# Mortality portfolio risk management.

ABSTRACT

We provide a new method, the "MV + CVaR approach," for managing unexpected mortality changes underlying annuities and life insurance. The MV + CVaR approach optimizes the mean-variance trade-off of an insurer's mortality portfolio, subject to constraints on downside risk. We apply the method of moments and the maximum entropy method to analyze the efficiency of MV + CVaR mortality portfolios relative to traditional Markowitz mean-variance portfolios. Our numerical examples illustrate the superiority of the MV + CVaR approach in mortality risk management and shed new light on natural hedging effects of annuities and life insurance.

INTRODUCTION

Life insurance companies sell a wide variety of life insurance and annuity products. The insurers' liabilities for these products depend on future mortality rates. During recent years, economic and demographic changes have made mortality projection and risk management more important than ever. On the one hand, life expectancy for ages 60 and older in the past two decades has improved at a much higher rate than what pension plans and annuity providers expected. Cowling and Dales (2008) find that companies in the United Kingdom FTSE100 index underestimated their aggregate pension liabilities by more than 40 billion pounds sterling]. If the firms do not take measures to control mortality downside risk, such longevity shocks are likely to cause serious financial consequences. For example, unanticipated mortality improvement was an important factor accounting for the failure of Equitable Life, once a highly regarded U.K. life insurer (Ombudsman, 2008). On the other hand, population growth, urbanization, and increased global mobility may lead to a more rapid and widespread disease. Genetic analysts recently confirmed that today's "bird flu" is similar to the 1918 "Spanish flu" that killed more than 40 million people. This finding spurs fears of a worldwide epidemic (Juckett, 2006). According to Toole (2007), losses due to a severe pandemic could amount to 25 percent of the U.S. life insurance industry's statutory capital. While the great majority of U.S. life insurance companies would weather such a pandemic, it is clear that these companies should be interested in mitigating the risk.

We propose a method that life insurance companies can use to alleviate extreme mortality outcomes while maintaining a relatively efficient mean-variance relationship for their mortality portfolios of life insurance and/or annuities. This method, the "MV+CVaR approach," combines Markowitz mean-variance (MV) portfolio theory and conditional value at risk (CVaR) by optimizing the trade-off between mean and variance subject to an upper bound on CVaR. Variance measures both positive and negative deviations of portfolio values from its expected level, while CVaR focuses on the portfolio tail loss caused by extreme events. Although the MV+CVaR portfolios are suboptimal relative to the Markowitz counterparts in terms of the mean-variance efficiency, they are attractive to insurers since the MV+CVaR portfolios have lower downside risk while achieving desirable risk-return trade-offs. In practice, life insurers are keenly interested in searching for an optimal risk-return relationship for their business. At the same time they are required to meet various solvency requirements for possible catastrophes such as flu epidemics. Therefore, incorporating both variance and CVaR as risk measures in business optimization such as the MV+CVaR approach should be appealing to life insurance companies. The risk control framework adopted in this article closely follows that of Rockafellar and Uryasev (2000) and Tsai, Wang, and Tzeng (2010). In their framework, firms minimize portfolio losses subject to CVaR constraints. In our context, we consider both variance and CVaR as risk measures by incorporating a CVaR constraint into the classical mean-variance setup.

We extend Rockafellar and Uryasev (2000) and Tsai, Wang, and Tzeng (2010) in one important dimension by applying the well-developed moments method to validate the quality of MV+CVaR portfolios. The existing literature on mortality models usually makes distributional assumptions. Since knowledge about the underlying mortality distributions may be limited, the assumed distributions may not represent the actual mortality dynamics. However, the moments method is solely based on moments, not a specific distribution. This method yields robust semiparametric upper and lower bounds that any reasonable model with the same moments must satisfy. Once a mortality portfolio has been constructed, the corresponding empirical benefit payments can be used to estimate moments of the benefit payment ratios. In this article, we show how such estimated moment information can be used to determine the bounds on the underlying portfolio benefit payment ratios. In particular, the moments method provides a mechanism to compare the downside risk of a MV+CVaR efficient mortality portfolio and its MV counterpart given their moments. Our results show the superiority of the MV+CVaR approach in mortality risk management and highlight the natural hedging effects of life insurance and annuities.

For the numerical examples, all optimization problems are solved with MATLAB (1) software. When computing the bounds of a particular portfolio, we solve the equivalent dual problem using built-in functions from the SOS programming solver. The SOS programming solver was developed by Prajna et al. (2004), which is a free toolbox written in MATLAB. Our implementation is fairly general and easy to use.

The remainder of this article is organized as follows. The first section describes how to calculate benefit payment ratios of mortality portfolios. The second section discusses the MV+CVaR mortality optimization model. We provide two numerical examples to illustrate the implementation of the approach. The third section describes the method of moments. We show how to compute the semiparametric upper and lower bounds for MV+CVaR efficient portfolios and then perform the bound analysis on those portfolios. The fourth section demonstrates the natural hedging effect when annuities and life insurance are considered jointly. The fifth section extends the analysis to efficient frontiers of MV+CVaR mortality portfolios by comparing bounds of MV and MV+CVaR optimal portfolios. In addition, we show to what extent downside risk is reduced by changing the CVaR constraint. The sixth section is our conclusion.

MORTALITY RISK PORTFOLIOS

Focus on Mortality

Consider a life insurer underwriting new business at the current moment 0, with the business sorted by underwriting class (x). The class symbol (x) represents the information at time 0 that the insurer uses to determine a mortality table for lives in the class. This information includes at least age, sex, and line of busines but could include other information such as height, weight, blood pressure, or tobacco use history. Actuarial textbooks usually work with a set of mortality tables that differ only by age. The other information is suppressed, and the tables are obtained by simply specifying the issue age x. However, in practice a company will have different tables for each age, sex, major line of business, and so on. Thus, in our notation (x) represents all the information the company needs to estimate a mortality table for this class at 0. Here are our assumptions:

1. All future cash flows are discounted at risk-free interest rates. These discount rates are known constants.

2. Life insurance policies and annuity contracts remain in force until settled at death. There are no policy lapses.

3. Expenses are known constant multiples of policy premiums.

4. The analysis is applied only to new business without regard to the insurer's book of business issued before time 0.

Under these assumptions, the present value of benefits (and related expenses) paid at times 1, 2, ... , PVB(x), and the present value of premiums (and related expenses) collected at 0, 1, 2, ... , PVP(x), are simple functions of the lifetime K(x). The randomness of the present values has two components. First, future mortality events will have an impact on annual survival rates for members of the class (x). That is, the annual survival rate P(x)+t for the year running from t to t + 1 is random for t = 1, 2, ... , Given a random path of the survival rates [pi] (x) = {P(x), P(x)+l, P(x)+2,...}, we can calculate the conditional distribution of the lifetime K (x). But some randomness still remains. However, the remaining randomness is subject to the law of large numbers; if the company writes a large enough number of policies in each underwriting class, it can be ignored. Therefore, we focus on the conditional expected values [??](x) = E[PVB(x)| [pi](x)] and [??](x) = E[PVP(x)|[pi](x)].

We assume the company uses the benefit payment ratio [??](x) = [??](x)/[??](x) + [phi] to manage its risk where [phi] is a constant expense ratio. Other measures are possible; for example, one might use [??](x) + [phi] [??](x) - [??](x). The benefit payment ratio is analogous to the loss ratio used in general (non-life) insurance where policies are much more often single premium (and single period) contracts.

The benefit payment ratio is a random variable as viewed from time 0, but only due to the uncertainty of future mortality rates. We give some examples below to illustrate the calculation of the benefit payment ratio. The company has a model for generating paths [pi] (x), and from them it generates a sample of values of [??](x). Moments of these sample distributions are inputs to the mortality risk management process. In our examples, we will use the Lee-Carter model (Lee and Carter, 1992) to generate future paths of the survival rates [P.sub.(x)+t], or equivalently the death rates [q.sub.(x)+t] = 1 - [P.sub.(x)+t]. In principle there are many methods that could be used; the company may have its own proprietary method.

In practice the insurance company will consider more than one line of business at 0; let ([x.sub.i]) denote a set of underwriting classes under consideration at 0. For example, ([X.sub.l]) might be male, 10-year term life insurance issued at age [x.sub.1] = 35; ([x.sub.2]) might be an immediate annuity issued to a female age [x.sub.2] = 65; and so on. The company jointly simulates paths for each class. This is much more difficult than simulating a single path. The Lee-Carter method has been extended to handle this situation (Li and Lee, 2005). Brouhns, Denuit, and Vermunt (2002) show how an extension of the Lee-Carter model can be used to jointly model annuitants and the general population to which they belong. Hyndman, Booth, and Yasmeen (2011) describe another generalization of Lee--Carter that allows for joint modeling of components of a population. However, these methods are still being developed. Moreover, there is very little publicly available data on insured lives and annuitants. Therefore, we will use a single mortality table forecast. There are circumstances where it is appropriate. For example, if we consider 10-year term insurance issued at 0 to a male age (35) and 20-year term insurance issued at 0 to a male age (45), then it is reasonable to use mortality rates from the same table that differs only by age. That is, we need to project only one table and then select values from it for the appropriate ages.

Examples

Recall that the benefit payment ratio [??](x) is the ratio, calculated at 0, of the conditional expected present value of benefits to the conditional expected present value of future premiums for the contract issued on (x), with the conditional distribution of K (x) calculated relative to a given future path of the underlying mortality table. For example, given a path of future survival rates [pi] (x) = {P(x), P(x)+l, P(x)+2,...} for (x), we can calculate the conditional survival function of K (x) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

with the last expression analogous to standard actuarial notation. It is analogous because in our setting the table is random so t [p.sub.(x)] is as random as the table. The probability density function of K (x) is

[f.sub.(x] (t) = Pr [K(x) = t |[pi] (x)] = [S.sub.(x)] (t) - [S.sub.(x)](t + 1) = t | q (x). (2)

Once we have the conditional pdf of K (x), we can calculate conditional moments. Here are some examples.

Life Insurance. For k-year term life insurance issued to (x) at 0 paying I at the end of the year of death, here is the present value of benefits and its conditional expected value:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [v.sup.(t)] is the t-year discount factor. For a large enough value of k, [??](x) is the same as the present value of benefits for a whole life insurance; in this case [A.sub.(x)] is analogous to the standard actuarial notation.

If the policy was issued at 0 and paid with an annual level premium [eta] per unit of benefit at that time, then the benefit payment ratio is simply [??](x) =[??](x)/[??](x) + [phi],

where the conditional expected value of premiums [??] (x) equals

[??] (X) = [k-1.summation over (t=0)] [eta][v.sup.(t)]

Immediate Annuity. This is the present value of benefits for a single premium immediate annuity of I per year paid at the end of the year issued on (x) at 0 and its conditional expected value:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The benefit payment ratio at 0 is [??] (x) = [??] (x)/P + [phi], where P is the single premium paid at 0 per unit of annual benefit.

In the next section we consider an insurer selling several lines of business at time 0 and optimize its portfolio. In order to do this we need statistical samples of the benefit payment ratios of lines of business under consideration, which we obtain by Monte Carlo simulation.

MORTALITY PORTFOLIO OPTIMIZATION

The original Markowitz portfolio optimization (Markowitz, 1952) maximizes an investor's expected portfolio return for a given level of risk, as measured by the variance of the return. More recent portfolio optimization techniques have been applied to corporate risk management to find optimal business strategies. For example, insurers are motivated to search for business compositions that not only achieve mean-variance efficiency but also minimize downside risk. In this section, we extend this line of research by adding capital constraints to the Markowitz problem for managing an insurer's mortality portfolio.

Portfolio Optimization With CVaR Constraints

The conditional value at risk (CVaR) is a risk measure, defined as the expected loss conditional on the benefit payment ratio being higher than a given value at risk (VaR). The existing literature has many CVaR-related portfolio management methodologies. For example, Rockafellar and Uryasev (2000) generate an efficient [CVaR.sub.[beta]]-mean frontier by minimizing the CVaR of portfolio's losses subject to an expected return requirement. Krokhmal, Palmquist, and Uryasev (2002) suggest minimizing the negative expected return with a CVaR constraint. In the context of mortality risk management, we explicitly consider the trade-off between mean and variance subject to CVaR constraints by adding a CVaR constraint to the traditional Markowitz problem. We call it the MV+CVaR approach.

The company's new business issued at 0 will have weight [w.sub.i] in line of business ([x.sub.i]). The weighted average of the benefit payment ratios is [??](w) = [[SIGMA].sup.n.sub.i=1] [w.sub.i] [??] ([x.sub.i]) Let [[sigma].sub.ij] = Cov ([??]([x.sub.i]), [??]([x.sub.j])) denote the covariance of benefit payment ratios of business lines I and j. Our MV+CVaR problem is to solve for portfolio weights w = [[w.sub.1], [w.sub.2], ... , [W.sub.n]] in terms of benefit payment ratios, so as to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [CVaR.sub.[beta]] ([??](w)) is the [beta]-level CVaR of [??](w), which in the case of a continuously distributed [??](w) can be calculated as

[CVaR.sub.[beta]] = ([??](w)) = E[[??](w)| [??] (w) [greater than or equal to] [VaR.sub.[beta]] ([??](w)))].

The [beta]-level value at risk, [VaR.sub.[beta]] ([??](w)), is the [bets]-quantile of the distribution of [??](w), or the smallest value of [alpha] such that Pr([??](w) [less than or equal to] [alpha]) [greater than or equal to] [beta]. That is,

[VaR.sub.[beta]] ([??](w)) = min{[alpha] | Pr([??](w) [less than nor equal to] [alpha]) [greater than or equal to [beta]}.

We enforce a right-tail constraint to manage the downside risk by setting [beta] at 0.95. The CVaR constraint in (6) ensures the tail expectation [CVaR.sub.[beta]] ([??](w)) is no higher than a prespecified value [zeta], thus reducing the downside risk. W is a subset of the mortality business feasible set. Specifically in our problem, W is the vector of weights to = [[w.sub.1], [w.sub.2], ... , [w.sub.n]] satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [??] ([x.sub.i]) is the random benefit payment ratio of line i, n is the number of business lines in the mortality portfolio, and [e.sub.0]is a prespecified level of the weighted average of the benefit payment ratios. We assume that [w.sub.i] [greater than or equal to] 0 for each i. Negative values would indicate the insurer buys insurance rather than selling. It is possible to do this in practice with reinsurance, but we are not allowing it in our model.

