Simpson’s Paradox

Simpson’s Paradox is something that every investor should know about. I had previously suspected something similar, when I wrote that investment decisions don’t really matter early in a person’s saving years, but become more important over time. However, I only learned that there is an associated paradox (with a name!) a few weeks ago. In essence, people tend to assume equal weight and random samples when they are presented with amalgamations of distinct performance results (or other measures). This can help explain why individual investors sometimes underperform the market or even their investment funds, when trying to time the market or rebalance.

Simpson’s Paradox explains that the outcome of a weighted average is not equivalent to an unweighted average, but people usually assume the latter. For example, an investor might use a buy and hold approach to owning a mutual fund. Imagine that each year he deposits $10,000. By the end of the first year, he makes his deposit and checks his performance: his fund has underperformed the market by the amount of fees. At the end of the second year, he makes his deposit and finds that his fund has underperformed the market by the fee amount. At the end of the third, fourth and fifth years, it’s the same story. Setting aside for the moment the problem with “closet-indexing”, the investor approaches his advisor and suggests they change their approach, since he has been underperforming the market. The advisor prepares a performance report for the five-year period and shows that the investor has in fact outperformed over the entire period. How could this be?

Performance Market Weight Fund Weight
Year 1 -10.00% 1 -12.00% $10,000.00
Year 2 5.00% 1 3.00% $20,000.00
Year 3 15.00% 1 13.00% $30,000.00
Year 4 12.00% 1 10.00% $40,000.00
Year 5 20.00% 1 18.00% $50,000.00
Weighted Average 8.40% 5 10.87% $150,000.00

This is where I assure you it’s not a trick. Look at the numbers. The investor underperformed the market by 2% year after year. But the market return is unweighted. The investor’s account, however, is weighted by the timing of his deposits. He was fortunate in that the returns were higher when his account balance was greater. In the end, he outperformed the market. (Yes, a market index would still have provided higher returns in his account, but that’s beside the point I’m trying to illustrate.) The first conclusion is that with weighted returns, accounting for deposits and withdrawals, timing matters. The sequence of returns doesn’t only matter for retirees, drawing on their funds, but also for savers, adding to their funds.

Further, “buy and hold” isn’t dead, just because performance could be increased by selling near the top and buying near the bottom. Simpson’s Paradox highlights the problems attendant to any time out of the market. Looking at the market crash of 2008, some people sold out near the bottom. (In fairness, the market fell so quickly, it was almost impossible to sell in the middle of the decline.) When do they buy back in? Suppose they sold in October 2008 and waited until March 2009, watching the market go nowhere. Being discouraged, they decide to wait until the market is obviously rising. At no point was there a smooth uptrend, but the market rose about 10% in a few days in March 2009. With no money in the market, anyone who sold at the bottom would permanently miss that return. It would be the opposite of the example above, where the investor would underperform, given the lighter weighting during good returns.

That is exactly what many investors do, whether they invest in actively managed funds or passive index funds. I don’t have a reference for the study, but I recall being shocked at an article that showed individual investor returns under 4% per year (mid-80s to late 90s?) while the market climbed by 10% per year. How could that happen? The investor puts her money in an index and, at the end of a year, switches it to last year’s best performer. It performs less well, and at the end of the year, she repeats her choice of the prior year’s star. Moving her money from index to index, she fails to capture the performance of the broad market, even if each index fund represents a sub-index of the overall market. This is one of my fundamental arguments why firing your advisor and buying “index funds” is not necessarily wise, even if it costs less.

Simpson’s Paradox helps explain why stock picking isn’t simply a matter of picking better companies, but weighting matters. It explains why performance becomes more important as an investment fund grows. It explains in part why individuals underperform their own investments. For a scholarly paper with a different illustration, see: (PDF file).


4 thoughts on “Simpson’s Paradox

  1. interesting paradaox. What’s the math behind it to get the weighted returns? I look at those numbers and fall for the obvious conclusion that the investor’s returns should be 2% less than the markets returns.

  2. Steve,

    The math for the weighted average of the “fund” is: (Fund*Weight+Fund*Weight+Fund*Weight…)/(Weight+Weight+Weight…) or, in this example, (-12*10,000+3*20,000+13*30,000+10*40,000+18*50,000)/150,000. That gives the dollar weighted returns, assuming evenly timed annual deposits. In Excel, I use the formula: =SumProduct(Fund1:Fund5,Weight1:Weight5)/Sum(Weight1:Weight5).

    Of course, if the order of returns had been reversed, the market would have trounced the fund, given that a larger fund ($50,000) would have experience the worst returns (-12%)

  3. Robert, Thanks for clarifying. That does make sense. Having a larger amount of money earning a higher return in year five should obviously have a greater weight on the averaged return over five years.

  4. Pingback: Passive vs. Active is Unimportant Compared to Fund Flows | Financial Thinking

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