Optimal spending under quadratic utility with mortality

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Beliavsky
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Joined: Sun Jun 29, 2014 10:21 am

Optimal spending under quadratic utility with mortality

Post by Beliavsky »

In the thread All the Personal-Finance Books are Wrong I advocated the use of a utility function of spending from savings, in particular quadratic utility,

u(x) = c^2 - (c-x)^2 for x < c
u(x) = c^2 for x >= c

to decide on a spending rate in retirement. The idea is that spending less than an amount c in a year is quadratically painful, but spending more than c does not give further enjoyment. You also need a probability distribution for how long you will live, since utility comes from spending while alive. The Gompertz distribution is used, which assumes that the probability of dying each year increases exponentially. For simplicity, inflation is assumed zero and a constant nominal rate of return is used. A floor annual spending level from savings up of $10K up to age 110 is specified, which constrains how much can be spent earlier -- at age 100 you must have at least $100K left. I assume that spending from savings is supplemented by spending of Social Security benefits. I find that an initial spending rate of 4% or $40K produces more utility than 3.5% or 4.5%, even though the spending declines to $36,000 at age 81 and to $10,000 by age 84. In general, in the absence of a bequest motive, spending a bit more than a sustainable amount in the early years maximizes utility.

The Excel spreadsheet, with mortality and spending parameters that can be changed, is here.
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