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## Abstract

The uncertainty in life expectancy plays a critical role in individual financial planning. Its impact is magnified during the retirement years, also called the wealth distribution stage of the life-cycle, as new sources of income are typically not available to individuals. Utilizing a multi-stage stochastic program, we model and solve the optimal asset allocation problem of a retired couple with uncertain life expectancy in the presence of a term life insurance policy. In the base case, we find optimal policies assuming no longevity risk (i.e., lifetime scenarios are uncertain although life expectancy is fixed on the retirement date). Next, we introduce longevity risk in the scenario generation stage through either a shift in the expected lifetimes or an unexpected cut in periodic retirement income. We find that optimal asset allocation policy depends on the presence and the type of these risks as well as the relative price of insurance and the percentage cut in pension benefits.

Individual financial planning is one of the most complex problems that involve financial decision making under uncertainty. The popular models dealing with individual financial planning are the life-cycle model and asset–liability management (ALM) for individual investors. The life-cycle model was developed to find optimal consumption and savings decisions over an individual’s lifetime. An ALM model for individual investors finds an optimal financial decision to meet investor’s financial goals or liabilities. In individual financial planning, there are several uncertain parameters, such as asset returns, labor incomes, and investor’s lifetime. Among these, the investor’s uncertain lifetime plays a critical role in decision making. However, when the life-cycle model and the individual ALM were first developed, researchers assumed a fixed horizon instead of uncertain lifetime (Ramsey [1928]; Phelps [1962]; Berger and Mulvey [1998]; Consigli et al. [2012]).

Yaari [1965] was the first to propose optimal consumption and saving decisions under lifetime uncertainty in a life-cycle model. This model has been extended in several directions by including risky assets, life insurance, different types of annuities, labor income uncertainty, and family instead of individual as a decision maker in life-cycle model (Merton [1969]; Richard [1975]; Fischer [1973]; Campbell [1980]; Bodie et al. [2004]; Cocco, Gomes, and Maenhout [2005]; Pliska and Ye [2007]; Huang and Milevsky [2008]). In addition, Geyer, Hanke, and Weissensteiner [2009] generalized the life-cycle model of Richard [1975] using stochastic linear programming.

In an ALM setting, Medova et al. [2008] and Dempster and Medova [2011] presented models with lifetime uncertainty for long-term financial planning using dynamic stochastic programming. However, it is hard to investigate the effect of longevity risk that is generated by stochastic properties of future mortality rates on optimal financial decisions in these works. In other words, lifetime uncertainty does not necessarily imply longevity risk as the life expectancy can be a fixed parameter even though the lifetime is stochastic.

In individual financial planning, one should take longevity risk into account because it increases the likelihood of outliving an investor’s wealth by creating a shift in the lifetime probability distribution. Many studies present life-cycle models addressing the effect of longevity risk on individual’s financial decisions, such as consumption, savings, time of retirement, and investment, on asset classes including annuities and longevity bonds in a life-cycle model. De Nardi, French, and Jones [2009] showed the effect of the changes in life expectancy with respect to income, gender, and health on individual’s savings. Huang, Milevsky, and Salisbury [2012] extended Yaari’s life-cycle model by including a stochastic force of mortality to show how individual’s consumption decision changes under longevity risk. Conversely, there are very few studies about longevity risk using an ALM model. Among these few, Konicz and Mulvey [2013, 2015] presented a modeling approach similar to ours, although they assume that the individuals annuitize all of their savings upon retirement.

Various studies focus on the optimal demand of a variety of annuities, such as immediate, deferred, inflation-indexed annuities and group self-annuitization for longevity risk management (Schulze and Post [2010]; Post [2012]; Stevens [2009]; Hanewald, Piggott, and Sherris [2013]). Menoncin [2008] introduced longevity bonds into individual’s financial planning of optimal consumption and portfolio to satisfy market completeness condition under longevity risk. Cocco and Gomes [2012] found that individual investors increase savings and postpone their retirement date to hedge longevity risk. They also show the utility gains by investing on longevity bonds. Kim et al. [2013] developed a simulation-based model to incorporate longevity risk in individual financial planning by modeling the optimal portfolio of traditional asset classes along with life insurance and pension benefits. Spaenjers and Spira [2015] found empirical evidence of increasing investment on risky assets with respect to subjective lifetime using data from U.S. household survey. Although they assumed the point forecasts of remaining lifespan instead of stochastic force of mortality, they showed that individual investors consider their subjective planning horizon to make financial decisions.

