfolio, the coefficients multiplying [DELTA]I[(t).sup.2] and [DELTA]K[(t).sup.2] are set to zero as well. Thus, the approximate instantaneous change of the unhedged portfolio is as follows:

[mathematical expression not reproducible]

and the change in the reserve of the DH portfolio is as follows:

[mathematical expression not reproducible]

The approximate change of the DGH portfolio is null: d[Z.sup.DGH.sub.H] = 0. Hence, we would expect the change in the reserve of a nonhedged portfolio to be larger than that of a delta-hedged portfolio, that we obviously expect to be larger than that of a delta--gamma-hedged portfolio, the last being null. To check this, we have simulated 100,000 realizations of the stochastic interest rate r(t + dt) and the stochastic mortality intensity [lambda](t + dt) using the calibrated parameters. In each of the 100,000 scenarios, we have computed the change in value of the reserve for the three portfolios over the time horizons of 3 months and 1 year. In Table 4, we collect the means and standard deviations of the absolute change in their portfolio reserves (hedging error).

As expected, over both horizons, the means and standard deviations are lowest when the portfolio is delta--gamma hedged. The reduction in the variability of the position after a 3-month period is substantial (0.69 in the NH portfolio, 0.20 in the DGH portfolio). The 1-year standard deviation of the DGH strategy is 0.64 versus 1.33 for the unhedged portfolio.

Because we use a continuous-time theoretical model, the results are likely to improve using a rebalancing interval shorter than 3 months. However, when rebalancing the strategy at intervals shorter than 1 year, seasonality effects, which are not captured by our mortality model, may intervene. The table shows that even if our strategy performs best when rebalanced at shorter time intervals, it still leads to relevant hedging gains when rebalanced at longer time spans.

The variability reduction obtained through the DGH portfolio, together with the lower average hedging error, is robust to calibration error in the mortality model. Table 5 shows the standard deviation of the strategy when the true value of each parameter is 10 percent above or below the value used in computing the strategy. The strategy is most sensitive to an error in the choice of [lambda](0), when the true initial instantaneous mortality intensity is higher than assumed when the hedge is calculated. Still, even if the error is as high as 10 percent, the DGH strategy outperforms the unhedged portfolio.

Cross-Generational Natural Hedging

We now consider the case in which the insurer has a portfolio including products issued on generations x and y and wants to hedge the longevity risk of the second generation using exclusively products on the first. In our example, the insurer has issued an annuity on the older cohort y ([n.sub.H] = -1) and wants to use life insurance policies with different maturities written on the lives of the younger generation x as hedging instruments.

Using cross-generational hedging, we exploit the correlation between the dynamics of the intensities of these two generations to hedge against the common risk factor. Table 6 reports the optimal hedging strategies to achieve different goals. As in the "Intragenerational Hedging" section, we assume that there are enough products to guarantee the existence and uniqueness of the hedging strategy.

To compare different possibilities associated with the same hedging objective, we identify and compare three types of hedging portfolios. The common hedging objective is delta hedging of mortality and financial risk. The first portfolio is reported in row 2 of Table 3 and includes only products written on the lives of individuals belonging to generation y. The second portfolio is reported in row 2 of Table 6 in which the annuity on y is hedged by LIs on x only. The third portfolio includes products on y and bonds, as described at the end of "Intragenerational Hedging" section. Idiosyncratic risk is involved, as the value of the second portfolio is affected by a change in the risk factor I'. The first and third portfolios are unaffected by idiosyncratic risk.

If enough products on generation y are available, namely, two LIs, there is no need to hedge cross-generationally. When no sale or reinsurance on LIs on generation y is feasible, the insurer can construct a delta-hedged portfolio using LIs written on generation x. He should issue 11.94 10-year LIs and buy reinsurance on 4.57 20-year LIs. In this example, the practical application of natural cross-generation hedging yields a portfolio whose value of liabilities is slightly higher than that of the intra-generational portfolio (15.44 vs. 13.42). To compare the effectiveness of the hedges, we compute the value change of the three portfolios after a shock on the three risk factors as large as their instantaneous volatility ([DELTA]I = -0.000198, [DELTA]I' = -0.000025, [DELTA]K = -0.0217). Table 7 reports the effects of such changes on the values of the portfolios and isolates the impact of each risk factor realization. In particular, the column labeled "Effect of [DELTA]I" reports the quantity [[DELTA].sup.M,j.sub.[PI]], [DELTA]I + 1/2 [[GAMMA].sup.M,j.sub.[PI]] [DELTA][I.sup.2], where [[DELTA].sup.M,j.sub.[PI]] and [[GAMMA].sup.M,j.sub.[PI]] are the delta and the gamma of the whole portfolio with respect to changes in I, and so on.

