Mortality Portfolio Risk Management

Focus on Mortality

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 , 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 and .

We assume the company uses the benefit payment ratio to manage its risk where is a constant expense ratio. Other measures are possible; for example, one might use . 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 , and from them it generates a sample of values of . 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_{(x)+ t}, or equivalently the death rates q_{(x)+ t} =1− p_{(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_i) denote a set of underwriting classes under consideration at 0. For example, (x₁) might be male, 10-year term life insurance issued at age x₁ =35; (x₂) might be an immediate annuity issued to a female age x₂ =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 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 for (x), we can calculate the conditional survival function of K(x) as

(1)

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

(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 1 at the end of the year of death, here is the present value of benefits and its conditional expected value:

(3)

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

If the policy was issued at 0 and paid with an annual level premium per unit of benefit at that time, then the benefit payment ratio is simply where the conditional expected value of premiums equals

(4)

Immediate Annuity

This is the present value of benefits for a single premium immediate annuity of 1 per year paid at the end of the year issued on (x) at 0 and its conditional expected value:

(5)

The benefit payment ratio at 0 is 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.

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 -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_i in line of business (x_i). The weighted average of the benefit payment ratios is . Let denote the covariance of benefit payment ratios of business lines i and j. Our MV+CVaR problem is to solve for portfolio weights in terms of benefit payment ratios, so as to

(6)

where is the -level CVaR of , which in the case of a continuously distributed can be calculated as

The -level value at risk, , is the -quantile of the distribution of , or the smallest value of such that . That is,

We enforce a right-tail constraint to manage the downside risk by setting at 0.95. The CVaR constraint in (6) ensures the tail expectation is no higher than a prespecified value , thus reducing the downside risk. is a subset of the mortality business feasible set. Specifically in our problem, is the vector of weights satisfying

(7)

where is the random benefit payment ratio of line i, n is the number of business lines in the mortality portfolio, and is a prespecified level of the weighted average of the benefit payment ratios. We assume that 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 -level CVaR is the optimal value of the following optimization problem

(8)

where the notation

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

(9)

If a pair achieves the minimization of (9), will return a -level CVaR and will give the corresponding -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., ) and the constraint 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 .

Compared with the CVaR constraint in Problem (6), has a tractable formulation. Using a well-known linearization technique, adding auxiliary variables y_k for , the constraint can be realized as

(10)

where the expected value of is calculated as the mean of K equiprobable observations,

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 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:

(11)

In addition, we assume an expense ratio of 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

(12)

where we have used in place of . We simulated a future path of mortality, for that path calculated the values of , , and A₍₄₀₎ using (3) and those of P₁, P₂, and P₃ 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).

Table 1. Sample Statistics for Benefit Payment Ratios of Three Types of Life Insurance

Line			Mean	Variance	Skewness	Kurtosis
i	Age	Type
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

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

(13)

The sample statistics of 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

and

(14)

where 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 . 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 changes the optimal solution. With (14) we get the following weights for the MV+CVaR optimal business composition:

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.

Table 2. Sample Statistics of Benefit Payment Ratios of Original and MV+CVaR 3-Line Portfolios

	Mean	Variance	Skewness	Kurtosis	Mode	CVaR 95%
Original	0.9567	0.0043	0.2912	−0.0182	0.9591	1.1047
MV+CVaR	0.9567	0.0038	0.1457	−0.2772	0.9755	1.0802

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 1 but they are very close and qualitatively the same.

Table 3. Sample Statistics for Benefit Payment Ratios of Nine Types of Life Insurance

Line			Mean	Variance	Skewness	Kurtosis
i	Age	Type
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

We assume the insurer originally has equal weights on each line of business (i.e., . 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.

Table 4. Sample Statistics of Benefit Payment Ratios of Original and MV+CVaR Optimal 9-Line Portfolios

	Mean	Variance	Skewness	Kurtosis	Mode
Original	0.9921	0.0055	−0.1264	−0.3395	0.9382	1.1301
MV+CVaR	0.9921	0.0036	−0.1950	−0.5096	0.9941	1.0960

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

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

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 _95percent (1.0960) than the original portfolio (see Table 4).

