Retirement Spending on Planet Vulcan: Longevity Risk and Withdrawal Rates
by Moshe A. Milevsky and Huaxiong Huang
Recommendations from the media and financial planners regarding retirement spending rates deviate considerably from utility maximization models (models that assume consumers optimize how they spend money).
We argue that wealth managers should advocate dynamic spending in proportion to survival probabilities, adjusted up for exogenous pension income and down for longevity risk aversion.
We conducted a study to attempt to derive, analyze, and explain the optimal retirement spending policy for a utility-maximizing consumer facing) an indeterminate lifetime. We deliberately ignored financial market risk by assuming that all investment assets are allocated to risk-free bonds (e.g., Treasury Inflation-Protected Securities [TIPS]). We made this simplifying assumption in order to focus attention on the role of longevity risk aversion in determining optimal consumption and spending rates during a retirement period of indeterminate length.
By longevity risk aversion, we mean that different people might have different attitudes toward the “fear” of living longer than anticipated and possibly depleting their financial resources. Some might respond to this economic risk by spending less early on in retirement, whereas others might be willing to take their chances and enjoy a higher standard of living while they are still able to do so.
In particular, we focused exclusively on the impact of life span uncertainty—longevity risk—on the optimal consumption and retirement spending policy. To isolate the impact of longevity risk on optimal portfolio retirement withdrawal rates, we placed our deliberations on Planet Vulcan, where investment returns are known and unvarying, the inhabitants are rational and utility-maximizing consumption smoothers, and only life spans are random.
The main results of our investigation are as follows: Counseling retirees to set initial spending from investable wealth at a constant inflation-adjusted rate (e.g., the widely popular 4% rule) is consistent with life-cycle consumption smoothing only under a very limited set of implausible preference parameters—that is, there is no universally optimal or safe retirement spending rate. Rather, the optimal forward-looking behavior in the face of personal longevity risk is to consume in proportion to survival probabilities—adjusted upward for pension income and downward for longevity risk aversion—as opposed to blindly withdrawing constant income for life.
Literature on Retirement Spending Rates
Within the community of retirement income planners, a frequently cited study is by W.P. Bengen (1994), in which he used historical equity and bond returns to search for the highest allowable spending rate that would sustain a portfolio for 30 years of retirement. Using a 50/50 equity/bond mix, Bengen settled on a spending rate between 4% and 5%. In fact, this rate has become known as the Bengen or 4% rule among retirement income planners and has caught on like wildfire. The rule simply states that for every $100 in the retirement nest egg, the retiree should withdraw $4 adjusted for inflation each year—forever, or at least until the portfolio runs dry or the retiree dies, whichever occurs first.
Indeed, it is hard to overestimate the influence of the Bengen study and its embedded “rule” on the community of retirement income planners. The 4% spending rule now seems destined for the same immortality enjoyed by other (unduly simplistic) rules of thumb, such as “buy term and invest the difference” and dollar cost averaging. And although numerous authors have extended, refined, and recalibrated these spending rules, the spirit of each rule remains intact across all versions.
We are not the first to point out that this “start by spending x%” strategy has no basis in economic theory. The goal of our study was to illustrate the solution to the life-cycle problem and demonstrate how longevity risk aversion—in contrast to financial risk aversion, so familiar to financial analysts—affects retirement spending rates.
Numerical Examples and Cases
Note that we use the following terms (somewhat loosely and interchangeably, depending on the context) throughout the article: (1) The retirement spending rate is an annualized dollar amount that includes withdrawals from the portfolio, as well as pension income, and is scaled to reflect an initial portfolio value of $100; (2) the portfolio withdrawal rate is the annualized ratio of the amount withdrawn from the portfolio divided by the value of the portfolio at that time; and (3) the initial portfolio withdrawal rate is the annualized ratio of the initial amount withdrawn from the portfolio divided by the initial value of the portfolio.
