The Impact of Longevity Risk on the Optimal Contribution Rate and Asset Allocation for Defined Contribution Pension Plans

Defined contribution (DC) pension plans have become popular in the global pension market because recent increases in longevity have increased pressures on defined benefit (DB) pension plan providers. The main difference between DB and DC pension plans pertains to the way they treat financial risk. In a DC plan, the member bears the financial risk, so whether that individual receives a good pension depends on his or her investment strategy. In contrast, for a DB plan, the retirement benefit is promised in advance according to some predetermined formulae. Thus, employers bear the risks of poor investment performance by the pension fund. The study of the asset allocation problem for DC pension plans has emerged as an important research topic.

Investment strategies to address financial risk in DC plans have been widely explored.¹ Blake et al.² study asset risk and estimate values-at-risk in the accumulation phase of DC pension plans by examining a range of asset-return models. Vigna and Haberman³ and Haberman and Vigna⁴ both study asset risk, and Haberman and Vigna⁵ further consider downside risk in the asset allocation strategy. Battocchi and Menoncin⁶ note two background risks: salary risk and inflation risk in pension management. Maurer et al.⁷ also study optimal life cycle strategies in the presence of interest rate and inflation risk under a stochastic framework. These studies all investigate the investment strategy during the accumulation phase and consider financial risks including asset risk, salary risk, inflation risk and interest rate risk. Other research pertaining to the decumulation period focus on the choice or time of annuitisation and thus neglect longevity risk.⁸

However, other than financial risk, longevity entails the greatest risk faced by individual pensioners in DC plans, because life expectancy is dramatically increasing.⁹ Willets et al.¹⁰ analyse changes in mortality and find that life expectancy at retirement in the U.K. is likely to increase rapidly in the 21st century. For a DC plan with the benefit paid in the form of an annuity, the benefits provided depend on not only the investment strategy but also the price of the annuity at retirement. Longevity risk is the risk of future uncertainty with a long-term trend of mortality improvement, which will lead to a low conversion rate used to buy the annuity at retirement. For the retiree, longevity risk may involve losing a financial resource because the retirement benefit might not be sufficient to last for the rest of his or her life.

Dealing with the uncertainty of future mortality trends in a pension fund or annuity products thus constitutes an important issue and the purpose of this work. The study of longevity risk and asset allocation has not received the same focus as financial risk, with a few exceptions.¹¹ Hainaut and Devolder¹² address mortality and financial risks for DB pension plans instead of DC plans, and Charupat and Milevsky¹³ analyse the interaction between financial risk and mortality risk and derive the optimal asset allocation in the pay-out phase for a variable annuity contract. Battocchio et al.¹⁴ study the asset allocation under mortality risk during the accumulation and decumulation phases. They assume a Gompertz-Makeham mortality model to represent the remaining lifetime of the member.

Unlike Charupat and Milevsky¹⁵ or Battocchio et al.,¹⁶ we consider a more realistic mortality experience to illustrate the impact of longevity risk on the optimal contribution rate and asset allocation for a DC plan. This investigation of longevity risk uses the U.K. mortality experience for pensioners. In addition to longevity risk, we consider asset risk, salary risk and inflation risk, and provide a stochastic framework to determine the optimal asset allocation and contribution rates. The optimal asset allocation depends on the asset liability matching criteria. The liability is determined by the pension fund target, measured using the income replacement ratio (RR). We set the objective function for the individual to minimise the variance of the simulated sample path with the asset liability matching criteria. In other words, we minimise the risk that a person will be unable to reach his or her retirement target.

In the U.K., increased longevity has been dramatic in recent decades and caused severe financial problems for pension fund providers and annuity providers, such as Equitable Life, that issue guaranteed annuity options.^17,18 In practice, the Continuous Mortality Investigation Bureau (CMIB), which has responsibility for the production and publication of standard mortality tables in the U.K., uses reduction factors to project future mortality.¹⁹ The actuary prices the annuity value according to the projected mortality table. However, there may be a risk that the mortality projections turn out to be wrong, and modelling uncertainty associated with future mortality trends has become very important for the retirement system. In addition, CMIB²⁰ conducted a follow-up study to investigate the stochastic projection methodologies of the Lee-Carter model, whereas Lee²¹ and Yang²² set out stochastic mortality rates using Kalman filter-like processes based on the projected mortality rates to analyse the problem of guarantee annuity options. Other stochastic mortality models have been widely explored.²³ Pitacco²⁴ describes some aspects of the development of survival modelling in actuarial mathematics, with a special emphasis on issues pertaining to longevity risk. Cairns et al.²⁵ in particular demonstrate that the inclusion of a cohort effect can provide a better fit, using data from England, Wales and the U.S.

