The cold harsh calculus of retirement income tells us with unwavering accuracy exactly how long a nest egg will last under fixed withdrawals and known returns. In a so-called deterministic world, it doesn’t require spinning roulette wheels or computer simulations to back out one’s date with ruin.

For example, if a current $100,000 portfolio is subjected to monthly withdrawals of $750 ($9,000 annually) and is earning a nominal rate of 7% a year (0.5833% a month), the nest egg will be exhausted within month No. 259. Start this doomed process at age 65 and ruin occurs halfway through age 86.

We know this inevitable date with destiny with absolute certainty, since the textbook equation (1-(1+l)-n) / k teaches that the present value of $750 n = 258.59 for periods under a periodic rate of k = 0.005833 is exactly $100,000. Ergo, the $100,000 will only last until age 86.5.

Of course, if the plan is to withdraw a lower $625/month ($7,500/year), the money runs out by month 466, and the nest egg lasts beyond the mythical age of 100 for the same 65-year-old retiree. The present value of $625 paid over 465.59 periods under a periodic rate of 0.5833% is also $100,000.

The question to investigate is, what happens if the hypothetical 65-year-old retiree does not earn a constant 7% each year but instead an arithmetic average 7% over retirement? How variable is the final outcome and what does it depend on?

To put some structure on the problem–since there are so many ways to generate an average return of 7%–imagine that the annual investment returns are generated in a cyclical, systematic manner. Think of a triangle with the corners represented by the numbers 7%, -13% and 27%. During the first year of retirement, the portfolio earns 7%. In year two, it earns -13% and in year three, 27%.

By construction, the arithmetic average of these numbers is exactly 7%. Assume the retiree plans to withdraw the same $750 a month as before and that in year four, the cycle starts again, with the portfolio earning 7%, then -13% and then 27%, a cyclical process that continues in three-year increments until the nest egg is exhausted.

Will ruin occur earlier or later than when returns were 7% each year?

The answer is “earlier.” Indeed, since retirement started on the “wrong foot,” ruin occurs a full three years earlier, or at age 83. The 27% return in the third, sixth, ninth, etc. years of retirement isn’t enough to offset the -13% returns in the second, fifth, eighth, etc. years of retirement. (This is akin to this year’s 20% bull market failing to undo the damage of last year’s 20% bear market.)