The Importance Of Avoiding Retirement Ruin

By Moshe A. Milevsky with Anna Abaimova

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, one doesn’t require spinning roulette wheels or computer simulations to back-out your 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 (a.k.a. 0.5833% a month), the nest egg will be exhausted within month number 259. Start this doomed process at age 65 and ruin strikes half-way through age 86. (See Figure 1.)

Figure 1

 

 

 

 

 

 

 

 

 

We know this inevitable date with destiny with absolute certainty since the textbook equation   teaches us that the present value of $750 for n = 258.59 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 withdrawals occur at a lower $625 a month--$7,500 a 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.

Figure 2

 

 

 

 

 

 

The question investigated here is: What happens if the hypothetical 65-year-old retiree does not earn a constant 7% every year but instead earns an arithmetic average 7% over his or her 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 and systematic manner, as in Figure 2. During the first three years of retirement, the portfolio earns 7%, -13%, and 27%, respectively, producing an arithmetic average of exactly 7% a year. Now, assume the monthly withdrawal is the same $750 as in the earlier case. Then, in the fourth year, the cycle starts again, with the portfolio earning 7%, then -13% and then 27%.

This cyclical process continues in three-year increments until the nest-egg is exhausted and the money runs out. Will ruin occur earlier or later than the prior case where returns were a smooth 7% each and every year?

The answer is, earlier.  Indeed, since retirement started on the “wrong foot,” the date with zero occurs a full 3 years earlier, or at age 83.The plus 27% return in the third, sixth, ninth, etc, years of retirement isn’t enough to offset the minus 13% returns in the second, fifth, eight, etc, years.

Note that the answer is obtained and can be computed with just as much accuracy as the previous case, although a simple formula can be used for the present value (which must be done manually or by hand). A simple spreadsheet in Excel will do the trick (and is available from www.ifid.ca). Figure 3 illustrates the result graphically.

Figure 3

 

 

 

 

 

 

 

 

 

 

 

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% for the next month. Do this for 12 months and then repeat for 12 months under an investment return of -1.0833% per month, which is nominal -13% per year. Finally, repeat for 12 months under an investment return of 2.2500% per month, which is a nominal 27% per year. Every 36 months the pattern repeats itself. Start with twelve 0.5833% numbers, then twelve -1.0833% numbers and finally twelve 2.2500% numbers. This should produce a very long column of returns that mimics Figure 3, with the account ultimately reaching zero shortly after the 83rd birthday. In this case, an average of 7% is worse than a 7% every year

Figure 4

 

 

 

 

 

 

 

 

Now, what happens if the triangle is reversed, starting in the other direction—i.e., the earnings are 7%, then 27% and then -13% over and over again? Figure 4 displays the same triangle, but with the arrows going in the other direction. The arithmetic average investment return is the same 7% regardless of what side of the triangle the retirement earnings and withdrawals start. However, this time around, the money runs out at age 89.5 as opposed to age 83.33 or 86.5  

 

In this case, an average of 7% is better than 7% every year. (See Figure 5).

Figure 5

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 to the account on an ongoing basis--it is impossible to exhaust the account no matter how poor the returns. Also, remember that one lump sum investment grows to the same value regardless of sequencing, since (1.07)(1.27)(0.87) = (1.07)(0.87)(1.27).

Table 1

 

 

 

 

 

 

 

 

Finally, Table 1 shows a summary of 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 and every 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%} versus {27%,7%,-13%}.

What do we learn? First of all, 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  

This provides a more pessimistic (but more accurate) indication of the risks that lie ahead. Remember that the greater the gap between your portfolio’s arithmetic and geometric mean the greater the chances of an early ruin, all else being equal.

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

Moshe A. Milevsky, Ph.D, is a finance professor at York University and executive director of The IFID Centre, both of Toronto, Ontario, Canada. Anna Abaimova is a research associate at The IFID Centre. Their e-mail addresses are, respectively: milevsky@yorku.ca and aabaimova@ifid.ca.