Risk and
Return: Past
and Prologue
5
Bodie, Kane and Marcus
Essentials of Investments
9th Global Edition
5.1 RATES OF RETURN
• Holding-Period Return (HPR)
• Rate of return over given investment period
• HPR= [PS − PB + CF] / PB
• PS = Sale price
• PB = Buy price
• CF = Cash flow during holding period
TABLE 5.1 QUARTERLY CASH FLOWS/RATES
OF RETURN OF A MUTUAL FUND
1st
2nd
3rd
4th
Quarter Quarter Quarter Quarter
Assets under management at start of
quarter ($ million)
1
1.2
2
0.8
Holding-period return (%)
10
25
−20
20
Total assets before net inflows
1.1
1.5
1.6
0.96
Net inflow ($ million)
0.1
0.5
−0.8
0.6
Assets under management at end of
quarter ($ million)
1.2
2
0.8
1.56
5.1 RATES OF RETURN
• Measuring Investment Returns 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
• Compound period-by-period returns; find per-period rate that
compounds to same final value
• Dollar-weighted average return
• Internal rate of return on investment
5.1 RATES OF RETURN

Conventions for Annualizing Rates of Return

APR = Per-period rate × Periods per year
1 + EAR = (1 + Rate per period)
APR
n
 1 + EAR = (1 + Rate per period) = (1 +
n
 APR = [(1 + EAR)1/n – 1]n
APR
 Continuous compounding: 1 + EAR = e