As in Table 4, the difference between unhedged and hedged portfolios is remarkable: the order of magnitude of the change in portfolio value of the unhedged portfolio is at least 10 times larger than in the hedged portfolios. In all portfolios, the changes are non-null because they are delta hedged, and second-order terms affect the change of the reserve. The table highlights that even if the forecast error on interest-rate risk is the main source of hedging error for the unhedged portfolio, unexpected changes in mortality rates can contribute substantially to portfolio variability. Realized mortality intensity is lower than predicted and interest rates decrease, leading to a substantial and unexpected 22.76 percent increase in the fair value of the annuity, mainly (approximately 90 percent) due to interest-rate forecast error. This is the change in the value of an unhedged position. In contrast, 96 percent of the variation of the single-generation portfolio is due to the change [DELTA]I. In the cross-generation portfolio, we see the effect of idiosyncratic risk: even if correlation is very close to 1, the effect of the change in I' (-0.46 percent) is notable, being (in absolute terms) nearly one-third of the change due to I (1.80 percent). The within-generation strategy performs best, as the change in the portfolio value is 0.31 percent. Compared to the unhedged portfolio, the cross-generation hedging strategy greatly reduces the value change to 2.54 percent with respect to the unhedged portfolio. The hedging strategy including the bond leads to a slightly higher percentage portfolio value change of -2.55 percent. The increase in the value of the long position on the bond is indeed offset by a larger increase in the value of the insurance liabilities. Consequently, it performs worse than the portfolios that include insurance liabilities only.

CONCLUSIONS

In this article, we have studied the natural hedging of financial and longevity risks using the delta--gamma technique. We have assumed a continuous-time, cohort-based model for longevity risk that generalizes the classic Gompertz law and a standard stochastic interest-rate model (Hull--White).

We have extended the existing literature by analyzing financial and longevity risks at the same time by providing closed-form hedges and by considering intrageneration as well as cross-generation hedges. We have obtained portfolios that are immunized to longevity and financial risks up to the second order, and we have clarified the role of natural hedging between annuities and life insurance policies. Our numerical application to UK data achieves different goals. First, it permits comparison of financial and longevity sensitivities (the Greeks) within and across generations. Second, it proves how to natural hedge up to the second order in closed form. Third, it allows discussion of intra- and cross-generational hedging. In particular, we demonstrate how to exploit the fact that survival probabilities of a cohort are correlated with the longevity-risk factors of another cohort; when there are not enough products written on one generation, products written on other cohorts help complete the longevity market and make natural hedging feasible. This is particularly important when hedging annuities written on older people, such as pensioners. Last but not least, we assess the effectiveness of natural hedging strategies; Monte Carlo simulations show that the change in the reserve of a nonhedged portfolio is higher than that of a delta- or delta--gamma-hedged portfolio.

APPENDIX A

This appendix obtains the risk factors against which to hedge in the presence of two correlated Brownian motions that affect the intensities of two generations/genders.

We can write the dynamics of the x and y generations' intensities in (12) and (13) in terms of two independent Brownian motions, [[??].sub.x] and [[??].sub.y]:

[mathematical expression not reproducible]

Because from (B1) d[[??].sub.x](t) = (d[[lambda].sub.x](t) - [a'.sub.x][[lambda].sub.x](t)dt)/[[sigma].sub.x], we can rewrite d[[lambda].sub.y] as follows:

[mathematical expression not reproducible]

The dynamics d[[lambda].sub.y](t) depend on d[[lambda].sub.x](t) = dI(t) and on d[[lambda]'.sub.y](t), defined as follows:

