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.)

[Note: The answer can be computed with just as much accuracy as previously. However, one can't use a simple formula for the present value. Instead, this must be done manually or by hand. A simple Excel spreadsheet--available from www.ifid.ca--will do the trick.]

Start with $100,000 and force it to earn 0.5833% in the first month. Then, withdraw $750 and have the remaining sum earn the same 0.5833% the next month. Do this for 12 months and then repeat for 12 months under an investment return of -1.0833% a month, which is a nominal -13% a year. Finally, repeat for 12 months under an investment return of 2.25% per month, which is a nominal 27% per year. Every 36 months, the pattern repeats. Start with twelve 0.5833% numbers, then twelve -1.0833% numbers and finally twelve 2.25% numbers. The very long column of returns that results shows the account ultimately reaching zero shortly after the 83rd birthday. In this case, an average of 7% is worse than getting 7% every year.

What happens if we reverse the imaginary triangle and instead start in the other direction–i.e., if the earnings are 7%, then 27% and then -13% cyclically? The arithmetic average investment return is the same, 7% regardless of what side of the triangle retirement earnings and withdrawals start. However, this time, ruin strikes at age 89.5–not age 83.33 or 86.5.

In this case, an average of 7% is better than 7% every year.

The variance in outcomes would have been even greater if starting with -13% or 27% as opposed to the same 7%. For example, if the sequence was -13%, 7% and then 27%, the age of ruin would be 81. This peculiar phenomenon is unique to the distribution phase of the lifecycle. In the accumulation phase–as money is being added on an ongoing basis–it is impossible to exhaust the account no matter how poor the returns. Also, remember that: (1.07)(1.27)(0.87)=(1.07)(0.87)(1.27).

Finally, Table 1 summarizes the impact of the various sequences on the ruin age as well as the variation in months between the given sequence and the baseline case of 7% each year of retirement. Note that this sequencing gap can get quite large. There is a 14-year gap between repeating the sequence {-13%, 7%, 27%} vs. {27%, 7%, -13%}.

Lessons learned: First, arithmetic averages can be a deceiving measure of central tendency when it comes to investment returns while withdrawing. The arithmetic average of -13%, 7% and 27% is exactly 7%. However, the geometric average of these three numbers is ((1-0.13)(1=0.07)(1+0.27))1/3 – 1 = 5.6%, which provides a more pessimistic (but more accurate) indication of the risks that lie ahead. Remember that the greater the gap between the portfolio’s arithmetic and geometric mean, the greater the chances of early ruin, all else being equal.

More importantly, this is yet another indication of how fragile the first few investment years of withdrawals really can be…and why they should be protected. Starting withdrawals (i.e., retirement) during a bull market vs. a bear market can cost you 14 years. So, don’t leave your retirement income at the mercy of a spinning merry-go-round.