As evident from these studies, longevity risk has a significant effect on individual financial planning. However, many studies have focused on the effect of longevity risk only on consumption, savings, and investment in annuities and longevity bonds instead of traditional asset allocation, such as stocks and bonds. One can argue that these longevity products may not necessarily be optimal for every individual and do not constitute a universal solution. In addition, with the financial technology (fintech) revolution underway, one may create traditional investment-based strategies that can be customized with respect to investor characteristics. Furthermore, as shown by Spaenjers and Spira [2015], subjective planning horizon is one of the factors affecting an individual’s financial decisions. Therefore, it is necessary to study how individual investors change their optimal asset allocation under longevity risk. Finally, as shown by Kim et al. [2013], it is not likely to find a generalized linear rule for optimal investment under longevity risk. Therefore, a systematic approach is required to address the longevity risk management problems.

In this article, we develop an asset–liability management model by including several risky assets and life insurance and utilize multistage stochastic programming to investigate the effect of longevity risk on optimal portfolio decisions. Taking Kim et al. [2013] as a starting point, we consider a retired couple who has a portfolio comprising several risky assets, life insurance, and pension plans to achieve the prespecified consumption plan with bequest motives. Due to the presence of bequest motives, annuities cannot be proposed as a simple solution to the couple’s problem, even though the pension income might be perceived as a fixed whole-life annuity. Because the investors have to meet liabilities in the future and need to invest on several risky assets, we develop our model using multistage stochastic programming (MSP). MSP is a widely used methodology to solve a multistage decision-making problem under uncertainty. It is easier to incorporate several uncertainties and such realistic features as taxes, transaction costs, and inflation, in financial decision making by using MSP. The effect of longevity risk on asset allocation is shown by changing several assumptions about life insurance and pension benefits. We find that this effect depends on how well an investor can hedge the risk by life insurance or pension plan.

The main contribution of this study is to show the relationship between a retired couple’s longevity risk and optimal asset allocation including life insurance. In addition, we contribute to the extant literature on retirement planning by extending Kim et al. [2013] in several dimensions. First, we propose an algorithm that can be used to convert their simulation-based scenarios to a scenario tree. Second, by using MSP, we are able to generalize the results obtained from their rule-based policy simulations. Third, we also address other practical considerations, such as the reduction of retirement benefits after the death of a spouse with pension plan.

There are several entities that are affected by the longevity risk in addition to the retirees. Pension plans, for example, have a short position on longevity risk. This means that their liability valuation increases as the life expectancy gets longer. Life insurance companies, for their part, have a long position on longevity risk because the value of their liabilities decrease as the life expectancy gets longer. It is possible to mitigate the longevity risk of pension plans and life insurance companies via properly designed financial securities, such as longevity bonds or longevity derivatives.

The retirees, conversely, face a different situation. Assuming that the surplus after their deaths does not have a significant value, change in life expectancy only plays a negative role. While pension plans could suffer from longevity risk, they could benefit if average lifespan decreases. Although insurance companies could benefit as average lifespan increases, they could suffer from shortened life expectancy. However, the investors do not benefit much from shorter life expectancies, because the surplus that is likely caused by the shortened life does not carry much value to them. Therefore, the investors are taking a short position on the longevity call option, as compared to the pensions, which are taking a short position on the longevity risk itself.

Another nontrivial issue is that the retirees cannot utilize the financial securities, such as longevity bonds or longevity derivatives, that are specifically designed to hedge these risks. Those securities are based on the average life expectancy of the whole population or specific groups, not for individuals. In other words, because retirees cannot use the “law of large numbers” principle, their problem related to longevity risk becomes more difficult than their counterparties, such as pension plans and insurance companies. Moreover, typical retirees do not have easy access to those securities due to various reasons such as lack of information, minimum required investment amount, margin requirements, and so on, even if those securities could be helpful to some extent.

The rest of the article is organized as follows. In the next section, we introduce the model and present the assumptions and the objective functions. The subsequent section provides the details of the scenario tree generation procedure, which is followed by numerical examples illustrating the benefits of our model. The final section presents the conclusions.