This study of the impact of longevity risk on asset allocation strategies for a DC pension plan considers the investment strategy during the accumulation phase that employees use to deal with longevity risk during the decumulation phase. We consider four assets in the asset allocation and observe four types of changes to the rebalancing strategies. To assess an investment strategy, we measure the tracking error. The asset dynamics are modelled using Wilkie's²⁶ investment model. We investigate the effects of longevity risk with three patterns of mortality: base mortality rate, projected mortality rate and stochastic mortality rate. The base rate and projected mortality derive from the recent CMIB's mortality tables for the pensioners. Finally, we incorporate Lee's²⁷ and Yang's²⁸ models to project the future stochastic mortality rate and model longevity risk.

The remainder of this article is organised as follows: the following section describes the model settings for a DC pension plan. We present the investment strategies and objective functions for finding the optimal asset allocation in the subsequent section then explain how to model longevity risk in the section following it. In the penultimate section, we compute the optimal contribution rate and asset allocation for three patterns of mortality. We conclude and suggest future research directions in the final section.

We design a DC pension plan that includes two phases: an accumulation phase and a decumulation phase. In the accumulation phase, the employee contributes a constant proportion of his or her salary to a personal pension fund. In the decumulation phase, the pension fund gets converted into an annuity at the time of retirement.

Accumulation phase

Assume that an employee joins the pension scheme at age x and retires at age x+n. The employee pays contributions yearly in advance, equal to a fixed percentage of his or her salary. The level of the employee’s pension fund at retirement depends on factors such as the level of contribution, age at entry, retirement age and investment earnings. Let F(t) denote the level of the pension fund at time t, expressed as follows:

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where C(t) is the contribution at time t, such that C(t)=c%·S_x+t−1, c% is the constant contribution rate, S_x+t is the annual salary at age x+t, p _i (t) is the proportion of funds invested in the ith asset during year [t, t+1], r _i (t) is the investment return of the ith asset in year [t, t+1], and k is the number of assets.

Decumulation phase

In the decumulation phase, the pension fund at the time of retirement is converted into a life annuity. According to the equivalence principle, the amount of annuity is calculated as:

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where ä_x+n is the value of an annuity of one per annum payable annually in advance for life, after the member attains retirement age x+n. The value of ä_x+n is determined by the assumptions of interest rates and mortality rates.

Pension fund target

We attempt to find the optimal asset allocation and contribution rate of a DC pension fund to meet the individual’s pension fund target and study the impact of longevity risk on the optimal asset allocation and contribution rate. The pension fund target is measured by the income RR, defined as the ratio of the annuity payment to the final salary for a person joining the pension plan at age x and retiring at age x+n, or

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where S_x+n−1 is the final year’s salary before retirement.

The pension fund target represents the percentage of income required to maintain a desired standard of living for the person during his or her retirement years. According to the pension fund target, we can calculate the corresponding required pension fund at retirement f̄(n) as follows:

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Therefore, we attempt to find the optimal asset allocation to meet a specific amount of f̄(n). To study the effect of longevity risk, we reflect the dynamics of mortality to calculate the annuity factor and the corresponding required pension fund.

To understand the impact of longevity risk on the contribution rate and asset allocation, we employ actuarial mortality experiences for pensioners from the U.K. According to the CMIB's mortality tables for the pensioners, we consider three patterns of mortality: base, projected and stochastic mortality rates. The base mortality rate represents the expected assumptions and does not consider the effect of mortality improvements, whereas the projected and stochastic mortality rates include the effect of mortality improvements. We use the stochastic mortality rate to investigate the longevity risk. In this research, the base mortality rates are demonstrated using the most recent mortality table for pensioners of PMA92.³⁷ The projected mortality rates are modelled using reduction factors based on CMIB,³⁸ which is denoted as PMA92C2020.³⁹ For the stochastic rates, we consider the uncertainty in modelling the reduction factor and use the Kalman filter-like process proposed by Lee⁴⁰ and Yang.⁴¹ The methodology for the construction of the base mortality rate (PMA92), projected mortality rates (PMA92C2020) and stochastic mortality rates appear next.

