Investments, tenth edition

Some Funds Stop Grading on the Curve

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, the nearly 

40% stock-market decline in 2008 would be more common 

than previously thought. These new assumptions present a 

far different picture of risk. Consider the 60% stock, 40% 

bond portfolio that fell about 20%. 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, The Wall Street Journal, September 8, 2009, 

p. C1. Reprinted with permission. © 2009 Dow Jones & Company, 

Inc. All Rights Reserved Worldwide.

WORDS FROM THE STREET

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 This measure can be quite informative about downside risk, but in practice is most use-

ful for large, high-frequency samples. Observe from  Figure 5.4  that the relative frequency 

of negative 3-sigma jumps in a standard normal distribution is only 0.13%, that is, 1.3 

observations per 1,000. Thus in a small sample, it is hard to obtain a representative out-

come, one that reflects true statistical expectations of extreme events. 

 In the analysis of the history of some popular investment vehicles in the next section 

we will show why practitioners need this plethora of statistics and performance measures 

to analyze risky investments. The nearby box discusses the growing popularity of these 

measures, and particularly the new focus on fat tails and extreme events.    

  We can now apply the analytical tools worked out in previous sections to six interesting 

risky portfolios. 

 Our base portfolio is the broadest possible U.S. equity portfolio, including all stocks 

listed on the NYSE, AMEX, and NASDAQ. We shall denote it as “All U.S.” Logic sug-

gests that an unmanaged (passive) portfolio should invest more in larger firms, and hence 

the natural benchmark is a value-weighted portfolio. Firm capitalization, or market cap, is 

highly skewed to the right, with many small but a few gigantic firms. Because it is value 

weighted, the All U.S. portfolio is dominated by the large-firm corporate sector. 

 The data include monthly excess returns on all U.S. stocks from July 1926 to September 

2012, a sample period spanning just over 86 years. We break this period into three subpe-

riods, different in length and economic circumstances. We would like to see how portfolio 

performance varies across the subperiods. 

    5.8 

Historic Returns on Risky Portfolios 

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P A R T   I I

  Portfolio Theory and Practice

 The early subperiod stretches over the second quarter of the 20th century, July 1926 to 

December 1949, and its 282 months span the Great Depression, WWII, and its immediate 

aftermath. 

 The second subperiod is the second half of the 20th century (January 1950 to December 

1999) and its 600 months cover a relatively stable period, albeit with three wars (Korea, 

Vietnam, and the first Gulf War) and eight relatively mild recessions and recoveries. It ends 

with the tech bubble of the late 1990s. 

 The third subperiod covers the first 153 months of the 21st century, a difficult period. 

It includes two deep recessions of different types: one following the bursting of the tech 

bubble in 2001, and the other following the bursting of the housing-price bubble beginning 

in 2007; each episode slashed stock values by about 40%. The prolonged Iraq war and the 

Afghan conflict put further strain on the U.S. economy in these years. 

 We also present four portfolios to compare with the benchmark All U.S portfolio. These 

comparison groups are motivated by empirical evidence that two variables (other than risk) 

have been associated with stock returns: firm size (measured by market capitalization) and 

the ratio of firm book value to market value of equity. 

 Average realized returns have generally been higher for stocks of small rather than large 

capitalization firms, other things equal. “Other things” in this context means risk as best as 

we can measure it. Hence, two of the four portfolios include stocks of firms in the top half 

of the distribution of market capitalization, while the other two portfolios include firms 

from the bottom half. 

 The accounting value of a firm reported on its balance sheet reflects the historical cost 

of its  past  investments in assets, often dubbed  assets in place  and therefore is a backward-

looking measure of value. Book value of equity equals total firm value minus the par value 

of outstanding debt. In contrast, the  market  value of equity reflects the present value of 

 future  cash flows from existing lines of business, expected growth in those businesses, 

as well as cash flows from projects yet to be started, often dubbed  growth opportunities.  

A good portion of the difference between the book value and market value of equity will 

depend on the relative proportions of assets in place versus growth opportunities. A low 

ratio of book-to-market value (B/M) is typical of firms whose market value derives mostly 

from growth prospects. A high B/M ratio is typical of “value” firms, whose market values 

derive mostly from assets in place. Realized average returns, other things equal, histori-

cally have been higher for value firms than for growth firms. 

 Eugene Fama and Kenneth French extensively documented the firm size and B/M regu-

larities, and these patterns have since been corroborated in stock exchanges around the 

world.  

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   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) and rebalances these six 

portfolios every midyear.  

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 We drop the medium B/M portfolios, identify high B/M firms as “Value firms” and low 

B/M firms as “Growth firms,” and thus obtain four comparison portfolios as Big/Value; 

Big/Growth; Small/Value; Small/Growth. While it is common to use value weighting, we 

will use equally weighted versions of these portfolios in order to give greater emphasis to 

smaller firms and thus sharpen the contrast with the large-cap heavy All U.S. benchmark. 

  Table 5.3  shows the number of firms, average firm size, and average B/M ratio for each 

portfolio at the outset and end of each subperiod. The number of stocks in the portfolios 

steadily increases, except during the 21st century, when bad times killed off a large num-

ber of small firms. The exit of so many small firms from the sample generally increased 

the average capitalization of the remaining firms, despite their loss of market value. The 

  

15

 This literature began in earnest with their 1992 publication: “The Cross Section of Expected Stock Returns,” 

 Journal of Finance  47, 427–465. 

   

16

 The database is available at:  http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html   

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  C H A P T E R  

5

  Risk, Return, and the Historical Record 

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average market cap of all firms grew from $57 million to $4.47 billion, an annual growth 

rate of 5.2% over the 86 years, slower than average growth of nominal GDP (6.7%), but 

2 percentage points higher than average inflation (3.2%).  

  Figure 5.6  presents six histograms of the 1,035 total monthly returns on T-bills and excess 

returns on the five risky portfolios.  

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   Bear in mind that even small differences in average 

monthly returns will have great impact on final wealth when compounded over long periods. 

The histogram in panel A shows T-bills rates in the range of  2 .05% to 1.5%,  

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    while  panel 

B shows excess monthly returns on the All U.S. stock portfolio lying in the range between 

2 20% and  1 20%; annualized, this is equivalent to a range of  2 93% to 891%! The vertical 

axis shows the fraction of returns in each bin. (The dark columns in the histograms are based 

on the historical sample while the light columns describe a normal distribution.) The bins 

are 2.5 basis points (.025%) wide for T-bill returns, and 50 basis points wide for the All U.S. 

portfolio. The extreme bins at the far right and left of each histogram are actually the sums 

of the frequencies of all returns beyond the reported range (less than  2 20% or greater than 

20%). Panels C and D show histograms of excess returns on the two “Big” or large-cap port-

folios (one for Big/Value stocks and the other for Big/Growth stocks) while panels E and F 

are excess return histograms on the two “Small” portfolios (Small/Value and Small/Growth).

 A first look at the actual excess returns on stocks verifies that the tails are uniformly fat-

ter than would be observed in a normal distribution, implying higher incidence of extreme 

results. Given the potential outsized impact of extreme returns, it is customary to fit a