CHAPTER 5
Introduction to Risk, Return, and
the Historical Record
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McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
Interest Rate Determinants
• Supply
– Households
• Demand
– Businesses
• Government’s Net Supply and/or
Demand
– Federal Reserve Actions
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5-2
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?)*
*The Big Mac Index is a different thing
Let rn = nominal rate,
rr = real rate and
i = inflation rate. Then:
𝑟𝑟 ≈ 𝑟𝑛 − 𝑖
More precisely:
1 + 𝑟𝑛
1 + 𝑟𝑟 =
1+𝑖
solve
𝑟𝑛 − 𝑖
𝑟𝑟 =
1+𝑖
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5-3
Fig 5.1: Real Rate of Interest Equilibrium
Determined by supply, demand, government actions,
expected rate of inflation
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5-4
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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5-5
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:
R1  t   i  r  i 1  t   i  r 1  t   i  t
• As intuition suggests, the after-tax, real rate
of return falls as the inflation rate rises.
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5-6
Rates of Return
for Different Holding Periods
•
•
•
•
•
Zero Coupon Bond
Par = $100
T = maturity
P = price
rf(T) = total risk free return
100
P
1  rf T 
100
rf T  
1
P
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5-7
Time Does Matter
Use Annualized Rates of Return
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5-8
Effective Annual Rate (EAR)
• Time matters → use EAR to annualize
• EAR definition: percentage increase in funds
invested over a 1-year horizon
1  rf T   1  EAR 
T


1  EAR  1  rf T 
1
T
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5-9
Equation 5.8 APR
• Annual Percentage Rate (APR): annualizing
using simple interest
1  APR  T  1  EAR 
T

1  EAR 
APR 
T
1
T
Q. You invest $1 for 30 years. Do you
prefer [A] 5% APR, or [B] 5% EAR?
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5.00
End Value with APR=5.0%
4.50
End Value with EAR=5.0%
Investment End Value
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0
5
10
15
(years)
20
25
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30
5-11
Table 5.1 APR vs. EAR
Hold the EAR fixed at 5.8%
and solve for APR
for each holding period
Hold the APR fixed at 5.8%
and solve for EAR
for each holding period
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1-12
Continuous Compounding
• Frequency of compounding matters
• At the limit to (compounding time)→0:
1  EAR  e
rcc
Q. You invest $1 for 30 years. Which
interest rate do you prefer?
A. 5% EAR
B. 5% Rcc
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5.00
End Value with APR=5.0%
End Value with EAR=5.0%
End Value with Rcc=5.0%
4.50
Investment End Value
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0
5
10
15
(years)
20
25
30
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How to derive Rcc
Let r=rate and
x=compounding time → T  N  x  N  T / x
End Value  1  r  x     1  r  x   1  r  x 


N
compounding N times
lim1  r  x  S lim e
Make x very
small. Then
use A=eln(A)
 lim e
N
x 0
T ln 1 r  x 
x
x 0
 lim e
x 0
1
T
r
1 r  x
1
ln 1 r  x N
x 0

Looks like 0/0.
Use de l’Hôpital
 lim e
x 0
e
rT
Q.E.D.
Substitute
N=T/x
 d


T
ln

1

r

x




dx


d


x


dx



Checks: r=0 →End Value=1
T=0 →End Value=1
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Table 5.2 Statistics for T-Bill Rates, Inflation
Rates and Real Rates, 1926-2012
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Bills and Inflation, 1926-2009
• Moderate inflation can offset most of the
nominal gains on low-risk investments.
• One dollar invested in T-bills from1926–2012
grew to $20.25, but with a real value of only
$1.55.
• Negative correlation between real rate and
inflation rate means the nominal rate doesn’t
fully compensate investors for increased in
inflation
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Fig 5.3: Interest Rates and Inflation
1926-2009
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Risk and Risk Premiums
Rates of Return: Single Period

P
1  P 0   D1
HPR 
P0
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
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Rates of Return: Single Period Example
• Ending Price =
110
• Beginning Price = 100
• Dividend =
4
• HPR = (110 - 100 + 4 ) / (100) = 14%
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Expected Return and Standard
Deviation
Expected (or mean) returns
E (r )   p ( s )r ( s )
s
s = state
p(s)= probability of a state
r(s) = return if a state occurs
Q. What is the
expected value
of rolling a die?
A. 1
B. Sqrt(6)
C. Pi
D. 3.5
E. 6
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Scenario Returns: Example
State
Prob. of state r for that 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
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Variance and Standard Deviation
Variance (VAR):
   p( s)  r ( s)  E (r ) 
2
2
s
Standard Deviation (STD):
STD 