Recall that we are spending our retirement on Vulcan, where only life spans are random. Our approach forced us to specify a real (inflation-adjusted) investment return. So, after carefully examining the real yield (a rate that has been adjusted to remove the effects of inflation) from U.S. TIPS over the last 10 years on the basis of data from the Federal Reserve, we found that the maximum real yield over the period was 3.15% for the 10-year bond and 4.24% for the five-year bond. The average yields were 1.95% and 1.50%, respectively. The longer-maturity TIPS exhibited higher yields but obviously entailed some duration (interest rate) risk. After much deliberation, we decided to assume a real interest rate of 2.5% for most of the numerical examples, even though current (fall 2010) TIPS rates were substantially lower.
As for longevity risk, we exercised a great deal of modeling caution because it was the impetus for our investigation. We assumed that the retiree’s remaining lifetime obeys a (unisex) biological law of mortality under which the hazard rate increases exponentially over time. This notion is known as the Gompertz assumption in the actuarial literature, and we calibrated this model to common pension annuitant mortality tables.
In most of our numerical examples, therefore, we assumed an 86.6% probability that a 65-year-old will survive to the age of 75, a 57.3% probability of surviving to 85, a 36.9% probability of reaching 90, a 17.6% probability of reaching 95, and a 5% probability of attaining 100. Again, note that we do not plan for a life expectancy or an ad hoc 30-year retirement. Rather, we account for the entire term structure of mortality.
Our main objective was to focus attention on the impact of risk aversion on the optimal portfolio withdrawal rate and especially the initial portfolio withdrawal rate. Therefore, we display results for a range of values—for example, for a retiree with a very low and a relatively high coefficient of relative risk aversion.
To aid a clear understanding of mortality risk aversion, we offer the following analogy to classical asset allocation models. An investor with medium risk aversion would invest 40% of her assets in a stock (equity) portfolio and 60% in a bond portfolio, assuming an equity risk premium of 5% and volatility of 18%. Our model does not have a risky asset and does not require an equity risk premium, but the idea is that the coefficient of relative risk aversion can be mapped onto more easily understood risk attitudes. Along the same lines, a very low risk-aversion value would lead to an equity allocation of 150%, and a high risk-aversion value implies an equity allocation of 20% (all rounded to the nearest 5%).
Finally, to complete the parameter values required for our model, we assume that the subjective discount rate, which is a proxy for personal impatience, is equal to the risk-free rate (mostly 2.5% in our numerical examples). This assumption suggests that the optimal retirement spending rates would be constant over time in the absence of longevity risk considerations. Again, our motivation for all these assumptions is to tease out the impact of pure longevity risk aversion.
In the language of economics, when the subjective discount rate in a life-cycle model is set equal to the constant and risk-free interest rate, a rational consumer will spend his total (human plus financial) capital evenly and in equal amounts over time. In other words, in a model with no horizon uncertainty, retirement spending rates and spending amounts are, in fact, constant, regardless of the consumer’s elasticity of intertemporal substitution.
The question is, what happens when lifetimes are indeterminate?
Let us take a look at some results. We will assume a 65-year-old with a (standardized) $100 nest egg. Initially, we allow for no pension annuity income; therefore, all consumption must be sourced to the investment portfolio that is earning a deterministic interest rate. On Planet Vulcan, financial wealth must be depleted at the very end of the life cycle (say, age 120), and bequest motives are nonexistent. So, the optimal retirement spending rate at retirement age 65 is $4.605 when the risk-aversion parameter is set to a medium level (see Table 1), and the optimal retirement spending rate is $4.121 when the risk-aversion parameter is set to a high level.
Note that these rates—perhaps surprisingly—are within the range of numbers quoted by the popular press for optimal portfolio withdrawal (spending) rates. Thus, at first glance, these numbers seem to suggest that simple 4% rules of thumb are consistent with a life-cycle model. Unfortunately, the euphoria is short-lived. The numbers) coincide only in the first year of withdrawals (at age 65) and for a limited range of risk-aversion coefficients (most importantly, no pension income). As retirees age, they rationally consume less each year—in proportion to their survival probability adjusted for risk aversion. For example, at our baseline medium level of risk aversion, the optimal retirement spending rate drops from $4.605 at age 65 to $4.544 at age 70, and then to $4.442 at age 75, $3.591 at 90, and $2.177 at 100, assuming the retiree is still alive.