## MODEL

Assumptions of our model are similar to those of Kim et al. [2013], who showed that it is hard to find generalized linear rules for optimal investment under longevity risk. Therefore, in this article, we attempt to find the impact of longevity risk on optimal portfolios using a more systematic approach.

We consider a couple’s personal financial planning problem during their retirement years. We assume that the husband, who carries a pension plan, is about to retire. His pension benefits, which are the only source of periodic income for this couple, will be received until his death. In addition, they have a certain amount of savings accumulated prior to the retirement. The couple’s objective is to cover their annual living expenses, which are projected for the rest of their lives.

We define the target planning horizon in multiyear stages denoted by *T* = {1, …, τ + 1}. Even though the uncertainty is modeled in annual granularity, the maximum life expectancy is too long to be assessed in a scenario tree that branches annually due to the curse of dimensionality. Therefore, we aggregate years into stages in an uneven way as explained in the next two sections. We denote the length of each stage in years with *pr*_{t}, *t* ∈ {1, …, τ}. Investment, insurance premium, and borrowing decisions occur at the first instant of each time stage. For convenience, dividends and interest payments are reinvested in the originating asset class. For simplicity, we employ after-tax real dollar returns. Asset classes are defined by set *A* = {1, 2, …, *I*}, with asset class 1 representing cash. The remaining asset classes can include broad investment groupings, such as stocks, long-term government or corporate bonds, and foreign equity. The asset classes should track well-defined market segments. Ideally, the co-movements between pairs of asset returns would be relatively low so that diversification can be done across the asset classes.

The couple could decide how much to invest in life insurance at the beginning of each time stage. For practical reasons, we assume that the length of insurance contract in stage *t*, denoted by *J _{t}*, is equal to the length of that stage, that is,

*J*=

_{t}*pr*

_{t}. If the policy owner dies within this coverage period, they will receive a designated amount of benefits at the end of stage. This amount that the beneficiary receives, also called the face value of the contract, depends on a factor called conversion ratio (CR) as follows:

In reality, the actual CR calculated by the insurance companies depends on different population-specific factors, such as smoking history, current wealth, presence of chronic diseases (diabetes, high blood pressure, etc.), and location (car accident rates, crime rates, etc.). However, a “fair” CR can always be calculated by setting the expected total premium payment equal to the expected benefit based on the prevailing mortality rates. Naturally, CR is expected to decrease at every policy renewal as the remaining life expectancy gets shorter, so this calculation needs to be updated at the beginning of each stage. For a coverage period of *n* years, assuming the premium is paid at once at the beginning of the stage, we consider a policy owner who is currently *x* years old and has a mortality probability of *q _{x}* over the following year (i.e., between ages

*x*and

*x*+ 1). For an annual discount factor of

*v*, the expected present value of the benefit payment for one unit of premium can be written as:

As a result, the “fair” CR is calculated as 1/*M*_{x}. In order to obtain the actual CR that reflects reality as explained above, we employ a linear adjustment approach:

In our numerical examples, we conduct most of our tests by carrying out sensitivity analyses on the value of this adjustment factor.

We assume that the couple is rational as investors. Because they don’t value surplus after their deaths, they buy insurance only for the husband. However, if the wife dies before the husband does, they no longer invest in insurance. Thus, we assume that insurance investment continues until either member of the family dies. Regarding the investment assets, we assume a buy-and-hold strategy within each multiyear stage. In other words, rebalancing within the stage is not allowed. The surplus will be reinvested in the portfolio of several asset classes and insurance. Conversely, in case of a deficit (i.e., their wealth becomes negative), they borrow money at a fixed rate and make no further investment on the asset classes, implying that their wealth growth is 0. However, we assume that they continue to pay the insurance premium.

The couple’s objective is to maximize their expected utility function, which is a convex combination of expected final wealth (*Z*_{1}, a performance measure) and the expected sum of lifetime borrowing to meet the living expenses (*Z*_{2}, a risk measure) as follows:

The value of α indicates the couple’s risk preference, so that the large value of α means that the couple is aggressive. Conversely, the conservative couple has small value of α. In the simulation study of Kim et al. [2013], the expected utility formulation is similar, but *Z*_{1} is the expected final surplus and *Z*_{2} is the expected lifetime average shortage in consumption. To obtain linearity in the MSP formulation, we adopt different risk and reward functions in this study. It should also be noted that the discount factor is not needed in the objective function because we assume after-tax real dollars in our model.

