Learning Objectives
Chapter 6
The Meaning and Measurement of
Risk and Return
§ Define and measure the expected rate of return
of an individual investor.
§ Define and measure the riskiness of an
individual investment.
§ Compare the historical relationship between risk
and return in the capital markets.
Learning Objectives
Principles Used in this Chapter
§ Explain how diversifying investments affects the
riskiness and expected rate of return of a
portfolio or combination of assets.
• Principle 1: The Risk-Return Trade-off –
We Won’t Take on Additional Risk Unless
We Expect to Be Compensated with
Additional Return.
§ Explain the relationship between an investor’s
required rate of return on an investment and the
riskiness of the investment.
Expected Cash Flow
• Weighted Average of the possible cash flows
outcomes such that the weights are the
probabilities of the occurrence of the various
states of the economy.
• Principle 3: Cash-Not Profits-Is King.
Expected Cash flow
• Conventionally, we measure the expected cash
flow as follows:
P(X1)X1 + P(X2)X2 + … + P(Xn)Xn
X =
where
n = the number of possible states of the economy
Xi =the cash flow in the ith state of the economy
P(Xi) = the probability of the ith cash flow
1
Expected Rate of Return
Measuring the Expected Return
State of the
economy
Probability of
the states
Cash flow
from the
investment
% Return (Cash
Flow/Inv. Cost)
Economic
Recovery
20%
$1,000
10%
($1,000/$10,000)
Moderate
Economic
Growth
30%
1,200
12%
($1,200/$10,000)
Strong
Economic
Growth
50%
1,400
14%
($1,400/$10,000)
• Weighted average of all the possible
returns, weighted by the probability that
each return will occur.
Expected Cash flow = 0.2*1000 + 0.3*1200 + 0.5* 1400 = 1260
Risk
Studying and Understanding Risk
• What is risk?
• How do we know the amount of risk
associated with a given investment; that is
how do we measure risk?
• If we choose to diversify our investments by
owning more that one asset, as most of us do,
will such diversification reduce the riskiness of
our combined portfolio of investments?
• Potential variability in future cash flows
• The wider the range of possible events
that can occur, the greater the risk
Standard Deviation of Return
• Square root of the weighted average
squared deviation of each possible
return from the expected return
• Quantitative measure of an asset ’s
riskiness
• Measures the volatility or riskiness of
portfolio returns
Standard Deviation of Return
−
n
∑ (k
σ=
i
− k )2 P( ki )
i =1
Where,
n
Ki
= the number of possible outcomes
= the value of ith possible rate of return
P(k i ) = Prob. that ith return wil l occur
-
k
= Expected value of the rate of return
2
Annual Rates of Return, 19262000
Total Risk or Variability
• Company -Unique Risk (Unsystematic)
• Market Risk (Systematic)
Diversification
• If we diversify investments across different
securities, the variability in the returns
declines
Company-Unique Risk
• Unsystematic risk
• Diversifiable
--Can be diversified away
• We can lower risk without sacrificing expected return,
and/or we can increase expected return without having to
assume more risk.
• Diversifying among different kinds of assets is called
asset allocation. Compared to diversification within the
different asset classes, the benefits received are far
greater through effective asset allocation.
Rates of Return: The Investors’ Experience
•
Data have been compiled by Ibbotson and Associates on the actual
returns for various portfolios of securities from 1926-1998.
•
The following portfolios were studied:
•
•
•
•
•
•
1.
2.
3.
4.
5.
6.
Common stocks of large firms
Common stocks of small firms
Corporate bonds
Long-term U.S. government bonds
Intermediate U.S. government bonds
U.S. Treasury bills
•
Investors historically have received greater returns for greater risktaking with the exception of the long -term U.S. government bonds.
•
The only portfolio with returns consistently exceeding the inflation
rate has been common stocks.
Total variability
• Total variability can be divided into:
– The variability of returns unique to the security
(diversifiable or unsystematic risk)
– The risk related to market movements
(nondiversifiable or systematic risk)
• By diversifying, the investor can eliminate the
"unique" security risk.
– The systematic risk, however, cannot be diversified
away.
3
Market Risk
• Systematic
• Non-diversifiable
• Cannot be eliminated through random
diversification
Measuring Market Risk
• Characteristic line
– The slope of the characteristic line
measures the average relationship between
a stock’s returns and those of the S&P 500
Index Returns.