[mathematical expression not reproducible]

Applying Ito's lemma and rearranging, we have the following expression for d[S.sub.y] as follows:

[mathematical expression not reproducible]

[S.sub.y] depends on the two risk factors identified above: change in the mortality intensity of cohort x, d[[lambda].sub.x], and change in the mortality intensity of [[lambda].sub.y], which is uncorrelated with the dynamics of the mortality of cohort x, d[[lambda]'.sub.y]. Notice also that dI(t) = d[[lambda].sub.x](t). Defining as Greeks the coefficients of the first- and second-order changes in the risk factors--according to (15), (16), (17), (18)--and setting dI' =d[[lambda]'.sub.y], we obtain d[S.sub.y] (t, T)as in (14) in the text.

APPENDIX B

The processes [[lambda].sub.x](t) and [[lambda].sub.y](t) can be reparametrized as follows:

d[[lambda].sub.x](t) = [a.sub.x][[lambda].sub.x](t)dt + [[sigma].sub.x]d[W.sub.x](t), d[[lambda].sub.y](t) = [a.sub.y][[lambda].sub.y](t)dt + [[sigma].sub.x]yd[W.sub.x](t) + [[sigma].sub.yy]d[W.sub.y](t). (B1)

Applying the framework of Duffie et al. (2000), it is possible to calculate the survival probabilities for generations x and y, yielding:

[mathematical expression not reproducible]

with

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

with

[mathematical expression not reproducible]

Finally, according to (B1), the instantaneous correlation between d[[lambda].sub.x] and d[[lambda].sub.y] is given by the following:

Corr (d[[lambda].sub.x](t), d[[lambda].sub.y] (t)) = [[rho].sub.xy] = [[sigma].sub.xy]/[square root of [[sigma].sup.2.sub.xy] + [[sigma].sup.2.sub.yy]].

REFERENCES

Blake, D., and W. Burrows, 2001, Survivor Bonds: Helping to Hedge Mortality Risk, Journal of Risk and Insurance, 68: 339-348.

Blake, D., A. Cairns, K. Dowd, and R. MacMinn, 2006, Longevity Bonds: Financial Engineering, Valuation, and Hedging, Journal of Risk and Insurance, 73(4): 647-672.

Cairns, A., D. Blake, and K. Dowd, 2006, Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk, Astin Bulletin, 36(1): 79-120.

Cairns, A., D. Blake, and K. Dowd, 2008, Modelling and Management of Mortality Risk: A Review, Scandinavian Actuarial Journal, (2): 79-113.

Cox, S., and Y. Lin, 2007, Natural Hedging of Life and Annuity Mortality Risks, North American Actuarial Journal, 11: 1-15.

Duffie, D., J. Pan, and K. Singleton, 2000, Transform Analysis and Asset Pricing for Affine Jump-Diffusions, Econometrica, 68(6): 1343-1376.

Gatzert, N., and H. Wesker, 2012, The Impact of Natural Hedging on a Life Insurer's Risk Situation, Journal of Risk Finance, 13: 396-423.

Gatzert, N., and H. Wesker, 2013, Mortality Risk and Its Effect on Shortfall and Risk Management in Life Insurance, Journal of Risk and Insurance, 81: 57-90.

Heath, D., R. Jarrow, and A. Morton, 1992, Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation, Econometrica, 60(1): 77-105.

Hull, J., and A. White, 1990, Pricing Interest-Rate-Derivative Securities, Review of Financial Studies, 3(4): 573-592.

Jarrow, R., and S. Turnbull, 1994, Delta, Gamma and Bucket Hedging of Interest Rate Derivatives, Applied Mathematical Finance, 1: 21-48.

Luciano, E., L. Regis, and E. Vigna, 2012, delta--gamma Hedging of Mortality and Interest Rate Risk, Insurance: Mathematics and Economics, 50(3): 402-412.

Luciano, E., and E. Vigna, 2008, Mortality Risk via Affine Stochastic Intensities: Calibration and Empirical Relevance, Belgian Actuarial Bulletin, 8(1): 5-16.

Stevens, R., A. De Waegenaere, and B. Melenberg, 2011, Longevity Risk and Natural Hedge Potential in Portfolios of Life Insurance Products: The Effect of Investment Risk, CentER Discussion Paper 2011-036.