As with most ALM models, uncertainty is depicted by a set of distinct realizations, called scenarios. The scenarios may reveal identical values for the uncertain quantities up to a certain period; that is, they share common information history up to that period by means of a scenario tree. There have been many successful efforts to generate representative scenarios for financial planning (Høyland and Wallace [2001]; Kouwenberg and Zenios [2006]; and Ziemba and Mulvey [1998]). The scenario tree generation procedure is explained in the next section while the MSP formulation is presented in the Appendix.

## SCENARIO TREE GENERATION

We consider two types of uncertainties: the couple’s lifetime and asset returns. Assuming that the realizations of the couple’s lifetime and asset returns are independent, we generate two separate scenario trees for these. We call two scenario trees of the lifetime and asset returns as marginal scenario trees.

For the marginal lifetime scenario trees, we make use of the mortality table from the National Vital Statistics Reports 2008. In order to improve the accuracy of the final scenario tree, we design the lifetime scenario tree to represent all possible cases of realization. For example, if the couple is alive at stage *t*, then there are four possible nodes on the scenario tree at stage *t* + 1. We can label these nodes as (husband, wife) = {(alive, alive), (alive, dead), (dead, alive), (dead, dead)}.

We then assign probabilities and values (i.e., remaining lifetime in this stage, ) for each node at stage *t* + 1. We assume that the couple is at the same age (denoted with *x*) in the beginning of stage *t* (whose length is *n* years). If the probability of dying between age *x* and *x* + 1 is denoted with for husband (wife), the probability of living until *x* + *n* (i.e., being alive for the next stage) is for husband (wife). As a result, the probability of dying before age *x* + *n* is for the husband (wife).

When the husband or the wife is alive until stage *t* + 1, then the node value for the husband or the wife’s lifetime is the length of stage *t*, that is, *n* years. If the husband or the wife dies between stage *t* and *t* + 1, the node of death has to contain information of all possible lifetimes, such as dying between age [*x*, *x* + 1], [*x* + 1, *x* + 2], …, [*x* + *n* – 1, *x* + *n*]. Therefore, we assign the weighted average value of all possible lifetimes to the node value. The possible cases of lifetimes and their probabilities are shown in Exhibit 1.

Finally, the node values in the marginal lifetime scenario tree are the weighted average of all possible lifetimes in the given stage, which can be calculated as follows:

After all these steps, probabilities on each node of the marginal lifetime scenario tree are computed as summarized in Exhibit 2.

In numerical analysis, we assume that there are two asset classes: stocks and bonds. Asset returns are generated using a two-regime model. Two regimes can occur: normal and crash. In order to be able to compare this MSP-based approach with the simulation-based one in Kim et al. [2013] and to generate a scenario tree with heavy tails of asset returns, we employ the clustering method that was introduced in Dupačová, Consigli, and Wallace [2000] and Gülpinar et al. [2004]. Gülpinar et al. [2004] presented two types of clustering methods: parallel and sequential. Among these two clustering methods, we use the sequential simulation method because our model has too many clustering constraints to apply the parallel method. If the parallel method is used, it is not likely to find a scenario tree satisfying clustering conditions.

We implement the following algorithm:

•

**Step 1.**Create a large one period scenario fan for asset returns with*N*nodes by using MC-simulation. Note that when the length of period is*t*(*t*> 1) years, we generate*t*annual returns and use the sum of these annual returns for clustering.•

**Step 2.**Randomly choose distinct seed scenarios. We group the remaining scenarios by these seeds.•

**Step 3.**Group each rest scenario with the seed scenario that is the most similar to the rest scenario. We use the Euclidean distance to compare the similarity between scenarios. The clusters should satisfy the following conditions.•

*Condition 1*. In the same cluster, the scenarios should have the same regime. Therefore, the cluster whose seed is a normal regime should have scenarios with a normal regime.•

*Condition 2*. The relative sizes of the clusters should not be extreme. We define that a ratio of largest/smallest cluster size should be smaller than the fixed ratio.•

*Condition 3*. After finding the representative value of each cluster, we should check for an arbitrage opportunity. The existence of the first and the second type of arbitrage opportunities are tested using the optimization method introduced in Klaassen [2002].

