Relative Frequency of Large, Negative 3-Sigma Returns
5.8 Historic Returns on Risky Portfolios
Figure 5.5 Frequency distribution of annual returns on U.S. Treasury bills, Treasury bonds, and common stocks 40
50 60 70
A: Treasury Bills
0 10 20 30
–45 to –40 –40 to –35 –35 to –30 –30 to –25 –25 to –20 –20 to –15 –15 to –10 –10 to –5 –5 to 0 0 to 5 5 to 10 10 to 15 15 to 20 20 to 25 25 to 30 30 to 35 35 to 40 40 to 45 45 to 50 50 to 55 55 to 60
Percent of Observations (%)
Annual return (%)
(continued)
13You might wonder about the negative T-bill rates that show up in the frequency distribution in Figure 5.5.
T-bills did not make their debut until the 1940s. For earlier dates, commercial paper is used as the closest approxi- mation to short-term risk-free rates. In a few instances they were issued slightly above par and thus yielded
Figure 5.5 (Concluded)
–45 to –40 –40 to –35 –35 to –30 –30 to –25 –25 to –20 –20 to –15 –15 to –10 –10 to –5 –5 to 0 0 to 5 5 to 10 10 to 15 15 to 20 20 to 25 25 to 30 30 to 35 35 to 40 40 to 45 45 to 50 50 to 55 55 to 60
30 25 20 15 10 5 0
Percent of Observations (%)
Annual return (%) B: Treasury Bonds
–45 to –40 –40 to –35 –35 to –30 –30 to –25 –25 to –20 –20 to –15 –15 to –10 –10 to –5 –5 to 0 0 to 5 5 to 10 10 to 15 15 to 20 20 to 25 25 to 30 30 to 35 35 to 40 40 to 45 45 to 50 50 to 55 55 to 60
20
15
10
5
0
Percent of Observations (%)
Annual return (%) C: U.S. Equity Market Index
reflect risk but rather changes in the risk-free rate over time.13 Anyone buying a T-bill knows exactly what the (nominal) return will be when the bill matures, so variation in the return is not a reflection of risk over that short holding period.
While the frequency distribution is a handy visual representation of investment risk, we also need a way to quantify that volatility; this is provided by the standard deviation of returns. Table 5.3 shows that the standard deviation of the return on stocks over this period, 20.28%, was about double that of T-bonds, 10.02%, and more than 6 times that of T-bills. Of course, that greater risk brought with it greater reward. The excess return on stocks (i.e., the return in excess of the T-bill rate) averaged 8.30% per year, providing a generous risk premium to equity investors.
Table 5.3 uses a fairly long sample period to estimate the average level of risk and reward. While averages may well be useful indications of what we might expect going forward, we nevertheless should expect both risk and expected return to fluctuate over time. Figure 5.6 plots the standard deviation of the market’s excess return in each year calculated from the 12 most recent monthly returns. While market risk clearly ebbs and flows, aside from its abnormally high values during the Great Depression, there does not seem to be any obvious trend in its level. This gives us more confidence that historical risk estimates provide useful guidance about the future.
Of course, as we emphasized in the previous sections, unless returns are normally dis- tributed, standard deviation is not sufficient to measure risk. We also need to think about
Table 5.3
Risk and return of investments in major asset classes, 1926–2015
T-Bills T-Bonds Stocks
Average 3.47% 6.00% 11.77%
Risk premium N/A 2.53 8.30
Standard deviation 3.13 10.02 20.28
Max 14.71 40.36 57.53
Min −0.02 −14.90 −44.04
Figure 5.6 Annualized standard deviation of the monthly excess return on the market index portfolio
Source: Authors’ calculations using data from Prof. Kenneth French’s Web site: http://mba.tuck.dartmouth.edu/
pages/faculty/ken.french/data_library.html.
0 10 20 30 40 50 60 70 80
1927 1932 1937 1942 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 2007 2012 2017
Annualized Standard Deviation of Monthly Excess Returns
Figure 5.7 Frequency distribution of monthly excess returns on the market index (the first bar in each set) versus predicted frequency from a normal distribution with matched mean and standard deviation (the second bar in each set)
Source: This frequency distribution is for monthly excess returns on the market index, obtained from Prof. Kenneth French’s Web site, http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. The returns are expressed as continuously compounded rates, as these are most appropriate for comparison with the normal distribution.
