Chapter
5
Risk, Return and the
Historical Record
Bodie, Kane, and Marcus
Essentials of Investments
12th Edition
5.1 Rates of Return
• Holding-Period Return (HPR)
• Rate of return over given investment period
HPR 
PEnding  PBeginning  DivCash
PBeginning
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5.1 Rates of Return: Example
• What is the HPR for a share of stock that was
purchase for $25, sold for $27 and distributed
$1.25 in dividends?
$27.00 – $25.00  $1.25
HPR 
 0.13  13.00%
$25.00
• The HPR is the sum of the dividend yield plus the
capital gains yield
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5.1 Rates of Return: Measuring over Multiple Periods
• Arithmetic average
• Sum of returns in each period divided by
number of periods
• Geometric average
• Single per-period return
• Gives same cumulative performance as
sequence of actual returns
• Dollar-weighted average return
• Internal rate of return on investment
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Table 5.1 Annual Cash Flows & Rates of Return of a
Mutual Fund
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5.1 Rates of Return: Measuring over Multiple Periods
• Arithmetic average: The sum of the returns
divided by the number of years.
rArithmetic
r1  r2  ...  rn

n
10  25  20  20

 .0875  8.75%
4
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5.1 Rates of Return: Measuring over Multiple Periods
• Geometric average: Single period return that gives
the same cumulative performance as the sequence
of actual returns
rGeometric  [(1  r1 )  (1  r2 )  ...  (1  rn )]1/ n  1
 1.10 1.25  .80 1.20  1  0.0719  7.19%
1/4
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5.1 Rates of Return
• Dollar-weighted average return
• The internal rate of return on an investment
• Annualizing Rates of Return
• APR = Annual Percentage Rate
• Per-period rate × Periods per year
• Ignores Compounding
• EAR = Effective Annual Rate
• Actual rate an investment grows
• Does not ignore compounding
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5.1 Rates of Return: EAR vs. APR
n-Periods of Compounding:
Continuous Compounding:
APR 