•

**Step 4.**Repeat steps 2 and 3 for*M*times to find optimal clusters having minimum distance between scenarios in each cluster.•

**Step 5.**After finding the optimal clusters in step 4, find the representative value of each cluster. We employ an average value of scenarios in the cluster. Note that we assign the annualized returns to node values.•

**Step 6.**Repeat steps 1 to 5 at every stage and at every node.

In our numerical examples, we implement the scenario tree procedure in the following structure. We divide the planning horizon into four uneven stages of length 5, 5, 10, and 20 years. Branching for marginal scenario tree of asset returns is 40 × 10 × 5 × 5 = 10,000 scenarios (assuming a symmetric tree for asset returns). The major clustering parameters for the marginal asset return scenario tree are given in Exhibit 3.

Two marginal trees are combined by the tensor product to generate a whole scenario tree as shown in Exhibit 4 (Hochreiter, Pflug, and Paulsen [2007]). The number of scenarios of marginal lifetime scenario tree is 25. Therefore, the combined tree has 10,000 × 25 = 250,000 scenarios.

## NUMERICAL EXAMPLES AND RESULTS

We solve our retirement model under different sets of assumptions to show the effect of longevity risk on the retiree’s optimal portfolio. In the base case, we simply adopt the following assumptions. The couple is both 60 years old. They can live up to 100 years old. Therefore, the maximum horizon of this problem is 40 years. The remaining lives of the husband and the wife are assumed to be independent. The life table in National Vital Statistics Reports 2008 is used to generate the couple’s lifetime without longevity risk. Annual projected expenses and annual pension benefit are both assumed to be $50 thousand. Their initial savings are $400 thousand. For pricing the insurance policy, we assume that CR adjustment factor is 0.8.

The couple can invest in two asset classes: bonds and stocks. Asset returns are generated using the two-regime model. Two regimes can occur: normal and crash (Exhibit 5). Although cash is not included in this numerical analysis, we assume that the cash flows occur in the bond account.

Based on these regimes, transition probability matrix is [0.9, 0.1; 0.4, 0.6] and the stationary distribution for normal regime is 0.8. Thus, the crash regime comes once every five years. Unconditional expected returns are 7% and 4.5% for normal and crash regimes, respectively. Finally, we assume that the annual rate of borrowing is 8%. To improve the accuracy of solutions, we generate 20 different asset return scenario trees and solve the MSP with these scenario trees. The following results are average values of these 20 solutions.

We first report the results of the base case, where the longevity risk is not present (i.e., the couple’s life expectancy follows the mortality table in the National Vital Statistics Reports 2008). For different levels of risk aversion, optimal expected utility values are displayed in Exhibit 6 as well as optimal investment in asset classes and insurance.

The expected utility and the investment in stocks increase for more risk-seeking couples. These aggressive couples optimally invest less in bonds. They also allocate less to insurance, however, this difference is insignificant compared with changes in asset allocation. Because the couple can hedge the lifetime uncertainty by purchasing an insurance policy, they maintain almost the same amount of insurance in their portfolio at all levels of risk aversion.

We solve the model under the following α values: α = 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.02, 0.03, 0.04, 0.05. Because there are two patterns of the effect of longevity risk on optimal portfolios, under small α and under large α, we show the optimization results for risk-averse investors with α = 0.0005 and risk-seeking couples with α = 0.02.

### Impact of Longevity Risk

The previous example illustrates the optimal investment policy for the couple when we assume that their expected lifespans are correct. The truth is, however, that the expected lifespans change over time. The historical data suggest that the couple is likely to live longer on average than the original estimates. In this subsection, we examine the effects of the changes in the remaining life parameters. We vary the life expectancy of husband within the interval [17, 27] years and that of the wife in the interval [20, 30] years. Exhibit 7 displays how optimal expected utility changes in this new setup. It should be noted that the longevity risk is induced by shorter life expectation for husband and/or longer life expectation for wife (left-most corner of the charts). Under the optimal policy, the expected utility decreases as the longevity risk increases (i.e., life expectation for husband decreases and/or life expectation for wife increases).