Rockafellar and Uryasev (2000, 2002) show that the [beta]-level CVaR is the optimal value of the following optimization problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where the notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Problem (6) is equivalent to the following Problem (9) in the sense that their objectives achieve the same minimum values.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

If a pair ([[alpha].sup.*], [[??]([w.sup.*])) achieves the minimization of (9), G([[alpha].sup.*], [??]([w.sup.*])) will return [[alpha].sup.*] level CVaR and [[alpha].sup.*] will give the corresponding [beta]-level VaR. For the detailed proof of the theorem, see Theorem 4 of Krokhmal, Palmquist, and Uryasev (2002).

Here, we present the equivalence of the CVaR constraint (i.e., [CVaR.sub.[beta]] ([??](w)) [less than or equal to] [zeta] if) and the constraint G([alpha], [??](w))[zeta] [less than or equal to] in the form shown in (9) that will be suitable for our purpose, instead of presenting it in its general form proposed by Krokhmal, Palmquist, and Uryasev (2002). In general, if the objective function is convex and the constraints (other than the CVaR constraint) are linear, one can replace the CVaR constraint in the optimization problem with G([alpha], [??](w)) [less than or equal to] [zeta].

Compared with the CVaR constraint in Problem (6), G([alpha] [??](w)) [less than or equal to] [zeta] has a tractable formulation. Using a well-known linearization technique, adding auxiliary variables [y.sub.k] for k = 1, ... K, the constraint [alpha] + 1/[1-[beta]] E[[[[??](w) - [alpha]].sup.+]] [less than or equal to] [zeta] can be realized as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where the expected value of E[[[??](w) - [alpha]]+] is calculated as the mean of K equiprobable observations,

1/K [k.summation over (k=1) [[[??].sub.k] (W) - [alpha]].sup.+].

Below we proceed with an example to illustrate our MV+CVaR approach.

Numerical Illustration

Now we conduct numerical experiments for a life insurer's business portfolio to illustrate the superiority of the MV+CVaR approach to the MV approach. Example 1 demonstrates the case of an insurance company considering three lines of business. Example 2 extends the analysis to a larger scale problem of an insurer considering nine business lines.

Example 1. Assume an insurer considers selling three types of life insurance in year 0: 10-year term life insurance (i = 1) on a male age (25), 5-year term life insurance (i = 2) on a male age (35), and whole life insurance (i = 3) on a male age (40). Further we assume:

* The annual level premiums [[eta].sub.i] are the average market prices in 2005 for i = 1, 2, 3.

* The insurer uses the Lee and Carter (1992) model to simulate 150 future mortality paths. Because we do not have a time series of insured life tables, we use population data. An insurer would very likely use its own data.

* The insurer uses the U.S. Treasury yield curve on December 28, 2005 to compute the benefit payment ratios based on Formula (3).

We obtained premium data from Compulife, a company that sells market data from over 100 U.S. and Canadian life insurers. (2) We calculated an industry-weighted average of U.S. term life insurance prices for male nonsmokers and smokers using weights based on the incidence of smoking in the 2005 U.S. population reported by the Centers for Disease Control. (3) These are the resulting conditional expected values of premiums offered by our insurer in year 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

In addition, we assume an expense ratio of [phi] = 12.5 percent for each line of business. This means expenses are 12.5 percent of premiums, paid at the same time as the premiums are collected. Therefore, the benefit payment ratios by line in year 0 are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where we have used [[??].sub.i] in place of [??]([x.sub.i]). We simulated a future path of mortality, for that path calculated the values of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , and [A.sub.(40)] using (3) and those of [P.sub.1], [P.sub.2], and [P.sub.3] using (11), and then calculated the benefit payment ratios using (12). One path of mortality yields one joint observation of the three benefit payment ratios. We repeat this 150 times. The Appendix shows some details on the Lee and Carter (1992) method we used to simulate those tables.

Table 1 shows the descriptive statistics for the benefit payment ratio of these three lines of life insurance based on 150 trials of simulation. It turns out that the 10-year term life insurance (i = 1) on a male age (25) has the lowest mean benefit payment ratio and the highest variance. The whole life insurance (i = 3) on a male age (40) has a higher mean benefit payment ratio and lower variance than the 5-year term life insurance (i = 2) on a male age (35).

Suppose the insurer's original portfolio has 10 percent of total premium from 10-year term life insurance, 10 percent from 5-year term life insurance, and 80 percent from whole life insurance. The original weighted average of benefit payment ratios of this portfolio at time 0 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The sample statistics of [??]([w.sub.0]) are shown in the row labeled "Original" in Table 2 for the same 150 simulations that we used in Table 1. To search for an optimum business strategy, we solve the MV+CVaR optimization Problem (9). We specify

[l.sub.0] = [3.summation over (i=j)] [w.sub.i] E[[[??].sub.i] = 0.9567

and

[zeta] = [CVaR.sub.95 percent] ([??]([w.sub.0])) - 0.05|1 - [CVaR.sub.95 percent] ([??]([w.sub.0]))|, (14)

where [CVaR.sub.95 percent] ([??]([w.sub.0])) is the 95 percent level CVaR of the original portfolio. The specification (14) controls the downside risk by reducing the original 95 percent level CVaR by 0.05 [absolute value of 1 - [CVaR.sub.95 percent] ([??]([W.sub.0]))]. The firm can set a different risk reduction level to fit its particular situation. Later we will show how an adjustment in risk reduction level [delta] changes the optimal solution. With (14) we get the following weights for the MV+CVaR optimal business composition:

[w.sub.MV+CVaR] = [0, 0.4660, 0.5340].

This means the insurance company could improve its portfolio of life insurance if it were to decline writing the 10-year term life insurance, put 46.60 percent of its business in the 5-year term life insurance and the remaining 53.40 percent in the whole life insurance. The row labeled "MV+CVaR" in Table 2 shows the summary statistics for the MV+CVaR efficient mortality portfolio. The MV+CVaR efficient portfolio reduces the variance from 0.0043 to 0.0038. Furthermore, Table 2 also shows that the new portfolio's skewness reduces from 0.2912 to 0.1457, and its 95 percent CVaR decreases from 1.1047 to 1.0802. In sum, we conclude the MV+CVaR portfolio returns a better mean-variance trade-off and a lower downside risk than the original one.

Example 2. Now we include more life insurance lines and analyze a larger scale problem with nine lines of life insurance: 5-year term, 10-year term, and whole life insurance issued at ages (25), (35), and (40). The sample statistics of these nine lines are summarized in Table 3. These sample statistics come from a different set of simulations from that in 1, so the sample statistics are not exactly the same as those in Table I but they are very close and qualitatively the same.

We assume the insurer originally has equal weights on each line of business (i.e., [w.sub.i] = 1/9, i = 1, 2, ... , 9). Given this assumption, the summary statistics of this evenly weighted portfolio are shown in the row labeled "Original" in Table 4. The expected benefit payment ratio of the original nine-line portfolio is 0.9921 and its variance is 0.0055.

To obtain the MV+CVaR efficient portfolio with the above nine life insurance lines, we solve Problem (9). We specify the mean of the optimal portfolio equal to that of the original one

[l.sub.0] = [9.summation over (i=1)] [w.sub.i] [[??].sub.i] = 0.9921

and the 95 percent level CVaR of the optimal portfolio is no higher than [zeta] determined by Equation (14). The MV+CVaR optimal business composition solution is

[w.sub.MV+CVaR] = [0.1467, 0, 0.2955, 0, 0, 0, 0, 0, 0.5578].

This suggests that the insurer could improve its life insurance portfolio if it were to underwrite only 5-year term life on age (25), 5-year term life on age (40), and whole life insurance on age (40) with weights 14.67 percent, 29.55 percent, and 55.78 percent, respectively. The optimal portfolio has lower variance (0.0036), lower skewness (-0.1950), and lower [CVaR.sub.95 percent] (1.0960) than the original portfolio (see Table 4).

BOUNDS ANALYSIS WITH THE METHOD OF MOMENTS AND MAXIMUM ENTROPY

How well does the optimal business strategy secure an insurer's financial position? In this section, we apply the method of moments to address this problem. In addition, we allow for higher order moment information to be included in the mortality risk management setup.

Here is the setting of a reasonably general moment problem. Given moments [[mu].sub.1], ... , [[mu].sub.n] and an interval [[b.sub.1], [b.sub.2]], let M denote the set of random variables Z with support in [[b.sub.1], [b.sub.2]] (i.e., [b.sub.1] [less than or equal to] Z [less than or equal to] [b.sub.2] with probability 1) and E([Z.sup.i]) = [[mu].sub.i] for i = 1, ... , n. Let [phi](z) be a well-behaved real-valued function defined on [[b.sub.1], [b.sub.2]]. The moment problem is to determine the best upper and lower bounds on E[[phi]b(Z)], over all random variables Z [member of] M, given E([Z.sup.i]) = [[mu].sub.i] for i = 1, ... , n. These ideas have roots in the work of Tchebyshev, Markov, and Stieltjes in the 1870s. Tian (2008) provides a detailed discussion of moment problems in portfolio risk management.

We will apply the moment technique to [[phi].sub.d] (z) = 0 for z [less than or equal to] d and [[phi].sub.d] (z) = 1 for z > d, so that E[[[phi].sub.d] (Z)] = Pr(Z > d) is the survival function of Z. The solution to the moment problem below produces bounds on Pr(Z > d):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where F represents a probability distribution on M satisfying [E.sub.F] ([Z.sup.i]) = [[mu].sub.i] for = 1, ... , n. Similarly, the problem for the lower bound of Pr(Z > d) is the solution of the following problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The bounds depend on the given moments and support, as well as d. They are called semiparametric bounds, the parameters being the given moments. We will refer to them as the "general bounds" because they are robust bounds that any feasible distribution with same moments must satisfy.

In our analysis, we are interested in the bounds on the survival function of the weighted average benefit payment ratios, that is, S(d)= 1 - F(d)= Pr([??](w)> d), where [??](w) is the weighted average benefit payment ratio of a mortality portfolio, defined previously.

This moment problem can be solved numerically with semidefinite programming solvers such as SOSTOOLS (Prajna et al., 2004). See Akhiezer (1965), Parrillo (2000), Popescu (2005), and Bertsimas and Popescu (2005) for the details on applying semidefinite programming solvers to the bound Problems (15) and (16).

Unimodal Bounds

Adding a distribution assumption such as unimodal to the constraints in (15) or (16) means that we optimize over a smaller set and thus obtain tighter bounds than the general bounds. Since many data sets have unimodal empirical distributions, we find the unimodal assumption appealing. However, one has to be aware that the resulting bounds are valid only if the variable (benefit payment ratio in our case) has a unimodal distribution. If the distribution is not unimodal, then implementing moment analysis with a unimodal assumption may eliminate important tail behavior.

We transfer the unimodal bounds problems to general bounds problems using the Khintchine (1938) representation of a unimodal random variable; it is also discussed by Feller (1971, p. 158). Feller credits this version to Shepp (without a citation): Z is unimodally distributed if and only if there are two independent random variables U and Y such that Z = m + UY, where U is uniformly distributed on (0,1) and m is the unique mode of Z. Tian (2008) and Brockett and Cox (1985) describe the details of this transfer procedure, originally due to Kemperman (1971). Popescu (2005) shows how to include assumptions such as symmetry, convexity, and smoothness as well as unimodality in the moment problem. Here we summarize the final results of the transfer: the interval [[b.sub.1],[b.sub.2]] is replaced by [[b.sub.1]-m, [b.sub.2]- m]. The function [[phi].sub.d] (Z) is replaced by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [d.sup.*] = d - m. And the bounds [[mu].sub.i] are replaced by [[mu].sup.*.sub.i] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The resulting problems (for upper and lower bounds) have the same form as (15) and (16) and also can be solved with semidefinite program solvers. For a given set of moments and support interval, we solve for many values of d and obtain the optimal unimodal bounds [[p.bar].sup.*.sub.d] and [[bar.p].sup.*.sub.d]. The general bounds on the survival function of [??] (w) always satisfy

[[p.bar].sub.d] [less than or equal to] Pr ([??](w) > d) [less than or equal to] [[bar.p].sub.d], (17)

and if [??] (w) has a unimodal distribution the following relationship applies

[[p.bar].sup.*.sub.d] [less than or equal to] Pr ([??](w) > d) [less than or equal to] [[bar.p].sup.*.sub.d]. (18)

If we compare the general bounds ([[p.bar].sub.d] and [[bar.p].sub.d]) with their unimodal counterparts, we have [[p.bar].sub.d] [less than or equal to] [[p.bar].sup.*.sub.d] [less than or equal to] [[bar.p].sup.*.sub.d] [less than or equal to] [[bar.p].sub.d]

Maximum-Entropy Distribution

The work of Shannon (1948a, 1948b) in information theory and Jaynes (1957) in statistical physics led eventually to the concept of entropy in probability theory. The entropy of a probability distribution is a measure of how much information it contains. The maximum-entropy distribution is the "most likely," "most unbiased," "least prejudiced," or "most uniform" distribution of a given class of distributions.

Consider the set F of distributions on [[b.sub.1], [b.sub.2]] with given moments [[mu].sub.i] for i = 1, ... , n; these are the distribution functions of the random variables Z in M give in the previous section. The entropy of a continuous type distribution f(z) [member of] F is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The definitions for discrete and mixed distributions are analogous. The distribution [f.sup.*] (z) [member of] F with maximum entropy is the solution to the following moment problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

The maximum-entropy distribution is sensitive to the support interval [[b.sub.1], [b.sub.2]]. For our applications, we used a sample to estimate moments so we simply set the support interval equal to the range of the sample.

We solved problem (19) by applying a modified Newton method (Luenberger, 1984) to its dual problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where [[lambda].sub.0], ... , [[lambda].sub.n] are Lagrange multipliers associated with the constraints defining F and

[f.sup.*] (z) =exp (-1 - [n.summation over (1=10)] [[lambda].sub.i] [Z.sup.i]

Tian (2008) describes the details. These maximum-entropy problems were solved with MATLAB software. The maximum-entropy distribution determined by the moments of the weighted average portfolio benefit ratio [??](w) is the distribution using as little information as possible (using only moments) to analyze the risk of MV+CVaR mortality portfolios.