University of California, Berkeley (USA) and Max Planck Institute for Demographic Research (Germany). Human Mortality Database, World Wide Web: www.mortality.org (accessed November 3, 2010).

Wang, J., H. Huang, S. Yang, and J. Tsai, 2010, An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach, Journal of Risk and Insurance, 77(2): 473-497.

Elisa Luciano is at the University of Torino, Collegio Carlo Alberto and NETSPAR. Luca Regis is at the IMT Institute for Advanced Studies Lucca and Collegio Carlo Alberto. Elena Vigna is at the University of Torino, Collegio Carlo Alberto and CeRP. Luciano can be contacted via e-mail: elisa.luciano@unito.it. Regis can be contacted via e-mail: luca.regis@imtlucca.it. Vigna can be contacted via e-mail: elena.vigna@unito.it. We thank Jeff Mulholland, An Chen, Nadine Gatzert, and participants at the VII Longevity Conference, MAF 2012 Conference, 1st EAJ Conference, Longevity VIII Conference, and XXXVI AMASES Conference and to seminar at IMT Lucca for helpful suggestions. We also thank three anonymous referees for valuable comments, in particular for suggesting the "Practical Issues" and the "Hedge Effectiveness" sections. Luca Regis gratefully acknowledges financial support from the Crisis Lab project funded by the Italian Ministry of Education. An incomplete version of the article has been circulated under the title "Natural Delta Gamma Hedging of Longevity and Interest Rate Risk" as ICER wp 21/2011.

(1) This filtration reflects both mortality and financial information. For a discussion of its relationship to the natural filtration of the mortality intensity and interest-rate processes as well as for the relevant change of measure, see Luciano et al. (2012).

(2) See Luciano and Vigna (2008). In that article, the authors argue that the OU model, along with other non-mean-reverting affine processes, meets all the criteria of a good mortality model developed by Cairns et al. (2006), with the exception of strictly positive intensity. Indeed, it fits historical data well; its long-term future dynamics are biologically reasonable; it is convenient for pricing, valuation, and hedging; and its long-term mortality improvements are not mean reverting. Most importantly for the case at hand, mortality-linked products can be priced using analytical methods.

(3) That measure is unique, with one Wiener and one risky bond with nondegenerate volatility.

(4) With a slight abuse of notation, we also use Q to denote this measure. A more detailed discussion of the change of measure is provided in Luciano et al. (2012).

(5) Notice that, given the absence of a rich market for longevity bonds, there are no standard choices to apply in the choice of [[theta].sub.x](t). See, for instance, the extensive discussion in Cairns et al. (2006).

(6) The horizon T depends on the type of annuity issued; that is, it is T for an annuity payable for T years and [omega] - x for a whole life annuity.

(7) Our results also apply when more life insurance contracts with different maturities [T.sub.i] are considered, as the calibrated example in the "UK-Calibrated Application" section shows.

(8) Alternatively, we can interpret positive positions as need for mortality-linked contracts, such as survivor bonds and other derivatives.

(9) Notice that the number of longevity risk factors to hedge against is the same as the number of generations in the portfolio.

(10) Because any bond can be stripped into zero-coupon bonds, with some additional notation, we can also include coupon bonds.

(11) This assumption will be easily removed by calibrating the model parameters to actual mortality derivative prices as soon as a liquid market for them exists. Alternatively, we could parametrize the results to a hypothetical, positive risk premium.

(12) It is well known that no parsimonious mortality model describes mortality over the whole life span. Hence, survival probabilities at older ages may not be forecasted properly. The use of a two-factor model is likely to improve on this point. Alternatively, the calibration of the parameters [a.sub.x] and [[sigma].sub.x] could be performed using projected mortality tables rather than observed figures, as in Luciano and Vigna (2008). This would imply calibrating the parameters of the OU mortality model using data forecasted by other mortality models (those underlying the projected tables).

