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
Topics
• Discussion of Risk and Risk Premiums
–
–
–
–
Determinants of the Level of Interest Rates
Bills and Inflation, 1926-2009
Risk and Risk Premiums
Time Series Analysis of Past Rates of Return
• Normality of Returns
– The Normal Distribution
– Deviations from Normality and Risk Measures
– Historic Returns on Risky Portfolios
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Real and Nominal Rates of Interest
• Nominal interest rate (R): Growth rate of your money
• Real interest rate (r): Growth rate of your purchasing
power
• Inflation rate (i): Growth rate of the price level
• Then,
1
1
1
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Equilibrium Real Rate of Interest
• Determined by:
– Supply (Households, Government)
– Demand (Businesses, Government)
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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:
R(1  t )  i  (r  i )(1  t )  i  r (1  t )  it
• The after-tax real rate of return falls as the
inflation rate rises.
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Table 5.2 Statistics for T‐Bill Rates, Inflation
Rates and Real Rates, 1926‐2009
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Figure 5.2 Nominal and Real Wealth Indexes
for Investments in T‐Bills, 1968‐2009
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Figure 5.3 Interest Rates and Inflation,
1926‐2009
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Correlation of Inflation with …
Nominal R = Real r + Exp. Inflation
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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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Interest Rates and Inflation in Korea
•
Source: ECOS (Economic Statistics System: 한국은행 경제통계시스템)
Monthly inflation data
measured by CPI, YoY basis
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Interest Rates and Inflation in Korea
•
Source: ECOS (Economic Statistics System: 한국은행 경제통계시스템)
Monthly inflation data
measured by CPI, YoY basis
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Interest Rates and Inflation in Korea
•
Source: ECOS (Economic Statistics System: 한국은행 경제통계시스템)
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Interest Rates and Inflation in Korea
•
Source: ECOS (Economic Statistics System: 한국은행 경제통계시스템)
30.00
무담보콜금리(1일물, 은행간직거래)
25.00
CD유통수익률(91일)
산금채(3년)
회사채(장외3년,AA- 등급)
20.00
소비자물가지수 (전년대비증감률)
15.00
10.00
5.00
0.00
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Holding Period Return
Rates of Return: Single Period
P
1  P 0  D1
HPR 
P0
= capital gains yield + dividend yield
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
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Excess Returns and Risk Premiums
• We measure the reward as the difference between the
expected HPR and the risk-free rate; we call this
difference the risk premium on common stocks.
• The difference in any particular period between the
actual rate of return on a risky asset and the actual riskfree rate is called excess return.
• Therefore, the risk premium is the expected value of the
excess return, and STD of the excess return is a
measure of its risk.
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The Reward‐to‐Volatility (Sharpe) Ratio
• Sharpe Ratio for Portfolios:
Risk Premium

SD of Excess Returns
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Expected Return (Definition)
Expected returns
p(s) = probability of a state s
r(s) = return if a state s occurs
s = state
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Scenario Returns: Example
State
Excellent
Good
Poor
Crash
Prob. of State
.25
.45
.25
.05
r in State
0.3100
0.1400
-0.0675
-0.5200
E(r) = (.25)(.31) + (.45)(.14) + (.25)(-.0675)
+ (0.05)(-0.52)
E(r) = .0976 or 9.76%
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Variance and Standard Deviation (Definition)
Variance (VAR):
Standard Deviation (STD):
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Scenario VAR and STD: Example
• 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:
  .038
 .1949
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Estimation of The Sharpe Ratio
• The both definitions of expected return (risk premium)
and standard deviation (risk or volatility) involve
expectation (E[ ]).
• Empirically, since we don’t know the true distribution of
an asset return, we cannot use the definitions of
expected return and standard deviation to estimate the
Sharpe ratio of an asset.
• In order to estimate an expected value of a random
variable, we need something called ‘Law of Large
Numbers.’
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Law of Large Numbers
• Idea: If you run many trials in a simulation or experiment,
the empirical probability will get closer and closer to the
actual or theoretical probability of the event.
• Example: Coin flipping
• Formal Statement (of a version of LLN):
,
,…
. . .
⋯
1
⋯
∞,
→
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Time Series Analysis of Past Rates of Return
The Arithmetic Average of Rate of Return:
1
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Time Series Analysis of Past Rates of Return
The Geometric Average Rate of Return
∙
…
or,
1
̅
1
1
… 1
1
̅ = geometric average rate of return
Example: Terminal Value of the Investment
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Variance and Standard Deviation Estimation
• Estimated Variance = expected value of
squared deviations
_ 2
1 

