We can now apply the analytical tools worked out in previous sections to five important risky portfolios:
1. World large stocks. This portfolio includes all available country index portfolios (48 countries in 2009). Each index is composed of that country’s large stocks (e.g., firms in the S&P 500 for the U.S.), each weighted by market capitalization (the market value of all outstanding shares). These country-index portfolios are then weighted by the country market capitalization to form a world stock portfolio.
2. U.S. large stocks. This portfolio is composed of the stocks in the S&P 500 index and is capitalization weighted. These corporations account for about 75% of the market value of all publicly traded stocks in the U.S., and an even larger share of the U.S. GDP.
3. U.S. small stocks. This index measures the return on the smallest quintile (i.e., smallest 20% ranked by capitalization) of U.S. publicly traded stocks.
4. U.S. long-term government bonds. Long-term T-bond returns are measured by the Barclays Capital (formerly Lehman Brothers) value-weighted index of all outstand- ing U.S. government bonds of 10 years and longer maturity.
In 2008 a typical investment portfolio of 60% stocks and 40% bonds lost roughly a fifth of its value. Standard portfolio-construction tools assume that will happen only once every 111 years. Though mathematicians and many investors have long known market behavior isn’t a pretty picture, standard portfolio construction assumes returns fall along a tidy, bell-curve-shaped distribution. With that approach, a 2008-type decline would fall near the skinny left tail, indicating its rarity.
Recent history would suggest such meltdowns aren’t so rare. In a little more than two decades, investors have been buffeted by the 1987 market crash, the implosion of hedge fund Long-Term Capital Management, the bursting of the tech-stock bubble, and other crises.
Many of Wall Street’s new tools assume market returns fall along a “fat-tailed” distribution, where, say, last year’s nearly 40% stock-market decline would be more com- mon than previously thought. These new assumptions present a far different picture of risk. Consider the 60%
stock, 40% bond portfolio that fell about 20% last year.
Under the fat-tailed distribution, that should occur once every 40 years, not once every 111 years as assumed under a bell-curve-type distribution. (The last year as bad as 2008 was 1931.)
One potential pitfall: Number-crunchers have a smaller supply of historical observations to construct models
focused on rare events. “Data are intrinsically sparse,” says Lisa Goldberg, executive director of analytic initiatives at MSCI Barra.
Many of the new tools also limit the role of conven- tional risk measures. Standard deviation, proposed as a risk measure by Nobel Prize–winning economist Harry Markowitz in the 1950s, can be used to gauge how much an investment’s returns vary over time. But it is equally affected by upside and downside moves, whereas many investors fear losses much more than they value gains. And it doesn’t fully gauge risk in a fat-tailed world.
A newer measure that has gained prominence in recent decades ignores potential gains and looks at downside risk.
That measure, called “value at risk,” might tell you that you have a 5% chance of losing 3% or more in a single day, but it doesn’t home in on the worst downside scenarios.
To focus on extreme risk, many firms have begun using a measure called “expected shortfall” or “conditional value at risk,” which is the expected portfolio loss when value at risk has been breached. Conditional value at risk helps estimate the magnitudes of expected loss on the very bad days. Firms such as J.P. Morgan and MSCI Barra are employing the measure.
Source: Eleanor Laise, “Some Funds Stop Grading on the Curve,”
The Wall Street Journal, September 8, 2009, p. C1. Reprinted by permission of The Wall Street Journal, © 2009.
WORDS FROM THE STREET
5. A diversified portfolio. Our most inclusive portfolio is a mixture of the other portfolios with weights of 50% in world large stocks, 20% in U.S. small stocks, and 30% in long-term U.S. government bonds. A common asset allocation s uggested for “average” investors is 70% in stocks and 30% in bonds. In our diversified portfolio, the 70% weight in stocks is placed primarily in world large stocks (of which about half are large U.S. stocks), with 20% allocated to the riskier portfolio of small stocks. Regardless of whether this is an optimal portfolio, it allows us to examine whether diversification beyond the first four portfolios, which are themselves already well diversified, can further enhance the risk–return trade-off.
