Investments, tenth edition

A1: 25 Years in Large Stocks

B2: 10 Years in Small Stocks and 15 Years in T-bills

A2: 10 Years in Large Stocks and 15 Years in T-bills

B1: 25 Years in Small Stocks

0

2

4

6

8

10

12

14

21521025 0 5 10 15 20 25 30 35 40 45

Avg. 25-year HPR

Terminal loss (%)

Probability (loss)

Avg. 1-year HPR

12.13

10.24

4.27

211.53

95

28.88

.0095

SD

Min

Max

Frequency

0

2

4

6

8

10

12

14

21521025 0 5 10 15 20 25 30 35 40 45

Avg. 25-year HPR

Terminal loss (%)

Probability (loss)

Avg. 1-year HPR

17.97

12.28

7.28

217.00

99

47.03

.0415

SD

Min

Max

Frequency

0

2

4

6

8

10

12

14

21521025 0 5 10 15 20 25 30 35 40 45

Avg. 25-year HPR

Terminal loss (%)

Probability (loss)

Avg. 1-year HPR

9.40

7.01

4.44

210.77

94

26.63

.0552

SD

Min

Max

Frequency

0

2

4

6

8

10

12

14

21521025 0 5 10 15 20 25 30 35 40 45

Avg. 25-year HPR

Terminal loss (%)

Probability (loss)

Avg. 1-year HPR

7.10

6.27

2.65

25.38

75

16.02

.0127

SD

Min

Max

Frequency

bod61671_ch05_117-167.indd   157

bod61671_ch05_117-167.indd   157

6/18/13   8:04 PM

6/18/13   8:04 PM

Final PDF to printer

---

158 

P A R T   I I

  Portfolio Theory and Practice

will use our entire historical sample so that we are more likely to include low-probability 

events of extreme value. 

 At this point, it is well to bring up again Nassim Taleb’s metaphor of the black swan.  

24

   

Taleb uses the black swan, once unknown to Europeans, as an example of events that may 

occur without any historical precedent. The black swan is a symbol of tail risk—highly 

unlikely but extreme and important outcomes that are all but impossible to predict from 

experience. The implication for bootstrapping is that limiting possible future returns to the 

range of past returns, or extreme returns to their historical frequency, may easily underes-

timate actual exposure to tail risk. Notice that when simulating from a normal distribution, 

we do allow for unbounded bad outcomes, although without allowing for fat tails, we may 

greatly underestimate their probabilities. However, using  any  particular probability dis-

tribution predetermines the shape of future events based on measurements from the past.

  

 The dilemma of how to describe uncertainty largely comes down to how investors 

should respond to the possibility of low-probability disasters. Those who argue that an 

investment is less risky in the long run implicitly downplay extreme events. The high price 

of portfolio insurance provides proof positive that a majority of investors certainly do not 

ignore them. As far as the present exercise is concerned, we show that even a simulation 

based on generally benign past U.S. history will produce cases of investor ruin. 

 An important objective of this exercise is to assess the potential effect of deviations 

from normality on the probability distribution of a long-term investment in U.S. stocks. For 

this purpose, we bootstrap 50,000 25-year simulated “histories” for large and small stocks, 

and produce for each history the average annual return. We contrast these samples to simi-

lar samples drawn from normal distributions that (due to compounding) result in lognor-

mally distributed long-term total returns. Results are shown in  Figure 5.10 . Panel A shows 

frequency distributions of large U.S. stocks, constructed by sampling both from actual 

returns and from the normal distribution. Panel B shows the same frequency distributions 

for small U.S. stocks. The boxes inside  Figure 5.10  show the statistics of the distributions. 

 We first review the results for large stocks in panel A. We see that the difference in 

frequency distributions between the simulated history and the normal curve is small but 

distinct. Despite the very small differences between the averages of 1-year and 25-year 

annual returns, as well as between the standard deviations, the small differences in 

skewness and kurtosis combine to produce significant differences in the probabilities of 

shortfalls and losses, as well as in the potential terminal loss. For small stocks, shown in 

panel B, the smaller differences in skewness and kurtosis lead to almost identical figures 

for the probability and magnitude of losses. 

 We should also consider the risk of long-term investments. The probability of ruin is 

miniscule, and indeed, as the following table indicates, the probability of  any  loss is less 

than 1% for large stocks and 5% for small stocks. This is in line with our calculations in 

Example 5.11 showing that shortfall probabilities fall as the investment horizon extends. 

But look at the top line of the table, showing the potential  size  of your loss in the (admit-

tedly unlikely) worst-case scenarios. Risk depends on both the probability and the size of 

the potential loss, and here that worst-case scenario is very bad indeed.