Reducing the Risk of Black Swans

by Larry Swedroe

  The Three Factors

  Before looking at the returns to the market beta, size and value factors, we need to go over a few key points. The first is that factor returns are always expressed in terms of annual average returns, not compound or annualized returns. If we have 90 years of data, we calculate a factor’s return over that period by taking the sum of the returns in each of those 90 years and dividing the total by 90. The second point is that factors are always considered long-short portfolios. Thus:

  The market beta factor is the average annual return on the total stock market (the long) minus the average annual return on Treasury bills (the short). From 1927 through 2016, the average annual return of market beta has been 8.4 percent.

  The size factor is the annual average return on small-cap stocks (the long) minus the average annual return on large-cap stocks (the short). Using the Center for Research in Securities Prices at the University of Chicago (CRSP) data, small stocks are those in deciles 6-10 of all stocks sorted by market capitalization and large stocks are those in deciles 1-5. From 1927 through 2016, small-cap companies have outperformed large-cap companies by an annual average of 3.3 percent per year.

  The value (or price) factor is the average annual return on value stocks (the long) minus the average annual return on growth stocks (the short). Ranking stocks by their book-to-market values, value stocks are the 30 percent of stocks with the highest book-to-market values and growth stocks are the 30 percent with the lowest book-to-market values. From 1927 through 2016, value companies have outperformed growth companies by an annual average of 5.1 percent per year.

  Independent Risk Factors

  An important contribution Fama and French made to the evolution of asset pricing models was to show that size and price (value) are independent (unique) risk factors in that they give investors exposure to different risks than exposure to market risk (market beta) does. Evidence of this independence can be seen when we examine the historical correlations of the size and price factors to the market factor. High correlations would mean the risk factors would be relatively good substitutes for each other. If that were the case, while investors could increase the expected return (and risk) of their portfolio by increasing their exposure to these risk factors, there would be little real diversification benefit. If the correlations are low, investors could both increase expected returns for a given level of risk and gain a diversification benefit. Thus, finding factors with low correlation provides valuable information we can use to build more efficient portfolios.

  Because most people have an incorrect understanding of the term correlation (even many of the professional advisors we have met), before exploring the data we need to make sure you know and fully comprehend the definitions of positive and negative correlation.

  A positive correlation exists between two assets when one asset produces above-average returns (relative to its average) and the other asset tends to also produce above-average returns (relative to its average). The stronger the tendency, the closer the correlation will be to +1.

  While most people seem to believe negative correlation means that when one asset increases in value the other falls in value, it actually means that when one asset produces above-average returns (relative to its average), the other asset tends to produce below-average returns (relative to its average). The stronger the tendency, the closer the correlation will be to -1.

  If the correlation is 0, two assets would be said to be uncorrelated. That means that when one asset produces above average-returns relative to its average, the other asset is just as likely to also produce above-average returns relative to its average as it is likely to produce below-average returns relative to its average.

  Some examples will help clarify this concept.

  Example 1:

  Consider two assets, A and B, and their returns over a 10-year period. Their return series is depicted in the following table.

  Both assets A and B have an annual average return of 10 percent. Whenever A’s return is above its average of 10 percent, B’s return is below its average of 10 percent. And whenever A’s return is below its average of 10 percent, B’s return is above its average of 10 percent. Thus, the assets are negatively correlated. Note that they are negatively correlated even though they both always produce positive returns.

  Example 2:

  Both assets A and B in this series have an average annual return of 0 percent. Whenever A’s return is above its average of 0 percent, B’s return is below its average of 0 percent. And whenever A’s return is below its average of 0 percent, B’s return is above its average of 0 percent. Thus, again we see that the two assets are negatively correlated.

  Now comes the fun part. We will string together the two examples so that we have a 20-year period. The first 10 years are the returns from Example 1, and the second 10 years are from Example 2. Thus, the return series looks like this:

  Asset A: 12, 8, 12, 8, 12, 8, 12, 8, 12, 8, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2

  Asset B: 8, 12, 8, 12, 8, 12, 8, 12, 8, 12, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2

  Recall that both A and B had average returns in the first 10 years of 10 percent, and average returns of 0 percent in the second 10 years. Thus, their average return for the full 20 years in both cases is 5 percent. Now, recall our definitions. If you are not sure, go back and read them again before attempting to answer this question: Are A and B positively or negatively correlated?

  With the definitions in mind, we see that whenever A’s return was above its average of 5 percent, B’s return was also above its average of 5 percent. And whenever A’s return was below its average of 5 percent, B’s return was also below its average of 5 percent. Thus, we see that, despite the fact A and B were negatively correlated over each of the two 10-year sub-periods, over the full 20-year period they were positively correlated. Besides illustrating the concepts of positive and negative correlation, we hope you also come away with the understanding that you need long-term data series for correlations to have any real meaning. In addition, it is important to understand that correlations of risky assets have a tendency to drift over time. As 2008 demonstrated, the correlation of all risky assets has a nasty tendency to move toward 1 during crises. Thus, when considering an asset for inclusion in a portfolio, you need not only to think about the asset’s long-term correlation with other portfolio assets, but also when the correlation tends to rise and when it tends to fall.