(13) For instance, consider a 1-year change in the two risk factors equal to one standard deviation: [DELTA]I = 0.0002 and [DELTA]K = 0.0217. The effect on the value of an annuity written on generation y is

[[DELTA].sup.M] [DELTA]I + 1/2 [DELTA][I.sup.2] + [[DELTA].sup.F] [DELTA]K + 1/2 [DELTA][K.sup.2] = -2.57.

The first two terms, which represent the change in annuity value due to mortality risk, sum to -0.28, while the last two terms, which represent the value change due to interest-rate risk, sum to -2. 29. Overall, the effect of mortality risk, though non-negligible, is smaller than that due to interest-rate risk despite the magnitude of the longevity Greeks.

DOI: 10.1111/jori.12104
TABLE 1 Calibrated Parameters

                               Mortality Intensity
        [a.sub.GEN]  [[sigma].sub.GEN]  [[lambda].sub.GEN](0)

GEN: x    0.0809         0.0000325           0.000396
GEN: y    0.0801         0.0001987           0.002919
GEN: z    0.0750         0.0005970           0.0087

        Mortality Intensity
        [[rho].sub.x,GEN]

GEN: x       1
GEN: y       0.9919
GEN: z       0.9971

                  Spot Interest Rate
g       [sigma]  [theta]   r(0)   Calibration Error

0.0244  0.0217   0.2432   0.0153   0.000346101

TABLE 2 Greeks for Annuities and Life Insurance Policies

                        Whole Life Ann.  10-Year LI   12-Year LI

Gen. 1973 (x)
 Price                                          2.52         3.14
 [[DELTA].sup.M.sub.i]                      6,282.31     7,793.43
 [[GAMMA].sup.M.sub.i]                   -457,732.60  -734,018.14
 [[DELTA].sup.F.sub.i]                        -13.49       -19.70
 [[GAMMA].sup.F.sub.i]                         87.79       150.17
Gen. 1943 (y)
 Price                          12.68          16.60        20.15
 [[DELTA].sup.M.sub.i]      -1,411.54       5,153.97     6,070.33
 [[GAMMA].sup.M.sub.i]     479,726.3     -365,650.98  -553,680.14
 [[DELTA].sup.F.sub.i]        -105.87         -86.49      -122.03
 [[GAMMA].sup.M.sub.i]       1,270.02         553.62       910.38
Gen. 1933 (z)
 Price                           8.87          36.88        42.81
 [[DELTA].sup.M.sub.i]        -494.96       3,276.24     3,525.16
 [[GAMMA].sup.M.sub.i]      65,834.52    -201,349.93  -274,028.32
 [[DELTA].sup.F.sub.i]         -57.37        -181.47      -240.85
 [[GAMMA].sup.F.sub.i]         542.04       1,121.42     1,716.88

                          15-Year LI    20-Year LI       25-YearLI

Gen. 1973 (x)
 Price                           4.13           5.98           8.08
 [[DELTA].sup.M.sub.i]      10,188.18      14,492.64      19,154.94
 [[GAMMA].sup.M.sub.i]  -1,344,487.87  -3,114,396.87  -6,385,279.86
 [[DELTA].sup.F.sub.i]         -31.50         -58.36         -95.47
 [[GAMMA].sup.F.sub.i]         290.55         683.80       1,339.05
Gen. 1943 (y)
 Price                          25.37          33.43          40.16
 [[DELTA].sup.M.sub.i]       7,250.65       8,532.00       8,799.38
 [[GAMMA].sup.M.sub.i]    -915,343.48  -1,689,235.25  -2,505,336.59
 [[DELTA].sup.F.sub.i]        -183.87        -300.95        -418.91
 [[GAMMA].sup.M.sub.i]       1,644.84       3,351.31       5,423.88
Gen. 1933 (z)
 Price                          50.27          58.77          63.07
 [[DELTA].sup.M.sub.i]       3,637.64       3,311.63       2,742.82
 [[GAMMA].sup.M.sub.i]    -377,954.11    -486,175.47    -473,502.73
 [[DELTA].sup.F.sub.i]        -328.95        -451.38        -526.27
 [[GAMMA].sup.F.sub.i]       2,760.36       4,531.13       5,837.12