Optimal allocations to bonds and stocks are displayed in Exhibit 8 and Exhibit 9, respectively. The investment in bonds decreases as the couple becomes more aggressive. When the couple is conservative (α ≤ 0.001), there is no significant change of investment in bonds and stocks with respect to the couple’s lifespan. Although bonds have lower returns than stocks, the returns are more stable. Therefore, higher weight on bonds gives more stable returns to couple. If the couple is conservative, they only want to minimize borrowing. This goal can be achieved by investing in assets with stable returns. Therefore, the couple does not change the investment policy with respect to the change of their lifespans. When the couple is aggressive (α ≥ 0.005), the investment in bonds increases as the longevity risk increases (expected life of husband ↑, wife ↓). The investment in stocks decreases as the longevity risk increases. If the longevity risk increases, the wife should live without any income after husband’s early death. Therefore, the couple chooses a more stable investment by betting more on bonds. Also, the change of investment in bonds and stocks is more sensitive to the change of husband’s lifespan than the change of wife’s lifespan. The reason is that the only income of the couple is husband’s pension benefit depending on the husband’s lifespan.

Optimal investment in insurance policy as a percentage of the overall wealth is displayed in Exhibit 10. The investment in insurance decreases as the couple becomes more aggressive. However, the amount of decrease is smaller than the change of other assets, stocks, and bonds. Also, the change of investment in insurance with respect to the couple’s risk preference becomes smaller as the longevity risk increases. Insurance is an asset that can hedge the risk caused by the husband’s early death. Therefore, the change of investment in insurance is small although the couple becomes more aggressive. When the couple is conservative (α ≤ 0.001), there is no significant change of investment on insurance with respect to the couple’s lifespan. If the couple is conservative, they focus on the minimization of borrowing. The sum of borrowing (not the expected sum of borrowing) does not depend on the couple’s lifespan. Therefore, the couple does not change the investment on insurance even though they live longer. For aggressive couples (α ≥ 0.005), the investment in insurance increases as the longevity risk increases. The life expectation parameter for the husband is the dominant factor. If the husband lives longer, the risk of borrowing and the income from the insurance go down. Therefore, the couple reduces the holdings of insurance as the longevity risk increases.

In this subsection, we have presented how the couple changes their portfolio with respect to the change of their expected lifespan. In our model, the hedging tools to reduce the longevity risk are the husband’s pension plan and the life insurance. Couple can achieve their consumption goal by the sustainable pension benefit until the husband is alive. Also, after the death of the husband, the wife can bear the consumption by the insurance claim although there is no income. Therefore, the pension benefit and the insurance are important assets to the couple to reduce the longevity risk. In the following two subsections, we show the change of optimal portfolios with respect to the change of lifespan when the couple cannot fully hedge their longevity risk with the insurance or the pension benefits.

### Impact of Availability and Price of Life Insurance

To demonstrate the effect of insurance on the couple’s optimal investment, we test the following two cases: 1) when the insurance becomes expensive and 2) when the couple cannot invest in the insurance. In the following four exhibits in this subsection, the left-hand chart repeats the results in the previous subsection, the middle chart corresponds to expensive insurance (with a lower CR), and the right-hand chart corresponds to the absence of insurance. Assuming the source of longevity risk is the husband, Exhibit 11 and Exhibit 12 show the results for conservative and aggressive couples, respectively. Exhibit 13 and Exhibit 14 do the same for the case of wife as the source of longevity risk.

As explained earlier, when the CR adjustment factor decreases, the insurance policy becomes more expensive. In the previous subsection, we assumed the adjustment factor as 0.8. We change the adjustment factor from 0.8 to 0.6 to show how the couple responds to more expensive insurance. Although the insurance becomes more expensive, the pattern of the optimal portfolio change with respect to the change of lifespan is same as the result in the previous subsection.

When the couple is conservative (α ≤ 0.001), there is no significant change of investment in stocks, bonds, and insurance with respect to the couple’s lifespan. When the couple is aggressive (α ≥ 0.005), as the longevity risk increases, the couple invests more in bonds and insurance and less in stocks. Also, the change of husband’s lifespan is a dominant factor.

As the price of insurance increases, however, expected utility diminishes, the couple invests more in stocks, and less in bonds. Even though the insurance becomes expensive, the couple does not reduce the investment on insurance. Because the couple can hedge the longevity risk with insurance, they maintain their investment policy on insurance. The income from insurance is not enough to achieve the couple’s consumption, however, so they invest more in stocks to get higher returns.

