CHAPTER 5

Investments

Learning About

Return and Risk

from the Historical

Record Slides by

Richard D. Johnson

Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/Irwin

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Factors Influencing Rates

Supply

–Households

Demand

–Businesses

Government’s Net Supply and/or Demand

–Federal Reserve Actions

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Figure 5.1 Determination of the Equilibrium

Real Rate of Interest

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Return for Holding Period – Zero Coupon

Bonds

1)(

100)(

TPTr f

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Example 5.2 Annualized Rates of Return

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Formula for EARs and APRs

T

EARAPR

TrEART

T

f

1)1(

1})(1{]/1[

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Table 5.1 Annual Percentage Rates (APR) and Effective Annual

Rates (EAR)

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Table 5.2 History of T-bill Rates, Inflation and Real Rates for

Generations, 1926-2005

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Figure 5.2 Interest Rates and Inflation, 1926-

2005

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Figure 5.3 Nominal and Real Wealth Indexes,

1966-2005

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P

DPPHPR

0

101

HPR = Holding Period Return

P0 = Beginning price

P1 = Ending price

D1 = Dividend during period one

Rates of Return: Single Period

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Ending Price = 48

Beginning Price = 40

Dividend = 2

HPR = (48 - 40 + 2 )/ (40) = 25%

Rates of Return: Single Period Example

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Subjective returns

p(s) = probability of a state

r(s) = return if a state occurs

1 to s states

Mean Scenario or Subjective Returns

rprE s

s

s1

)(

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State Prob. of State r in State

.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

Scenario or Subjective Returns: Example

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Standard deviation = [variance]1/2

Subjective or Scenario

Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]

Var= .01199

S.D.= [ .01199] 1/2 = .1095

Using Our Example:

Variance or Dispersion of Returns

rErp s

s

s 2

2

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Mean and Variance of Historical Returns

n

s

n

ssr

nsrsprE

11)(

1)()()(

n

s rsrn 1 )(

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Arithmetic average or rates of

return

Average return is arithmetic

average

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Geometric Average Returns

)1()1)(1(21 rrrTV nn

TV = Terminal Value of the

Investment

1/1TVg n

g= geometric average rate of

return

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Sharpe Ratio

Sharpe Ratio for Portfolios Risk Premium

SD of Excess Return

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Figure 5.4 The Normal Distribution

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Figure 5.5A Normal and Skewed

(mean = 6% SD = 17%)

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Figure 5.5B Normal and Fat Tails Distributions

(mean = .1 SD =.2)

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Figure 5.6 Histograms of Rates of Return

for 1926-2005

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Table 5.3 History of Rates of Returns of Asset

Classes for Generations, 1926- 2005

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Table 5.4 History of Excess Returns of Asset

Classes for Generations, 1926- 2005

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Figure 5.7 Nominal and Real Equity Returns

Around the World, 1900-2000

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Figure 5.8 Standard Deviations of Real Equity and Bond

Returns Around the World, 1900-2000

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Figure 5.9 Probability of Investment Outcomes

After 25 Years with A Lognormal Distribution

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Terminal Value with Continuous Compounding

erETgTT 22/1

)](1[

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Figure 5.10 Annually Compounded, 25-Year HPRs from

Bootstrapped History and A Normal Distribution

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Figure 5.11 Annually Compounded, 25-Year

HPRs from Bootstrapped History

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Figure 5.12 Wealth Indexes of Selected Outcomes of

Large Stock Portfolios and the Average T-bill Portfolio

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Table 5.5 Risk Measures for Non-Normal

Distributions