)n
5.2 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 of distribution of
HPR
• Variance: Expected value of squared deviation
from mean
• Standard deviation: Square root of variance
SPREADSHEET 5.1 SCENARIO ANALYSIS
FOR THE STOCK MARKET
5.2 RISK AND RISK PREMIUMS
FIGURE 5.1 NORMAL DISTRIBUTION WITH MEAN
RETURN 10% AND STANDARD DEVIATION 20%
5.2 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
5.2 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
• Using Time Series of Return
• Scenario analysis derived from sample history of
returns
• Variance and standard deviation estimates from tim
series of returns:
5.2 RISK AND RISK PREMIUMS
Suppose you’ve estimated that the fifthpercentile value at risk of a portfolio is –
30%. Now you wish to estimate the
portfolio’s first-percentile VaR (the value
below which lie 1% of the returns). Will the
1% VaR be greater or less than –30%?
5.2 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 (A): Ratio of risk premium to
variance
5.2 RISK AND RISK PREMIUMS
• The Sharpe (Reward-to-Volatility) Ratio
• Ratio of portfolio risk premium to standard
deviation
• Mean-Variance Analysis
• Ranking portfolios by Sharpe ratios
5.2 RISK AND RISK PREMIUMS
11. Consider a risky portfolio. The end-of-year cash flow derived from
the portfolio will be either $ 70,000 or $ 195,000, with equal
probabilities of .5. The alternative riskless investment in T-bills pays
4%.
a. If you require a risk premium of 8%, how much will you be willing
to pay for the portfolio?
b. Suppose the portfolio can be purchased for the amount you found
in (a). What will the expected rate of return on the portfolio be?
c. Now suppose you require a risk premium of 11%. What is the price
you will be willing to pay now?
d. Comparing your answers to (a) and (c), what do you conclude about
the relationship between the required risk premium on a portfolio and
the price at which the portfolio will sell?
5.3 THE HISTORICAL RECORD
• World and U.S. Risky Stock and Bond
Portfolios
• World Large stocks: 24 developed countries, about
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
FIGURE 5.3 FREQUENCY DISTRIBUTION OF ANNUAL,
CONTINUOUSLY COMPOUNDED RATES OF RETURN,
1926-2010
FIGURE 5.4 RATES OF RETURN ON STOCKS,
BONDS,
AND BILLS
TABLE 5.2 ANNUAL RATE-OF-RETURN
STATISTICS FOR DIVERSIFIED PORTFOLIOS FOR
1926-2010 AND THREE SUBPERIODS (%)
5.4 INFLATION AND REAL RATES OF
RETURN
• Equilibrium Nominal Rate of Interest
• Fisher Equation
• R = r + E(i)
• E(i): Current expected inflation
• R: Nominal interest rate
• r: Real interest rate
5.4 INFLATION AND REAL RATES OF
RETURN
• 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 5.5 INTEREST RATES, INFLATION,
AND REAL INTEREST RATES 1926-2010
5.5 ASSET ALLOCATION ACROSS
PORTFOLIOS
• Asset Allocation
John Bogle of Vanguard Says:
• Portfolio choice among broad investment
The most fundamental decision of investing is the
classesallocation of your assets: How much should you own in
stock? How much should you own in bonds? How much
• Complete
Portfolio
should
you own in cash reserves? . . . That decision [has
been shown to account] for an astonishing 94% of the
• Entire differences
portfolio,
including risky and risk-free
in total returns achieved by institutionally
assetsmanaged pension funds. . . . There is no reason to believe
that the same relationship does not also hold true for
• Capital Allocationindividual investors.
• Choice between risky and risk-free assets
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
5.5 ASSET ALLOCATION ACROSS
PORTFOLIOS
• Portfolio Expected Return and Risk
P: portfolio composition
y: proportion of investment budget
rf: rate of return on risk-free asset
rp: actual rate of return
E(rp): expected rate of return
σp: standard deviation
E(rC): return on complete portfolio
E(rC) = yE(rp) + (1 − y)rf
σC = yσrp + (1 − y) σrf
FIGURE 5.6 INVESTMENT OPPORTUNITY
SET
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
5.5 ASSET ALLOCATION ACROSS
PORTFOLIOS
5.5 ASSET ALLOCATION ACROSS
PORTFOLIOS
13) Assume that you manage a risky portfolio with an
expected rate of return of 12% and a standard deviation
of 28% (Weights: Stock A 20%; Stock B 30%; Stock C
50%(. The T-bill rate is 4%. Suppose your client decides
to invest in your risky portfolio a proportion (y) of his
total investment budget so that his overall portfolio will
have an expected rate of return of 11%.
a)
What is the proportion, y?
b)
What are your client’s investment proportions in
your 3 stocks and the T-bill fund?
c)
What is the standard deviation of the rate of return
on your client’s portfolio?
5.5 ASSET ALLOCATION ACROSS
PORTFOLIOS
14) Assume that you manage a risky portfolio
with an expected rate of return of 12% and a
standard deviation of 28%. The T-bill rate is 4%.
Suppose your client prefers to invest in your
portfolio a proportion (y) that maximizes the
expected return on the overall portfolio subject
to the constraint that the overall portfolio’s
standard deviation will not exceed 20%.
a) What is the investment proportion, y?
b) What is the expected returns of the overall
portfolio?
5.5 ASSET ALLOCATION ACROSS
PORTFOLIOS
17) Assume that you manage a risky portfolio with an
expected rate of return of 12% and a standard deviation
of 28%. There is also a passive fund that mimics the S&P
500 Index with an expected return of 13% and a
standard deviation of 25%. The T-bill rate is 7%.
a)
Explain to your client the disadvantage of the switch.
b)
Show your client the maximum fee you could charge
(as a percent of the investment in your fund deducted
at the end of the year) that would still leave him at
least as well off investing in your fund as in the
passive one.
5.5 ASSET ALLOCATION ACROSS
PORTFOLIOS
You’ve just decided upon your capital allocation
for the next year, when you realize that you’ve
underestimated both the expected return and
the standard deviation of your risky portfolio by
4%. Will you increase, decrease, or leave
unchanged your allocation to risk-free T-bills?
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
TABLE 5.4 EXCESS RETURN STATISTICS
FOR S&P 500
Excess Return (%)
Averag
e
Std
Dev.
Sharpe
Ratio
5% VaR
1926-2010
8.00
20.70
.39
−36.86
1926-1955
11.67
25.40
.46
−53.43
1956-1985
5.01
17.58
.28
−30.51
1986-2010
7.19
17.83
.40
−42.28
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