Next, we assume that the couple cannot purchase insurance. Therefore, the wife should bear the consumption without any income after the husband’s death. When the couple cannot invest in insurance, their expected utility is reduced and the effect of longevity risk on the couple’s optimal investment is changed. When the couple is conservative (α ≤ 0.001), as the longevity risk increases, the couple invests more in stocks and less in bonds. It is the opposite result of the optimization allowing investment on insurance. The role of insurance is to reduce the possibility of the shortage of consumption after the husband’s death. Therefore, the wife does not need to take risk to have high returns by betting heavily on stocks if she can hedge the longevity risk by insurance. However, if the investment on insurance is not allowed, the couple should protect themselves from the longevity risk by investing only in stocks and bonds. Therefore, as the longevity risk increases, the couple seeks for higher returns by heavily betting on stocks. When the couple is aggressive (α ≥ 0.005), they invest their whole wealth only in stocks regardless of their expected lifespan. As the couple becomes aggressive, they do not worry about the shortage of consumption and seek higher final wealth by betting their whole savings on stocks.

### Impact of Changes in Pension Benefits

Pension benefit is the only income that helps the couple in meeting the consumption targets without any shortage as long as the husband is alive. Many pension plans have recently decided to raise the contributions or to cut down the pension benefits to reduce their exposure to longevity risk. However, some pension plans in Europe continue giving a percentage of pension benefits to the wife after the husband’s death. Therefore, we investigate the impact of pension benefits in optimal investment decisions through two cases. In the first case, the pension benefits are reduced by 20% and are therefore insufficient to meet the living expenses. In the second case, the wife keeps receiving the pension benefits after the husband’s death with a 50% cut. As in the previous subsection, the left-hand chart in Exhibits 15–18 repeats the base case results in the presence of longevity risk. The middle chart corresponds to the case of benefit cuts, and the right-hand chart is when the wife continues to receive reduced benefits. Assuming the source of longevity risk is the husband, Exhibit 15 and Exhibit 16 show the results for conservative and aggressive couples, respectively. Exhibit 17 and Exhibit 18 do the same for the case of the wife as the source of longevity risk.

If the pension benefits are reduced, expected utility goes down as well as investment in bonds and insurance. Stock investment, conversely, increases. Because the pension benefits cannot fully cover the couple’s annual living expenses, there exists a possibility of consumption shortage even when the husband is alive. Therefore, to hedge the risk of borrowing before the husband’s death, the couple seeks higher returns by investing more in stocks and less in bonds and insurance. For all conservative and aggressive couples, as the longevity risk increases, the optimal strategy becomes more aggressive. It is the opposite result of the optimization in which the couple can achieve their consumption goal by pension benefits. If the pension benefits can cover the couple’s consumption, they do not need to consider the consumption shortage until the husband’s death. When the benefit cut occurs, however, the couple should use their savings for consumption. Therefore, the wealth after the husband’s death becomes smaller than the wealth without benefit cut. Therefore, the longevity risk becomes too significant to be hedged by bonds, so the couple attempts to reduce the shortage by betting more on stocks.

Although the wife receives the pension benefits on behalf of the husband after his death, the effect of longevity risk on the optimal portfolios does not change. When the couple can hedge more the longevity risk by the pension plan, expected utility increases, portfolio becomes more aggressive, and the investment in insurance goes down. Because the wife has income after the husband’s death through his pension benefits, she does not need to invest heavily in bonds to get stable returns. She does not also need large insurance claims anymore, so she decreases the investment in insurance. When the couple is conservative (α ≤ 0.0005), there is no significant change of investment in stocks, bonds, and insurance with respect to the couple’s expected lifespan. When the couple is aggressive (α ≥ 0.001), as the longevity risk increases, the couple invests more in bonds and insurance and less in stocks. After the husband’s death, the wife can bear almost all consumption with his pension benefits and insurance claims. Therefore, she does not need large returns from optimal portfolios to remove the possibility of the shortage of consumption. Thus, as the longevity risk increases, she seeks for stable returns by betting more on bonds despite getting small returns. Also, we can observe that the variability across optimal portfolios with respect to expected lifespans is smaller than the case with the assumption that only husband receives pension benefits.