Figure 5.6 gives a pictorial view of the dispersion of returns of each portfolio, as well as T-bills, over the 84-year period ending in 2009. Clearly, the risk of large losses progres- sively increases as we move from T-bills to long-term bonds, to large stocks, and finally to small stocks. The further diversified Portfolio 5 already emerges as offering a better risk-return trade-off. Interestingly, the superimposed graphs of normal distributions with the same mean and standard deviation as each portfolio shows that exposure to extreme losses greater than those consistent with normality, if they exist at all, are too minor to be discerned with a naked eye.
Table 5.3 summarizes all the relevant statistics for the five risky portfolios from the entire 84-year period (84 observations), as well as for the 42 observations from the more recent period (1968–2009) and earlier (1926–1967) subperiods. The table is quite dense, but we’ll highlight its major implications section by section.
Total Returns
One notable feature of total return is the difference between the arithmetic and geometric averages. Recall that Equation 5.15 tells us that for normally distributed returns, the geo- metric average return equals the arithmetic average minus ẵ 2 . With only one exception (small stocks), this relationship is nearly satisfied. While the difference between the geo- metric return and the arithmetic return minus ẵ 2 for small stocks ranges from .29% to 1.91%, for no other portfolio does this difference exceed .17%. This is the first clue that despite the issues raised by the financial crisis of 2008, we will find no strong evidence that the normal distribution fails to adequately describe portfolio returns.
Excess Returns
Our major focus is on excess returns. Average excess returns over the recent half of the history (1968–2009) appear quite different from those of the earlier half period. Except for U.S. long-term government bonds, recent averages are lower, despite the 10 years in the earlier period spanned by the Great Depression. For example, the average excess return of world large stocks fell from 9.02% in the earlier half of the sample to 6.02% in the more
Figure 5.6 Frequency distribution of annual rates of return, 1926–2009
Source: Prepared from data in Table 5.3.
T-Bills
0−90 −60 −30 0 30 60 90
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Long-Term T-Bonds
0−90 −60 −30 0 30 60 90
0.1 0.2 0.3 0.4
0 0.1 0.2 0.3 0.4
0 0.1 0.2 0.3 0.4
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
World Stock Portfolio
−90 −60 −30 0 30 60 90
Large U.S. Stocks
−90 −60 −30 0 30 60 90
Small U.S. Stocks
−90 −60 −30 0 30 60 90
Diversified Portfolio
−90 −60 −30 0 30 60 90
Arithmetic Mean = 3.71 Standard Deviation = 3.09
Arithmetic Mean = 11.23
Standard Deviation = 19.27 Arithmetic Mean = 11.63
Standard Deviation = 20.58
Arithmetic Mean = 10.77 Standard Deviation = 16.02 Arithmetic Mean = 17.43
Standard Deviation = 37.18
Arithmetic Mean = 5.60 Standard Deviation = 8.01
Statistic Period
World Large Stocks
U.S. Large Stocks
U.S. Small Stocks
Long-Term
U.S. T-Bonds Diversified*
Total return
Arithmetic average 1926–2009 11.23 11.63 17.43 5.69 10.81
1968–2009 11.77 10.89 13.47 8.44 11.11
1926–1967 10.69 12.38 21.39 2.95 10.51
SD 1926–2009 19.27 20.56 37.18 8.45 15.79
1968–2009 19.36 17.95 27.41 10.34 14.10
1926–1967 19.41 23.07 44.89 4.70 17.49
Geometric average 1926–2009 9.43 9.57 11.60 5.37 9.66
1968–2009 9.90 9.32 10.00 7.96 10.18
1926–1967 8.97 9.82 13.22 2.84 9.14
Arithmetic average minus
1/2 variance 1926–2009 9.37 9.52 10.52 5.34 9.56
1968–2009 9.90 9.27 9.72 7.90 10.12
1926–1967 8.81 9.71 11.31 2.84 8.98
Excess return
Average 1926–2009
1968–2009 1926–1967