Note how a lower real interest rate (e.g., 0.5% in Table 1) leads to a reduced optimal retirement consumption/spending rate. Indeed, in the yield curve and TIPS environment of fall 2010, our model offered an important message for Baby Boomers: Your parents’ retirement plans might not be sustainable anymore.
The first insight in our model is that a fully rational plan is for retirees to spend less as they progress through retirement. Life-cycle optimizers (i.e., “consumption smoothers” on Vulcan) spend more at earlier ages and reduce spending as they age, even if their subjective discount rate is equal to the real interest rate in the economy.
Intuitively, they deal with longevity risk by setting aside a financial reserve and by planning to reduce consumption (if that risk materializes) in proportion to their survival probability adjusted for risk aversion—all without any pension annuity income.
Including Pension Annuities
Let us now use the same model to examine what happens when the retiree has access to a defined-benefit pension income annuity, which provides a guaranteed lifetime cash flow.
|Real Interest Rates|
|Retire at age 65||$3.330||$3.94||$4.61||$5.32|
|Five years later||3.286||3.888||4.544||5.247|
|10 years later||3.212||3.801||4.442||5.130|
|20 years later||2.898||3.429||4.007||4.627|
|30 years later||2.156||2.552||2.982||3.444|
Notes: The initial portfolio (nest egg) is worth $100 and is invested at the indicated rates.
There is a 5% probability of survival to age 100. No pension income is assumed.
All consumption spending is from the investment portfolio.
Table 2 displays the optimal policy for four different retirees with varying degrees of longevity risk aversion, each with $1 million in investable retirement assets. The first retiree has no pension, the second has an annual pension of $10,000, the third has an annual pension of $20,000, and the fourth has a pension of $50,000.
Table 2 shows the net initial portfolio withdrawal rate (i.e., the optimal amount withdrawn from the investment portfolio) as a function of the risk-aversion values and pre-existing pension income. Thus, for example, when the medium-risk-aversion retiree has $1 million in investable assets and is entitled to a real lifetime pension of $50,000—which, in our language, is a scaled nest egg of $100 and a pension of $5—the optimal total retirement spending rate is $10.551 in the first year. Of that amount, $5.00 obviously comes from the pension and $5.551 is withdrawn from the portfolio. The net initial portfolio withdrawal rate is thus 5.551%.
In contrast, if the retiree has the same $1 million in assets but is entitled to only $10,000 in annual lifetime pension income, the optimal total retirement spending rate is $5.873 per $100 of assets at age 65, of which $1.00 comes from the pension and $4.873 is withdrawn from the portfolio. Hence, the initial portfolio withdrawal rate is 4.873%.
The main point of our study can be summarized in one sentence: The optimal portfolio withdrawal rate depends on longevity risk aversion and the level of pre-existing pension income. The larger the amount of the pre-existing pension income, the greater the optimal retirement spending rate and the greater the portfolio withdrawal rate.
The pension acts primarily as a buffer and allows the retiree to consume more from discretionary wealth. Even at high levels of longevity risk aversion, the risk of living a long life does not “worry” retirees too much because they have pension income to fall back on should that chance (i.e., a long life) materialize. We believe that this insight is absent from most of the popular media discussion (and practitioner implementation) of optimal spending rates. If an individual has substantial income from a defined-benefit pension or Social Security, she can afford to withdraw more—percentage-wise—than her neighbor, who is relying entirely on his investment portfolio to finance his retirement income needs.
|Investable Assets = $1 million (scaled to $100)|
|Annual Pension Income|
|Longevity Risk Aversion Level|
Notes: The mortality assumption is that there is a 5% probability of survival to age 100.
The Gompertz mortality parameters are m = 89.335 and b = 9.5. The interest rate is 2.5%.