Numerical Illustrations

Now we apply the bound and maximum entropy analysis to the numerical examples in the previous section.

Example 1. Figure 1 shows the histograms of the original and the MV+CVaR three-line portfolio benefit payment ratios based on eight bins. The histograms suggest that both the original and optimal portfolio benefit payment ratios have unimodal distributions. Figure 2 shows the graphs of the bounds in (17) and (18), which we computed for a large numbers of values of d. If the distribution is unimodal with business composition [w.sub.0] = [0.10, 0.10, 0.80], its four-moment bounds are significantly narrowed. These graphs can be used to derive bounds on the value at risk. Consider the 95 percent VaR, which corresponds to the 0.05 level of the survival function. Draw a horizontal line through the survival probability of 0.05 on the vertical axis. It intersects the solid curves at d values of 1.038 and 1.087. Therefore, if the original portfolio has unimodal distribution, what we can say is that

1.038 [less than or equal to] [VaR.sub.95 percent] [les than or equal to] 1.087. (21)

Without the unimodal assumption, the 95 percent VaR range is much wider:

0.985 [less than or equal to] [VaR.sub.95 percent] [less than or equal to] 1.122, (22)

which is determined by the intersection of the horizontal line with the o curves. The normal distribution (the dotted lines in Figure 2) with the same mean and variance as the original mortality portfolio must fall between the two-moment upper and lower unimodal bounds. In this example, the normal curve is even within the four-moment unimodal bounds. Under the normal assumption, the business's [VaR.sub.95 percent] is 1.065.

Figure 3 shows the four-moment general and unimodal bounds of the original three-line life insurance business with [w.sub.0] = [0.10, 0.10,0.80] and its MV+CVaR optimum with [w.sub.MV+CVaR] = [0, 0.4660, 0.5340]. The MV+CVaR approach reduces the downside risk for both sets of bounds, general and unimodal. To illustrate, just as what we did in Figure 2, we draw a horizontal line through the 0.05 level of the survival probability on the vertical axis to analyze the 100 percent confidence bounds on the 95 percent VaR. If we assume the optimized portfolio is also unimodal, the bounds on the [VaR.sub.95 percent] of the portfolio benefit payment ratio drops to

1.030 [less than or equal to] [VaR.sub.95 percent] [less than or equal to] 1.075, (23)

by adding 95 percent CVaR constraint to the traditional Markowitz optimization. Thus, both the upper and lower bounds are improved. For example, the lower bound indicates that there is only 5 percent probability that the portfolio annual benefit payment ratio will rise above 1.030.

To take our discussion one step further, we solve the maximum-entropy Problem (19). The results are shown in Figure 4, which compares the maximum-entropy distributions of the original and the MV+CVaR mortality portfolios. In particular, the right tail of the MV+CVaR three-line portfolio shifts to the left of the original portfolio distribution. Therefore, the optimized portfolio has lower downside risk than the original one, which is consistent with the conclusion of the bound analysis.

Example 2. Continuing with Example 2, Figure 5 shows the histograms of the nine-line original and MV+CVaR optimal portfolios with eight bins. Since neither portfolio distribution is unimodal, we only compute their four-moment general bounds. Figure 6 illustrates their four-moment general bounds and maximum-entropy survival functions. Again the right tail of the MV+CVaR optimal portfolio lies on the left of that of the original portfolio in the bound and maximum-entropy analyses, implying lower shortfall risk after optimization.

NATURAL HEDGING EFFECTS

Cox and Lin (2007) argue that an insurer selling both life insurance and annuities is exposed to lower one-directional changes in mortality. This effect is called "natural hedging." That is, adding annuities to a life insurance portfolio may lower the portfolio's mortality risk. In this section, we examine how natural hedging improves the MV+CVaR optimal business composition and decreases downside risk with the method of moments.

Example 1

To illustrate the natural hedging effect, we add an annuity to the three-line life insurance portfolio analyzed in Example 1. Specifically, in addition to selling the three life policies, the insurer also sells a single premium immediate life annuity on male age (65). Assume that the insurer sells the annuity at the average market quote of 2005, which charges a monthly payout rate of $6.56 per lump-sum premium $1,000 (Stern, 2008). We further assume that the insurer uses the same mortality data and method as those for life insurance to forecast the future survival payments of annuities.

As we mentioned earlier, we are using a single times series of mortality, the U.S. male population data, as the basis for the mortality forecasts. The summary statistics for the benefit payment ratio of this annuity, [[??].sub.10], with an expense ratio of [phi] = 12.5 percent, are shown in Table 5.

Assume that the insurer is considering a portfolio of 10.00 percent 10-year term life insurance, 10.00 percent the 5-year term life insurance, 40.00 percent the whole life insurance, and 40.00 percent the annuity, that is,

[w.sub.0] = [0.10, 0.10, 0.40, 0.40]

in the original portfolio. To optimize this mortality portfolio, we set the objective benefit payment ratio [l.sub.0] at 0.9652, same as the expected benefit payment ratio of the original portfolio, and construct a 95 percent CVaR constraint with [zeta] in (6) determined by Equation (14). We get the MV+CVaR efficient four-line mortality portfolio with the weights,

[w.sub.MV+CVaR] = [0.2028, 0, 0, 0.7972]. (24)

That is, the insurer should not sell either the 5-year life insurance on male age (35) or the whole life insurance on male age (40), and adjust its business to make 20.28 percent of its total premium from the 10-year life insurance on male age (25), and 79.72 percent from the single premium immediate life annuity on male age (65). Table 6 compares the original and MV+CVaR mortality portfolios. The MV+CVaR portfolio has a much lower variance and 95 percent CVaR. Figure 7 shows the histograms of the original and the MV+CVaR optimal four-line portfolios based on eight bins. Notably, in Figure 7, the distribution of the MV+CVaR portfolio is tightly grouped around the mean relative to that of the original portfolios, indicating that the MV+CVaR portfolio has a much lower risk. The histograms also show that both the original and the optimized four-line portfolios are unimodally distributed.

Figure 8 shows the four-moment unimodal bounds and the maximum-entropy distributions of the original and MV+CVaR portfolios with four lines of business. The probability density function of the MV+CVaR optimal four-line portfolio is much tighter than that of the original portfolio, so the upper and lower unimodal bounds of the MV+CVaR portfolio are narrower.

To explore the natural hedging effect, we compare the bounds of the MV+CVaR three-line and four-line portfolios. If we assume both the optimized portfolios have unimodal distributions, the right plot in Figure 9 shows how the 100 percent confidence interval of the 95 percent level VaR improves by adding an annuity to the three-line pure life insurance portfolio. The confidence interval of [VaR.sub.95percent] of the four-line portfolio stays at a much lower level of benefit payment ratio range than that of [VaR.sub.95percent] of the three-line portfolio. It highlights the benefits of natural hedging: by including both annuity and life insurance in a portfolio, natural hedging reduces benefit payment and decreases mortality risk.

Example 2

We add three immediate annuities on male with issue ages (65), (70), and (75) to the nine-line life insurance portfolio we discussed in previously. The sample statistics of these three annuities are given in Table 7. The sample statistics of the immediate life annuity on male age (65) come from a different set of simulations from Example 1. So its moments are not exactly the same as those in Table 5 but they are very close and qualitatively the same.

We assume the insurer earns the same amount of premium from each of its 12 lines of business (i.e., [w.sub.i] = 1/12, i = 1, 2,...,12). By adding 95 percent CVaR constraint to the Markowitz problem and specifying the expected benefit payment ratio

[l.sub.0] = [12.summation over (i=1)] [w.sub.i]E[[[??].sub.i]] = 0.9895, i=1

we get the weights of the MV+CVaR portfolio as follows:

[w.sub.MV+CVaR] = [0, 0, 0, 0.0437, 0, 0.0564, 0, 0, 0.0649, 0, 0, 0.8350]. (25)

Therefore, the MV+CVaR optimal portfolio is composed of 4.37 percent 10-year term life on a male age (25), 5.64 percent 10-year term life on a male age (40), 6.49 percent whole life insurance on a male age (40), and 83.50 percent single premium immediate life annuity on a male age (75). Table 8 shows that given the same expected benefit payment ratio as that of the original portfolio (0.9895), the MV+CVaR optimal portfolio has a much lower variance than the original portfolio.

The histograms of the original and the MV+CVaR 12-line life insurance portfolios in Figure 10 suggest that both portfolios have unimodal distributions. Similar to Figure 8 of Example 1, Figure 11 illustrates a lower risk of the MV+CVaR 12-line portfolio than its original 12-line counterpart: it has much narrower bounds and lower values of the right tails.

We further perform the bound analysis to compare the MV+CVaR efficient 9-line portfolio without annuities and the MV+CVaR efficient 12-line portfolio with annuities. As expected, Figure 12 suggests a much lower downside risk of the MV+CVaR efficient 12-line portfolio, again, supporting the natural hedging benefits between life insurance and annuities.

In practice, it may be difficult for an insurer to directly accomplish an MV+CVaR optimal business composition in Portfolio (25). For example, it may not be easy for an insurer specializing in life insurance to enter the annuity business (Cox and Lin, 2007). To solve this problem, the insurer has at least two possible solutions. First, the insurer can buy or sell reinsurance to rebalance its weights in various business lines. Second, the insurer can issue or purchase mortality or longevity securities in the capital markets. The mortality-linked securities are new in the financial markets but have attracted a lot of attention from insurers, investors, pension plans, and academia (Blake and Burrows, 2001; Cowley and Cummins, 2005; Lin and Cox, 2005, 2008; Cairns, Blake, and Dowd, 2006 Cox, Lin, and Wang, 2006). As the mortality-linked security markets develop, the insurer may be able to cede or assume risk to realize the MV+CVaR efficient mortality portfolio at a low cost.

FRONTIERS OF EFFICIENT MORTALITY PORTFOLIOS

Our analysis has focused on improving an insurer's existing mortality portfolio with the MV+CVaR approach and applying the method of moments to examine whether and how the MV+CVaR portfolio could control downside risk. Can these techniques be applied to any arbitrary insurance business composition? How does the MV+CVaR approach compare to the traditional Markowitz optimization method? How does natural hedging reduce mortality risk for different MV+CVaR portfolios? To answer these questions, in this section, we extend our analysis to a range of efficient portfolios, not just a particular efficient portfolio given a certain level of benefit payment ratio. Specifically, we compare

1. the two- or three-dimensional representations of the Markowitz and MV+CVaR optimal portfolios;

2. the two- or three-dimensional representations of the 9-line and 12-line optimal portfolios.

Frontiers of Markowitz and MV+CVaR Optimized Portfolios We write Equation (14) in the following general form:

[zeta] = [CVaR.sub.95percent] ([??]([w.sub.0])) - [delta] [absolute value of 1 - [CVaR.sub.95percent]([??]([w.sub.0]))]. (26)

In Equation (14), [delta] = 0.05 and [CVaR.sub.95percent]([??]([w.sub.0])) corresponds to the 95 percent level CVaR of the original portfolio. To compare the MV+CVaR optimal portfolios with their Markowitz counterparts, we set the benchmark [CVaR.sub.95%]([??]([w.sub.0])) at the 95 percent CVaR of the Markowitz portfolio given the benefit payment ratio [??]([w.sub.0]). This setting guarantees that the 95 percent CVaR of the MV+CVaR portfolio is no higher than that of its Markowitz counterpart. With [zeta] specified as (26), the upper limit of 95 percent level CVaR constraint imposed on the traditional Markowitz problem changes with the risk reduction level [delta]. In practice, the insurer can choose a level of [delta] that fits its specific risk management needs. In our illustration, we choose the highest possible [delta] for each portfolio that keeps problem (6) feasible.

For the nine lines of business mentioned earlier, we solve (1) the traditional Markowitz portfolio problem and (2) the MV+CVaR Problem (6) with a 95 percent CVaR constraint specified by (26). We obtain a set of optimal portfolios with different benefit payment ratios. We plot the variance-mean, skewness-mean, [CVaR.sub.5percent]-mean, and [CVaR.sub.95percent]-mean graphs for the MV+CVaR optimal portfolios and compare them to their Markowitz counterparts. Each graph in Figure 13 is a piecewise linear interpolation based on 20 solved efficient portfolios. Note that although our MV+CVaR problem is not a typical multiple objective optimization problem, loosely speaking, it minimizes both variance and CVaR subject to a preset expected benefit payment ratio. Therefore, the efficient frontier for a MV+CVaR optimization problem is a surface in the three-dimensional space (mean, variance, CVaR). Any two-dimensional profile shown in Figure 13 is a cross-section of the three-dimensional representation with the value of the third fixed.

While the upper left graph of Figure 13 shows that the MV+CVaR efficient frontier somewhat deviates from the Markowitz efficient frontier in terms of the mean-variance trade-off, the MV+CVaR approach effectively decreases the skewness of relatively high variance portfolios shown in the skewness-mean profile. A lower skewness is desirable because it decreases the likelihood of obtaining higher benefit payment ratios. The [CVaR.sub.95percent]-mean curves in the bottom right graph demonstrates that for the same mean, the portfolios constructed from the MV+CVaR approach are able to reach a lower 95 percent CVaR, implying a lower downside mortality risk. However, the impact of adding CVaR constraint to the Markowitz model on the portfolios with low benefit payment ratio is not as significant as that on the relatively high benefit payment ratio portfolios.

It is worth noting that by adding a large percentile CVaR constraint, the MV+CVaR approach aims at reshaping the right tail of the distribution, which corresponds to high benefits. The approach has little impact on the left tail that represents low benefits. This is confirmed by the bottom left graph in Figure 13, which shows that the [CVaR.sub.5percent] curve of the MV+CVaR portfolios just barely differs from that of its Markowitz counterparts.

Natural Hedging for Efficient Portfolios

We investigate the natural hedging effect for various MV+CVaR efficient portfolios to extend our previous analysis. Given the 95 percent CVaR constraint specification (26), Figure 14 shows that the 12-line MV+CVaR portfolios ([nabla]) composed of both life insurance and annuity lines outperform those containing only life insurance ([omicron]). Given the same means, the 12-line efficient portfolios achieve lower variance, and lower [CVaR.sub.95percent] than their 9-line counterparts. Thus, inclusion of annuities reduces the potential mortality risk (measured by variance) and downside risk (measured by [CVaR.sub.95percent]) of life insurance portfolios. This provides a new evidence to support the natural hedging effects. Notice that adding annuities to a life insurance portfolio does not necessarily decrease the portfolio skewness, as shown in the upper right plot of Figure 14.