0 5
Excess Monthly Return (%) 10
15 20 25 30 35 40
Percent (%)
45 50
–35 to –30 –30 to –25 –25 to –20 –20 to –15 –15 to –10 –10 to –5 –5 to 0 0 to 5 5 to 10 10 to 15 15 to 20 20 to 25 25 to 30 30 to 35
Monthly Excess Returns, Market Index Frequency for a Normal Distribution with Mean and SD Equal to Market Index
“tail risk,” that is, our exposure to unlikely but very large outcomes in the left tail of the probability distributions. Figure 5.7 provides some evidence of this exposure. It shows a fre- quency distribution of monthly excess returns on the market index since 1926. The first bar in each set shows the historical frequency of excess returns falling within each range, while
the second bar shows the frequencies that we would observe if these returns followed a nor- mal distribution with the same mean and variance as the actual empirical distribution. You can see here some evidence of a fat-tailed distribution: The actual frequencies of extreme returns, both high and low, are higher than would be predicted by the normal distribution.
Further evidence on the distribution of excess equity returns is given in Table 5.4.
Here, we use monthly data on both the market index and, for comparison, several “style”
portfolios. You may remember from Chapter 4, Figure 4.5, that the performance of mutual funds is commonly evaluated relative to other funds with similar investment
“styles.” (See the Morningstar style box in Figure 4.5.) Style is defined along two dimensions: size (do the funds invest in large cap or small cap firms?) and value vs.
growth. Firms with high ratios of market value to book value are viewed as “growth firms” because, to justify their high prices relative to current book values, the market must anticipate rapid growth.
The use of style portfolios as a benchmark for performance evaluation traces back to influential papers by Eugene Fama and Kenneth French, who extensively documented
Table 5.4
Statistics for monthly excess returns on the market index and four “style” portfolios
Sources: Authors’ calculations using data from Prof. Kenneth French’s Web site: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/
data_library.html.
Market Index Big/Growth Big/Value Small/Growth Small/Value A. 1926−June 2016
Mean excess return (annualized) 08.30 7.98 11.67 8.79 15.56
Standard deviation (annualized) 18.64 18.50 24.62 26.21 28.36
Sharpe ratio 0.45 0.43 0.47 0.34 0.55
Lower partial SD (annualized) 19.49 18.57 22.78 25.92 25.98
Skew 0.20 −0.10 1.70 0.70 2.19
Kurtosis 7.77 5.55 19.05 7.83 22.21
VaR (1%) actual (monthly) returns −13.95 −14.68 −19.53 −20.59 −20.47
VaR (1%) normal distribution −11.87 −11.80 −15.63 −16.92 −17.87
% of monthly returns more than 3 SD below mean
0.94% 0.75% 0.94% 0.75% 0.57%
Expected shortfall (monthly) −20.14 −20.33 −24.30 −25.02 −25.76
B. 1952−June 2016
Mean excess return (annualized) 7.52 7.18 9.92 7.05 13.34
Standard deviation (annualized) 14.89 15.54 15.95 22.33 18.42
Sharpe ratio 0.50 0.46 0.62 0.32 0.72
Lower partial SD (annualized) 16.51 15.67 16.01 23.79 19.36
Skew −0.52 −0.36 −0.29 −0.36 −0.35
Kurtosis 1.90 1.81 2.26 2.17 3.48
VaR (1%) actual (monthly) returns −10.80 −10.90 −11.94 −16.93 −15.21
VaR (1%) normal distribution −9.37 −9.84 −9.89 −14.41 −11.26
% of monthly returns more than 3 SD below mean
0.66% 0.66% 0.80% 0.93% 1.19%
Expected shortfall (monthly) −18.85 −17.99 −21.30 −24.66 −28.33
that firm size and the book value-to-market value ratio predict average returns; these patterns have since been corroborated in stock exchanges around the world.14 A high book-to-market (B/M) ratio is interpreted as an indication that the value of the firm is driven primarily by assets already in place, rather than the prospect of high future growth. These are called “value” firms. In contrast, a low book-to-market ratio is typical of firms whose market value derives mostly from ample growth opportunities. Realized average returns, other things equal, historically have been higher for value firms than for growth firms and for small firms than for large ones. The Fama-French database includes returns on portfolios of U.S. stocks sorted by size (Big; Small) and by B/M ratios (High; Medium; Low).15
Following the Fama-French classifications, we drop the medium B/M portfolios and identify firms ranked in the top 30% of B/M ratio as “value firms” and firms ranked in the bottom 30% as “growth firms.” We split firms into above and below median levels of mar- ket capitalization to establish subsamples of small versus large firms. We thus obtain four comparison portfolios: Big/Growth, Big/Value, Small/Growth, and Small/Value.
