Sunday, April 18, 2021

An investment puzzle and speculation as to why some think its hard

 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. 



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%


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?



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years near the beginning before much money was withdrawn.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


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.



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.


  1. This all requires some thought.
    Just wanted to understand we are on the same page here.

    (1) The balances are always referring to end of year status?
    (2) Statement : "Portfolio A has a final balance that is 25,000 more than Portfolio B's ..." :: 1271765 - 1246433 = 25332 ... What am I missing? Did we mean "about 25,000 more"?

    1. 1) Yes. 2) Yes, the numbers in that sentence are approximate.

    2. Thank you! This makes all sense now.
      Yes, nice puzzle! I appreciate now.
      Here is an issue that might not be one.

      What if strategies in portfolio B would have done astronomically(*) well in a year not shown, let's say for instance 2018. (*well enough to counter balance the counter intuitive point made in this post. Is this possible?)

    3. We know the product of the gains, so if the gain for portfolio B in 2018 was astronomical, the gains in some other years for portfolio B must have been poor enough to counter balance it.

  2. Great topic. You make a great point that it is hard to know what someone is thinking before they cross over to understanding the example. This is a dilemma for teachers. I'm a big fan of Mr Barton Maths, and one of the admirable things that he does is his Diagonostic Questions web site. More explanation is on the brief video at but in short, besides serving up the multiple choice questions, he also asks students to briefly say why they gave the answer. He's got thousands of replies. What people say is often eye-opening.

  3. I have to quibble slightly with my friend David. Although the analysis is spot on, the last sentence, "...recognize that the gain factors are the true measures of how the strategies perform" is only true for a FIXED investment/analysis period, and does NOT apply to the strategy itself. For example, it's pretty clear that investment strategy for Portfolio B was way, way better during the market boom years of 2010 through 2020 -- Portfolio A lost ~3% (after withdrawals) and Portfolio B gained a whopping 53% (after withdrawals).

    1. By "FIXED", do you mean that if you choose a different period to compare the portfolios, you might come to a different conclusion? That's certainly true.

  4. Seems like you made a mistake (or at least 1) copying over the values for B. In 2015 the 0.99% return should have resulted in a small decrease in the balance (after accounting for the 5% withdrawal) instead of the large increase seen that year.

  5. A good way to compare the strategies is to graph f(n)=\prod_{j=1}^n A_j and g(n)=\prod_{j=1}^n B_j on a log scale.