May 31, 2022

Spending at 63 using life expectancy

Based on an email from David C, I had forgotten that a rule of thumb for spending is 1/e where e is remaining lifetime. The super quick look here is from a page from Gordon Irlam's aacalc.com site: 

where the operative text is this:

Perhaps less well known than Markowitz's modern portfolio theory (MPT) is the subsequent work of Merton and Samuelson. This is a shame because while MPT only concerns itself with optimizing investing in a single time period, Merton's portfolio model concerns itself with optimizing over time, where it is possible to change asset allocation and consumption in response to portfolio performance. This is far closer to the problem faced by most investors. Unfortunately the math involved is quite complex. I've been trying to derive some very simple rules of thumb for stock/bond asset allocation and consumption planning using Merton's portfolio model and the current returns environment as a guide. Here is what I came up with: 

stocks percent of portfolio = 50% with future income treated like bonds = 50% * (1 + If / W)
(but not to exceed your risk tolerance)

annual retirement portfolio consumption = W / e

... and ...

I am being lazy in not discounting estimated future income. I can get away with this because the discount rate is small. If nu is small enough, and life expectancy is fixed, the optimal consumption is W / e. In reality we should consume more because the portfolio experiences growth over time (ignoring withdrawals), but we should consume less to set aside funds for the possibility that we live longer than our life expectancy. On balance it is thought these two largely cancel out and the simple rule, W / e, seems to work quite well.

A more sophisticated approach of using the complex formula was tried, but while it performed 3% better for large portfolios it performed 4% worse for small portfolios (where there was a shorting constraint). Lacking a consistent advantage for the complex formula, I elected to use the simple formula, W / e. (Using VPW is another possibility. If you get the parameters right, it performed 1% better than the simple formula for affluent and constrained investors alike. W / max(e, 8) was also found to be a contender: 3% better for the affluent, 1% better for the constrained.)

These two rules are very simple, but they have strong theoretical underpinnings, and they work well empirically. Like all rules of thumb there are limits. These limits are likely to be hit when gamma is not a constant but drops as consumption becomes satiated, if real interest rates ever rise significantly and we are no longer able to ignore the need for time discounting of future income, or there is interest in leaving a bequest. In these cases it may be better to perform stochastic dynamic programming (SDP) to compute the optimal asset allocation and consumption.

When applied to actuarial life tables it might look like this though I don't know if Gordon intended to limit "e" to mean, median, or something else. Me? I just wanted to see what 1/e looked like for someone my age for a range of "e"s. 


No conclusions or discussion.




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