  With these important understandings, we now turn to examining the long-term correlation of the three Fama-French factors. The following data shows the annual correlations of returns to the market beta, size and value factors from 1927 through 2016.

  What we find is that the size factor has a correlation of about 0.3 to market beta. Recalling our definitions, there is a tendency, though not a very strong one, that whenever the market beta factor (the return of stocks minus the return of one-month Treasury bills) produces a return greater than 8.4 percent (its average premium), small-cap stocks will outperform large-cap stocks by more than the average premium of 3.3 percent a year. On the other hand, there will also be a significant number of years when the return on market beta is greater than its average of 8.4 percent and the size premium will be below 3.3 percent, including many years when the size premium will be negative (large-cap stocks will outperform small-cap stocks).

  We also see that the value factor has a correlation to the market beta factor of -0.2. Again recalling our definitions, whenever the market beta factor is greater than 8.4 percent, there is a small tendency for the value premium to be less than its average of 5.1 percent, and vice versa.

  Finally, the correlation of the size factor to the value factor was 0.0—their returns are uncorrelated. Thus, when the size premium is greater than its average of 3.3 percent, the value premium is just as likely to be greater than its average of 5.1 percent as it is to be below it.

  The low to negative correlations of these factors become apparent when we look at their returns in 1998 and 2001. The
following table shows the returns of the S&P 500 Index and two Dimensional Fund Advisors (DFA) asset class mutual funds, the DFA U.S. Small Cap Portfolio and the DFA U.S. Small Cap Value Portfolio. The table also shows the return of a portfolio holding one-third of its assets in each of the three.

  Note that the gap between the good and bad years was 40.5 percentage points for the S&P 500 Index, 18.2 percentage points for the DFA U.S. Small Cap Portfolio and 30 percentage points for the DFA U.S. Small Cap Value Portfolio, but just 2.5 percentage points for the equal-weighted portfolio. This simple example conveys the benefits of diversifying across factors that have low correlation—you dampen the volatility of the portfolio. Dampening volatility is especially important to those in the withdrawal phase of their investment lifecycle, when the order of returns matters a great deal because higher volatility can greatly affect the odds of outliving your assets.

  The bottom line is that our three Fama-French factors have low correlations to each other. In fact, the size and value factors have been uncorrelated (their correlation was 0). That is good news, which we will use to build more efficient portfolios.

  Achieving Your Goals in a CAPM World

  In a CAPM (or one-factor) framework, the only ways to increase the expected return of your portfolio are to increase your allocation to stocks or to purchase higher-market-beta stocks. In either case, you are not diversifying the sources of the portfolio’s return—just adding more market beta to it.

  An example can illustrate this point. Let’s assume equities (as represented by a total stock market fund) are expected to return 7 percent and bonds (as represented by the yield on, say, a five-year Treasury bond) are expected to return 5 percent. We’ll use these figures to keep the math simple. Based upon your ability and willingness to take risk, you decide that a portfolio with an allocation of 50 percent stocks and 50 percent bonds would be appropriate. Such a portfolio would have an expected return of 6 percent. However, the desired spending component of your long-term financial plan requires a 6.5 percent return. In a one-factor world, to achieve the expected return of 6.5 percent, you have to increase your equity allocation to 75 percent.

  Portfolio 1: (75% x 7%) + (25% x 5%) = 6.5%

  The only other alternative would be to increase the market beta of the stocks in your portfolio. The math works out in such a way that you would be left owning a portfolio consisting basically of only very high-market-beta stocks. In either case, you would be adding more of the same market beta risk already in your portfolio. Instead putting all your eggs in one risk basket (market beta), would it not be better to diversify your sources of risk across other baskets with unique risks? Another important consideration is that the portfolio with 75 percent stocks is riskier than the one with just 50 percent stocks—the allocation you felt was appropriate based on your risk tolerance.

  Achieving Your Goals in a Three-Factor World

  The science of investing offers an alternative way to increase the expected return of your portfolio. Since 1927, small value stocks have outperformed the market (as represented by the S&P 500 Index) by an annualized 3.8 percentage points. Thus, if we assume stocks will return 7 percent, we might assume small value stocks will return an additional 3.8 percentage points a year for a total return of 10.8 percent.

  Recall our initial example of a 50 percent stock/50 percent bond allocation. Instead of increasing your stock allocation to achieve the higher 6.5 percent return, you decide to divide your 50 percent stock allocation equally between the S&P 500 Index and small value stocks—25 percent each. Now the expected return is almost 7 percent.