Base mortality rates

The methodology to calculate the base mortality rates (q _x ) for pensioners in CMIB is as follows:

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where μ _x and p _x denote the force of mortality and the survival probability for a person aged x, respectively. According to the PMA92 mortality table CMIB,⁴² μ _x can be calculated as:

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Projected mortality rates

In U.K. actuarial practice, the CMIB uses the reduction factor to model the future mortality rate. Thus, the base table must be calculated first, then applied to the improvement factor to obtain the projected mortality rates, given by:

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where q _x denotes the mortality rate for age x according to the base table, q _x,t is the projected probability of death within 1 year for a person of age x at time t, and RF _x,t is the projected mortality improvement factor for age x at time t. The CMIB's⁴³ calculation of the reduction factors is based on a 20-year time span, according to the mortality experience of 1991–1994,⁴⁴ which is

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where

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and

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The corresponding parameters are c=0.13, h=0.55 and k=0.29.

In this research, we use the reduction factor to project mortality rates for the calendar year 2020 from the PMA92 tables, which is denoted as PMA92C2020. In Figure 1, we depict the survival probability trend for pensioners aged 65 years, according to the a(55),⁴⁵ PA90M, PMA80C2010 and PMA92C2020 mortality tables. The most recent table, PMA92C2020, reveals a significant improvement over the a(55) table, which provides an upper bound on the mortality improvements. In addition, the significant projection changes between the 1967 and 1970 mortality rates and that based on 1991 and 1994 data indicates that even the best-constructed forecasts may prove wrong. To deal with this issue, we might allow future mortality rates to be random. We employ the Lee and Yang model, which we discuss next.

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Lee and Yang's stochastic mortality model

Due to the problem of the deterministic mortality table, Lee⁴⁶ and Yang⁴⁷ derive stochastic mortality rates to model actual mortality rates, given the expected mortality rates from the projected tables. Specifically, Lee⁴⁸ develops a feasible stochastic mortality model, and Yang⁴⁹ modifies and uses that model to analyse a guaranteed annuity option. Cairns et al.⁵⁰ name this stochastic mortality model the Lee and Yang model.

The Lee and Yang model assumes two random variables: X(t) and Y(t). X(t) captures the permanent changes in mortality and performs a random walk with a zero mean, whereas Y(t) describes the transient changes in mortality in each year and depends on X(t). That is,

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where Z _X (t) and Z _Y (t) are independent and follow ND(0,σ _X ²) and ND(0,σ _Y ²), respectively. In the Lee and Yang model, the actual mortality rates (q̂ _x,t ) for a person aged x in year t are assumed to be

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This assumption makes the projected mortality rate (q _x,t ) an unbiased estimate of the future actual mortality rate (q̂ _x,t ), which is provided by CMIB. To determine the impact of longevity risk on the optimal contribution rate and asset allocation, we calculate the value of a life annuity under a stochastic mortality rate environment to reflect the level of pension funds needed to meet the pension fund target.

In the numerical observation, we investigate the investment strategy in a DC pension plan and the impact of longevity risk on optimal asset allocations and optimal contribution rates. The asset allocation suggested by the base mortality rate represents the benchmark and does not consider the effect of mortality improvements, whereas the asset allocations that rely on projected and stochastic mortality rates include mortality improvements. We use the stochastic mortality rate to investigate the longevity risk. The parameters used for the stochastic mortality model are those estimated by Yang,⁵¹ namely σ _x =0.044 and σ _y =0. We also consider the dynamics of asset returns and the salary growth rate using Wilkie's investment model; the parameters are those estimated by Wilkie⁵² (see ). In the following investigation, we set 80 per cent of the income RR as the pension fund target. Assume the employee starts working at age 25 and retires at age 65 for the base illustration case. The initial yearly salary is one unit and the salary growth rate follows the setting of Wilkie’s model.

Impact of longevity risk on optimal asset allocation

In this subsection, we consider how to change the optimal asset allocation to deal with longevity risk when the contribution rate is predetermined.⁵⁸ To investigate the effect of longevity, we set the same contribution rate of 16.55 per cent⁵⁹ but with different mortality assumptions. Figure 2 presents the optimal asset allocations with different mortality assumptions for investment strategy 1 during the accumulation period (40 years). The optimal asset allocation changes from the base to projecting the stochastic mortality rate and suggests putting more money in riskier assets. For example, the proportion invested in stocks obviously increases with the stochastic mortality assumption. Thus, the individual hedges life uncertainty with riskier assets. The effects of investment strategy 1 also apply to investment strategies 2–4. For simplicity, we do not repeat the figures for these investment strategies.