EAR  1 
 1
n 

EAR  e APR  1
APR  [( EAR  1)1/ n  1]  n
APR  ln( EAR  1)
n
where
n  compounding per period
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5.2 Inflation and The Real Rates of Interest
• Nominal Interest and Real Interest
1  rReal 
1  rNom
1 i
where
rReal  Real Interest Rate
rNom  Nominal Interest Rate
i  Inflation Rate
• Example: What is the real return on an investment that
earns a nominal 10% return during a period of 5% inflation?
1  .10
 1.048
1  .05
r  .048 or 4.8%
1  rReal 
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5.2 Inflation and The Real Rate of Interest
• Equilibrium Nominal Rate of Interest
• Fisher Equation (5.9)
rNom  rReal  E (i )
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5.2 Inflation and The Real Rate of Interest
• U.S. History of Interest Rates, Inflation, and
Real Interest Rates
• Since the 1950s, nominal rates have increased
roughly in tandem with inflation
• 1930s/1940s: Volatile inflation affects real rates
of return
• Figure 5.1
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Figure 5.1 Inflation and Interest rates (1927-2018)
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5.3 Risk and Risk Premiums
• Scenario Analysis and Probability Distributions
• Scenario analysis: Possible economic scenarios;
specify likelihood and HPR
• Probability distribution: Possible outcomes with
probabilities
• Expected return: Mean value
• Variance: Expected value of squared deviation from
mean
• Standard deviation: Square root of variance
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Spreadsheet 5.1 Scenario Analysis for a Stock Index Fund
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5.3 Risk and Risk Premiums
• The Normal Distribution
• Transform normally distributed return into
standard deviation score:
𝑟𝑖 − 𝐸(𝑟𝑖 )
𝑠𝑟𝑖 =
𝜎𝑖
• Original return, given standard normal return:
𝑟𝑖 = 𝐸 𝑟𝑖 + 𝑠𝑟𝑖 × 𝜎𝑖
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Figure 5.3 Normal Distribution r = 10% and σ = 20%
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5.3 Risk and Risk Premiums
• Normality over Time
• When returns over very short time periods are
normally distributed, HPRs up to 1 month can be
treated as normal
• Use continuously compounded rates where
normality plays crucial role
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5.3 Risk and Risk Premiums: Value at Risk
• Value at risk (VaR):
•Measure of downside risk
•Worst loss with given probability, usually 1%
or 5%
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5.3 Risk and Risk Premiums
• Deviation from Normality and Value at Risk
• Kurtosis: Measure of fatness of tails of probability
distribution; indicates likelihood of extreme
outcomes
• Skew: Measure of asymmetry of probability
distribution
• The Sharpe (Reward-to-Volatility) Ratio
• Ratio of portfolio risk premium to standard
deviation
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5.3 Risk and Risk Premiums
• Risk Premiums and Risk Aversion
• Risk-free rate: Rate of return that can be earned
with certainty
• Risk premium: Expected return in excess of that
on risk-free securities
• Excess return: Rate of return in excess of risk-
free rate
• Risk aversion: Reluctance to accept risk
• Price of risk: Ratio of risk premium to variance
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5.3 Risk and Risk Premiums
• Mean-Variance Analysis
• Ranking portfolios by Sharpe ratios
E (rp )  rf
Portfolio Risk Premium
SP 
Standard Deviation of Excess Returns
P
where
E (rp )  Expected Return of the portfolio
rf  Risk Free rate of return
 P  Standard Deviation of portfolio excess return
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5.4 The Historical Record
• Using Time Series of Return
• Scenario analysis derived from sample history of returns
• Variance and standard deviation estimates from time
series of returns:
1
2
Var (rt ) 
   rt  rt 
n 1
SD(rt )  Var (rt )
1
rt   rt
n
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5.4 The Historical Record: World Portfolios
• World Large stocks: 24 developed countries,
~6000 stocks
• U.S. large stocks: Standard & Poor's 500 largest
cap
• U.S. small stocks: Smallest 20% on NYSE,
NASDAQ, and Amex
• World bonds: Same countries as World Large
stocks
• U.S. Treasury bonds: Barclay's Long-Term
Treasury Bond Index
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Table 5.3: Historical Return and Risk
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Figure 5.4: Treasury Bills
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Figure 5.4: 30-year Treasury Bonds
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Figure 5.4: Common Stocks
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5.5 Asset Allocation across Portfolios
• Asset Allocation
• Portfolio choice among broad investment
classes
• Complete Portfolio
• Entire portfolio, including risky and risk-free
assets
• Capital Allocation
• Choice between risky and risk-free assets
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5.5 Asset Allocation across Portfolios
• The Risk-Free Asset
• Treasury bonds (still affected by inflation)
• Price-indexed government bonds
• Money market instruments effectively risk-free
• Risk of CDs and commercial paper is miniscule
compared to most assets
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5.5 Portfolio Asset Allocation: Expected Return and Risk
Expected Return of the Complete Portfolio
E (rC )  y  E (rp )  (1  y )  r f
where E (rC )  Expected Return of the complete portfolio
E (rp )  Expected Return of the risky portfolio
rf  Return of the risk free asset
y  Percentage assets in the risky portfolio
Standard Deviation of the Complete Portfolio
 C  y  p
where
 C  Standard deviation of the complete portfolio
 P  Standard deviation of the risky portfolio
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Figure 5.7 Investment Opportunity Set
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5.5 Asset Allocation across Portfolios
• Capital Allocation Line (CAL)
• Plot of risk-return combinations available by
varying allocation between risky and risk-free
• Risk Aversion and Capital Allocation
• y: Preferred capital allocation
Available risk premium to variance ratio
y
Required risk premium to variance ratio

[ E (rP )  rf ] /  P2
A

[ E (rP )  rf ]
A P2
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5.6 Passive Strategies and the Capital Market Line
• Passive Strategy
• Investment policy that avoids security analysis
• Capital Market Line (CML)
• Capital allocation line using market-index
portfolio as risky asset
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Table 5.5: Excess Returns Statistics for the Market Index
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5.6 Passive Strategies and the Capital Market Line
• Cost and Benefits of Passive Investing
• Passive investing is inexpensive and simple
• Expense ratio of active mutual fund averages 1%
• Expense ratio of hedge fund averages 1%-2%,
plus 10% of returns above risk-free rate
• Active management offers potential for higher
returns
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