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.
5-2
Interest Rate Determinants
• Supply
– Households
• Demand
– Businesses
• Government’s Net Supply and/or
Demand
– Federal Reserve Actions
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5-3
Real and Nominal Rates of Interest
Let R = nominal rate,
• Nominal interest
r = real rate and
rate: Growth rate of
i = inflation rate. Then:
your money
• Real interest rate:
Growth rate of your
purchasing power
(how many Big Macs
can I buy with my
money?)
r  R i
1 R
1 r 
1 i
Solve:
R i
r
1 i
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5-4
Equilibrium Real Rate of Interest
• Determined by:
–Supply
–Demand
–Government actions
–Expected rate of inflation
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5-5
Figure 5.1 - Real Rate of Interest Equilibrium
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5-6
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:
R1  t   i  r  i 1  t   i  r 1  t   i  t

after tax



inflation -adjusted
• As intuition suggests, the after-tax, real rate
of return falls as the inflation rate rises.
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5-8
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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5-9
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
1  rf T   1  EAR 
T


1  EAR  1  rf T 
1
T
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5-11
Equation 5.8 APR
• Annual Percentage Rate (APR):
annualizing using simple interest
1  APR  T  1  EAR 
T

1  EAR 
APR 
T
1
T
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5.00
5-12
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
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(years)
5-13
Table 5.1 APR vs. EAR
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5-14
Continuous Compounding
• Frequency of compounding matters
• At the limit to (compounding time)→0:
1  EAR  e
rcc
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5.00
5-15
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
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(years)
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
Substitute
N=T/x
 d

 T ln 1 r * x  
 dx

d


x


dx


x 0
e
rT

Checks: r=0 →End Value=1
Q.E.D.
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T=0 →End Value=1
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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5-19
Figure 5.3 Interest Rates and Inflation,
1926-2009
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5-20
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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5-21
Rates of Return: Single Period Example
Ending Price =
Beginning Price =
Dividend =
110
100
4
HPR = (110 - 100 + 4 )/ (100) = 14%
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5-22
Expected Return and Standard
Deviation
Expected (or mean) returns
E (r )   p ( s )r ( s )
s
p(s) = probability of a state
r(s) = return if a state occurs
s = state
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5-23
Scenario Returns: Example
State
Excellent
Good
Poor
Crash
Prob. of State
0.25
0.45
0.25
0.05
r in State
0.3100
0.1400
-0.0675
-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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5-24
Variance and Standard Deviation
Variance (VAR):
   p( s)  r ( s)  E (r ) 
2
2
s
Standard Deviation (STD):
STD 

2
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5-25
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:
  0.038  0.1949
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5-26
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
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5-27
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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5-28
Geometric Variance and Standard
Deviation Formulas
Recall the definition of variance
   p( s)  r ( s)  E (r ) 
2
2
s
Estimated Variance = expected value of
squared deviations (from the mean)
n
2
1
ˆ   r s   r 
n s 1
2
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5-29
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*:
n
2
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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5-30
The Reward-to-Volatility (Sharpe)
Ratio
• Sharpe Ratio for Portfolios:
Risk Premium

SD of Excess Returns
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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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5-32
Figure 5.4 The Normal Distribution
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5-33
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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5-34
Skew and Kurtosis
this is zero for symmetric distributions


3
 R  R  
skew  average 

3
 ˆ

 R  R 4 
kurtosis  average 
 3
4
 ˆ



this equals 3 for a Normal distribution
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5-35
Figure 5.5A Normal and Skewed
Distributions
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5-36
Figure 5.5B Normal and Fat-Tailed
Distributions (mean = 0.1, SD =0.2)
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5-37
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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5-38
Normal Distribution and VaR
2.5
Percentile
2
1.5
1
VaR
0.5
0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
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0.2
0.4
0.6
0.8
1.0
5-39
Expected Shortfall (ES)
• a.k.a. Conditional Tail Expectation (CTE)
• More conservative measure of downside
risk than VaR:
– VaR takes the highest return from the worst
cases
– Real life distributions are asymmetric and
have fat tails
– ES takes an average return of the worst cases
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Normal Distribution, VaR, and Expected Shortfall
5-40
2.5
The area is the
percentile
2
1.5
1
VaR
Expected
Shortfall
0.5
0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
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0.2
0.4
0.6
0.8
1.0
5-41
Lower Partial Standard Deviation (LPSD)
and the Sortino Ratio
• Issues with real life returns:
– Need to look at negative returns separately to
account for asymmetry and fat tails
– Need to consider excess returns: 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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5-42
A game with a coin
Let’s play a game: flip a (non-fair) 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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5-43
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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5-44
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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5-45
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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5-46
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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5-47
1926-2009
S
(A)
(B)
(D)
(C)
(E)
30% (A)
50% (B)
20% (D)
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5-48
Historic Returns on Risky Portfolios
• Returns appear approximately 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
• Better diversified portfolios have higher
Sharpe Ratios
• Negative skew
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5-49
Figure 5.7 Nominal and Real Equity
Returns Around the World, 1900-2000
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5-50
Figure 5.8 Standard Deviations of Real Equity and
Bond Returns Around the World, 1900-2000
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5-51
Figure 5.9 Probability of Investment Outcomes
After 25 Years with a Lognormal Distribution
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5-52
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  1 / 2
so
m  g  1 / 2
2
2
• The Terminal Value will then be:
1  EAR
T

 e
g 1 / 2 2
 e
T
Tg T / 2 2
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5-53
Figure 5.10 Annually Compounded,
25-Year HPRs
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5-54
Figure 5.11 Annually Compounded,
25-Year HPRs
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5-55
Figure 5.12 Wealth Indices of Selected Outcomes of Large
Stock Portfolios and the Average T-bill Portfolio
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