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
INVESTMENTS | BODIE, KANE, MARCUS
5-2
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 (how many Big Macs
can I buy with my
money?)
Let R = nominal rate,
r = real rate and
i = inflation rate. Then:
iRr
i
Rr
1
11
i
iRr
1
Solve:
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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 - Real Rate of Interest Equilibrium
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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 + expected inflation
( )R r E i
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5-7
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 of return is:
titritiritR 111
adjusted-inflation
after tax
• As intuition suggests, the after-tax, real rate
of return falls as the inflation rate rises.
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Rates of Return for Different Holding Periods
Zero Coupon Bond
Par = $100
T = maturity
P = price
rf(T) = total risk free return
TrP
f
1
100 1
100
PTrf
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Example 5.2 Time Does Matter: Use Annualized Rates of Return
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5-10
Equation 5.7 EAR
• Time matters → use EAR to annualize
• Effective Annual Rate definition:
percentage increase in funds invested
over a 1-year horizon
Tf EARTr 11
Tf TrEAR
1
11
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Equation 5.8 APR
• Annual Percentage Rate (APR): annualizing using simple interest
TEARTAPR 11
T
EARAPR
T11
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1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
0 5 10 15 20 25 30
Inve
stm
en
t En
d V
alu
e
(years)
End Value with APR=5.0%
End Value with EAR=5.0%
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5-13
Table 5.1 APR vs. EAR
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Continuous Compounding
• Frequency of compounding matters
• At the limit to (compounding time)→0:
ccreEAR 1
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1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
0 5 10 15 20 25 30
Inve
stm
en
t En
d V
alu
e
(years)
End Value with APR=5.0%
End Value with EAR=5.0%
End Value with Rcc=5.0%
INVESTMENTS | BODIE, KANE, MARCUS
S
xTNxNT /* Let r =rate and
x =compounding time →
Nxrxrxr *1*1*1 Value End
timesN gcompoundin
NxrNexr *1ln
0x0x lim*1lim
How to derive Rcc
x
xrT
e
*1ln
0xlim
xdx
d
xrTdx
d
e
*1ln
0xlim
rT
rxr
T
ee
1
*1
1
0xlim
Looks like 0/0.
Use de l’Hôpital
Q.E.D.
Make x very
small. Then
use A=eln(A)
Checks: r=0 →End Value=1
T=0 →End Value=1
Substitute
N=T/x
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5-17
Table 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates, 1926-2009
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5-18
Bills and Inflation, 1926-2009
• Moderate inflation can offset most of the
nominal gains on low-risk investments.
• One 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
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 = 110
Beginning Price = 100
Dividend = 4
HPR = (110 - 100 + 4 )/ (100) = 14%
Rates of Return: Single Period Example
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5-22
Expected (or mean) 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 0.25 0.3100
Good 0.45 0.1400
Poor 0.25 -0.0675
Crash 0.05 -0.5200
E(r) = (0.25)(0.31) + (0.45)(0.14) + (0.25)(-0.0675)
+ (0.05)(-0.52)
= 0.0976
= 9.76% (think of a probability-weighted avg) NOTE: use decimals instead of percentages to be safe
Scenario Returns: Example
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5-24
Variance (VAR):
Variance and Standard Deviation
22 ( ) ( ) ( )
s
p s r s E r
2STD
Standard Deviation (STD):
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Scenario VAR and STD
• Example VAR calculation:
σ2 = 0.25(0.31 - 0.0976)2 +
0.45(0.14 - 0.0976)2 +
0.25(-0.0675 - 0.0976)2 +
0.05(-0.52 - 0.0976)2 =
= 0.038
• Example STD calculation:
1949.0038.0
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Time Series Analysis of Past Rates of Return
n
s
n
ssr
nsrsprE
11)(
1)()()(
The Arithmetic Average of historical rate
of return as an estimator of the expected
rate of return
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5-27
Geometric Average Return
1/1 1 TVggTV nn
TV = Terminal Value of the Investment
g = geometric average rate of return
)1)...(1)(1( 21 nn rrrTV
Solve for a rate g that, if compounded n
times, gives you the same TV
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Geometric Variance and Standard Deviation Formulas
Estimated Variance = expected value of
squared deviations (from the mean)
2
1
2 1ˆ
n
s
rsrn
22 ( ) ( ) ( )
s
p s r s E r
Recall the definition of variance
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Geometric Variance and Standard Deviation Formulas
• Using the estimated ravg instead of the real E(r)
introduces a bias:
– we already used the n observations to estimate ravg
– we really have only (n-1) independent observations
– correct by multiplying by n/(n-1)
• When eliminating the bias, Variance and
Standard Deviation become*:
2
11
1ˆ
n
j
rsrn
* More at http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation
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The Reward-to-Volatility (Sharpe) Ratio
• Sharpe Ratio for Portfolios:
Returns Excess of SD
PremiumRisk
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5-31
The Normal Distribution
• Investment management math 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
– Assuming Normality, future scenarios can be
estimated using just mean and 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
3
3
RRaverageskew
3
ˆ
ondistributi Normal afor 3 equals this
4
4
RRaveragekurtosis
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Figure 5.5A Normal and Skewed Distributions
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Figure 5.5B Normal and Fat-Tailed Distributions (mean = 0.1, SD =0.2)