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 [b₁, b₂ ] is replaced by [b₁ − m, b₂ − m]. The function is replaced by

where . And the bounds are replaced by where

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 and . The general bounds on the survival function of always satisfy

(17)

and if has a unimodal distribution the following relationship applies

(18)

If we compare the general bounds ( and ) with their unimodal counterparts, we have .

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 of distributions on [b₁, b₂ ] with given moments for ; these are the distribution functions of the random variables Z in give in the previous section. The entropy of a continuous type distribution is defined as

The definitions for discrete and mixed distributions are analogous. The distribution with maximum entropy is the solution to the following moment problem

(19)

The maximum-entropy distribution is sensitive to the support interval [b₁, b₂]. 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:

(20)

where are Lagrange multipliers associated with the constraints defining and

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 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 w₀=[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

(21)

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

(22)

which is determined by the intersection of the horizontal line with the 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 _95percent is 1.065.

Figure 3 shows the four-moment general and unimodal bounds of the original three-line life insurance business with w₀=[0.10, 0.10, 0.80] and its MV+CVaR optimum with w_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 _95percent of the portfolio benefit payment ratio drops to

(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.

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, , with an expense ratio of , 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,

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

(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.

Table 6. Sample Statistics of Benefit Payment Ratios of Original and MV+CVaR Optimal 4-Line Portfolios

	Mean	Variance	Skewness	Kurtosis	Mode	CVaR 95percent
Original	0.9652	0.0011	0.2671	−0.1178	0.9636	1.0383
MV+CVaR	0.9652	9.11E-05	0.7142	1.0956	0.9608	0.9886

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 _95percent of the four-line portfolio stays at a much lower level of benefit payment ratio range than that of VaR _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., ). By adding 95 percent CVaR constraint to the Markowitz problem and specifying the expected benefit payment ratio

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

(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.

Table 8. Sample Statistics of Benefit Payment Ratios of Original and MV+CVaR Optimal 12-Line Portfolios

	Mean	Variance	Skewness	Kurtosis	Mode	CVaR 95percent
Original	0.9895	0.0027	−0.1069	−0.3555	0.9521	1.0865
MV+CVaR	0.9895	3.89E-06	0.5264	0.8525	0.9880	0.9940

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 Markowitz and MV+CVaR Optimized Portfolios

We write Equation (14) in the following general form:

(26)

In Equation (14), and 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 at the 95 percent CVaR of the Markowitz portfolio given the benefit payment ratio . This setting guarantees that the 95 percent CVaR of the MV+CVaR portfolio is no higher than that of its Markowitz counterpart. With specified as (26), the upper limit of 95 percent level CVaR constraint imposed on the traditional Markowitz problem changes with the risk reduction level . In practice, the insurer can choose a level of that fits its specific risk management needs. In our illustration, we choose the highest possible 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 _5percent–mean, and CVaR _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_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_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 () composed of both life insurance and annuity lines outperform those containing only life insurance (). Given the same means, the 12-Line efficient portfolios achieve lower variance, and lower CVaR _95percent than their 9-Line counterparts. Thus, inclusion of annuities reduces the potential mortality risk (measured by variance) and downside risk (measured by CVaR _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

A change in in (26) alters the upper limit of 95 percent level CVaR constraint . The bigger the , 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 . Specifically, we solve the MV+CVaR problem (6), subject to a 95 percent CVaR constraint with a different levels of , 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 . For the MV model, is set at 0. We show the percentage difference of 95 percent CVaR between MV and MV+CVaR optimal portfolios

(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 specified in (26) to the traditional Markowitz problem reduces the downside risk. In general, given a specified objective benefit payment ratio , the percentage difference (27) increases with . As increases, we impose a more stringent CVaR constraint so we observe lower downside risk of MV+CVaR portfolios and bigger difference between and .

Figure 16 shows the mean–variance–CVaR _95percent graphs for the 9-Line portfolios (left) and the 12-Line portfolios (right), respectively. In each graph, the CVaR _95percent of each MV+CVaR portfolio changes with the upper limit of the CVaR _95percent constraint , thus forming a three-dimensional surface. As for the MV portfolios, each optimal mean and variance relationship uniquely determines a CVaR _95percent. Thus, in each graph, the mean–variance–CVaR _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,

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.