    r s   r 
n s 1 

^ 2
where
n
is the arithmetic average
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Variance and Standard Deviation Estimation
• When eliminating the (degrees of freedom)
bias, Variance and Standard Deviation
become:
_ 2
1



r s   r 


n  1 j 1 

^
n
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How Precise is The Estimate?
• Once we estimate the risk premium (expected excess
return) of an asset, we would naturally be interested in
how precise the estimate is.
• In order to find out the precision of the estimate, we need
something called ‘Central Limit Theorem.’
• The ‘Central Limit Theorem’ gives us the distribution of
the sample mean estimated.
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Central Limit Theorem
• Idea: The ‘Central Limit Theorem’ says that the
distribution of a sample mean
, for sufficiently large
sample, will be normal, no matter what distribution the
original data have.
• Formal Statement:
,
,…
∞
. . .
.
,
0,
1
→
0,
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Back to The Estimation Precision…
• According to the CLT, the estimated risk premium
(sample mean ) will have the following distribution:
For large enough n,
,
,
0,1
• Hence,
1.96
1.96
95%
2.58
2.58
99%
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Figure 5.4 The Normal Distribution
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Back to The Estimation Precision…
• However, the problem is that we don’t know the true
of ; what we have is a sample variance
variance
.
• Then, if is replaced with its estimate , for large
enough n,
1
• Hence,
, 0.975
, 0.975
95%
, 0.995
, 0.995
99%
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Back to The Estimation Precision…
•
, 0.975 is the critical value for
which makes
, 0.975
distribution
0.975
• However, we know that
distribution converges to
normal distribution as → ∞.
• Hence, as far as this course is concerned, we will use
–
–
, 0.975
, 0.995
1.96
2.58
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Student t‐dist’n vs. Normal dist’n
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t‐test for Sample Mean
• Now that we know how precise the estimate for the risk
premium is, we can test if the risk premium is zero or
not.
• Let’s assume that the ‘true’ risk premium is zero for an
asset. In other words, our null hypothesis is
:
0
• Then, under the null hypothesis,
0
1.96
1.96
1.96
95%,
5%
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t‐test for Sample Mean
• If we assume that 5% event is a rare/outlier event, we
formally say that the level of significance is set at 5%.
• Then, if the absolute value of the t-stat is greater than
1.96, we conclude that the observed t-stat is in the
rare/outlier region under the null hypothesis.
• In other words, we conclude that the observed t-stat
cannot be justified by the null hypothesis and we
REJECT the null hypothesis; we say the risk premium is
statistically significant or statistically different from zero.
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The Normal Distribution
• Investment management is easier when returns are
normal.
– Symmetricity: Standard deviation is a good measure of risk when
returns are symmetric.
– Stability: If security returns are normal, portfolio returns will be,
too.
– Simplicity: Future scenarios can be estimated using only the
mean and the standard deviation.
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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
_ 3


 R  R  
skew  average   ^  


3
 



Kurtosis
• Equation 5.20
_ 4


 R  R  
kurtosis  average  ^    3


4
 



3
3
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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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Extreme Negative Return Risk Measures
• Higher frequency of extreme negative returns may result
from negative skew and/or kurtosis (fat tails).
• Therefore, we would like a risk measure that indicates
vulnerability to extreme negative returns:
– Value at Risk
– Expected Shortfall
– Lower Partial STD
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Value at Risk (VaR)
•
A measure of loss most frequently associated with extreme
negative returns
•
VaR is written into regulation of banks and closely watched by
risk managers.
•
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 with STD as a risk measure when the return distribution is
nonnormal:
– 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 (rather than negative deviations from the sample
average)
•
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 (19862009)
•
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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Historic Returns on Risky Portfolios
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Historic Returns on Risky Portfolios
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Historic Returns on Risky Portfolios
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