A Sustainable Spending Rate without Simulation

What we want to know is the actual shape of figure 2

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sustain the desired consumption level. What we want to know is the actual shape of Figure 2. The analytic contribution of this article is implementation of a closed-form expression for the SPV defined by Equation 2b under the assumption that the total investment return, R t , is generated by a lognormal distribution—that is, an exponential Brownian motion. This classical assumption has many supporters—from Merton (1975) to Rubin- stein (1991). But even from an empirical perspec- tive, Levy and Duchin (2004) found that the lognormal assumption “won” many of the “horse races” when plausible distributions for historical returns were compared. Furthermore, many popu- lar optimizers, many asset allocation models, and much oft-quoted common advice are based on the classical Markowitz–Sharpe assumptions of log- normal returns. Therefore, for the remainder of this article, we follow this tradition. 4 Analytic Formula for Sustainable Spending Three important probability distributions allow us to derive a closed-form solution for sustainability. The first is the ubiquitous lognormal distribution, the second is the exponential lifetime distribution, and the last one is the—perhaps lesser-known— reciprocal gamma distribution. These three distri- butions merge together in the SPV. They allow us to solve the problem when investment return and date of death are risky or stochastic variables. Figure 2. SPV of Retirement Consumption SPV T t R dt t = > ( ) prob ± 1 0 , 20 PDF Su s t a in able R uin 0.04 0.10 0.12 0.08 0.06 0.02 0 0 35 10 5 15 25 30 C u rre nt Dollars ( n es t e gg ) S PV: Woma n , A g e 65 S PV: Woma n , A g e 50
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A Sustainable Spending Rate without Simulation November/December 2005 pubs .org 93 Lognormal Random Variable. The invest- ment total return , R t , between time t 0 and time t is said to be lognormally distributed with a mean of μ and standard deviation of σ if the expected total return is (3a) the expected log return is E [ln( R t )] = ( μ – 0.5 σ 2 ) t , (3b) the log volatility is (3c) and the probability law can be written as (3d) where N ( . ) denotes the cumulative normal distri- bution. For example, a mutual fund or portfolio that is expected to earn an inflation-adjusted continuously compounded return of μ = 7 percent a year with a logarithmic volatility of σ = 20 percent has a N (0.05,0.20,0) = 40.13 percent chance of earning a negative return in any given year. But if the expected return is a more optimistic 10 percent a year, the chances of losing money are reduced to N (0.08,0.20,0) = 34.46 percent. Note that the expected value of lognormal random variable R t is e μ t but the median value (that is, geometric mean) is a lower e ( μ –0.5 σ 2 ) t . By definition, the probability that a lognormal random variable is less than its median value is precisely 50 percent. The gap between expected value e μ t and median value e ( μ –0.5 σ 2 ) t is always greater than zero, proportional to the vola- tility, and increasing in time. We will return to the mean versus median distinction later.
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