In our example, the improvement of the frontier by adding an annuity can be explained by the negative correlation between the annuities and the lines of life insurance, as shown in Table 9.

Comparison of Markowitz and MV+CVaR Optimal Portfolios With Different values of [zeta]

A change in [delta] in (26) alters the upper limit of 95 percent level CVaR constraint [zeta]. The bigger the [delta], the more stringent the 95 percent level CVaR constraint in (6). The MV+CVaR optimal portfolios shown in Figures 13 and 14 are determined by adding the most stringent feasible 95 percent CVaR constraints. In this section, we illustrate how the 95 percent CVaR of a MV+CVaR portfolio improves relative to its MV counterpart with various feasible risk reduction levels of [delta]. Specifically, we solve the MV+CVaR problem (6), subject to a 95 percent CVaR constraint with a different levels of [zeta], to obtain a set of optimal portfolios given a preset expected benefit payment ratios.

Figure 15 shows how the 95 percent CVaR of the MV+CVaR portfolio improves compared to its MV counterpart at each risk reduction level [delta]. For the MV model, [delta] is set at 0. We show the percentage difference of 95 percent CVaR between MV and MV+CVaR optimal portfolios

[CVaR.sup.MV.sub.95percent] - [CVaR.sup.MV+CVaR.sub.95percent]/[CVaR.sup.Mv.sub.95percent] (27)

on the vertical axis. For both 9-line and 12-line portfolios, the percentage differences are always positive. This means that adding a 95 percent CVaR constraint with [zeta] specified in (26) to the traditional Markowitz problem reduces the downside risk. In general, given a specified objective benefit payment ratio [l.sub.0], the percentage difference (27) increases with [delta]. As [delta] increases, we impose a more stringent CVaR constraint so we observe lower downside risk of MV+CVaR portfolios and bigger difference between [CVaR.sup.MV.sub.95percent] and [CVaR.sup.MV+CVaR.sub.95percent]

Figure 16 shows the mean-variance-[CVaR.sub.95percent] graphs for the 9-line portfolios (left) and the 12-line portfolios (right), respectively. In each graph, the [CVaR.sub.95percent] of each MV+CVaR portfolio changes with the upper limit of the [CVaR.sub.95percent] constraint [zeta], thus forming a three-dimensional surface. As for the MV portfolios, each optimal mean and variance relationship uniquely determines a [CVaR.sub.95percent]. Thus, in each graph, the mean-variance-[CVaR.sub.95percent] frontier of the MV optimal portfolios is a curve represented by the dotted line. This dotted line lies above the three-dimensional surface of the MV+CVaR portfolios, suggesting the higher downside risk of the optimized MV portfolios.

Comparison of Markowitz and MV+CVaR Optimal Portfolios Using Method of Moments

As a posterior check, we compare the maximum-entropy survival probabilities of the 20 MV optimal portfolios and their MV+CVaR counterparts illustrated in Figures 13 and 14. We call them counterparts since they have the same mean, that is, the same target benefit payment ratio. The difference between maximum-entropy survival probabilities of the 20 optimized MV and MV+CVaR portfolios on the right tail (65-100 percentiles) of survival function,

Pr [([??](w) > d).sup.MV] - Pr [([??](w) > d).sup.MV+CVaR],

are shown in Figure 17. (4) The left plot shows the difference for the 9-line portfolios, and the right one is for the 12-line portfolios. The difference of each pair is always positive on the right tail, demonstrating the robustness of the MV+CVaR approach. That is, the MV+CVaR portfolios consistently have lower downside risk than the MV portfolios. Notice that compared to the 9-line portfolios, the 12-line portfolios provide more room to reduce downside risk since the difference on the right plot is bigger than that on the left plot.

CONCLUSION

We applied the method of moments and portfolio optimization to mortality risk management, making two contributions to the mortality literature. First, we propose the MV+CVaR approach to manage mortality portfolio risk with a reasonable sacrifice of mean-variance efficiency. The MV+CVaR approach controls the downside risk by specifying the mean value of the benefit payment ratio above the [beta]-level VaR to be no more than some prespecified value. This method is effective in obtaining an optimal mortality portfolio while controlling its downside risk.

Second, we applied the method of moments to mortality risk management by calculating the semiparametric upper and lower bounds on the survival function of benefit payment ratios for mortality portfolios. The bounds are used to illustrate the 100 percent confidence interval of the downside risk, which is measured by [beta]-level VaR. We propose how to use the moments method to investigate downside risk of MV+CVaR efficient mortality portfolios. In addition, as an extension to the moments method, we derive the maximum-entropy distribution of mortality portfolios. We use the maximum-entropy approach to conduct a robustness check for the moments method because the maximum-entropy approach provides a representative distribution that is the most unbiased, given the moment information.

We illustrate our examples with up to 12 lines of business with annual observations, but these methods can be extended to more lines of business and higher frequency data. Furthermore, we would obtain more practical results by using joint forecasts of several mortality models (recognizing at least mortality for male, female, life insurance, and annuities). In addition, our results could be generalized by developing a joint model of investment and mortality risk. Such a model could be applied to a life insurer's asset liability management problem. Finally, policyholder behavior (such as surrendering the policy) is important but incorporating it will require development of new models and access to industry data. We leave these questions for future research.

APPENDIX: ESTIMATING BENEFIT PAYMENT RATIOS SUMMARIZED IN TABLE 1

We use the Lee and Carter (1992) model to forecast the insurer's future mortality rates. This model incorporates both the age variation and the underlying time trend of death rates. The logarithm form of the 1-year death rate [q.sub.x,t] of age x in year t in Lee and Carter (1992) is modeled as

1n [q.sub.x,t] = [a.sub.x] + [b.sub.x][[theta].sub.t] + [[member of].sub.x,t], (A1)

where [a.sub.x] and [b.sub.x] are the age-specific parameters and [[theta].sub.t] is the time-series common risk factor that drives the mortality rates of all age cohorts. The transitory shock [[member of].sub.x,t] is normally distributed with zero mean. Lee and Carter model the central death rate, but usually the 1-year probability of death [q/sub.x,t] is very close to the central death rate. Thus, we model [q.sub.x,t] directly rather than model the central death rate, which would then have to be transformed to a probability.

Model (A1) is not unique hence two constraints are imposed to ease the estimation:

[summation over (x)] [b.sub.x] = 1, [summation over (t)] [[theta].sub.t] = 0 (A2)

These two constraints imply that the intercept ax simplifies to the empirical average of age x over time:

[a.sub.x] = [summation over (t)]ln [q.sub.x,t]/t (A3)

We follow a two-step procedure in Lee and Carter (1992) to estimate Model (A1). In the first step, singular value decomposition of the matrix is applied to obtain estimates for [b.sub.x], (x - 0,1, 2 ....) and [[theta].sub.t], (t = 0, 1, 2, ...). In the second step, the time-series common risk factor [[theta].sub.t] in year t is recalculated based on the actual number of deaths.

We assume the insurer has the same mortality experience as that of the U.S. population. The insurer uses [q.sub.x,t] in the U.S. male population mortality tables, observed each year from 1901 to 2005 from the Human Life Table Database and the Human Mortality Database, to estimate Model (A1). The tables for years 1901-1999 are from the Human Life Table Database and the tables for 2000-2005 are from the Human Mortality Database, published by the University of California, Berkeley and Max Planck Institute for Demographic Research. (5) The age range is x = 0,1, 2,..., 103 for U.S. males from t = 1901 to t = 2005. The parameter estimates of ax and [b.sub.x] are given in Table A1 and the estimated [[theta].sub.t] is shown in Figure A1.

For mortality projection, [[theta].sub.t] is assumed to follow a random walk with drift c,

[[theta].sub.t] = [[theta].sub.t-1] + c + [e.sub.t], (A4)

where the error term [e.sub.t] is normally distributed with a zero mean and a variance [[sigma].sub.[theta]]. Based on the time series of [[theta].sub.t] where t = 1901, 1902, ... , 2004, 2005, we obtain c = -0.2032 and [[sigma].sub.[theta]] = 0.6491.

When producing forecasts for the k-year ahead mortality rates, we first simulate [e.sub.2005+j] (j = 1, 2,...,k) each year for k years. Then the common risk factor [[theta].sub.2005+j] in year 2005 + j is calculated by adding the constant c = -0.2032 to each simulated error term [e.sub.2005+j]. Given the estimated [a.sub.x]s and [b.sub.x]s in Table A1, the simulated [[theta].sub.2005+j]s, and the U.S. Treasury yield curve on December 28, 2005, we use Model (A1) to calculate simulated future mortality rates for all ages and years.

DOI: 10.1111/j.1539-6975.2012.01469.x

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Shannon, C. E., 1948a, A Mathematical Theory of Communication, Bell System Technical Journal, 27(4): 593-622.

Shannon, C. E., 1948b, A Mathematical Theory of Communication, Bell System Technical Journal, 27(3): 379-423.

Stern, H. L., 2008, Annuity Shopper. World Wide Web: www.annuityshopper.com.

Tian, R., 2008, Moment Problems With Applications to Value-at-Risk and Portfolio Management, Ph.D. thesis, Georgia State University.

Toole, J., 2007, Potential Impact of Pandemic Influenza on the U.S. Life Insurance Industry, Research report, Society of Actuaries.

Tsai, J. T., J. L. Wang, and L. Y. Tzeng, 2010, On the Optimal Product Mix in Life Insurance Companies Using Conditional Value at Risk, Insurance: Mathematics and Economics (46): 235-241.

(1) http://www.mathworks.com/

(2) See the company's website for additional information: http://www.compulife.ca/historicaldata.php.

(3) Source: http://www.cdc.gov/media/pressrel/r061026a.htm. The percentage of U.S. adults who smoked in 2005 was 20.9 percent.

(4) The percentage difference of each pair in Figure 17,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is always greater than 10 percent.

(5) Available at www.mortality.org or www.humanmortality.de (data downloaded on June 8, 2008).

Samuel H. Cox is the L. A. H. Warren Chair Professor at the Asper School of Business, University of Manitoba. Yijia tin is in the Department of Finance, College of Business Administration, University of Nebraska-Lincoln. Ruilin Tian is in the Department of Accounting, Finance, and Information System, College of Business, North Dakota State University. Luis F. Zuluaga is at the Faculty of Business Administration, University of New Brunswick. The authors can be contacted via e-mail: scox@cc.umanitoba.ca, yijialin@unl.edu, ruilin.tian@ndsu.edu, and lzuluaga@unb.ca, respectively. This article was presented at the 2009 American Risk and Insurance Association annual meeting and the 2010 Financial Management Association annual meeting. We appreciate helpful comments from Alexander Kling and other participants at the meetings. The authors also thank two anonymous referees for their very helpful suggestions and comments during the revision process.

We provide a new method, the "MV + CVaR approach," for managing unexpected mortality changes underlying annuities and life insurance. The MV + CVaR approach optimizes the mean-variance trade-off of an insurer's mortality portfolio, subject to constraints on downside risk. We apply the method of moments and the maximum entropy method to analyze the efficiency of MV + CVaR mortality portfolios relative to traditional Markowitz mean-variance portfolios. Our numerical examples illustrate the superiority of the MV + CVaR approach in mortality risk management and shed new light on natural hedging effects of annuities and life insurance.

INTRODUCTION

Life insurance companies sell a wide variety of life insurance and annuity products. The insurers' liabilities for these products depend on future mortality rates. During recent years, economic and demographic changes have made mortality projection and risk management more important than ever. On the one hand, life expectancy for ages 60 and older in the past two decades has improved at a much higher rate than what pension plans and annuity providers expected. Cowling and Dales (2008) find that companies in the United Kingdom FTSE100 index underestimated their aggregate pension liabilities by more than 40 billion pounds sterling]. If the firms do not take measures to control mortality downside risk, such longevity shocks are likely to cause serious financial consequences. For example, unanticipated mortality improvement was an important factor accounting for the failure of Equitable Life, once a highly regarded U.K. life insurer (Ombudsman, 2008). On the other hand, population growth, urbanization, and increased global mobility may lead to a more rapid and widespread disease. Genetic analysts recently confirmed that today's "bird flu" is similar to the 1918 "Spanish flu" that killed more than 40 million people. This finding spurs fears of a worldwide epidemic (Juckett, 2006). According to Toole (2007), losses due to a severe pandemic could amount to 25 percent of the U.S. life insurance industry's statutory capital. While the great majority of U.S. life insurance companies would weather such a pandemic, it is clear that these companies should be interested in mitigating the risk.

We propose a method that life insurance companies can use to alleviate extreme mortality outcomes while maintaining a relatively efficient mean-variance relationship for their mortality portfolios of life insurance and/or annuities. This method, the "MV+CVaR approach," combines Markowitz mean-variance (MV) portfolio theory and conditional value at risk (CVaR) by optimizing the trade-off between mean and variance subject to an upper bound on CVaR. Variance measures both positive and negative deviations of portfolio values from its expected level, while CVaR focuses on the portfolio tail loss caused by extreme events. Although the MV+CVaR portfolios are suboptimal relative to the Markowitz counterparts in terms of the mean-variance efficiency, they are attractive to insurers since the MV+CVaR portfolios have lower downside risk while achieving desirable risk-return trade-offs. In practice, life insurers are keenly interested in searching for an optimal risk-return relationship for their business. At the same time they are required to meet various solvency requirements for possible catastrophes such as flu epidemics. Therefore, incorporating both variance and CVaR as risk measures in business optimization such as the MV+CVaR approach should be appealing to life insurance companies. The risk control framework adopted in this article closely follows that of Rockafellar and Uryasev (2000) and Tsai, Wang, and Tzeng (2010). In their framework, firms minimize portfolio losses subject to CVaR constraints. In our context, we consider both variance and CVaR as risk measures by incorporating a CVaR constraint into the classical mean-variance setup.