### Summary of Results

The composition of the optimal portfolio with respect to longevity risk depends on how well the couple can hedge this risk using pension benefits or life insurance (Exhibit 19). If the couple is conservative (lower bequest motive) and the longevity risk hedge is effective, optimal portfolios are stable. If the couple is aggressive (higher bequest motive), they increase the purchase of life insurance and the portfolio becomes less risky as longevity risk increases. If the couple cannot effectively hedge their longevity risk, the proportion of life insurance increases and the portfolio becomes riskier as longevity risk increases.

## CONCLUSIONS

In this article, we develop an asset–liability management model for individual investors and solve it via the methodology of multistage stochastic programming to investigate the effect of longevity risk on the optimal portfolio decisions during the retirement stage of the life cycle. To illustrate the model, we consider a recently retired couple who carries a portfolio comprising several risky assets, term life insurance, and a pension plan in order to achieve a pre-specified consumption plan with bequest motives. We assume that the husband is the sole pension plan beneficiary and introduce longevity risk through an increased life expectancy for wife and/or a decreased life expectancy for the husband. We then solve this model under five different assumptions to analyze the effect of longevity risk on the retiree’s optimal portfolio. We find that this effect depends on how well investor can hedge the risk by life insurance or pension plan. In other words, whether or not the longevity risk changes the expected utility depends on the availability and price of the life insurance and the uncertainty around the pension benefits. Furthermore, we show that the benefits provided by life insurance are minimal for a couple with higher risk aversion and lower bequest motive.

We contribute to the literature by providing a systematic analysis of the relationship between a retired couple’s longevity risk and optimal asset allocation. Although a significant portion of the literature focuses on the use of annuities as the investment solution for this final stage of the life cycle, we take the opposite route partly due to the recent popularity of the robo-advising segment of the fintech industry. In this respect, our recommendations are not addressed at the individuals but to those fintech companies who propose retirement products that can be mass-customized and managed in traditional asset classes. The novelties of our model lie in the scenario generation procedure as well as the usage of multistage stochastic programming in a setting that integrates insurance decisions into the asset allocation problem.

## APPENDIX

This technical appendix explains the formulation of multi-stage stochastic program used in this article. For each *i* ∈ *A*, *t* ∈ *T*, and *s* ∈ *S*, we define the following parameters and decision variables:

### Parameters

### Decision Variables

We present the deterministic equivalent of the stochastic program in terms of these parameters and variables.

### Model MSP

A-1Subject to

A-2 A-3 A-4 A-5 A-6 A-7 A-8The couple’s objective in Equation A-1 is to maximize their expected utility function, which is a convex combination of expected final wealth (*Z*_{1}) and the expected sum of lifetime borrowing to meet the living expenses (*Z*_{2}) as follows:

The value of α indicates the couple’s risk preference, so that the large value of α means that the couple is aggressive. Conversely, the conservative couple has small value of α.

Cash flow balance constraint for non-cash asset classes are given in constraint A-2 whereas the cash account is tracked in constraint A-3. Cash account shows inflows due to sale of other assets, pension benefit if the husband is alive at stage *t* under scenario s, insurance premium if the husband died between stage *t* − 1 and *t* under scenario *s*, and borrowing if the total asset value becomes negative without borrowing. Outflows are due to purchase of other assets, living expenses, insurance premium if husband and wife are both alive at stage *t* under scenario s, and debt payment at stage *t* − 1 under scenario s. Cash flow balance constraints are modified for the final stage as in constraints A-4 and A-5. Constraint A-7 sets an upper bound on the insurance investment. The fact that the insurance premium will no longer be paid if one of the couple dies is handled through constraint A-8. Finally, constraints A-6 are the non-anticipativity constraints typically required due to the tree structure of stochastic programs. All decision variables are non-negative except for . It should also be noted that the constraints do not have to be satisfied after both husband and wife die before the end of horizon τ. Also, if the couple dies between stage *t* and *t* + 1, liquidation occurs at stage *t* + 1.

A generalized network graph of the model appears in Exhibit A1. This graph depicts, for a given scenario, cash flows across time (Mulvey et al. [2008]). While not all constraints can be incorporated, the graphical form allows readers to readily comprehend the model’s structure.

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