2
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Scenario VARiance and STD
• Example VARiance 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:
  0.038  0.1949
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Time Series Analysis of Past Rates of Return
The Arithmetic Average of historical
rate of return as an estimator of the
expected rate of return
n
1 n
E (r )   p( s )r s    r s 
n s 1
s 1
Q. What assumptions are we implicitly
making?
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Geometric Average Return
TVn  (1  r1 )(1  r2 )...(1  rn )
TV = Terminal Value of the Investment
Solve for a rate g that, if compounded n
times, gives you the same TV
TV  1 g   g  TV
n
1/ n
1
g = geometric average rate of return
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Estimating
Variance and Standard Deviation
• Estimated Variance = expected value of
squared deviations (from the mean)
   p( s)  r ( s)  E (r ) 
2
2
s
Recall the definition of variance
n
2
1
ˆ   r s   r 
n s 1
2
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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
n
1
r s   r 
ˆ 

n  1 j 1
* More at http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation
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The Reward-to-Volatility (Sharpe) Ratio
• Excess Return
• The difference in any particular period between
the actual rate of return on a risky asset and the
actual risk-free rate
• Risk Premium
• The difference between the expected HPR on a
risky asset and the risk-free rate
• Sharpe Ratio
Risk Premium

SD of Excess Returns
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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 would not be a complete measure of
portfolio performance
– Need to consider higher moments, like skew and
kurtosis
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Skew and Kurtosis
this is zero for symmetric distributi ons


 R  R 3 
skew  average

3
 ˆ

 R  R  
kurtosis  average
 3
4
 ˆ



4
this equals 3 for a Normal distributi on
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Fig.5.5A
Normal and Skewed Distributions
Mean = 6%
SD = 17%
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Fig 5.5B
Normal & Fat-Tailed Distributions
Mean = 0.1
SD = 0.2
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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.
Also referred to as 95%-ile (depends on
perspective)
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Normal Distribution and VaR
2.5
2
The area is
the percentile
1.5
1
VaR
0.5
0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
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Expected Shortfall (ES)
• a.k.a. Conditional Tail Expectation (CTE)
• More conservative measure of downside risk
than VaR:
– VaR = highest return from the worst cases
– Real life distributions are asymmetric and have
fat tails
– ES = average return of the worst cases
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Normal Distribution, VaR, and Expected Shortfall
2.5
2
The area is
the percentile
1.5
1
Expected
Shortfall
0.5
VaR
0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
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1.0
A game with a coin
• Let’s play a game: flip one coin, and receive
$1 if heads
• Assume Pr[Heads]= p (for example p=50%)
• Q. What is the game’s expected outcome?
• Q. What is the Variance?
• Q. What is the St.Dev?
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A game with two coins
• Let’s play a game: flip 2 fair coins, and
receive $1 for each head
• Q. What is the portfolio expected return?
• Q. What is the portfolio Variance?
• Q. What is the portfolio St.Dev?
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A lot more coins
• Let’s play a game: flip 30 fair coins, and
receive $1 for each head.
• Q. What is the portfolio expected return?
• Q. What is the portfolio Variance?
• Q. What is the portfolio St.Dev?
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A Portfolio of 2 stocks
• Portfolio = 0.5 * A + 0.5 * B
• A: rA = 0.08 StDevA = 0.1
• B: rB = 0.10 StDevB = 0.1
• Q. What is the portfolio Expected Return?
• Q. What is the portfolio Variance?
• Q. What is the portfolio Standard Deviation?
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A Portfolio of 3 stocks
• Portfolio = 𝑤𝐴 × 𝐴 + 𝑤𝐵 × 𝐵 + 𝑤𝐶 × 𝐶
• Q. What is the portfolio expected return?
• Q. What is the portfolio Variance?
• Q. What is the portfolio Standard Deviation?
• Q. What is if you have N stocks?
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(A)
(B)
(D)
(C)
(E)
Q. Which
portfolio
has best
Sharpe?
30% (A)
50% (B)
20% (D)
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Historic Returns on Risky Portfolios
• Normal distribution is generally a good
approximation of portfolio returns
– VaR indicates no greater tail risk than is
characteristic of the equivalent normal
– The ES does not exceed 0.41 of the monthly SD,
presenting no evidence against the normality
• However
– Negative skew is present in some of the
portfolios some of the time, and positive kurtosis
is present in all portfolios all the time
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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
When the continuously compounded rate of
return on an asset is normally distributed, the
effective rate of return will be lognormally
distributed. Remember:
E Geom. Avg  E Arithm. Avg  0.5
so
m  g  0.5
2
2
The Terminal Value will then be:
1 + 𝐸𝐴𝑅
𝑇
= 𝑒
𝑔+0.5 𝜎 2
𝑇
=𝑒
𝑇𝑔+0.5𝑇𝜎 2
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