A Sustainable Spending Rate without Simulation

# Where gammadist α β denotes the cumulative

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where GammaDist( α , β| . ) denotes the cumulative distribution function of the gamma distribution (in Microsoft Excel notation) evaluated at the parame- ter pair ( α , β ). The familiar μ and σ are the return and volatility parameters from the investment portfolio, and λ is the mortality rate. The expected value of the SPV is ( μ σ 2 + λ ) –1 . For example, start with an investment (endowment, nest egg) of \$20 that is expected to earn a 7 percent real return in any given year with a volatility (standard deviation) of 20 percent a year. 5 A 50-year-old (of any gender) with a median future life span of 28.1 years intends to consume \$1 after inflation a year for the rest of his or her life. If the median life span is 28.1 years, then by defi- nition, the probability of survival for 28.1 years is exactly 50 percent; so, the “implied mortality rate” parameter is λ = ln(2)/28.1 = 0.0247. According to Equation 8, the probability of retirement ruin, which is the probability that the stochastic present value of \$1 consumption is greater than \$20, is 26.8 percent. In the language of Figure 2, if we evaluate the SPV at w = 20, the area to the right has a mass of 0.268 units. The area to the left—the probability of sustainability—has a mass of 0.732 units. 6 In the random life span of λ > 0, our result is approximate, albeit correct to within two moments of the true SPV density. We will show that this issue is not significant. In the infinite horizon case of λ = 0, our result is not an approximation . It is a theorem that the SPV defined by Equation 2b is, in fact, the reciprocal gamma distributed. 7 Numerical Examples Our base case is a newly retired 65-year-old who has a nest egg of \$1,000,000 which must last for the remainder of this individual’s life. In addition to pensions, the retiree wants \$60,000 a year in real dollars from this nest egg (which is \$6 per \$100 in the terms commonly used in practice). The \$60,000 is to be created via a systematic withdrawal plan that sells off the required number of shares/units each month in a reverse dollar-cost-average strat- egy. These numbers are prior to any income taxes, and our results are for pretax consumption needs; in addition, we are not distinguishing between tax- sheltered and taxable plans, which is a different important issue. The retiree wants to know whether the stochas- tic present value of the desired \$60,000 income a year is probabilistically less than the initial nest egg of \$1,000,000. If it is, the retiree’s standard of living is sustainable. If the SPV of the consumption plan is larger than \$1,000,000, however, the retirement plan is unsustainable and the individual will be “ruined” at some point, unless of course, he or she reduces consumption. Table 2 provides an extensive combination of consumption/withdrawal rates for various ages based on our model in Equation 8 and based on exact mortality rates instead of the exponential approximation. The rates assume an all-equity portfolio with expected return of 7 percent and volatility of 20 percent. The time variable is deter- mined by the first columns in Table 2—retirement

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