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In addition to the impact on optimal asset allocations, we address the tracking error of different investment strategies in Figure 3. When considering the longevity risk in asset allocation, employees become more aggressive, which results in larger tracking error. According to the stochastic mortality table, the tracking error increases from 963.0 to 2,208.64 for investment strategy 1, from 1,011.7 to 2,207.58 for investment strategy 2, from 1,055.9 to 2,242.1 for investment strategy 3 and from 1,097.6 to 2,306.3 for investment strategy 4.

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Impact of the working period on the results

In the previous investigation, we assumed the employee started working at age 25 and retired at age 65, which means the working period was of 40 years. In this section, we assume the employee starts working at age 45 and retires at age 65 to assess the impact of a shorter working period (i.e. 20 years).

The corresponding results of the contribution rates for employees starting at age 45 appear in Figure 4. The contribution rate increases from 16.55 per cent to 40.70 per cent according to the base mortality rates, from 19.76 per cent to 46.48 per cent with the forecasting mortality rates, and from 24.26 per cent to 51.92 per cent under the stochastic mortality assumption. These results are intuitive: the employee must raise the contribution rate to meet the pension retirement target faster for a shorter working period.

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The impacts of the working period on the optimal asset allocation appear in Figure 5. Similar to our preceding conclusion, the employee increases the contribution rate to accelerate the pension fund accumulation to meet the target faster. In addition, the optimal asset allocation for a shorter working period with the different mortality assumptions increases the proportion invested in stocks compared with results for a 40-year duration.

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Asset allocation for DC pension plans is critical to employees, because the pension they receive depends entirely on investment performance. For example, the recent global financial crisis had great impacts on employees with DC pension plans, especially those approaching retirement age. The crisis has highlighted the dangers of leaving pension money invested freely in financial markets. Thus, the asset allocation to deal with the financial risk should be considered for employees in the DC pension plan.

Significant literature has addressed financial risk, including asset, salary and inflation risks in the asset allocation problem. However, longevity risk has become ever more important due to increasing life expectancy trends worldwide. The effect of longevity risk is systematic, so it has caused significant financial pressures on pension funds and annuity providers. This pressure can explain the dramatic increase in DC pension plans in recent years. For the employee, longevity risk might mean running out of available resources during retirement. In recent years, research on the asset allocation problem has gained greater attention from researchers who investigate the interaction between longevity risk and asset allocation for pension funds.⁶⁰

In this research, in addition to asset, salary and inflation risk, we consider longevity risk in the asset allocation problem. Unlike current literature, we consider a more realistic mortality experience to study the impact of longevity risk. We also demonstrate the longevity risk using actuarial mortality experience for pensioners from the U.K. The effect of longevity risk is illustrated with three patterns: base, projected and stochastic mortality rates, according to the structure of CMIB mortality tables for pensioners. We develop a stochastic framework that includes asset, salary and mortality dynamics to determine the optimal asset allocation and contribution rates and to study the effect of longevity risk. The objective function meets the employee's income RR, which enables us to minimise the variance of the simulated sample path with the asset liability matching criteria. We consider four assets in the asset allocation and observe four rebalancing strategies.

The results show that longevity risk can be hedged by either raising the contribution rate or setting more aggressive asset allocations. With the objective function to minimise the variance with the asset liability matching criteria, we find that the employee shows a life cycle investment strategy. To compare the performance of different investment strategies, more frequent rebalancing can improve target liability matching. These findings apply both when we consider and when we ignore longevity risk in the asset allocation. However, the effect is more significant when we consider longevity risk. Specifically, when we take longevity risk into account, the asset allocation is more aggressive when we do not increase the contribution rate.

This article therefore responds to two issues in DC pension research. First, we propose a stochastic framework that considers future uncertainty to find the optimal asset allocation. Second, the proposed life cycle investment strategy can protect the employee from both financial risk and longevity risk when approaching retirement in the DC plan. In addition, our research might be extended in several ways. First, other objective functions could be investigated, or the employee's attitude toward risk might be taken into account. Second, the development of stochastic mortality models is very important to deal with longevity risk. A wide range of mortality models have been proposed and discussed.⁶¹ The effect of model risk on the result is worth determining. Third, we note that the methodology of analysing longevity risk is applicable to other life annuity products and DB pension plans, so further research should attempt such applications.