– Indicates the average movement in a
stock’s price to a movement in the S&P 500
Price Index.
Measuring Market Returns
Monthly Holding-Period Returns of Barnes & Noble and the S&P 500
Index, December 2002 to November 2004
Market Risk
Events that affect market risk
Changes in the general economy, major political
events, sociological changes
Examples:
*Interest rates
*General economic conditions
*Changes in tax legislation that
affect all companies
*War
Measuring Market Risk….
• The characteristic line tells us the average movement
in a firm's stock price in response to a movement in the
general market, such as the S&P 500 Index.
– The slope of the characteristic line, which has come to be called
beta, is a measure of a stock's systematic or market risk. The
slope of the line is merely the ratio of the "rise" of the line relative
to the "run" of the line.
– If a security's beta equals one, a 10 percent increase (decrease)
in market returns will produce on average a 10 percent increase
(decrease) in security returns.
– A security having a higher beta is more volatile and thus more
risky than a security having a lower beta value.
Beta
• Average relationship between a stock’s returns
and the market’s returns
• Slope of the characteristic line—or the line that
measures the average relationship between a
stock’s returns and the market
• Measure of a firm’s market risk or systematic
risk that remains for a company even after
diversified our portfolio.
4
Beta
• A stock with a Beta of 0 has no systematic risk
• A stock with a Beta of 1 has systematic risk
equal to the “typical” stock in the marketplace
• A stock with a Beta exceeding 1 has
systematic risk greater than the “typical” stock
• Most stocks have betas between .60 and 1.60
Portfolio Beta
Holding-Period Returns: High- and Low-Beta Portfolios and the
S&P 500 Index
Asset Allocation
• Diversification among different kinds of
asset types:
T Bills
Long-Term Government Bonds
Common Stocks
Portfolio Beta
• Weighted average of the individual securities’
betas, with the weights being equal to the
proportion of the portfolio invested in each
security
• Portfolio beta indicates the percentage
change on average of the portfolio for every 1
percent change in the general market
Risk and Diversification
• The market rewards diversification
• We can lower risk without sacrificing
expected returns
• We can increase expected returns without
having to assume more risk
Required Rate of Return
• Minimum rate of return necessary to
attract an investor to purchase or hold a
security
• Considers the opportunity cost of funds
– The next best investment
5
Real Average Annual Rate of
Return
• Nominal rate of return less the inflation
rate
Required Rate of Return
k=kf r + krp
Where:
k = required rate of return
kfr = Risk-Free Rate
krp = Risk Premium
Risk-Free Rate
Risk Premium
• Required rate of return or discount rate
for risk-less investments
• Typically measured by U.S. Treasury
Bill Rate
• Additional return we must expect to
receive for assuming risk
• As risk increases, we will demand
additional expected returns
Measuring the Required Rate
of Return
Measuring the Required Rate
of Return
• Systematic risk is the only relevant riskthe rest can be diversified away
• The required rate of return, k, equals
the risk free rate, k rf, plus a risk
premium, k rp
Risk Premium = Required Return – RiskFree rate
krp = k - kf r
Where:
k = required rate of return
k fr = Risk-Free Rate
k rp = Risk Premium
6
Capital Asset Pricing Model
CAPM
• Equation that equates the expected rate of
return on a stock to the risk-free rate plus a
risk premium for the systematic risk.
If required return is 15% and the risk-free
rate is 5%, then the risk premium is
10%.
If the required rate of return for the market
portfolio k m is 12%, and the k rf is 5%,
the risk premium krp for the market
would be 7%.
• CAPM provides for an intuitive approach for
thinking about the return that an investor
should require on an investment, given the
asset’s systematic or market risk.
CAPM
CAPM
This 7% risk premium would apply to any
security having systematic
(nondiversifiable) risk equivalent to the
general market, or beta of 1.
In the same market, a security with Beta
of 2 would provide a risk premium of
14%.
CAPM
Example:
Market risk = 12%
Risk-free rate = 5%
5% + B(12% - 5%)
If B = 0
Required rate = 5%
If B = 1
Required rate = 12%
If B = 2
Required rate = 19%
• CAPM suggests that Beta is a
factor in required returns
kj = krf + B(market rate – risk-free rate)
The Security Market Line
• Graphic representation of the CAPM,
where the line shows the appropriate
required rate of return for a given stock’s
systematic risk.
7
The Security Market Line
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