7.52 7.92 13.72 1.99 7.10
6.02 5.14 7.73 2.69 5.36
9.02 10.71 19.72 1.28 8.84
SD 1926–2009 19.54 20.81 37.75 8.24 16.19
1968–2009 19.69 17.93 27.76 10.49 14.49
1926–1967 19.52 23.23 45.17 5.13 17.73
LPSD 1926–2009 19.43 18.43 26.86 6.64 16.40
1968–2009 21.09 17.82 25.15 7.92 16.51
1926–1967 17.30 19.07 28.47 4.92 16.25
Skew 1926–2009 0.18 0.27 0.78 0.52 0.15
1968–2009 0.64 0.57 0.03 0.39 0.45
1926–1967 0.29 0.29 0.70 0.25 0.39
Kurtosis 1926–2009 0.98 0.12 1.57 0.67 1.42
1968–2009 0.03 0.29 0.41 0.43 0.27
1926–1967 2.01 0.22 0.76 0.18 1.92
Serial correlation 1926–2009 0.04 0.02 0.14 0.11 0.05
1968–2009 0.06 0.04 0.09 0.16 0.11
1926–1967 0.12 0.02 0.16 0.09 0.14
Performance
Sharpe ratio 1926–2009 0.38 0.38 0.36 0.24 0.44
1968–2009 0.31 0.29 0.28 0.26 0.37
1926–1967 0.46 0.46 0.44 0.25 0.50
Sortino ratio 1926–2009 0.39 0.43 0.51 0.30 0.43
1968–2009 0.29 0.29 0.31 0.34 0.32
1926–1967 0.52 0.56 0.69 0.26 0.48
VaR actual 1926–2009 25.56 33.19 49.68 11.30 21.97
1968–2009 32.71 33.34 41.12 12.92 24.32
1926–1967 24.33 35.52 54.14 7.48 32.11
continued Table 5.3
Statistics from the history of portfolio returns, 1926–2009
recent half. Is this decline statistically significant? In other words, could the difference be caused by statistical fluctuation, or is it more likely to reflect a true shift in mean? We need to compare the difference in average return to likely sampling error.
Recall from your statistics class that the standard error of the estimate of a mean return is the standard deviation across observations divided by the square root of the number of observations. As a rough cut, we may take the SD of the world large stock portfolio as 19.6%, implying that the SD of the average excess return over a 42-year period would be 19.6/"4253%. Therefore, the difference of average excess return across the two subperiods, 6.02% versus 9.02%, is exactly 1 SD. Such an outcome could easily be due to chance. Similarly, none of the differences in average excess returns across the two subpe- riods for the other portfolios is significant.
Another noticeable difference between the subperiods is in the SD of small stocks and long-term bonds. For small stocks, the marked decline in SD, from 45.17% to 27.76%, is largely explained by the fact that market capitalizations of “small” stocks in the recent period are much larger than in the earlier one. The lower SD is also consistent with the decline in the small stock risk premium, from 19.72% to 7.73%. The reverse happened to long-term bonds. The standard deviation of long-term rates almost doubled, and so did the average risk premium.
Turning next to deviations from normality, we first compare the conventional standard deviation with the lower partial standard deviation, or LPSD (which measures volatility using only negative deviations from the risk-free rate). We see that, generally speaking, the LPSD is roughly the same as the conventional standard deviation, which does not suggest fat lower tails. Similarly, kurtosis (which directly measures fat tails) is near zero (in fact is negative) for all portfolios in the recent subperiod. Kurtosis was indeed positive for large world stocks and small stocks in the earlier subperiod. 11 We do observe negative skewness in the recent period, albeit the greatest skewness (0.64 for world large stocks) is still well below the level used in Figure 5.5 A, suggesting the skews are not all that large. In sum, we detect no serious deviations from normality in these historical returns.
11 Merging time series with different averages can significantly increase apparent kurtosis. This explains the posi- tive kurtosis for bonds in the overall period, despite it being negative in each of the subperiods.
