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years near the beginning before much money was withdrawn.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.
This all requires some thought.
ReplyDeleteJust 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) Yes. 2) Yes, the numbers in that sentence are approximate.
DeleteThank you! This makes all sense now.
DeleteYes, 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?)
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.
DeleteGreat 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 http://www.mrbartonmaths.com/blog/diagnostic-questions/ 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.
ReplyDeleteI 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).
ReplyDeleteBy "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.
DeleteSeems 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.
ReplyDeleteSome years are not shown.
DeleteA 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.
ReplyDelete