TABLE 3 Hedging Strategies for an Annuity on Generation y

Strategy/Instrument  10-Year LI  12-Year LI  15-Year LI  20-Year LI

D M                     -0.27
D MF                    -7.84        6.43
DG M                     3.36       -3.09
DG MF                  -27.04                   59.54      -52.09
D M SF                 -34.49       29.05
DMF SF                  26.25                  -36.98       15.40
DG M SF                603.32     -807.15      246.70
DG MF SF              -197.89      322.69     -132.32      -12.02

Strategy/Instrument  25-Year LI  Value

D M                              17.22
D MF                             13.42
DG M                             23.25
DG MF                   17.12     5.00
D M SF                            0
DMF SF                            0
DG M SF                           0
DG MF SF                13.82     0

Note: In the table, D stands for delta hedging, DG for delta--gamma
hedging, M for longevity only, MF for both longevity and financial risk
neutralization, and SF for self-financing. Empty cells refer to
instruments that are not used in the strategy; their holding is set to
zero. The "Value" column refers to the fair value of the liability
portfolio, which equals the premiums collected when setting up the
position.

TABLE 4 Mean and Standard Deviation of the Hedging Error After 3 Months
and 1 Year

Strategy            Mean (Std Deviation)
                    3 Months      1 Year

Portfolio 1 (NH)   0.90 (0.69)  1.88 (1.33)
Portfolio 2 (dh)   0.29 (0.32)  0.91 (1.28)
Portfolio 3 (DGH)  0.25 (0.20)  0.83 (0.64)

TABLE 5 Stress Test on the Means and Standard Deviations of Hedging
Error After 1 Year

                              a                     [sigma]
Strategy              +10%         -10%         +10%        -10%
                                                Mean (Std Deviation)

Portfolio 1 (NH)   1.88 (1.33)  1.89 (1.33)  1.89 (1.33)  1.88 (1.33)
Portfolio 3 (DGH)  0.88 (0.67)  0.79 (0.62)  0.86 (0.69)  0.80 (0.60)

                          [lambda](0)
Strategy              +10%         -10%

Portfolio 1 (NH)   1.96 (1.30)  1.86 (1.41)
Portfolio 3 (DGH)  1.66 (0.96)  0.71 (0.72)

TABLE 6 Hedging Strategies for an Annuity on Generation y Using
Products Written on Generation x

Strategy/Instrument  10-Year [LI.sub.x]  12-Year[LI.sub.x]

D M
D MF                          -11.94
DG M                           18.33
DG MF                       2,202.73         -3,874.76
D M SF                       -233.75
D MF SF                     1,196.14

Strategy/Instrument  15-Year[LI.sub.x]  20-Year [LI.sub.x]  Value

D M                                           - 0.45        15.30
D MF                                            4.57        15.44
DG M                                           -8.55        17.54
DG MF                     2,007.37           -282-95        17.80
D M SF                                        100.73         0
D MF SF                  -1,497.71            533.77         0

Note: In the table, D stands for delta hedging, DG for delta--gamma
hedging, M for longevity only, MF for both longevity and financial risk
neutralization, and SF for self-financing. Empty cells refer to
instruments that are not used in the strategy; their holding is set to
zero. The "Value" column refers to the fair value of the liability
portfolio, which equals the premiums collected when setting up the
position.

TABLE 7 Effect of a Realized Change in the Risk Factors on Three
Portfolios

Portfolio/Risk Factor  Effect of [DELTA]I  Effect of [DELTA]I'

Unhedged                    -2.29%               0
Single-Generation            0.30%               0
Cross-Generation             1.80%              -0.46%
Bond                        -0.43%               0

Portfolio/Risk Factor  Effect of [DELTA]K    Total

Unhedged                    -20.48%        -22.76%
Single-Generation             0.01%          0.31%
Cross-Generation              1.19%          2.54%
Bond                         -2.11%         -2.55%

Note: The table shows the percentage change in the unhedged portfolio
and three hedging portfolios following realized changes in the risk
factors equal to [DELTA]I =-0.000198,[DELTA]I' = -0.000025, and
[DELTA]K = -0.0217.
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Author:Luciano, Elisa; Regis, Luca; Vigna, Elena
Publication:Journal of Risk and Insurance
Date:Sep 1, 2017
Words:10935
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