We extend Rockafellar and Uryasev (2000) and Tsai, Wang, and Tzeng (2010) in one important dimension by applying the well-developed moments method to validate the quality of MV+CVaR portfolios. The existing literature on mortality models usually makes distributional assumptions. Since knowledge about the underlying mortality distributions may be limited, the assumed distributions may not represent the actual mortality dynamics. However, the moments method is solely based on moments, not a specific distribution. This method yields robust semiparametric upper and lower bounds that any reasonable model with the same moments must satisfy. Once a mortality portfolio has been constructed, the corresponding empirical benefit payments can be used to estimate moments of the benefit payment ratios. In this article, we show how such estimated moment information can be used to determine the bounds on the underlying portfolio benefit payment ratios. In particular, the moments method provides a mechanism to compare the downside risk of a MV+CVaR efficient mortality portfolio and its MV counterpart given their moments. Our results show the superiority of the MV+CVaR approach in mortality risk management and highlight the natural hedging effects of life insurance and annuities.

For the numerical examples, all optimization problems are solved with MATLAB (1) software. When computing the bounds of a particular portfolio, we solve the equivalent dual problem using built-in functions from the SOS programming solver. The SOS programming solver was developed by Prajna et al. (2004), which is a free toolbox written in MATLAB. Our implementation is fairly general and easy to use.

The remainder of this article is organized as follows. The first section describes how to calculate benefit payment ratios of mortality portfolios. The second section discusses the MV+CVaR mortality optimization model. We provide two numerical examples to illustrate the implementation of the approach. The third section describes the method of moments. We show how to compute the semiparametric upper and lower bounds for MV+CVaR efficient portfolios and then perform the bound analysis on those portfolios. The fourth section demonstrates the natural hedging effect when annuities and life insurance are considered jointly. The fifth section extends the analysis to efficient frontiers of MV+CVaR mortality portfolios by comparing bounds of MV and MV+CVaR optimal portfolios. In addition, we show to what extent downside risk is reduced by changing the CVaR constraint. The sixth section is our conclusion.

MORTALITY RISK PORTFOLIOS

Focus on Mortality

Consider a life insurer underwriting new business at the current moment 0, with the business sorted by underwriting class (x). The class symbol (x) represents the information at time 0 that the insurer uses to determine a mortality table for lives in the class. This information includes at least age, sex, and line of busines but could include other information such as height, weight, blood pressure, or tobacco use history. Actuarial textbooks usually work with a set of mortality tables that differ only by age. The other information is suppressed, and the tables are obtained by simply specifying the issue age x. However, in practice a company will have different tables for each age, sex, major line of business, and so on. Thus, in our notation (x) represents all the information the company needs to estimate a mortality table for this class at 0. Here are our assumptions:

1. All future cash flows are discounted at risk-free interest rates. These discount rates are known constants.

2. Life insurance policies and annuity contracts remain in force until settled at death. There are no policy lapses.

3. Expenses are known constant multiples of policy premiums.

4. The analysis is applied only to new business without regard to the insurer's book of business issued before time 0.

Under these assumptions, the present value of benefits (and related expenses) paid at times 1, 2, ... , PVB(x), and the present value of premiums (and related expenses) collected at 0, 1, 2, ... , PVP(x), are simple functions of the lifetime K(x). The randomness of the present values has two components. First, future mortality events will have an impact on annual survival rates for members of the class (x). That is, the annual survival rate P(x)+t for the year running from t to t + 1 is random for t = 1, 2, ... , Given a random path of the survival rates [pi] (x) = {P(x), P(x)+l, P(x)+2,...}, we can calculate the conditional distribution of the lifetime K (x). But some randomness still remains. However, the remaining randomness is subject to the law of large numbers; if the company writes a large enough number of policies in each underwriting class, it can be ignored. Therefore, we focus on the conditional expected values [??](x) = E[PVB(x)| [pi](x)] and [??](x) = E[PVP(x)|[pi](x)].

We assume the company uses the benefit payment ratio [??](x) = [??](x)/[??](x) + [phi] to manage its risk where [phi] is a constant expense ratio. Other measures are possible; for example, one might use [??](x) + [phi] [??](x) - [??](x). The benefit payment ratio is analogous to the loss ratio used in general (non-life) insurance where policies are much more often single premium (and single period) contracts.

The benefit payment ratio is a random variable as viewed from time 0, but only due to the uncertainty of future mortality rates. We give some examples below to illustrate the calculation of the benefit payment ratio. The company has a model for generating paths [pi] (x), and from them it generates a sample of values of [??](x). Moments of these sample distributions are inputs to the mortality risk management process. In our examples, we will use the Lee-Carter model (Lee and Carter, 1992) to generate future paths of the survival rates [P.sub.(x)+t], or equivalently the death rates [q.sub.(x)+t] = 1 - [P.sub.(x)+t]. In principle there are many methods that could be used; the company may have its own proprietary method.

In practice the insurance company will consider more than one line of business at 0; let ([x.sub.i]) denote a set of underwriting classes under consideration at 0. For example, ([X.sub.l]) might be male, 10-year term life insurance issued at age [x.sub.1] = 35; ([x.sub.2]) might be an immediate annuity issued to a female age [x.sub.2] = 65; and so on. The company jointly simulates paths for each class. This is much more difficult than simulating a single path. The Lee-Carter method has been extended to handle this situation (Li and Lee, 2005). Brouhns, Denuit, and Vermunt (2002) show how an extension of the Lee-Carter model can be used to jointly model annuitants and the general population to which they belong. Hyndman, Booth, and Yasmeen (2011) describe another generalization of Lee--Carter that allows for joint modeling of components of a population. However, these methods are still being developed. Moreover, there is very little publicly available data on insured lives and annuitants. Therefore, we will use a single mortality table forecast. There are circumstances where it is appropriate. For example, if we consider 10-year term insurance issued at 0 to a male age (35) and 20-year term insurance issued at 0 to a male age (45), then it is reasonable to use mortality rates from the same table that differs only by age. That is, we need to project only one table and then select values from it for the appropriate ages.

Examples

Recall that the benefit payment ratio [??](x) is the ratio, calculated at 0, of the conditional expected present value of benefits to the conditional expected present value of future premiums for the contract issued on (x), with the conditional distribution of K (x) calculated relative to a given future path of the underlying mortality table. For example, given a path of future survival rates [pi] (x) = {P(x), P(x)+l, P(x)+2,...} for (x), we can calculate the conditional survival function of K (x) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

with the last expression analogous to standard actuarial notation. It is analogous because in our setting the table is random so t [p.sub.(x)] is as random as the table. The probability density function of K (x) is

[f.sub.(x] (t) = Pr [K(x) = t |[pi] (x)] = [S.sub.(x)] (t) - [S.sub.(x)](t + 1) = t | q (x). (2)

Once we have the conditional pdf of K (x), we can calculate conditional moments. Here are some examples.

Life Insurance. For k-year term life insurance issued to (x) at 0 paying I at the end of the year of death, here is the present value of benefits and its conditional expected value:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [v.sup.(t)] is the t-year discount factor. For a large enough value of k, [??](x) is the same as the present value of benefits for a whole life insurance; in this case [A.sub.(x)] is analogous to the standard actuarial notation.

If the policy was issued at 0 and paid with an annual level premium [eta] per unit of benefit at that time, then the benefit payment ratio is simply [??](x) =[??](x)/[??](x) + [phi],

where the conditional expected value of premiums [??] (x) equals

[??] (X) = [k-1.summation over (t=0)] [eta][v.sup.(t)]

Immediate Annuity. This is the present value of benefits for a single premium immediate annuity of I per year paid at the end of the year issued on (x) at 0 and its conditional expected value:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The benefit payment ratio at 0 is [??] (x) = [??] (x)/P + [phi], where P is the single premium paid at 0 per unit of annual benefit.

In the next section we consider an insurer selling several lines of business at time 0 and optimize its portfolio. In order to do this we need statistical samples of the benefit payment ratios of lines of business under consideration, which we obtain by Monte Carlo simulation.

MORTALITY PORTFOLIO OPTIMIZATION

The original Markowitz portfolio optimization (Markowitz, 1952) maximizes an investor's expected portfolio return for a given level of risk, as measured by the variance of the return. More recent portfolio optimization techniques have been applied to corporate risk management to find optimal business strategies. For example, insurers are motivated to search for business compositions that not only achieve mean-variance efficiency but also minimize downside risk. In this section, we extend this line of research by adding capital constraints to the Markowitz problem for managing an insurer's mortality portfolio.

Portfolio Optimization With CVaR Constraints

The conditional value at risk (CVaR) is a risk measure, defined as the expected loss conditional on the benefit payment ratio being higher than a given value at risk (VaR). The existing literature has many CVaR-related portfolio management methodologies. For example, Rockafellar and Uryasev (2000) generate an efficient [CVaR.sub.[beta]]-mean frontier by minimizing the CVaR of portfolio's losses subject to an expected return requirement. Krokhmal, Palmquist, and Uryasev (2002) suggest minimizing the negative expected return with a CVaR constraint. In the context of mortality risk management, we explicitly consider the trade-off between mean and variance subject to CVaR constraints by adding a CVaR constraint to the traditional Markowitz problem. We call it the MV+CVaR approach.

The company's new business issued at 0 will have weight [w.sub.i] in line of business ([x.sub.i]). The weighted average of the benefit payment ratios is [??](w) = [[SIGMA].sup.n.sub.i=1] [w.sub.i] [??] ([x.sub.i]) Let [[sigma].sub.ij] = Cov ([??]([x.sub.i]), [??]([x.sub.j])) denote the covariance of benefit payment ratios of business lines I and j. Our MV+CVaR problem is to solve for portfolio weights w = [[w.sub.1], [w.sub.2], ... , [W.sub.n]] in terms of benefit payment ratios, so as to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [CVaR.sub.[beta]] ([??](w)) is the [beta]-level CVaR of [??](w), which in the case of a continuously distributed [??](w) can be calculated as

[CVaR.sub.[beta]] = ([??](w)) = E[[??](w)| [??] (w) [greater than or equal to] [VaR.sub.[beta]] ([??](w)))].

The [beta]-level value at risk, [VaR.sub.[beta]] ([??](w)), is the [bets]-quantile of the distribution of [??](w), or the smallest value of [alpha] such that Pr([??](w) [less than or equal to] [alpha]) [greater than or equal to] [beta]. That is,

[VaR.sub.[beta]] ([??](w)) = min{[alpha] | Pr([??](w) [less than nor equal to] [alpha]) [greater than or equal to [beta]}.

We enforce a right-tail constraint to manage the downside risk by setting [beta] at 0.95. The CVaR constraint in (6) ensures the tail expectation [CVaR.sub.[beta]] ([??](w)) is no higher than a prespecified value [zeta], thus reducing the downside risk. W is a subset of the mortality business feasible set. Specifically in our problem, W is the vector of weights to = [[w.sub.1], [w.sub.2], ... , [w.sub.n]] satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [??] ([x.sub.i]) is the random benefit payment ratio of line i, n is the number of business lines in the mortality portfolio, and [e.sub.0]is a prespecified level of the weighted average of the benefit payment ratios. We assume that [w.sub.i] [greater than or equal to] 0 for each i. Negative values would indicate the insurer buys insurance rather than selling. It is possible to do this in practice with reinsurance, but we are not allowing it in our model.

Rockafellar and Uryasev (2000, 2002) show that the [beta]-level CVaR is the optimal value of the following optimization problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where the notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Problem (6) is equivalent to the following Problem (9) in the sense that their objectives achieve the same minimum values.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

If a pair ([[alpha].sup.*], [[??]([w.sup.*])) achieves the minimization of (9), G([[alpha].sup.*], [??]([w.sup.*])) will return [[alpha].sup.*] level CVaR and [[alpha].sup.*] will give the corresponding [beta]-level VaR. For the detailed proof of the theorem, see Theorem 4 of Krokhmal, Palmquist, and Uryasev (2002).

Here, we present the equivalence of the CVaR constraint (i.e., [CVaR.sub.[beta]] ([??](w)) [less than or equal to] [zeta] if) and the constraint G([alpha], [??](w))[zeta] [less than or equal to] in the form shown in (9) that will be suitable for our purpose, instead of presenting it in its general form proposed by Krokhmal, Palmquist, and Uryasev (2002). In general, if the objective function is convex and the constraints (other than the CVaR constraint) are linear, one can replace the CVaR constraint in the optimization problem with G([alpha], [??](w)) [less than or equal to] [zeta].

Compared with the CVaR constraint in Problem (6), G([alpha] [??](w)) [less than or equal to] [zeta] has a tractable formulation. Using a well-known linearization technique, adding auxiliary variables [y.sub.k] for k = 1, ... K, the constraint [alpha] + 1/[1-[beta]] E[[[[??](w) - [alpha]].sup.+]] [less than or equal to] [zeta] can be realized as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where the expected value of E[[[??](w) - [alpha]]+] is calculated as the mean of K equiprobable observations,

1/K [k.summation over (k=1) [[[??].sub.k] (W) - [alpha]].sup.+].

Below we proceed with an example to illustrate our MV+CVaR approach.

Numerical Illustration

Now we conduct numerical experiments for a life insurer's business portfolio to illustrate the superiority of the MV+CVaR approach to the MV approach. Example 1 demonstrates the case of an insurance company considering three lines of business. Example 2 extends the analysis to a larger scale problem of an insurer considering nine business lines.

Example 1. Assume an insurer considers selling three types of life insurance in year 0: 10-year term life insurance (i = 1) on a male age (25), 5-year term life insurance (i = 2) on a male age (35), and whole life insurance (i = 3) on a male age (40). Further we assume:

* The annual level premiums [[eta].sub.i] are the average market prices in 2005 for i = 1, 2, 3.

* The insurer uses the Lee and Carter (1992) model to simulate 150 future mortality paths. Because we do not have a time series of insured life tables, we use population data. An insurer would very likely use its own data.

* The insurer uses the U.S. Treasury yield curve on December 28, 2005 to compute the benefit payment ratios based on Formula (3).

We obtained premium data from Compulife, a company that sells market data from over 100 U.S. and Canadian life insurers. (2) We calculated an industry-weighted average of U.S. term life insurance prices for male nonsmokers and smokers using weights based on the incidence of smoking in the 2005 U.S. population reported by the Centers for Disease Control. (3) These are the resulting conditional expected values of premiums offered by our insurer in year 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

In addition, we assume an expense ratio of [phi] = 12.5 percent for each line of business. This means expenses are 12.5 percent of premiums, paid at the same time as the premiums are collected. Therefore, the benefit payment ratios by line in year 0 are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where we have used [[??].sub.i] in place of [??]([x.sub.i]). We simulated a future path of mortality, for that path calculated the values of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , and [A.sub.(40)] using (3) and those of [P.sub.1], [P.sub.2], and [P.sub.3] using (11), and then calculated the benefit payment ratios using (12). One path of mortality yields one joint observation of the three benefit payment ratios. We repeat this 150 times. The Appendix shows some details on the Lee and Carter (1992) method we used to simulate those tables.