VaR normal 1926–2009 24.62 26.31 48.37 11.56 19.53
1968–2009 26.36 24.35 37.94 13.58 19.37
1926–1967 23.08 27.51 54.57 7.16 20.32
Expected shortfall actual 1926–2009 35.94 38.68 53.19 12.32 27.51
1968–2009 37.28 35.40 42.62 13.15 26.70
1926–1967 30.13 39.50 54.58 8.90 25.40
Expected shortfall normal 1926–2009 30.57 32.40 49.60 13.28 25.44
1968–2009 33.19 30.36 43.25 16.33 25.69
1926–1967 27.95 34.10 54.64 9.07 25.29
* Portfolio constructed of world large stocks (50%), U. S. small stocks (20%), and long-term U.S. government bonds (30%).
Sources: World portfolio: Datastream; small stocks: Fama & French 1st quantile; large stocks: S&P 500; long-term government bonds:
Barclay Capital Long-term Treasury index.
Table 5.3 —concluded
Statistics from the history of portfolio returns, 1926–2009
Serial correlation in excess returns (either positive or negative) would suggest that investors can predict future from past returns and profit from reallocating funds across portfolios on the basis of past returns. Hence, we expect serial correlation of returns in well-functioning capital markets to be negligible. Indeed, the low level of serial correlation in the data assures us that investors cannot expect to produce higher returns on the basis of a rule as simple as serial correlation.
Performance
When asset returns are normally distributed, investors may assess performance using the reward-to-volatility ratio, that is, the Sharpe ratio, which divides average excess returns by standard deviation. The investments we are looking at are all passive, broad asset-class portfolios, that is, they are constructed without any special information about potential superior performance of any individual asset or portfolio. Under these circumstances, we would expect better-diversified portfolios to perform better. The Sharpe ratios of the five portfolios live up to this expectation; for the overall period as well as for each subperiod, the “diversified” portfolio shows the highest Sharpe ratio.
The bond portfolio is least diversified, forgoing all stocks. Thus, despite being the least risky, with the lowest SD by far, it has the lowest Sharpe ratio. The small-stock portfolio, with the highest average excess return but also the highest SD, is the next-worst performer since it is not sufficiently diversified. The bond and small-stock portfolios do improve performance, however, when added to the world large-stock portfolio to achieve broader diversification.
The question is whether deviations from normality would reverse this conclusion.
Because the LPSD is generally lower than SD, particularly over the earlier period, replac- ing the Sharpe ratio with the Sortino ratio (the ratio of average excess return to LPSD) actually increases the performance measure of the stock portfolios. This leaves us with a conundrum: the recent financial crisis has led some to argue that deviations from normality render portfolios riskier and performance inferior to what we measure with the assumption of normality. But, if anything, these results suggest the opposite.
Nevertheless, turning to risk measures that directly focus directly on extreme outcomes, value at risk (VaR) and expected shortfall (ES), we see from the following tables that esti- mates of these measures for large stocks from the historical record display more extreme losses than would be expected under the assumption of normality. That is, actual VaR values as well as expected shortfall are higher than would be observed under the normal distribution. As neither LPSD nor kurtosis is out of line with normality, it seems that these results must be due to the negative skew of historical returns. We must treat these conclu- sions as tentative, however, since these estimates are measured with considerable impreci- sion. Notice also that for the most diversified portfolio, the differences are smaller, since they are driven by the large-stock portfolios.
Difference between Expected VaR Assuming a Normal Distribution and the Historical VaR
World Large Stocks
U.S. Large Stocks
U.S. Small Stocks
Long-Term
U.S. T-Bonds Diversified
1926–2009 0.94 6.88 1.32 0.26 2.45
1968–2009 6.35 8.99 3.18 0.67 4.95
1926–1967 1.24 8.02 0.43 0.33 1.06
Difference between Expected Shortfall Assuming a Normal Distribution and the Historical ES
World Large Stocks
U.S. Large Stocks
U.S. Small Stocks
Long-Term
U.S. T-Bonds Diversified
1926–2009 5.37 6.28 3.59 0.95 2.07
1968–2009 4.09 5.04 0.63 3.18 1.01
1926–1967 2.18 5.40 0.06 0.17 0.11
Our conclusion from the historical record is that the data do not decisively refute portfo- lio management techniques that assume normality of returns. Still, there is some evidence of greater exposure to extreme negative outcomes than would be the case under the normal distribution. The possibility of extreme negative values should lead investors to lower their allocation to risky assets in favor of risk-free vehicles.
