CHAPTER 5
Learning About
Return and Risk
from the
Historical Record
Factors Influencing Rates
• Supply
– Households
• Demand
– Businesses
• Government’s Net Supply and/or Demand
– Federal Reserve Actions
5-2
Figure 5.1 Determination of the
Equilibrium Real Rate of Interest
5-3
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)
5-4
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
5-5
Comparing Rates of Return for Different
Holding Periods
Zero Coupon Bond
100
rf (T ) 
1
P(T )
5-6
Formula for EARs and APRs
1
EAR  {1 r f (T ) }T 1
1
(1

EAR
)
APR 
T
T
5-7
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
5-8
Table 5.2 History of T-bill Rates, Inflation
and Real Rates for Generations, 1926-2005
5-9
Figure 5.2 Interest Rates and Inflation,
1926-2005
5-10
Figure 5.3 Nominal and Real Wealth
Indexes for Investment in Treasury Bills,
1966-2005
5-11
Risk and Risk Premiums
Rates of Return: Single Period
P
1  P0  D1
HPR 
P0
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
5-12
Rates of Return: Single Period Example
Ending Price =
Beginning Price =
Dividend =
48
40
2
HPR = (48 - 40 + 2 )/ (40) = 25%
5-13
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
5-14
Scenario Returns: Example
State
1
2
3
4
5
Prob. of State
.1
.2
.4
.2
.1
r in State
-.05
.05
.15
.25
.35
E(r) = (.1)(-.05) + (.2)(.05)… + (.1)(.35)
E(r) = .15
5-15
Variance or Dispersion of Returns
Variance:
   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
5-16
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
5-17
Geometric Average Return
TV
n
 (1 r1 )(1 r2 ) x
x  (1 rn )
TV = Terminal Value of the
Investment
g  TV
1/ n
1
g= geometric average
rate of return
5-18
Variance and Standard Deviation
Formulas
• Variance = expected value of squared
deviations
2
n
1
2
    r ( s)  r 
n s 1
• When eliminating the bias, Variance and
Standard Deviation become:
n
1
r (s)  r 



n  1 j 1 
2
5-19
The Reward-to-Volatility (Sharpe) Ratio
Risk Premium
Sharpe Ratio for Portfolios =
SD of Excess Return
5-20
Figure 5.4 The Normal Distribution
5-21
Figure 5.6 Frequency Distributions of
Rates of Return for 1926-2005
5-22
Table 5.3 History of Rates of Returns of Asset
Classes for Generations, 1926- 2005
5-23
Table 5.4 History of Excess Returns of Asset
Classes for Generations, 1926- 2005
5-24
Figure 5.7 Nominal and Real Equity
Returns Around the World, 1900-2000
5-25
Figure 5.8 Standard Deviations of Real Equity
and Bond Returns Around the World, 1900-2000
5-26