Table 1 shows the descriptive statistics for the benefit payment ratio of these three lines of life insurance based on 150 trials of simulation. It turns out that the 10-year term life insurance (i = 1) on a male age (25) has the lowest mean benefit payment ratio and the highest variance. The whole life insurance (i = 3) on a male age (40) has a higher mean benefit payment ratio and lower variance than the 5-year term life insurance (i = 2) on a male age (35).

Suppose the insurer's original portfolio has 10 percent of total premium from 10-year term life insurance, 10 percent from 5-year term life insurance, and 80 percent from whole life insurance. The original weighted average of benefit payment ratios of this portfolio at time 0 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The sample statistics of [??]([w.sub.0]) are shown in the row labeled "Original" in Table 2 for the same 150 simulations that we used in Table 1. To search for an optimum business strategy, we solve the MV+CVaR optimization Problem (9). We specify

[l.sub.0] = [3.summation over (i=j)] [w.sub.i] E[[[??].sub.i] = 0.9567

and

[zeta] = [CVaR.sub.95 percent] ([??]([w.sub.0])) - 0.05|1 - [CVaR.sub.95 percent] ([??]([w.sub.0]))|, (14)

where [CVaR.sub.95 percent] ([??]([w.sub.0])) is the 95 percent level CVaR of the original portfolio. The specification (14) controls the downside risk by reducing the original 95 percent level CVaR by 0.05 [absolute value of 1 - [CVaR.sub.95 percent] ([??]([W.sub.0]))]. The firm can set a different risk reduction level to fit its particular situation. Later we will show how an adjustment in risk reduction level [delta] changes the optimal solution. With (14) we get the following weights for the MV+CVaR optimal business composition:

[w.sub.MV+CVaR] = [0, 0.4660, 0.5340].

This means the insurance company could improve its portfolio of life insurance if it were to decline writing the 10-year term life insurance, put 46.60 percent of its business in the 5-year term life insurance and the remaining 53.40 percent in the whole life insurance. The row labeled "MV+CVaR" in Table 2 shows the summary statistics for the MV+CVaR efficient mortality portfolio. The MV+CVaR efficient portfolio reduces the variance from 0.0043 to 0.0038. Furthermore, Table 2 also shows that the new portfolio's skewness reduces from 0.2912 to 0.1457, and its 95 percent CVaR decreases from 1.1047 to 1.0802. In sum, we conclude the MV+CVaR portfolio returns a better mean-variance trade-off and a lower downside risk than the original one.

Example 2. Now we include more life insurance lines and analyze a larger scale problem with nine lines of life insurance: 5-year term, 10-year term, and whole life insurance issued at ages (25), (35), and (40). The sample statistics of these nine lines are summarized in Table 3. These sample statistics come from a different set of simulations from that in 1, so the sample statistics are not exactly the same as those in Table I but they are very close and qualitatively the same.

We assume the insurer originally has equal weights on each line of business (i.e., [w.sub.i] = 1/9, i = 1, 2, ... , 9). Given this assumption, the summary statistics of this evenly weighted portfolio are shown in the row labeled "Original" in Table 4. The expected benefit payment ratio of the original nine-line portfolio is 0.9921 and its variance is 0.0055.

To obtain the MV+CVaR efficient portfolio with the above nine life insurance lines, we solve Problem (9). We specify the mean of the optimal portfolio equal to that of the original one

[l.sub.0] = [9.summation over (i=1)] [w.sub.i] [[??].sub.i] = 0.9921

and the 95 percent level CVaR of the optimal portfolio is no higher than [zeta] determined by Equation (14). The MV+CVaR optimal business composition solution is

[w.sub.MV+CVaR] = [0.1467, 0, 0.2955, 0, 0, 0, 0, 0, 0.5578].

This suggests that the insurer could improve its life insurance portfolio if it were to underwrite only 5-year term life on age (25), 5-year term life on age (40), and whole life insurance on age (40) with weights 14.67 percent, 29.55 percent, and 55.78 percent, respectively. The optimal portfolio has lower variance (0.0036), lower skewness (-0.1950), and lower [CVaR.sub.95 percent] (1.0960) than the original portfolio (see Table 4).

BOUNDS ANALYSIS WITH THE METHOD OF MOMENTS AND MAXIMUM ENTROPY

How well does the optimal business strategy secure an insurer's financial position? In this section, we apply the method of moments to address this problem. In addition, we allow for higher order moment information to be included in the mortality risk management setup.

Here is the setting of a reasonably general moment problem. Given moments [[mu].sub.1], ... , [[mu].sub.n] and an interval [[b.sub.1], [b.sub.2]], let M denote the set of random variables Z with support in [[b.sub.1], [b.sub.2]] (i.e., [b.sub.1] [less than or equal to] Z [less than or equal to] [b.sub.2] with probability 1) and E([Z.sup.i]) = [[mu].sub.i] for i = 1, ... , n. Let [phi](z) be a well-behaved real-valued function defined on [[b.sub.1], [b.sub.2]]. The moment problem is to determine the best upper and lower bounds on E[[phi]b(Z)], over all random variables Z [member of] M, given E([Z.sup.i]) = [[mu].sub.i] for i = 1, ... , n. These ideas have roots in the work of Tchebyshev, Markov, and Stieltjes in the 1870s. Tian (2008) provides a detailed discussion of moment problems in portfolio risk management.

We will apply the moment technique to [[phi].sub.d] (z) = 0 for z [less than or equal to] d and [[phi].sub.d] (z) = 1 for z > d, so that E[[[phi].sub.d] (Z)] = Pr(Z > d) is the survival function of Z. The solution to the moment problem below produces bounds on Pr(Z > d):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where F represents a probability distribution on M satisfying [E.sub.F] ([Z.sup.i]) = [[mu].sub.i] for = 1, ... , n. Similarly, the problem for the lower bound of Pr(Z > d) is the solution of the following problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The bounds depend on the given moments and support, as well as d. They are called semiparametric bounds, the parameters being the given moments. We will refer to them as the "general bounds" because they are robust bounds that any feasible distribution with same moments must satisfy.

In our analysis, we are interested in the bounds on the survival function of the weighted average benefit payment ratios, that is, S(d)= 1 - F(d)= Pr([??](w)> d), where [??](w) is the weighted average benefit payment ratio of a mortality portfolio, defined previously.

This moment problem can be solved numerically with semidefinite programming solvers such as SOSTOOLS (Prajna et al., 2004). See Akhiezer (1965), Parrillo (2000), Popescu (2005), and Bertsimas and Popescu (2005) for the details on applying semidefinite programming solvers to the bound Problems (15) and (16).

Unimodal Bounds

Adding a distribution assumption such as unimodal to the constraints in (15) or (16) means that we optimize over a smaller set and thus obtain tighter bounds than the general bounds. Since many data sets have unimodal empirical distributions, we find the unimodal assumption appealing. However, one has to be aware that the resulting bounds are valid only if the variable (benefit payment ratio in our case) has a unimodal distribution. If the distribution is not unimodal, then implementing moment analysis with a unimodal assumption may eliminate important tail behavior.

We transfer the unimodal bounds problems to general bounds problems using the Khintchine (1938) representation of a unimodal random variable; it is also discussed by Feller (1971, p. 158). Feller credits this version to Shepp (without a citation): Z is unimodally distributed if and only if there are two independent random variables U and Y such that Z = m + UY, where U is uniformly distributed on (0,1) and m is the unique mode of Z. Tian (2008) and Brockett and Cox (1985) describe the details of this transfer procedure, originally due to Kemperman (1971). Popescu (2005) shows how to include assumptions such as symmetry, convexity, and smoothness as well as unimodality in the moment problem. Here we summarize the final results of the transfer: the interval [[b.sub.1],[b.sub.2]] is replaced by [[b.sub.1]-m, [b.sub.2]- m]. The function [[phi].sub.d] (Z) is replaced by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [d.sup.*] = d - m. And the bounds [[mu].sub.i] are replaced by [[mu].sup.*.sub.i] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The resulting problems (for upper and lower bounds) have the same form as (15) and (16) and also can be solved with semidefinite program solvers. For a given set of moments and support interval, we solve for many values of d and obtain the optimal unimodal bounds [[p.bar].sup.*.sub.d] and [[bar.p].sup.*.sub.d]. The general bounds on the survival function of [??] (w) always satisfy

[[p.bar].sub.d] [less than or equal to] Pr ([??](w) > d) [less than or equal to] [[bar.p].sub.d], (17)

and if [??] (w) has a unimodal distribution the following relationship applies

[[p.bar].sup.*.sub.d] [less than or equal to] Pr ([??](w) > d) [less than or equal to] [[bar.p].sup.*.sub.d]. (18)

If we compare the general bounds ([[p.bar].sub.d] and [[bar.p].sub.d]) with their unimodal counterparts, we have [[p.bar].sub.d] [less than or equal to] [[p.bar].sup.*.sub.d] [less than or equal to] [[bar.p].sup.*.sub.d] [less than or equal to] [[bar.p].sub.d]

Maximum-Entropy Distribution

The work of Shannon (1948a, 1948b) in information theory and Jaynes (1957) in statistical physics led eventually to the concept of entropy in probability theory. The entropy of a probability distribution is a measure of how much information it contains. The maximum-entropy distribution is the "most likely," "most unbiased," "least prejudiced," or "most uniform" distribution of a given class of distributions.

Consider the set F of distributions on [[b.sub.1], [b.sub.2]] with given moments [[mu].sub.i] for i = 1, ... , n; these are the distribution functions of the random variables Z in M give in the previous section. The entropy of a continuous type distribution f(z) [member of] F is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The definitions for discrete and mixed distributions are analogous. The distribution [f.sup.*] (z) [member of] F with maximum entropy is the solution to the following moment problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

The maximum-entropy distribution is sensitive to the support interval [[b.sub.1], [b.sub.2]]. For our applications, we used a sample to estimate moments so we simply set the support interval equal to the range of the sample.

We solved problem (19) by applying a modified Newton method (Luenberger, 1984) to its dual problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where [[lambda].sub.0], ... , [[lambda].sub.n] are Lagrange multipliers associated with the constraints defining F and

[f.sup.*] (z) =exp (-1 - [n.summation over (1=10)] [[lambda].sub.i] [Z.sup.i]

Tian (2008) describes the details. These maximum-entropy problems were solved with MATLAB software. The maximum-entropy distribution determined by the moments of the weighted average portfolio benefit ratio [??](w) is the distribution using as little information as possible (using only moments) to analyze the risk of MV+CVaR mortality portfolios.

Numerical Illustrations

Now we apply the bound and maximum entropy analysis to the numerical examples in the previous section.

Example 1. Figure 1 shows the histograms of the original and the MV+CVaR three-line portfolio benefit payment ratios based on eight bins. The histograms suggest that both the original and optimal portfolio benefit payment ratios have unimodal distributions. Figure 2 shows the graphs of the bounds in (17) and (18), which we computed for a large numbers of values of d. If the distribution is unimodal with business composition [w.sub.0] = [0.10, 0.10, 0.80], its four-moment bounds are significantly narrowed. These graphs can be used to derive bounds on the value at risk. Consider the 95 percent VaR, which corresponds to the 0.05 level of the survival function. Draw a horizontal line through the survival probability of 0.05 on the vertical axis. It intersects the solid curves at d values of 1.038 and 1.087. Therefore, if the original portfolio has unimodal distribution, what we can say is that

1.038 [less than or equal to] [VaR.sub.95 percent] [les than or equal to] 1.087. (21)

Without the unimodal assumption, the 95 percent VaR range is much wider:

0.985 [less than or equal to] [VaR.sub.95 percent] [less than or equal to] 1.122, (22)

which is determined by the intersection of the horizontal line with the o curves. The normal distribution (the dotted lines in Figure 2) with the same mean and variance as the original mortality portfolio must fall between the two-moment upper and lower unimodal bounds. In this example, the normal curve is even within the four-moment unimodal bounds. Under the normal assumption, the business's [VaR.sub.95 percent] is 1.065.

Figure 3 shows the four-moment general and unimodal bounds of the original three-line life insurance business with [w.sub.0] = [0.10, 0.10,0.80] and its MV+CVaR optimum with [w.sub.MV+CVaR] = [0, 0.4660, 0.5340]. The MV+CVaR approach reduces the downside risk for both sets of bounds, general and unimodal. To illustrate, just as what we did in Figure 2, we draw a horizontal line through the 0.05 level of the survival probability on the vertical axis to analyze the 100 percent confidence bounds on the 95 percent VaR. If we assume the optimized portfolio is also unimodal, the bounds on the [VaR.sub.95 percent] of the portfolio benefit payment ratio drops to

1.030 [less than or equal to] [VaR.sub.95 percent] [less than or equal to] 1.075, (23)

by adding 95 percent CVaR constraint to the traditional Markowitz optimization. Thus, both the upper and lower bounds are improved. For example, the lower bound indicates that there is only 5 percent probability that the portfolio annual benefit payment ratio will rise above 1.030.

To take our discussion one step further, we solve the maximum-entropy Problem (19). The results are shown in Figure 4, which compares the maximum-entropy distributions of the original and the MV+CVaR mortality portfolios. In particular, the right tail of the MV+CVaR three-line portfolio shifts to the left of the original portfolio distribution. Therefore, the optimized portfolio has lower downside risk than the original one, which is consistent with the conclusion of the bound analysis.

Example 2. Continuing with Example 2, Figure 5 shows the histograms of the nine-line original and MV+CVaR optimal portfolios with eight bins. Since neither portfolio distribution is unimodal, we only compute their four-moment general bounds. Figure 6 illustrates their four-moment general bounds and maximum-entropy survival functions. Again the right tail of the MV+CVaR optimal portfolio lies on the left of that of the original portfolio in the bound and maximum-entropy analyses, implying lower shortfall risk after optimization.

NATURAL HEDGING EFFECTS

Cox and Lin (2007) argue that an insurer selling both life insurance and annuities is exposed to lower one-directional changes in mortality. This effect is called "natural hedging." That is, adding annuities to a life insurance portfolio may lower the portfolio's mortality risk. In this section, we examine how natural hedging improves the MV+CVaR optimal business composition and decreases downside risk with the method of moments.

