INVESTMENTS | BODIE, KANE, MARCUS
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin
CHAPTER 5
Introduction to Risk, Return, and
the Historical Record
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Interest Rate Determinants
• Supply
– Households
• Demand
– Businesses
• Government’s Net Supply and/or
Demand
– Federal Reserve Actions
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Real and Nominal Rates of Interest
• Nominal interest
rate: Growth rate of
your money
• Real interest rate:
Growth rate of your
purchasing power
• Let R = nominal
rate, r = real rate
and I = inflation
rate. Then:
iRr 
i
iR
r



1
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Equilibrium Real Rate of Interest
• Determined by:
– Supply
– Demand
– Government actions
– Expected rate of inflation
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Figure 5.1 Determination of the Equilibrium
Real Rate of Interest
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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:
• Nominal rate = real rate + inflation
forecast
( )R r E i 
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Taxes and the Real Rate of Interest
• Tax liabilities are based on nominal
income
– Given a tax rate (t) and nominal
interest rate (R), the Real after-tax
rate is:
• The after-tax real rate of return falls as
the inflation rate rises.
(1 ) ( )(1 ) (1 )R t i r i t i r t it        
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Rates of Return for Different Holding
Periods
100
( ) 1
( )
fr T
P T
 
Zero Coupon Bond, Par = $100, T=maturity,
P=price, rf(T)=total risk free return
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Example 5.2 Annualized Rates of Return
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Equation 5.7 EAR
• EAR definition: percentage increase in
funds invested over a 1-year horizon
   Tf TrEAR
1
11 
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Equation 5.8 APR
• APR: annualizing using simple interest
 
T
EAR
APR
T
11 

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Table 5.1 APR vs. EAR
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Table 5.2 Statistics for T-Bill Rates, Inflation
Rates and Real Rates, 1926-2009
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Bills and Inflation, 1926-2009
• Moderate inflation can offset most of the
nominal gains on low-risk investments.
• A dollar invested in T-bills from1926–2009
grew to $20.52, but with a real value of only
$1.69.
• Negative correlation between real rate and
inflation rate means the nominal rate
responds less than 1:1 to changes in
expected inflation.
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Figure 5.3 Interest Rates and Inflation,
1926-2009
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Risk and Risk Premiums
P
DPP
HPR
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 = 110
Beginning Price = 100
Dividend = 4
HPR = (110 - 100 + 4 )/ (100) = 14%
Rates of Return: Single Period Example
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Expected returns
p(s) = probability of a state
r(s) = return if a state occurs
s = state
Expected Return and Standard
Deviation
( ) ( ) ( )
s
E r p s r s
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State Prob. of State r in State
Excellent .25 0.3100
Good .45 0.1400
Poor .25 -0.0675
Crash .05 -0.5200
E(r) = (.25)(.31) + (.45)(.14) + (.25)(-.0675)
+ (0.05)(-0.52)
E(r) = .0976 or 9.76%
Scenario Returns: Example
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Variance (VAR):
Variance and Standard Deviation
 
22 ( ) ( ) ( )
s
p s r s E r  
2STD
Standard Deviation (STD):
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Scenario VAR and STD
• Example VAR calculation:
σ2 = .25(.31 - 0.0976)2+.45(.14 - .0976)2
+ .25(-0.0675 - 0.0976)2 + .05(-.52 -
.0976)2
= .038
• Example STD calculation:
1949.
038.


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Time Series Analysis of Past Rates of
Return
  
n
s
n
s
sr
n
srsprE
11
)(
1
)()()(
The Arithmetic Average of rate of return:
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Geometric Average Return
1
/1
TVg
n
TV = Terminal Value of the
Investment
g= geometric average
rate of return
)1)...(1)(1( 21 nn rrrTV 
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Geometric Variance and Standard
Deviation Formulas
• Estimated Variance = expected value of
squared deviations
 
2
1
_2^ 1









n
s
rsr
n

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Geometric Variance and Standard
Deviation Formulas
• When eliminating the bias, Variance and
Standard Deviation become:

 
2
1
_^
1
1











n
j
rsr
n

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The Reward-to-Volatility (Sharpe)
Ratio
• Sharpe Ratio for Portfolios:
Returns Excess of SD
PremiumRisk

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The Normal Distribution
• Investment management is easier when
returns are normal.
– Standard deviation is a good measure of risk
when returns are symmetric.
– If security returns are symmetric, portfolio
returns will be, too.
– Future scenarios can be estimated using only
the mean and the standard deviation.
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Figure 5.4 The Normal Distribution
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Normality and Risk Measures
• What if excess returns are not normally
distributed?
– Standard deviation is no longer a complete
measure of risk
– Sharpe ratio is not a complete measure of
portfolio performance
– Need to consider skew and kurtosis
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Skew and Kurtosis
Skew
Equation 5.19
Kurtosis
• Equation 5.20




















3
^
3
_

RR
averageskew 3
4
^
4
_






















RR
averagekurtosis
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Figure 5.5A Normal and Skewed
Distributions
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Figure 5.5B Normal and Fat-Tailed
Distributions (mean = .1, SD =.2)
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Value at Risk (VaR)
• A measure of loss most frequently
associated with extreme negative
returns
• VaR is the quantile of a distribution
below which lies q % of the possible
values of that distribution
– The 5% VaR , commonly estimated in
practice, is the return at the 5th percentile
when returns are sorted from high to low.
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Expected Shortfall (ES)
• Also called conditional tail expectation
(CTE)
• More conservative measure of
downside risk than VaR
– VaR takes the highest return from the
worst cases
– ES takes an average return of the worst
cases
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Lower Partial Standard Deviation (LPSD)
and the Sortino Ratio
• Issues:
– Need to consider negative deviations
separately
– Need to consider deviations of returns
from the risk-free rate.
• LPSD: similar to usual standard
deviation, but uses only negative
deviations from rf
• Sortino Ratio replaces Sharpe Ratio
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Historic Returns on Risky Portfolios
• Returns appear normally distributed
• Returns are lower over the most recent half of
the period (1986-2009)
• SD for small stocks became smaller; SD for
long-term bonds got bigger
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Historic Returns on Risky Portfolios
• Better diversified portfolios have higher
Sharpe Ratios
• Negative skew
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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
2 2
1 1
20 20[1 ( )]
T
g gT TT
e eE r
  
  
  

•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:
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Figure 5.10 Annually Compounded,
25-Year HPRs
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Figure 5.11 Annually Compounded,
25-Year HPRs
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Figure 5.12 Wealth Indexes of Selected Outcomes of Large
Stock Portfolios and the Average T-bill Portfolio