Table 2 confirms a number of other important results. Note that the optimal portfolio withdrawal rate—for a range of risk-aversion and pension income levels—is between 8% and 4%, but only when the inflation-adjusted interest rate is assumed to be a rather generous 2.5%. Adding another 100 basis points to the investment return assumption raises the initial portfolio withdrawal rate by 60 to 80 basis points. Reducing interest assumptions, however, will have the opposite effect.
Longevity risk aversion manifests itself by (essentially) assuming that retirees will live longer than the biological/medical estimate. Only extremely risk-tolerant retirees behave as if their modal life spans were the true (biological) modal value. Note that this behavior is not risk neutrality, which would ignore longevity risk altogether.
In the asset allocation literature, the closest analogy to these risk-adjusted mortality rates is the concept of risk-adjusted investment returns. A risk-averse investor observes a 10% expected portfolio return and adjusts it downward on the basis of the volatility of the return and her risk aversion. If the (subjectively) adjusted investment return is less than the risk-free rate, the investor shuns the risky asset. Of course, this analogy is not quite correct because retirees cannot shun longevity risk, but the spirit is the same. The longevity probability they see is not the longevity probability they feel.
Retiree With Medium
Longevity Risk Aversion
|Retiree With High|
|Longevity Risk Aversion|
at Age 65
at Age 80
at Age 65
|at Age 80|
Notes: Assumes a fairly priced pension annuity that pays $1 of lifetime income per
$15.7971 premium under a 2.5% interest rate. Real-world pricing will differ depending on market conditions.
Again, an important take-away is the impact of pension annuities on retirement spending.
Table 3 reports the optimal retirement spending rate at various ages, assuming that a fixed percentage of the retirement nest egg is used to purchase a pension annuity (“annuitized”).
Note that the pricing of pension (income) annuities by private sector insurance companies usually involves mortality rates that differ from population rates owing to anti-selection concerns. This factor could be easily incorporated by using different mortality parameters, but we will keep things simple to illustrate the impact of lifetime income on optimal total spending rates.
Results are reported for a retirement age of 65 and planned consumption 15 years later (assuming the retiree is still alive), at age 80. A variety of scenarios are illustrated in which 0%, 20%, 40%, 60% or 100% of initial wealth is annuitized—that is, a nonreversible pension annuity is purchased on the basis of the going market rate.
Table 3 shows total dollar retirement spending rates, including the corresponding pension annuity income. These rates are not the portfolio withdrawal rates that are reported in percentages in Table 2. For example, if the retiree with medium risk aversion allocates $20 (from the $100 available) to purchase a pension annuity that pays $1.2661 for life, optimal consumption will be $1.2661 + $3.9976 = $5.2637 at age 65. Note that the $3.9976 withdrawn from the remaining portfolio of $80 is equivalent to an initial portfolio withdrawal rate of 4.996%.
In contrast, the retiree with a high degree of longevity risk aversion will receive the same $1.2661 from the $20 that has been annuitized but will optimally spend only $3.5352 from the portfolio (a withdrawal rate of 4.419%), for a total retirement spending rate of $4.8013 at age 65.
If the entire nest egg is annuitized at 65, leading to $6.3303 of annual lifetime income, the retirement spending rate is constant for life—and independent of risk aversion—because there is no financial capital from which to draw down any income. This example is yet another way to illustrate the benefit of converting financial wealth into an income flow. The $6.3303 of annual consumption is the largest of all the consumption plans. Thus, most financial economists are strong advocates of annuitizing a portion of one’s retirement nest egg.
Reacting to Financial Shocks
Using our methodology, one can examine the optimal reaction to financial shocks over the retirement horizon. Take someone who experiences a 30% loss in his investment portfolio and wants to rationally reduce spending to account for the depleted nest egg. The rule of thumb suggesting that retirees spend 4% to 5% says nothing about how to update the rule in response to a shock to wealth.