A Global View of the Historical Record
As financial markets around the world grow and become more transparent, U.S. inves- tors look to improve diversification by investing internationally. Foreign investors that t raditionally used U.S. financial markets as a safe haven to supplement home-country investments also seek international diversification to reduce risk. The question arises as to how historical U.S. experience compares with that of stock markets around the world.
Figure 5.7 shows a century-long history (1900–2000) of average nominal and real returns in stock markets of 16 developed countries. We find the United States in fourth place in terms of average real returns, behind Sweden, Australia, and South Africa.
Figure 5.7 Nominal and real equity returns around the world, 1900–2000
Source: Elroy Dimson, Paul Marsh, and Mike Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton:
Princeton University Press, 2002), p. 50. Reprinted by permission of the Princeton University Press.
Annualized Percentage Return
15
Real Nominal
Bel
2.5 8.2
2.7 12.0
3.6 9.7
3.6 10.0
3.8 12.1
4.5 12.5
4.6 8.9
4.8 9.5
5.0 7.6
5.8 10.1
5.8 9.0
6.4 9.7
6.7 10.1
6.8 12.0
7.5 11.9
7.6 11.6
Ita Ger Spa Fra Jap Den Ire Swi U.K. Neth Can U.S. SAf Aus Swe 12
9
6
3
0
Figure 5.8 shows the standard deviations of real stock and bond returns for these same countries. We find the United States tied with four other countries for third place in terms of lowest standard deviation of real stock returns. So the United States has done well, but not abnormally so, compared with these countries.
One interesting feature of these figures is that the countries with the worst results, mea- sured by the ratio of average real returns to standard deviation, are Italy, Belgium, Germany, and Japan—the countries most devastated by World War II. The top-performing countries are Australia, Canada, and the United States, the countries least devastated by the wars of the twenti- eth century. Another, perhaps more telling feature is the insignificant difference between the real returns in the different countries. The difference between the highest average real rate (Sweden, at 7.6%) and the average return across the 16 countries (5.1%) is 2.5%. Similarly, the difference between the average and the lowest country return (Belgium, at 2.5%) is 2.6%. Using the aver- age standard deviation of 23%, the t -statistic for a difference of 2.6% with 100 observations is
t-Statistic5 Difference in mean
Standard deviation/"n5 2.6
23/"10051.3
which is far below conventional levels of statistical significance. We conclude that the U.S.
experience cannot be dismissed as an outlier. Hence, using the U.S. stock market as a yard- stick for return characteristics may be reasonable.
These days, practitioners and scholars are debating whether the historical U.S. average risk-premium of large stocks over T-bills of 7.92% ( Table 5.3 ) is a reasonable forecast for the long term. This debate centers around two questions: First, do economic factors that prevailed over that historic period (1926–2009) adequately represent those that may prevail over the forecasting horizon? Second, is the arithmetic average from the available history a good yardstick for long-term forecasts?
Figure 5.8 Standard deviations of real equity and bond returns around the world, 1900–2000
Source: Elroy Dimson, Paul Marsh, and Mike Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton:
Princeton University Press, 2002), p. 61. Reprinted by permission of the Princeton University Press.
Standard Deviation of Annual Real Return (%)
35
Equities Bonds
Can
17
11 18
13 20
15 20
12 20
10 20
8 21
9 22
12 22
13 23
12 23
11 23
13 23
14 29
14 30
21 32
16
Aus U.K. Den U.S. Swi Neth Spa Ire Bel SAf Swe Fra Ita Jap Ger 30
25 20 15 10 5 0