This is a Guest Post by David Marcus. He gives a puzzle and its solution, which is interesting, and then speculates as to why some people get it wrong.

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THE PROBLEM:

Investing Puzzle or Arithmetic Can Be Useful

The following is an example of investment results that I saw in an

investment newsletter. There are two portfolios that use different

strategies. Both portfolios start with $1 million twenty years ago and

withdraw 5% each year. The idea is that you are retired and withdrawing

money to spend. Not all years are shown in the tables.

Portfolio A

Year Return Withdrawal Balance

2000 15.31% 57,655 1,095,445

2005 1.81% 59,962 1,139,273

2008 -12.65% 51,000 969,004

2009 34.26% 65,049 1,235,936

2010 11.94% 69,175 1,314,331

2015 -2.48% 64,935 1,233,764

2020 10.27% 66,935 1,271,765

Total Withdrawal: 1,685,190

Change in Balance: 27.18%

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Portfolio B

Year Return Withdrawal Balance

2000 -0.95% 49,524 940,956

2005 3.80% 44,534 846,154

2008 -20.11% 35,922 682,523

2009 18.27% 40,360 766,833

2010 11.57% 42,777 812,764

2015 0.99% 50,767 964,567

2020 13.35% 65,602 1,246,433

Total Withdrawal: 1,425,573

Change in Balance: 24.64%

Portfolio A has a final balance that is 25,000 more than Portfolio B's and

had about 260,000 more in withdrawals. Does the example lend credence to

the Portfolio A strategy being better than the Portfolio B strategy?

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THE ANSWER:

Investing Puzzle or Arithmetic Can Be Useful: Analysis

Summary: The two portfolios have about the same performance over the 20

years. The difference is mainly due to Portfolio A having a good year or

years near the beginning before much money was withdrawn. The example

merely shows that it is better to withdraw money after a gain rather than

before.

Detailed Analysis:

The scenario is: Start with X = $1 million. Withdraw 5% a year.

Define "gain factor" to be 1 plus the percentage return. For example, if a

portfolio returns 5%, then the gain factor is 1.05.

Let A_j, j = 1, ..., 20, be the gain factors each year for portfolio A.

Let B_j, j = 1, ..., 20 be the gain factors each year for portfolio B.

The final amount in portfolio A is

F = X * A_1 * 0.95 * A_2 * 0.95 * ... * A_20 * 0.95 .

The final amount in portfolio B is

G = X * B_1 * 0.95 * B_2 * 0.95 * ... * B_20 * 0.95 .

From the "Change in Balance" values or the balances for year 2020, we see

that F and G are almost the same:

F = 1.271865 * X,

G = 1.246433 * X.

But, as we learned in elementary school, multiplication is commutative, so

F = X * 0.95^20 * \prod_{j=1}^20 A_j,

G = X * 0.95^20 * \prod_{j=1}^20 B_j.

Since F and G are almost the same, the total gains (product of the gain

factors) for the two portfolios are almost the same, i.e.,

\prod_{j=1}^20 A_j \approx \prod_{j=1}^20 B_j.

Then what accounts for the big difference in the amounts withdrawn?

Portfolio A must have had some good years near the beginning. (We see in

the tables that Portfolio A did better in 2000 than Portfolio B.) So, all

the example shows is that it is better to withdraw your money after your

gains rather than before.

To take an extreme example, suppose an investment is going to go up 100%

this year. It is better to take your money out at the end of the year

(after the gain) than at the beginning of the year (before the gain). This

is a triviality.

The example tells us nothing useful about the two strategies.

Note: The total gains aren't exactly the same, but the timing of the yearly

gains is what is driving the results. We have (rounding off)

( F - G ) / 0.95^20 = 70942.81 .

So, if there had been no withdrawals, the difference in the portfolio

balances would have been about $71,000, much less than the $260,000 +

$25,000 difference with the withdrawals.

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WHY IS THIS HARD FOR PEOPLE?

Many people have trouble with this puzzle. The difficulty may be that such

people don't make a mental model (or a written model) of the process that

is producing the balance. If you write down (or have in your head) a

formula for the balance, then you see that the gain factors are independent

of the withdrawal factors. That is, we could withdraw more or less money,

or even deposit money, without affecting the gain factors we would use in

the model. This then leads us to consider the gain factors on their own,

and to recognize that the gain factors are the true measures of how the

strategies perform.