Financial Thinking

We looked last week at Simpson’s Paradox to explain why individual investors can outperform or underperform the investment funds that they own. This causes some real cognitive dissonance, because we expect statistical samples to be formed from randomly selected, equally weighted groups. In fact, investment funds cannot control the timing of inflows or outflows and studies have shown that retail investor fund flows are a contrary indicator, ie. higher deposits take place when the market level is high and likely to fall and higher withdrawals take place when the market value is low and likely to rise. It becomes apparent then the investment vehicle chosen is somewhat less important than ordinarily assumed in comparison to other aspects of investing which tend to be dismissed as unimportant.

The timing of deposits and withdrawals has a material effect on performance. Due to the resulting weighting of returns, deposits and withdrawals have the result of placing additional weight on certain market periods. Most professional investors agree that it’s next to impossible to time the market. Maybe this is the reason that they place little emphasis on the timing of fund flows. Besides this difficulty, mutual fund managers have no control over cash flows into and out of their fund. These typically poorly-timed retail fund flows may also explain the difficulty that mutual funds have in matching the performance of their benchmarks: their performance is fine, but the retail fund flows overweight “toppy” and falling markets and underweight “cheap” and recovering markets.

The sequence of returns, normally believed to have no impact during saving years, actually matters. In a presentation about the importance of safety for portfolios producing retirement income, the idea of the “sequence of returns” was described as follows. During the saving years, it was explained, the sequence of returns don’t matter because the average of: 2, 7, 15, -8, 20, 8 is the same as the average of: -8, 8, 20, 7, 15, 2. (This is true if we assume an unweighted average.) Withdrawals place far more importance on the sequence of returns, however. When a retiree begins taking withdrawals of 7% (let’s suppose), there is less money left to experience the recovery if early years are negative. The weighted average should be: (100% x -8 + 86% x 8 + 79% x 20 + 72% x 7 + 65% x 15 + 58% x 2)/6. (This is true.) However, what was ignored is that people make deposits during saving years. These deposits weight the returns during this period in the same way, especially when lump sum deposits are made. Suppose an extreme example of a young professional who wants to retire at age 35. He works 8 years, over the course of which he receives bonuses. In the early years, there is little income and next to no deposits due to mortgage payments and the cost of living for a young family. But in years 4, 5, 6 and 7, there are large deposits. Finally, in year 8, there is a very large deposit worth 50% of the account value due to taking a package. The weighted average may look like this: ($5k x 8% + $30k x 10% + $55k x -40%, + $80k x 41% + $150k x 15% + $500k x ??%)/6. These performance numbers are based on the TSX over the past five years. It should be clear why the final year’s performance would be far more important to this investor than any year up to now.

Most investors diversify their holdings in some way. But with diversification comes rebalancing. Besides the cognitive bias that causes investors to cut their winners and let their losses run, rebalancing is another act that impacts the weighted average of returns. An investor who owns 50% stocks and 50% bonds might rebalance annually. When moving funds from stocks to bonds, the investor reduces their exposure to the outperforming asset class (eg. stocks) and increases their exposure to the underperformer (eg. bonds). If and when performance reverts to the mean (eg. stocks crash), this will appear wise, because the poor performance will now affect a smaller proportion of the portfolio. However, if outperformance comes in runs of longer than a year (it has been suggested that four years is more realistic, but this isn’t consistent over time), then increasing the weighting of the underperforming asset class (eg. bonds) will reduce the final dollar-weighted return. The sequence of returns, then, also plays a role in final performance.

How much value can professional investment advice bring to this dilemma? It is a fact that no one knows which investment or asset class will perform best in the future. advisors are no better than individual investors at timing the market. What advisors do well, however, is talk investors out of selling at the bottom. An especially good advisor will caution an investor against overoptimism at the top. A good advisor should also encourage investors to make dollar-weighted returns work for them by recommending deposits soon after market crashes. While we can’t divine future performance, it’s relatively easy to recognize a market crash after-the-fact. Advisors aren’t all worth their fees, but the ones who help investors resist the impulses of fear and greed can add value that far outstrips their cost.

Opposed to conventional wisdom, the sequence of returns is as important, or more important, depending on the size of deposits, for savers as compared to income-seekers in determining the dollar-weighted (real life) returns of their investments. This affects not only the timing of deposits, but also the choice to rebalance between assets and asset classes. Without the ability to predict the future, investors may want to rely on a third party who can help avoid emotional errors, such as reducing risk or withdrawals after market crashes, and instead encourage the opposite behaviour.

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?

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: http://tigger.uic.edu/~gib/simp.pdf (PDF file).

Investing in the stock market requires a person to take a view on the future. Because we are investing for growth, we want to buy where we believe prices will rise. There are two ways of looking at this problem: either looking backward and buying where prices have recently fallen or remained steady, or looking forward and buying where prices are expected to rise. It’s a fairly simple matter to look backward and find out where prices appear low in relation to the past. This is usually considered to be the basis for a “value” style of investing. It’s more difficult to look forward and divine where prices appear low in relation to their future. This is more like the “growth” style of investing.

The future is opaque. While we can make predictions such as the probable result of rising affluence in hugely populous developing nations, the timing and the actual results are impossible to predict. Although the past doesn’t repeat itself, it often rhymes, giving people reason to rely on past experience when making decisions about the future. When trying to determine whether the stock market appears over-valued or under-valued, I can either look backward and determine the apparent value in relation to the recent past, or I can look to accumulated market experience. While the latter approach makes sense, it could be classified as superstition.

Superstitions are widely held beliefs about intractable realities. It is usually impossible to collect evidence to prove or disprove popular superstitions, such as the seven years of “bad luck” brought by breaking a mirror. Perhaps when mirrors were more costly and annual income was lower, there would be a negative financial impact with seven years’ worth of repercussions. For our purposes, however, let’s consider some commonly held beliefs about stock market performance.

I will briefly review three examples of stock market beliefs that may have some ability to detect future conditions. First, the January effect suggests that the performance of the stock market over the course of the year can be predicted by the performance of the market over the first five days or over the entire month of January. This factor has been correct a little over half the time. A notable failure was the conflicting signals given in 2007 and 2009. Second, there is a market dictum that states: “sell in May and go away.” This implies that stock performance is strongest over the course of November to May and weakest over the months of June to October. This has been true more often than not over the last three centuries. When the markets varied from this pattern, however, it was for a period of over 20 years (pre-1920 to 1950).

The last example is technical analysis. Divining patterns in charts, especially if they show only the result of random movements is of questionable value. At best, the charts may show the result of human behavior, which can be expected to follow a familiar pattern. However, when the pattern may develop at varying magnitudes and speeds, I remain extremely skeptical that there chart shapes, even if they do repeat themselves, can be effectively recognized ahead of time, allowing anyone to accurately predict future movements. I am unable to find a study which shows the level of effectiveness of this type of prediction.

Rational decisions are made by considering all known values, not by trying to predict the future. Any endeavor that relies on divining what may happen will tend toward superstition. Beliefs about stock market behavior in the future, either popularly-held or academically researched, hold many similarities with superstition. While predictions of future performance are interesting, they should not be used as a basis for investing. This includes, of course, my weekly stock market outlook, which I produce in an order to gauge what is happen, not predict what will happen.