Investments, tenth edition

 Figure 5.10 

Annually compounded, 25-year HPRs from bootstrapped history and a normal  distribution 

(50,000 observations)  

SD

Min 25-year HPR

Avg. 1-year HPR

Avg. 25-year HPR

Worst-case 

  terminal loss (%)

Max

Probability (loss)

Probability (shortfall)

Skew

Kurtosis

*Bootstrapped history

Actual*

12.13

10.24

10.29

4.27

211.53

95

28.88

.0095

.1044

2.0854

.0040

Normal

12.15

4.32

26.67

82

29.32

.0064

.0603

.1135

2.0121

SD

Min 25-year HPR

Avg. 1-year HPR

Avg. 25-year HPR

Worst-case 

  terminal loss (%)

Max

Probability (loss)

Probability (shortfall)

Skew

Kurtosis

*Bootstrapped history

Actual*

17.97

12.28

12.21

7.28

217

99

47.03

.0415

.1178

.1362

.0678

Normal

17.95

7.41

214.9

98

48.34

.0428

.1232

.2000

.0598

0

1

2

3

4

5

6

7

8

9

10

2

15

2

10

2

5

0

5

10

15

20

25

30

35

40

45

Frequency

Rate of Return

B: Small Stocks

0

1

2

3

4

5

6

7

8

9

10

2

15

2

10

2

5

0

5

10

15

20

25

30

35

40

45

Frequency

A: Large Stocks

History

Normal

History

Normal

probability of possible losses and the magnitude of these losses. We show in later chap-

ters how the fair market price of portfolio insurance can be estimated from option-pricing 

models. 

 Despite the low probability that a portfolio insurance policy would have to pay up (only 

2.9% for the 25-year policy), the magnitude and timing  

23

   of possible losses would make 

such long-term insurance surprisingly costly. For example, standard option-pricing models 

suggest that the value of insurance against shortfall risk over a 10-year horizon would cost 

nearly 20% of the initial value of the portfolio. And contrary to any intuition that a longer 

horizon reduces shortfall risk, the value of portfolio insurance increases dramatically with 

the maturity of the contract. For example, a 25-year policy would be about 50% more 

costly, or about 30% of the initial portfolio value.

  

23

 By “timing,” we mean that a decline in stock prices is associated with a bad economy when extra income would 

be most important to an investor. The fact that the insurance policy would pay off in these scenarios contributes 

to its market value. 

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  C H A P T E R  

5

  Risk, Return, and the Historical Record 

157

  Simulation of Long-Term Future Rates of Return 

 The frequency distributions in  Figure 5.6  provide only rough descriptions of the nature of 

the return distributions and are even harder to interpret for long-term investments. One way 

to learn from history about the distribution of long-term future returns is to simulate these 

future returns from that history. The method to accomplish this task is called  bootstrapping.  

 Bootstrapping is a procedure that avoids any assumptions about the return distribution, 

except that all rates of return in the sample history are equally likely. For example, we 

could simulate 25 years of possible future returns by sampling (with replacement) 25 ran-

domly selected annual returns from our available history. We compound those 25 returns 

to obtain one possible 25-year holding-period return. This procedure is repeated thousands 

of times to generate a probability distribution of long-term total returns that is anchored in 

the historical frequency distribution. 

 The cardinal decision when embarking on a bootstrapping exercise is the choice of 

how far into the past we should go to draw observations for “future” return sequences. We 

 Figure 5.11 

Annually compounded, 25-year HPRs from bootstrapped history (50,000 observations)