This document is a chapter about learning about return and risk from analyzing historical data. It discusses factors that influence interest rates and defines real and nominal rates. It explains how the equilibrium real rate of interest is determined and shows the relationship between nominal interest rates and expected inflation. The chapter also covers topics like taxes and interest rates, comparing rates of return over different holding periods, and defining expected returns and standard deviation. It provides examples of analyzing the historical record of returns on Treasury bills, stocks, and other asset classes.

1. CHAPTER 5 Learning About
Return and Risk
from the
Historical Record
Investments, 8th edition
Bodie, Kane and Marcus
Slides by Susan Hine
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Factors Influencing Rates
• Supply
– Households
• Demand
– Businesses
• Government’s Net Supply and/or Demand
– Federal Reserve Actions
5-2

3. Real and Nominal Rates of Interest
• Nominal interest rate
– Growth rate of your money
• Real interest rate
– Growth rate of your purchasing power
• If R is the nominal rate and r the real rate and
i is the inflation rate:
r = R−i
5-3

4. Equilibrium Real Rate of Interest
• Determined by:
– Supply
– Demand
– Government actions
– Expected rate of inflation
5-4

5. Figure 5.1 Determination of the
Equilibrium Real Rate of Interest
5-5

6. Equilibrium Nominal Rate of Interest
• As the inflation rate increases, investors will
demand higher nominal rates of return
• If E(i) denotes current expectations of
inflation, then we get the Fisher Equation:
R = r + E (i )
5-6

7. Taxes and the Real Rate of Interest
• Tax liabilities are based on nominal income
– Given a tax rate (t), nominal interest rate
(R), after-tax interest rate is R(1-t)
– Real after-tax rate is:
R (1 − t ) − i = (r + i )(1 − t ) − i = r (1 − t ) − it
5-7

8. Comparing Rates of Return for Different
Holding Periods
Zero Coupon Bond
100
rf (T ) = −1
P (T )
5-8

9. Example 5.2 Annualized Rates of Return
5-9

10. Formula for EARs and APRs
1
EAR ={1+r f (T ) }T −1
(1+ EAR) −1
T
APR =
T
5-10

11. Table 5.1 Annual Percentage Rates
(APR) and Effective Annual Rates (EAR)
5-11

12. Bills and Inflation, 1926-2005
• Entire post-1926 history of annual rates:
– www.mhhe.com/bkm
• Average real rate of return on T-bills for the
entire period was 0.72 percent
• Real rates are larger in late periods
5-12

13. Table 5.2 History of T-bill Rates, Inflation
and Real Rates for Generations, 1926-2005
5-13

14. Figure 5.2 Interest Rates and Inflation,
1926-2005
5-14

15. Figure 5.3 Nominal and Real Wealth
Indexes for Investment in Treasury Bills,
1966-2005
5-15

16. Risk and Risk Premiums
Rates of Return: Single Period
HPR = P1 − P0 + D1
P0
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
5-16

17. Rates of Return: Single Period Example
Ending Price = 48
Beginning Price = 40
Dividend = 2
HPR = (48 - 40 + 2 )/ (40) = 25%
5-17

18. Expected Return and Standard Deviation
Expected returns
E (r ) = ∑ p ( s )r ( s )
s
p(s) = probability of a state
r(s) = return if a state occurs
s = state
5-18

19. Scenario Returns: Example
State Prob. of State r in State
1 .1 -.05
2 .2 .05
3 .4 .15
4 .2 .25
5 .1 .35
E(r) = (.1)(-.05) + (.2)(.05)… + (.1)(.35)
E(r) = .15
5-19

20. Variance or Dispersion of Returns
Variance:
σ = ∑ p ( s ) [ r ( s ) − E (r ) ]
2 2
s
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2…+ .1(.35-.15)2]
Var= .01199
S.D.= [ .01199] 1/2 = .1095
5-20

21. Time Series Analysis of Past Rates of
Return
Expected Returns and
the Arithmetic Average
1 n
E (r ) = ∑s =1 p ( s )r ( s ) = ∑s =1 r ( s )
n
n
5-21

22. Geometric Average Return
TV n
= (1 + r1 )(1 + r2 ) xK x = (1 + rn )
TV = Terminal Value of the
Investment
g = TV 1/ n
−1
g= geometric average
rate of return
5-22

23. Geometric Variance and Standard
Deviation Formulas
• Variance = expected value of squared
deviations n 2
1
σ = ∑ r (s) − r 
2
n s =1  
• When eliminating the bias, Variance and
Standard Deviation become:
n 2
1
σ= ∑ r (s) − r 
n − 1 j =1  
5-23

24. The Reward-to-Volatility (Sharpe) Ratio
Risk Premium
Sharpe Ratio for Portfolios =
SD of Excess Return
5-24

25. Figure 5.4 The Normal Distribution
5-25

26. Figure 5.5A Normal and Skewed Distributions
(mean = 6% SD = 17%)
5-26

27. Figure 5.5B Normal and Fat-Tailed
Distributions (mean = .1, SD =.2)
5-27

28. Figure 5.6 Frequency Distributions of
Rates of Return for 1926-2005
5-28

29. Table 5.3 History of Rates of Returns of Asset
Classes for Generations, 1926- 2005
5-29

30. Table 5.4 History of Excess Returns of Asset
Classes for Generations, 1926- 2005
5-30

31. Figure 5.7 Nominal and Real Equity
Returns Around the World, 1900-2000
5-31

32. Figure 5.8 Standard Deviations of Real Equity
and Bond Returns Around the World, 1900-2000
5-32

33. Figure 5.9 Probability of Investment Outcomes
After 25 Years with A Lognormal Distribution
5-33

34. Terminal Value with Continuous
Compounding
When the continuously compounded rate of
return on an asset is normally distributed, the
effective rate of return will be lognormally
distributed
The Terminal Value will then be:
T
 g+ 1 
2 2
1
gT +
[1+ E (r )]
T T
= e 20  = e 20

 

5-34

35. Figure 5.10 Annually Compounded, 25-Year
HPRs from Bootstrapped History and
A Normal Distribution (50,000 Observation)
5-35

36. Figure 5.11 Annually Compounded,
25-Year HPRs from Bootstrapped
History(50,000 Observation)
5-36

37. Figure 5.12 Wealth Indexes of Selected
Outcomes of Large Stock Portfolios and
the Average T-bill Portfolio
5-37

38. Table 5.5 Risk Measures for Non-Normal
Distributions
5-38