Example 1

To illustrate the natural hedging effect, we add an annuity to the three-line life insurance portfolio analyzed in Example 1. Specifically, in addition to selling the three life policies, the insurer also sells a single premium immediate life annuity on male age (65). Assume that the insurer sells the annuity at the average market quote of 2005, which charges a monthly payout rate of $6.56 per lump-sum premium $1,000 (Stern, 2008). We further assume that the insurer uses the same mortality data and method as those for life insurance to forecast the future survival payments of annuities.

As we mentioned earlier, we are using a single times series of mortality, the U.S. male population data, as the basis for the mortality forecasts. The summary statistics for the benefit payment ratio of this annuity, [[??].sub.10], with an expense ratio of [phi] = 12.5 percent, are shown in Table 5.

Assume that the insurer is considering a portfolio of 10.00 percent 10-year term life insurance, 10.00 percent the 5-year term life insurance, 40.00 percent the whole life insurance, and 40.00 percent the annuity, that is,

[w.sub.0] = [0.10, 0.10, 0.40, 0.40]

in the original portfolio. To optimize this mortality portfolio, we set the objective benefit payment ratio [l.sub.0] at 0.9652, same as the expected benefit payment ratio of the original portfolio, and construct a 95 percent CVaR constraint with [zeta] in (6) determined by Equation (14). We get the MV+CVaR efficient four-line mortality portfolio with the weights,

[w.sub.MV+CVaR] = [0.2028, 0, 0, 0.7972]. (24)

That is, the insurer should not sell either the 5-year life insurance on male age (35) or the whole life insurance on male age (40), and adjust its business to make 20.28 percent of its total premium from the 10-year life insurance on male age (25), and 79.72 percent from the single premium immediate life annuity on male age (65). Table 6 compares the original and MV+CVaR mortality portfolios. The MV+CVaR portfolio has a much lower variance and 95 percent CVaR. Figure 7 shows the histograms of the original and the MV+CVaR optimal four-line portfolios based on eight bins. Notably, in Figure 7, the distribution of the MV+CVaR portfolio is tightly grouped around the mean relative to that of the original portfolios, indicating that the MV+CVaR portfolio has a much lower risk. The histograms also show that both the original and the optimized four-line portfolios are unimodally distributed.

Figure 8 shows the four-moment unimodal bounds and the maximum-entropy distributions of the original and MV+CVaR portfolios with four lines of business. The probability density function of the MV+CVaR optimal four-line portfolio is much tighter than that of the original portfolio, so the upper and lower unimodal bounds of the MV+CVaR portfolio are narrower.

To explore the natural hedging effect, we compare the bounds of the MV+CVaR three-line and four-line portfolios. If we assume both the optimized portfolios have unimodal distributions, the right plot in Figure 9 shows how the 100 percent confidence interval of the 95 percent level VaR improves by adding an annuity to the three-line pure life insurance portfolio. The confidence interval of [VaR.sub.95percent] of the four-line portfolio stays at a much lower level of benefit payment ratio range than that of [VaR.sub.95percent] of the three-line portfolio. It highlights the benefits of natural hedging: by including both annuity and life insurance in a portfolio, natural hedging reduces benefit payment and decreases mortality risk.

Example 2

We add three immediate annuities on male with issue ages (65), (70), and (75) to the nine-line life insurance portfolio we discussed in previously. The sample statistics of these three annuities are given in Table 7. The sample statistics of the immediate life annuity on male age (65) come from a different set of simulations from Example 1. So its moments are not exactly the same as those in Table 5 but they are very close and qualitatively the same.

We assume the insurer earns the same amount of premium from each of its 12 lines of business (i.e., [w.sub.i] = 1/12, i = 1, 2,...,12). By adding 95 percent CVaR constraint to the Markowitz problem and specifying the expected benefit payment ratio

[l.sub.0] = [12.summation over (i=1)] [w.sub.i]E[[[??].sub.i]] = 0.9895, i=1

we get the weights of the MV+CVaR portfolio as follows:

[w.sub.MV+CVaR] = [0, 0, 0, 0.0437, 0, 0.0564, 0, 0, 0.0649, 0, 0, 0.8350]. (25)

Therefore, the MV+CVaR optimal portfolio is composed of 4.37 percent 10-year term life on a male age (25), 5.64 percent 10-year term life on a male age (40), 6.49 percent whole life insurance on a male age (40), and 83.50 percent single premium immediate life annuity on a male age (75). Table 8 shows that given the same expected benefit payment ratio as that of the original portfolio (0.9895), the MV+CVaR optimal portfolio has a much lower variance than the original portfolio.

The histograms of the original and the MV+CVaR 12-line life insurance portfolios in Figure 10 suggest that both portfolios have unimodal distributions. Similar to Figure 8 of Example 1, Figure 11 illustrates a lower risk of the MV+CVaR 12-line portfolio than its original 12-line counterpart: it has much narrower bounds and lower values of the right tails.

We further perform the bound analysis to compare the MV+CVaR efficient 9-line portfolio without annuities and the MV+CVaR efficient 12-line portfolio with annuities. As expected, Figure 12 suggests a much lower downside risk of the MV+CVaR efficient 12-line portfolio, again, supporting the natural hedging benefits between life insurance and annuities.

In practice, it may be difficult for an insurer to directly accomplish an MV+CVaR optimal business composition in Portfolio (25). For example, it may not be easy for an insurer specializing in life insurance to enter the annuity business (Cox and Lin, 2007). To solve this problem, the insurer has at least two possible solutions. First, the insurer can buy or sell reinsurance to rebalance its weights in various business lines. Second, the insurer can issue or purchase mortality or longevity securities in the capital markets. The mortality-linked securities are new in the financial markets but have attracted a lot of attention from insurers, investors, pension plans, and academia (Blake and Burrows, 2001; Cowley and Cummins, 2005; Lin and Cox, 2005, 2008; Cairns, Blake, and Dowd, 2006 Cox, Lin, and Wang, 2006). As the mortality-linked security markets develop, the insurer may be able to cede or assume risk to realize the MV+CVaR efficient mortality portfolio at a low cost.

FRONTIERS OF EFFICIENT MORTALITY PORTFOLIOS

Our analysis has focused on improving an insurer's existing mortality portfolio with the MV+CVaR approach and applying the method of moments to examine whether and how the MV+CVaR portfolio could control downside risk. Can these techniques be applied to any arbitrary insurance business composition? How does the MV+CVaR approach compare to the traditional Markowitz optimization method? How does natural hedging reduce mortality risk for different MV+CVaR portfolios? To answer these questions, in this section, we extend our analysis to a range of efficient portfolios, not just a particular efficient portfolio given a certain level of benefit payment ratio. Specifically, we compare

1. the two- or three-dimensional representations of the Markowitz and MV+CVaR optimal portfolios;

2. the two- or three-dimensional representations of the 9-line and 12-line optimal portfolios.

Frontiers of Markowitz and MV+CVaR Optimized Portfolios We write Equation (14) in the following general form:

[zeta] = [CVaR.sub.95percent] ([??]([w.sub.0])) - [delta] [absolute value of 1 - [CVaR.sub.95percent]([??]([w.sub.0]))]. (26)

In Equation (14), [delta] = 0.05 and [CVaR.sub.95percent]([??]([w.sub.0])) corresponds to the 95 percent level CVaR of the original portfolio. To compare the MV+CVaR optimal portfolios with their Markowitz counterparts, we set the benchmark [CVaR.sub.95%]([??]([w.sub.0])) at the 95 percent CVaR of the Markowitz portfolio given the benefit payment ratio [??]([w.sub.0]). This setting guarantees that the 95 percent CVaR of the MV+CVaR portfolio is no higher than that of its Markowitz counterpart. With [zeta] specified as (26), the upper limit of 95 percent level CVaR constraint imposed on the traditional Markowitz problem changes with the risk reduction level [delta]. In practice, the insurer can choose a level of [delta] that fits its specific risk management needs. In our illustration, we choose the highest possible [delta] for each portfolio that keeps problem (6) feasible.

For the nine lines of business mentioned earlier, we solve (1) the traditional Markowitz portfolio problem and (2) the MV+CVaR Problem (6) with a 95 percent CVaR constraint specified by (26). We obtain a set of optimal portfolios with different benefit payment ratios. We plot the variance-mean, skewness-mean, [CVaR.sub.5percent]-mean, and [CVaR.sub.95percent]-mean graphs for the MV+CVaR optimal portfolios and compare them to their Markowitz counterparts. Each graph in Figure 13 is a piecewise linear interpolation based on 20 solved efficient portfolios. Note that although our MV+CVaR problem is not a typical multiple objective optimization problem, loosely speaking, it minimizes both variance and CVaR subject to a preset expected benefit payment ratio. Therefore, the efficient frontier for a MV+CVaR optimization problem is a surface in the three-dimensional space (mean, variance, CVaR). Any two-dimensional profile shown in Figure 13 is a cross-section of the three-dimensional representation with the value of the third fixed.

While the upper left graph of Figure 13 shows that the MV+CVaR efficient frontier somewhat deviates from the Markowitz efficient frontier in terms of the mean-variance trade-off, the MV+CVaR approach effectively decreases the skewness of relatively high variance portfolios shown in the skewness-mean profile. A lower skewness is desirable because it decreases the likelihood of obtaining higher benefit payment ratios. The [CVaR.sub.95percent]-mean curves in the bottom right graph demonstrates that for the same mean, the portfolios constructed from the MV+CVaR approach are able to reach a lower 95 percent CVaR, implying a lower downside mortality risk. However, the impact of adding CVaR constraint to the Markowitz model on the portfolios with low benefit payment ratio is not as significant as that on the relatively high benefit payment ratio portfolios.

It is worth noting that by adding a large percentile CVaR constraint, the MV+CVaR approach aims at reshaping the right tail of the distribution, which corresponds to high benefits. The approach has little impact on the left tail that represents low benefits. This is confirmed by the bottom left graph in Figure 13, which shows that the [CVaR.sub.5percent] curve of the MV+CVaR portfolios just barely differs from that of its Markowitz counterparts.

Natural Hedging for Efficient Portfolios

We investigate the natural hedging effect for various MV+CVaR efficient portfolios to extend our previous analysis. Given the 95 percent CVaR constraint specification (26), Figure 14 shows that the 12-line MV+CVaR portfolios ([nabla]) composed of both life insurance and annuity lines outperform those containing only life insurance ([omicron]). Given the same means, the 12-line efficient portfolios achieve lower variance, and lower [CVaR.sub.95percent] than their 9-line counterparts. Thus, inclusion of annuities reduces the potential mortality risk (measured by variance) and downside risk (measured by [CVaR.sub.95percent]) of life insurance portfolios. This provides a new evidence to support the natural hedging effects. Notice that adding annuities to a life insurance portfolio does not necessarily decrease the portfolio skewness, as shown in the upper right plot of Figure 14.

In our example, the improvement of the frontier by adding an annuity can be explained by the negative correlation between the annuities and the lines of life insurance, as shown in Table 9.

Comparison of Markowitz and MV+CVaR Optimal Portfolios With Different values of [zeta]

A change in [delta] in (26) alters the upper limit of 95 percent level CVaR constraint [zeta]. The bigger the [delta], the more stringent the 95 percent level CVaR constraint in (6). The MV+CVaR optimal portfolios shown in Figures 13 and 14 are determined by adding the most stringent feasible 95 percent CVaR constraints. In this section, we illustrate how the 95 percent CVaR of a MV+CVaR portfolio improves relative to its MV counterpart with various feasible risk reduction levels of [delta]. Specifically, we solve the MV+CVaR problem (6), subject to a 95 percent CVaR constraint with a different levels of [zeta], to obtain a set of optimal portfolios given a preset expected benefit payment ratios.

Figure 15 shows how the 95 percent CVaR of the MV+CVaR portfolio improves compared to its MV counterpart at each risk reduction level [delta]. For the MV model, [delta] is set at 0. We show the percentage difference of 95 percent CVaR between MV and MV+CVaR optimal portfolios

[CVaR.sup.MV.sub.95percent] - [CVaR.sup.MV+CVaR.sub.95percent]/[CVaR.sup.Mv.sub.95percent] (27)

on the vertical axis. For both 9-line and 12-line portfolios, the percentage differences are always positive. This means that adding a 95 percent CVaR constraint with [zeta] specified in (26) to the traditional Markowitz problem reduces the downside risk. In general, given a specified objective benefit payment ratio [l.sub.0], the percentage difference (27) increases with [delta]. As [delta] increases, we impose a more stringent CVaR constraint so we observe lower downside risk of MV+CVaR portfolios and bigger difference between [CVaR.sup.MV.sub.95percent] and [CVaR.sup.MV+CVaR.sub.95percent]

Figure 16 shows the mean-variance-[CVaR.sub.95percent] graphs for the 9-line portfolios (left) and the 12-line portfolios (right), respectively. In each graph, the [CVaR.sub.95percent] of each MV+CVaR portfolio changes with the upper limit of the [CVaR.sub.95percent] constraint [zeta], thus forming a three-dimensional surface. As for the MV portfolios, each optimal mean and variance relationship uniquely determines a [CVaR.sub.95percent]. Thus, in each graph, the mean-variance-[CVaR.sub.95percent] frontier of the MV optimal portfolios is a curve represented by the dotted line. This dotted line lies above the three-dimensional surface of the MV+CVaR portfolios, suggesting the higher downside risk of the optimized MV portfolios.

Comparison of Markowitz and MV+CVaR Optimal Portfolios Using Method of Moments

As a posterior check, we compare the maximum-entropy survival probabilities of the 20 MV optimal portfolios and their MV+CVaR counterparts illustrated in Figures 13 and 14. We call them counterparts since they have the same mean, that is, the same target benefit payment ratio. The difference between maximum-entropy survival probabilities of the 20 optimized MV and MV+CVaR portfolios on the right tail (65-100 percentiles) of survival function,

Pr [([??](w) > d).sup.MV] - Pr [([??](w) > d).sup.MV+CVaR],

are shown in Figure 17. (4) The left plot shows the difference for the 9-line portfolios, and the right one is for the 12-line portfolios. The difference of each pair is always positive on the right tail, demonstrating the robustness of the MV+CVaR approach. That is, the MV+CVaR portfolios consistently have lower downside risk than the MV portfolios. Notice that compared to the 9-line portfolios, the 12-line portfolios provide more room to reduce downside risk since the difference on the right plot is bigger than that on the left plot.