  Portfolio 2: (25% x 7%) + (25% x 10.8%) + (50% x 5%) = 6.95%

  Without increasing your stock allocation, you have increased the portfolio’s expected return to more than the required 6.5 percent. We did this by adding an allocation to the higher expected returning small value stocks. It is important to recognize that while your exposure to stocks basically remained unchanged (the allocation of small value stocks, while riskier, did add two new unique risk factors, providing some diversification benefit), the expected return of the portfolio increased by more than the risk (the expected standard deviation of return). However, you don’t need to earn 6.95 percent. Your plan only requires a return of 6.5 percent. With that in mind, we can try lowering the stock allocation to 40 percent, again splitting it equally between the S&P 500 Index and small value stocks. Now the expected return is:

  Portfolio 3: (20% x 7%) + (20% x 10.8%) + (60% x 5%) = 6.56%

  You have now achieved your goal of an expected return of 6.5 percent. And you did so with an allocation to stocks of only 40 percent. Using your intuition, which portfolio, Portfolio 1 or Portfolio 3, do you think is riskier? Which portfolio would you expect to perform worse in a bear market? Intuitively, most people will say Portfolio 1.

  While Portfolio 3 has the same expected return as the 75 percent equity portfolio, the risks are completely different. Portfolios with higher equity allocations have greater potential for losses. The tradeoff is that the potential upside of the portfolios with higher equity allocations is much greater. For investors for whom the pain of a loss is greater than the benefit of an equal-sized gain (probably you), reducing downside risk as the price of reducing upside potential is a good tradeoff.

  Let’s tackle another consideration especially important to loss-averse investors (which most are). Because bonds are safer investments than stocks, in a severe bear market the portfolio’s maximum loss would likely be far lower with a 40 percent equity allocation than with a 75 percent one. 2008 provided a great example, at least if the fixed income investments you owned were limited to Treasuries and other high-quality bonds. While the market fell 37 percent, five-year Treasury bonds rose about 13 percent. And Portfolio 3 not only owned less of the losing stocks, but far more of the winning bonds.

  Using the S&P 500 Index, the Russell 2000 Value Index (for small-cap value stocks) and the five-year Treasury, we see that in 2008, Portfolio 1 would have lost 24.5 percent, Portfolio 2 would have lost 9.9 percent and Portfolio 3 would have lost just 5.3 percent.

  Thus, while the expected returns of Portfolio 1 and Portfolio 3 are the same, the portfolio with the lower equity allocation has much less downside risk. Of course, the upside potential during a strong bull market is correspondingly lower. For example, in 2003, Portfolio 1 gained 35.1 percent, Portfolio 2 gained 19.9 percent and Portfolio 3 gained just 16.4 percent.

  To illustrate why reducing downside risk at the price of limiting upside potential is likely a good tradeoff, examine the chart below.

  As you can see in the following illustration, Portfolio 1 experienced more years in which returns fall on the left-hand side of the chart, as well as more years in which they fall on the right-hand side of the chart—the tails were bigger.

  Considerations

  When constructing a portfolio—deciding on the right asset allocation—you should carefully consider several factors. For example, you should consider how your labor capital correlates with the greater economic cycle risks of small-cap and value stocks compared to large-cap and growth stocks.

  Another important consideration is a psychological one referred to as the risk of tracking error regret. Tracking error is the amount by which a portfolio’s performance varies from that of the total market or other broad market benchmark, such as the S&P 500 Index. By diversifying across risk factors, investors take on increased tracking error regret risk because their portfolios look less like the market. While very few investors care when tracking error is positive (their portfolio beats the benchmark), many care very much when it is negative. Misery loves company. In other words, if your portfolio performs poorly because “the market” has performed poorly, at least you have company. On the other hand, if your portfolio is underperforming the market, you might begin to question your strategy, wondering why you are doing relatively poorly.

  1998 provided the perfect example of this, as the DFA Small Value Portfolio underperformed the S&P 500
Index by about 36 percentage points. Such underperformance can test the mettle of even the most disciplined investors. Compounding the problem is that, to adhere to your asset allocation plan, you would be required to rebalance the portfolio—selling some of the S&P 500 Index allocation (having just seen it gain almost 29 percent) to buy some of the poorly performing DFA Small Value Portfolio (which just lost more than 7 percent). Because most investors chase returns, they would be far more likely to increase their allocation to the S&P 500 Index and get rid of their small value allocation. Of course, doing so would be pursuing a strategy of buying high and selling low—not exactly a prescription for investment success. It is why the evidence shows that investors, on average, underperform the very mutual funds in which they invest. On the other hand, investors who remained disciplined and rebalanced would have bought more small value stocks at low prices and sold some of their S&P 500 Index holdings at high prices, a far more successful strategy. It also would have left them much better prepared for 2001, because they already would have sold some of their S&P 500 Index holdings (which performed poorly then) and increased their small value holdings (which performed well).