Table 5.4, Panel A, presents results using monthly data for the full sample period, July 1926–June 2016. The top two lines show the annualized average excess return and stan- dard deviation of each portfolio. The broad market index outperformed T-bills by an aver- age of 8.30% per year, with a standard deviation of 18.64%, resulting in a Sharpe ratio (third line) of 8.30/18.64 = .45. In line with the Fama-French analysis, small/value firms had the highest average excess return and the best risk–return trade-off with a Sharpe ratio of .55. However, Figure 5.5 warns us that actual returns may have fatter tails than the nor- mal distribution, so we need to consider risk measures beyond just the standard deviation.
The table therefore also presents several measures of risk that are suited for non-normal distributions.
Several of these other measures actually do not show meaningful departures from the symmetric normal distribution. Skew is generally near zero; if downside risk were substan- tially grater than upside potential, we would expect skew to be generally negative, which it is not. Along the same lines, the lower partial standard deviation is generally quite close to the conventional standard deviation. Finally, while the actual 1% VaR of these portfolios are uniformly higher than the 1% VaR that would be predicted from normal distributions with matched means and standard deviations, the differences between the empirical and predicted VaR statistics are not large. By this metric as well, the normal appears to be a decent approximation to the actual return distribution.
However, there is other evidence suggesting fat tails in the return distributions of these portfolios. To begin, note that kurtosis (the measure of the “fatness” of both tails of the distribution) is uniformly high. Investors are, of course, concerned with the lower (left) tail of the distribution; they do not lose sleep over their exposure to extreme good returns!
Unfortunately, these portfolios suggest that the left tail of the return distribution is over- represented compared to the normal. If excess returns were normally distributed, then only .13% of them would fall more than 3 standard deviations below the mean. In fact, the actual incidence of excess returns below that cutoff are at least a few multiples of .13% for each portfolio.
The expected shortfall (ES) estimates show why VaR is only an incomplete measure of downside risk. ES in Table 5.4 is the average excess return of those observations that fall
14This literature began in earnest with their 1992 publication: “The Cross Section of Expected Stock Returns,”
Journal of Finance 47, 427–465.
15We use the Fama-French data to construct Figures 5.4 and 5.5 and Tables 5.3 and 5.4. The database is available at: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
in the extreme left tail, specifically, those that fall below the 1% VaR. By definition, this value must be worse than the VaR, as it averages among all the returns that are below the 1% cutoff. Because it uses the actual returns of the “worst-case outcomes,” ES is by far a better indicator of exposure to extreme events.
Figure 5.2 showed us that the post-war years (more accurately, the years after 1951) have been far more predictable, at least with respect to interest rates. This suggests that it may be instructive to examine stock returns in the post-1951 period as well to see if risk and return characteristics for equity investments have changed meaningfully in the more recent period. The relevant statistics are given in Panel B of Table 5.4. Perhaps not surpris- ingly in light of the history of inflation and interest rates, the more recent period is in fact less risky. Standard deviation for all five portfolios is noticeably lower in recent years, and kurtosis, our measure of fat tails, drops dramatically. VaR also falls. While the number of excess returns that are more than 3 SD below the mean changes inconsistently, because SD is lower in this period, those negative returns are also less dramatic: Expected shortfall generally is lower in the latter period.
The frequency distribution in Figure 5.5 and the statistics in Table 5.4 for the market index as well as the style portfolios tell a reasonably consistent story. There is some, admit- tedly inconsistent, evidence of fat tails, so investors should not take normality for granted.
On the other hand, extreme returns are in fact quite uncommon, especially in more recent years. The incidence of returns on the market index in the post-1951 period that are worse than 3 SD below the mean is .66%, compared to a prediction of .13% for the normal dis- tribution. The “excess” rate of extreme bad outcomes is thus only .53%, or about once in 187 months (15 1 ⁄ 2 years). So it is not unreasonable to accept the simplification offered by normality as an acceptable approximation as we think about constructing and evaluating our portfolios.