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INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
Risk, Return, and the Historical Record
Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
5-2
• Interest rate determinants• Rates of return for different holding periods• Risk and risk premiums• Estimations of return and risk• Normal distribution
• Deviation from normality and risk estimation• Historic returns on risky portfolios
Chapter Overview
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Supply• Households
• Demand• Businesses
• Government’s net demand• Federal Reserve actions
Interest Rate Determinants
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Nominal interest rate (rn): • Growth rate of your money
• Real interest rate (rr): • Growth rate of your purchasing power
• Where i is the rate of inflation
Real and Nominal Rates of Interest
rr rn i 1
rn irr
i
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Figure 5.1 Determination of the Equilibrium Real Rate of Interest
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• 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:
Equilibrium Nominal Rate of Interest
rn rr E i
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Tax liabilities are based on nominal income• Given a tax rate (t) and nominal interest rate (rn),
the real after-tax rate is:
• The after-tax real rate of return falls as the inflation rate rises
Taxes and the Real Rate of Interest
1 1 1rn t i rr i t i rr t it
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Zero Coupon Bond:• Par = $100• Maturity = T• Price = P• Total risk free return
Rates of Return for Different Holding Periods
100( ) 1
( )fr T P T
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Example 5.2 Annualized Rates of Return
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• EAR: Percentage increase in funds invested over a 1-year horizon
Effective Annual Rate (EAR)
1
1 EAR 1 Tfr T
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• APR: Annualizing using simple interest
Annual Percentage Rate (APR)
1 EAR 1APR
T
T
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Table 5.1 APR vs. EAR
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Table 5.2 T-Bill Rates, Inflation Rates,
and Real Rates, 1926-2012
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Moderate inflation can offset most of the nominal gains on low-risk investments
• A dollar invested in T-bills from 1926–2012 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
Bills and Inflation, 1926-2012
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Figure 5.3 Interest Rates and Inflation, 1926-2012
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Rates of return: Single period
• HPR = Holding period return
• P0 = Beginning price
• P1 = Ending price
• D1 = Dividend during period one
Risk and Risk Premiums
1 0 1
0
HPR P P D
P
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Ending Price = $110Beginning Price = $100Dividend = $4
Rates of Return: Single Period Example
$110 $100 $4HPR .14, or 14%
$100
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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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
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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State Prob. of State r in State Excellent .25 0.3100Good .45 0.1400Poor .25 -0.0675Crash .05 -0.5200
E(r) = (.25)(.31) + (.45)(.14) + (.25)(−.0675) + (0.05)(− 0.52) E(r) = .0976 or 9.76%
Scenario Returns: Example
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Variance (VAR):
• Standard Deviation (STD):
Expected Return and Standard Deviation
2STD
s
rEsrsp 22
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• 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:
Scenario VAR and STD: Example
σ .038
.1949
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• True means and variances are unobservable because we don’t actually know possible scenarios like the one in the examples
• So we must estimate them (the means and variances, not the scenarios)
Time Series Analysis of Past Rates of Return
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Arithmetic Average
• Geometric (Time-Weighted) Average
= Terminal value of the investment
Returns Using Arithmetic and Geometric Averaging
n
s
n
s
srn
srsprE11
)(1
)()()(
)1)...(1)(1( 21 nn rrrTV
1/ 1ng TV
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Estimated Variance• Expected value of squared deviations
• Unbiased estimated standard deviation
Estimating Variance and Standard
Deviation
n
s
rsrn 1
22 1̂
2
11
1ˆ
n
j
rsrn
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• 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
The Reward-to-Volatility (Sharpe) Ratio
Risk premium
SD of excess returns
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• 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 as well• Future scenarios can be estimated using only the
mean and the standard deviation• The dependence of returns across securities can be
summarized using only the pairwise correlation coefficients
The Normal Distribution
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Figure 5.4 The Normal Distribution
Mean = 10%, SD = 20%
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• 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 skewness and kurtosis
Normality and Risk Measures
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Figure 5.5A Normal and Skewed Distributions
Mean = 6%, SD = 17%
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Figure 5.5B Normal and Fat-Tailed Distributions
Mean = .1, SD = .2
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Value at Risk (VaR)• Loss corresponding to a very low percentile of the
entire return distribution, such as the fifth or first percentile return
• Expected Shortfall (ES)• Also called conditional tail expectation (CTE),
focuses on the expected loss in the worst-case scenario (left tail of the distribution)
• More conservative measure of downside risk than VaR
Normality and Risk Measures
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• Lower Partial Standard Deviation (LPSD)and the Sortino Ratio• Similar to usual standard deviation, but uses only
negative deviations from the risk-free return, thus, addressing the asymmetry in returns issue
• Sortino Ratio (replaces Sharpe Ratio)• The ratio of average excess returns to LPSD
Normality and Risk Measures
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• The second half of the 20th century, politically and economically the most stable sub-period, offered the highest average returns
• Firm capitalization is highly skewed to the right: Many small but a few gigantic firms
• Average realized returns have generally been higher for stocks of small rather than large capitalization firms
Historic Returns on Risky Portfolios
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• 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
Historic Returns on Risky Portfolios
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Figure 5.8 SD of Real Equity & Bond Returns Around the World
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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Figure 5.9 Probability of Investment with a Lognormal Distribution
INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
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• When the continuously compounded rate of return on an asset is normally distributed, the effective rate of return will be lognormally distributed
Terminal Value with Continuous Compounding