To understand how this approach would work in practice, let us begin with a medium-risk-adverse retiree who has $100 in investable assets and is entitled to $2 of lifetime pension income. With a real interest rate of 2.5%, the optimal policy is to consume a total of $7.078 at age 65 ($2 from the pension and $5.078 from the portfolio) and adjust withdrawals downward over time in proportion to the survival probability to the power of the risk-aversion coefficient. The wealth depletion time is at age 105.
Under this dynamic policy, the expectation is that at age 70, the financial capital trajectory will be $86.668 and total consumption will be $6.984 if the retiree follows the optimal consumption path for the next five years.
Now let us assume that the retiree survives the next five years and experiences a financial shock that reduces the portfolio value from the expected $86.668 to $60 at age 70, which is 31% less than planned. In this case, the optimal plan is to reduce consumption to $5.583, which is obtained by solving the problem from the beginning, but with a starting age of 70. This result is a reduction of approximately 20% compared with the original plan.
Of course, this scenario is a bit of an apples-to-oranges comparison because (1) a shock is not allowed in our model, and (2) the time-zero consumption plan is based on a conditional probability of survival that could change on the basis of realized health status. The problem of indeterminate versus deterministic investment returns and mortality hazard rates obviously takes us far beyond the simple agenda of our study.
In sum, a rational response to an x% drop in one’s retirement portfolio is not to reduce consumption and spending by the same x%. Consumption smoothing in the life-cycle model is about amortizing unexpected losses and gains over the remaining lifetime horizon, adjusted for survival probabilities.
To a financial economist, the optimal retirement spending rate, asset allocation (investments), and product allocation (insurance) are a complicated function of mortality expectations, economic forecasts, and the trade-off between the preference for retirement sustainability and the desire to leave a financial legacy (bequest motive).
Retirees can afford to spend more if they are willing to leave a smaller financial legacy and risk early depletion times. They should spend less if they desire a larger legacy and greater sustainability. Optimization of investments and insurance products occurs on this retirement income frontier. Ergo, a simple rule that advises all retirees to spend x% of their nest egg adjusted up or down in some ad hoc manner is akin to the broken clock that tells time correctly only twice a day.
We are not the first authors—and we will certainly not be the last—to criticize the “spend x%” approach to retirement income planning.
Conclusion: Back to Planet Earth
How might a full indeterminate model—with possible shocks to health and their related expenses—change optimal spending policies? Assuming agreement on a reasonable model and parameters for long-term portfolio returns, the risk-averse retiree would be exposed to the risk of a negative (shock and would plan for this risk by consuming less. With a full menu of investment assets and products available, however, the retiree would be free to optimize around annuities and other downside-protected products, in addition to long-term-care insurance and other retirement products. In other words, even the formulation of the problem itself becomes much more complex.
More importantly, the optimal allocation depends on the retiree’s preference for personal consumption versus bequest. A product and asset allocation suitable for a consumer with no bequest or legacy motives is quite different from the optimal portfolio for someone with strong legacy preferences. In our study, we assumed that the retiree’s objective is to maximize utility of lifetime consumption without any consideration for the value of bequest or legacy.
Although some have argued that a behavioral explanation is needed to rationalize the desire for a constant consumption pattern in retirement, we note that very high longevity risk aversion leads to relatively constant spending rates and might “explain” these fixed rules. In other words, we do not need a behavioral model to justify constant 4% spending. Extreme risk aversion does that for us.
That said, we believe that another important take-away from our study is that offering the following advice to retirees is internally inconsistent: “You might live a very long time, so you better make sure to own a lot of stocks and equity.” The first part of the sentence implies longevity risk aversion, while the second part is suitable only for risk-tolerant retirees. Risk is risk.
One thing seems clear: Longevity risk aversion and pension annuities remain very important factors to consider with regard to optimal portfolio withdrawal rates. That is the main message of our study, a message that does not change here on Planet Earth.
Copyright 2011, CFA Institute. Edited and republished from the March/April 2011 Financial Analysts Journal with permission from CFA Institute. All rights reserved.
Huaxiong Huang is a professor of mathematics at York University in Toronto.