A Sustainable Spending Rate without Simulation

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

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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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