CONCLUSION

We applied the method of moments and portfolio optimization to mortality risk management, making two contributions to the mortality literature. First, we propose the MV+CVaR approach to manage mortality portfolio risk with a reasonable sacrifice of mean-variance efficiency. The MV+CVaR approach controls the downside risk by specifying the mean value of the benefit payment ratio above the [beta]-level VaR to be no more than some prespecified value. This method is effective in obtaining an optimal mortality portfolio while controlling its downside risk.

Second, we applied the method of moments to mortality risk management by calculating the semiparametric upper and lower bounds on the survival function of benefit payment ratios for mortality portfolios. The bounds are used to illustrate the 100 percent confidence interval of the downside risk, which is measured by [beta]-level VaR. We propose how to use the moments method to investigate downside risk of MV+CVaR efficient mortality portfolios. In addition, as an extension to the moments method, we derive the maximum-entropy distribution of mortality portfolios. We use the maximum-entropy approach to conduct a robustness check for the moments method because the maximum-entropy approach provides a representative distribution that is the most unbiased, given the moment information.

We illustrate our examples with up to 12 lines of business with annual observations, but these methods can be extended to more lines of business and higher frequency data. Furthermore, we would obtain more practical results by using joint forecasts of several mortality models (recognizing at least mortality for male, female, life insurance, and annuities). In addition, our results could be generalized by developing a joint model of investment and mortality risk. Such a model could be applied to a life insurer's asset liability management problem. Finally, policyholder behavior (such as surrendering the policy) is important but incorporating it will require development of new models and access to industry data. We leave these questions for future research.

APPENDIX: ESTIMATING BENEFIT PAYMENT RATIOS SUMMARIZED IN TABLE 1

We use the Lee and Carter (1992) model to forecast the insurer's future mortality rates. This model incorporates both the age variation and the underlying time trend of death rates. The logarithm form of the 1-year death rate [q.sub.x,t] of age x in year t in Lee and Carter (1992) is modeled as

1n [q.sub.x,t] = [a.sub.x] + [b.sub.x][[theta].sub.t] + [[member of].sub.x,t], (A1)

where [a.sub.x] and [b.sub.x] are the age-specific parameters and [[theta].sub.t] is the time-series common risk factor that drives the mortality rates of all age cohorts. The transitory shock [[member of].sub.x,t] is normally distributed with zero mean. Lee and Carter model the central death rate, but usually the 1-year probability of death [q/sub.x,t] is very close to the central death rate. Thus, we model [q.sub.x,t] directly rather than model the central death rate, which would then have to be transformed to a probability.

Model (A1) is not unique hence two constraints are imposed to ease the estimation:

[summation over (x)] [b.sub.x] = 1, [summation over (t)] [[theta].sub.t] = 0 (A2)

These two constraints imply that the intercept ax simplifies to the empirical average of age x over time:

[a.sub.x] = [summation over (t)]ln [q.sub.x,t]/t (A3)

We follow a two-step procedure in Lee and Carter (1992) to estimate Model (A1). In the first step, singular value decomposition of the matrix is applied to obtain estimates for [b.sub.x], (x - 0,1, 2 ....) and [[theta].sub.t], (t = 0, 1, 2, ...). In the second step, the time-series common risk factor [[theta].sub.t] in year t is recalculated based on the actual number of deaths.

We assume the insurer has the same mortality experience as that of the U.S. population. The insurer uses [q.sub.x,t] in the U.S. male population mortality tables, observed each year from 1901 to 2005 from the Human Life Table Database and the Human Mortality Database, to estimate Model (A1). The tables for years 1901-1999 are from the Human Life Table Database and the tables for 2000-2005 are from the Human Mortality Database, published by the University of California, Berkeley and Max Planck Institute for Demographic Research. (5) The age range is x = 0,1, 2,..., 103 for U.S. males from t = 1901 to t = 2005. The parameter estimates of ax and [b.sub.x] are given in Table A1 and the estimated [[theta].sub.t] is shown in Figure A1.

For mortality projection, [[theta].sub.t] is assumed to follow a random walk with drift c,

[[theta].sub.t] = [[theta].sub.t-1] + c + [e.sub.t], (A4)

where the error term [e.sub.t] is normally distributed with a zero mean and a variance [[sigma].sub.[theta]]. Based on the time series of [[theta].sub.t] where t = 1901, 1902, ... , 2004, 2005, we obtain c = -0.2032 and [[sigma].sub.[theta]] = 0.6491.

When producing forecasts for the k-year ahead mortality rates, we first simulate [e.sub.2005+j] (j = 1, 2,...,k) each year for k years. Then the common risk factor [[theta].sub.2005+j] in year 2005 + j is calculated by adding the constant c = -0.2032 to each simulated error term [e.sub.2005+j]. Given the estimated [a.sub.x]s and [b.sub.x]s in Table A1, the simulated [[theta].sub.2005+j]s, and the U.S. Treasury yield curve on December 28, 2005, we use Model (A1) to calculate simulated future mortality rates for all ages and years.

TABLE A1 Model (A1) Fits the U.S. Male Population Data From 1901 to 2005 Age [a.sub.x] [b.sub.x] 0 -3.4280 0.1709 1 -5.6995 0.2618 2 -6.2021 0.2383 3 -6.4978 0.2296 4 -6.7050 0.2279 5 -6.8580 0.2181 6 -6.9871 0.2039 7 -7.1118 0.1918 8 -7.2246 0.1829 9 -7.3294 0.1829 10 -7.3991 0.1887 11 -7.3766 0.1892 12 -7.2279 0.1745 13 -6.9995 0.1503 14 -6.7722 0.1284 15 -6.5673 0.1118 16 -6.3911 0.0997 17 -6.2538 0.0939 18 -6.1434 0.0907 19 -6.0631 0.0915 20 -5.9922 0.0948 21 -5.9248 0.0963 22 -5.8865 0.0990 23 -5.8748 0.1010 24 -5.8818 0.1025 25 -5.8969 0.1043 26 -5.9044 0.1055 27 -5.9035 0.1071 28 -5.8871 0.1087 29 -5.8555 0.1096 30 -5.8209 0.1105 31 -5.7849 0.1104 32 -5.7455 0.1103 33 -5.7013 0.1095 34 -5.6540 0.1083 35 -5.6011 0.1064 36 -5.5465 0.1046 37 -5.4875 0.1020 38 -5.4285 0.0993 39 -5.3688 0.0969 40 -5.3043 0.0936 41 -5.2384 0.0906 42 -5.1698 0.0874 43 -5.0994 0.0843 44 -5.0278 0.0814 45 -4.9547 0.0785 46 -4.8791 0.0753 47 -4.8039 0.0722 48 -4.7275 0.0687 49 -4.6521 0.0655 50 -4.5753 0.0621 51 -4.4986 0.0592 52 -4.4214 0.0569 53 -4.3449 0.0553 54 -4.2685 0.0543 55 -4.1907 0.0531 56 -4.1127 0.0519 57 -4.0341 0.0504 58 -3.9552 0.0486 59 -3.8772 0.0468 60 -3.7976 0.0450 61 -3.7190 0.0437 62 -3.6404 0.0420 63 -3.5650 0.0406 64 -3.4897 0.0391 65 -3.4141 0.0379 66 -3.3373 0.0370 67 -3.2595 0.0365 68 -3.1830 0.0370 69 -3.1056 0.0377 70 -3.0273 0.0385 71 -2.9482 0.0390 72 -2.8689 0.0390 73 -2.7897 0.0386 74 -2.7116 0.0382 75 -2.6329 0.0372 76 -2.5543 0.0366 77 -2.4736 0.0362 78 -2.3915 0.0365 79 -2.3026 0.0354 80 -2.2265 0.0371 81 -2.1464 0.0369 82 -2.0703 0.0361 83 -1.9986 0.0345 84 -1.9298 0.0327 85 -1.8621 0.0309 86 -1.7936 0.0293 87 -1.7243 0.0281 88 -1.6541 0.0273 89 -1.5838 0.0267 90 -1.5141 0.0261 91 -1.4449 0.0254 92 -1.3773 0.0244 93 -1.3120 0.0235 94 -1.2491 0.0223 95 -1.1920 0.0225 96 -1.1357 0.0218 97 -1.0807 0.0215 98 -1.0282 0.0213 99 -0.9796 0.0210 100 -0.9305 0.0208 101 -0.8823 0.0209 102 -0.8352 0.0192 103 -0.7830 0.0226

DOI: 10.1111/j.1539-6975.2012.01469.x

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(1) http://www.mathworks.com/

(2) See the company's website for additional information: http://www.compulife.ca/historicaldata.php.

(3) Source: http://www.cdc.gov/media/pressrel/r061026a.htm. The percentage of U.S. adults who smoked in 2005 was 20.9 percent.

(4) The percentage difference of each pair in Figure 17,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is always greater than 10 percent.

(5) Available at www.mortality.org or www.humanmortality.de (data downloaded on June 8, 2008).

Samuel H. Cox is the L. A. H. Warren Chair Professor at the Asper School of Business, University of Manitoba. Yijia tin is in the Department of Finance, College of Business Administration, University of Nebraska-Lincoln. Ruilin Tian is in the Department of Accounting, Finance, and Information System, College of Business, North Dakota State University. Luis F. Zuluaga is at the Faculty of Business Administration, University of New Brunswick. The authors can be contacted via e-mail: scox@cc.umanitoba.ca, yijialin@unl.edu, ruilin.tian@ndsu.edu, and lzuluaga@unb.ca, respectively. This article was presented at the 2009 American Risk and Insurance Association annual meeting and the 2010 Financial Management Association annual meeting. We appreciate helpful comments from Alexander Kling and other participants at the meetings. The authors also thank two anonymous referees for their very helpful suggestions and comments during the revision process.

TABLE 1 Sample Statistics for Benefit Payment Ratios of Three Types of Life Insurance Line i Age Type Mean Variance Skewness Kurtosis 1 (25) 10-year term 0.8587 0.0083 0.3576 0.2481 2 (35) 5-year term 0.9403 0.0055 0.2997 0.6804 3 (40) Whole life 0.9710 0.0045 0.3631 0.1024 TABLE 2 Sample Statistics of Benefit Payment Ratios of Oriqinal and MV+CVaR 3-Line Portfolios [??](w) Mean Variance Skewness Original 0.9567 0.0043 0.2912 MV+CVaR 0.9567 0.0038 0.1457 [??](w) Kurtosis Mode [CvaR.sup.95%] Original -0.0182 0.9591 1.1047 MV+CVaR -0.2772 0.9755 1.0802 TABLE 3 Sample Statistics for Benefit Payment Ratios of Nine Types of Life Insurance Line i Age Type Mean Variance Skewness Kurtosis 1 (25) 5-year term 0.8369 0.0044 0.1810 -0.1170 2 (35) 5-year term 0.9346 0.0054 0.1444 -0.1219 3 (40) 5-year term 1.1238 0.0060 0.1032 -0.1131 4 (25) 10-year term 0.8534 0.0073 0.0500 -0.1606 5 (35) 10-year term 1.2023 0.0131 -0.0274 -0.0724 6 (40) 10-year term 1.4035 0.0128 -0.0888 -0.0200 7 (25) Whole life 0.7274 0.0068 0.0497 -0.0961 8 (35) Whole life 0.8842 0.0054 -0.0873 -0.2010 9 (40) Whole life 0.9632 0.0045 -0.1202 -0.2478 TABLE 4 Sample Statistics of Benefit Payment Ratios of Original and MV+CVaR Optimal 9-Line Portfolios [??](W) Mean Variance Skewness Original 0.9921 0.0055 -0.1264 MV+CVaR 0.9921 0.0036 -0.1950 [bar.l](W) Kurtosis Mode [CvaR.sub.95%] Original -0.3395 0.9382 1.1301 MV+CVaR -0.5096 0.9941 1.0960 TABLE 5 Sample Statistics of Benefit Payment Ratios of Single Premium Immediate Life Annuity Line i Age Type Mean Variance Skewness Kurtosis 10 (65) Immediate 0.9924 0.0002 -0.1071 -0.0074 annuity TABLE 6 Sample Statistics of Benefit Payment Ratios of Original and MV+CVaR Optimal 4-Line Portfolios [??](w) Mean Variance Skewness Original 0.9652 0.0011 0.2671 MV+CVaR 0.9652 9.11E-05 0.7142 [??](w) Kurtosis Mode [CvaR.sub.95 percent] Original -0.1178 0.9636 1.0383 MV+CVaR 1.0956 0.9608 0.9886 TABLE 7 Sample Statistics for Benefit Payment Ratios of Three Life Annuities Line i Age Type Mean Variance Skewness Kurtosis 10 (65) Immediate annuity 0.9938 0.0002 0.3246 -0.0847 11 (70) Immediate annuity 0.9801 0.0003 0.3797 0.0553 12 (75) Immediate annuity 0.9707 0.0003 0.4179 0.2097 TABLE 8 Sample Statistics of Benefit Payment Ratios of Original and MV+CVaR Optimal 12-Line Portfolios [??](w) Mean Variance Skewness Original 0.9895 0.0027 -0.1069 MV+CVaR 0.9895 3.89E-06 0.5264 [??](w) Kurtosis Mode [CvaR.sub.95 percent] Original -0.3555 0.9521 1.0865 MV+CVaR 0.852 0.9880 0.9940 TABLE 9 Correlation of Annuities (Lines 10-12) With Life Insurance (Lines 1-9) 5-Year Term 10-Year Term Life (25) (35) (40) (25) (35) (40) Annuities Insurance 1 2 3 4 5 6 10 (65) -0.63 -0.64 -0.64 -0.86 -0.87 -0.87 11 (70) -0.70 -0.71 -0.71 -0.91 -0.92 -0.92 12 (75) -0.78 -0.79 -0.79 -0.96 -0.96 -0.96 Whole Life Life (25) (35) (40) Annuities Insurance 7 8 9 10 (65) -0.89 -0.92 -0.93 11 (70) -0.84 -0.87 -0.89 12 (75) -0.78 -0.81 -0.83

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Author: | Cox, Samuel H.; Lin, Yijia; Tian, Ruilin; Zuluaga, Luis F. |
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Publication: | Journal of Risk and Insurance |

Geographic Code: | 1USA |

Date: | Dec 1, 2013 |

Words: | 11933 |

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