CHAPTER 5 Risk and Rates of Return
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Stand-alone risk
Portfolio risk
Risk & return: CAPM / SML
1
Investment returns
The rate of return on an investment can be
calculated as follows:
For example, if $1,000 is invested and $1,100 is
returned after one year, the rate of return for this
investment is:
($1,100 - $1,000) / $1,000 = 10%.
2
What is investment risk?

Investment risk is related to the probability of
earning a low or negative actual return. The
greater the chance of lower than expected or
negative returns, the riskier the investment.
Two types of investment risk
 Stand-alone risk: all our money is tied to a single
asset
 Portfolio risk : Asset is held as one of many assets
in the portfolio
3
Probability distributions
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A listing of all
possible outcomes,
and the probability
of each occurrence.
Can be shown
graphically.
4
Selected Realized Returns, 1926 – 2001
5
T-bills
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T-bills will return the promised return,
regardless of the economy.
T-bills do not provide a risk-free return, as
they are still exposed to inflation. Although,
very little unexpected inflation is likely to
occur over such a short period of time.
T-bills are also risky in terms of reinvestment
rate risk.
T-bills are risk-free in the default sense of the
word.
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Example: Investment alternatives
7
How do the returns of HT and Coll. behave in
relation to the market?
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HT – Moves with the economy, and has
a positive correlation. This is typical.
Coll. – Is countercyclical with the
economy, and has a negative
correlation. This is unusual.
8
Expected Return
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Weighted average
Weights are probabilities
Weights add up to 1
9
Measuring Stand-alone Risk
Calculating the standard deviation for each alternative
standard deviation is the square root of variance. It has
same unit as expected return
10
Comparing standard deviations
11
Comments on standard deviation as a measure of
risk
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Standard deviation (σi) measures total, or
stand-alone, risk.
The larger σi is, the lower the probability that
actual returns will be closer to expected
returns.
Larger σi is associated with a wider
probability distribution of returns.
Difficult to compare standard deviations,
because return has not been accounted for.
12
Investor attitude towards risk
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Risk aversion – assumes investors dislike
risk and require higher rates of return to
encourage them to hold riskier securities.
Risk premium – the difference between the
return on a risky asset and less risky asset,
which serves as compensation for investors
to hold riskier securities.
13
Portfolio construction: Risk and return
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Assume a two-stock portfolio is created with
$50,000 invested in HT and $50,000 invested
in Collections.
Expected return of a portfolio is a weighted
average of each of the component assets of
the portfolio.
Standard deviation is a little more tricky and
requires that a new probability distribution for
the portfolio returns be devised.
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portfolio expected return
15
An alternative method for determining
portfolio expected return
In recession,
$ return from HT $50,000*(-22%)= -$11,000
$ return from Coll $50,000*(28%)= $14,000
Total $ return is $3,000 so % return is 3%
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Calculating portfolio standard deviation
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Comments on portfolio risk measures
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σp = 3.3% is much lower than the σi of either stock
(σHT = 20.0%; σColl. = 13.4%).
σp = 3.3% is lower than the weighted average of HT
and Coll.’s σ (16.7%).
Portfolio provides average return of component
stocks, but lower than average risk.
Why? Negative correlation between stocks.
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Calculation of covariance and correlation
19
Combination line
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The risk of a portfolio is very different from a simple
average of the risk of individual assets in the
portfolio. There is a risk reduction from holding a
portfolio of assets if assets do not move in perfect
unison.
Combination line shows how the expected return and
risk of a two-security portfolio changes as weights
are changed.
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Combination line
PARAMETERS
12.00%
0.1
0.0028
0.07
0.0049
-0.5
10.00%
expected return
E.RETURN A
VARIANCE A
E.RETURN B
VARIANCE B
CORRELATION A,B
A
8.00%
B
6.00%
4.00%
2.00%
0.00%
0.00%
COVAB
-0.002
2.00%
4.00%
6.00%
8.00%
standard deviation
21
General comments about risk
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Most stocks are positively correlated
with the market (ρk,m  0.65).
σ  35% for an average stock.
Combining stocks in a portfolio
generally lowers risk.
22
Illustrating diversification effects of a stock
portfolio
To examine the relationship between portfolio size and portfolio
risk, consider average annual standard deviations for equallyweighted portfolios that contain different numbers of randomly
selected NYSE securities.
Number of stocks
in portfolio
Average standard deviation
Of annual portfolio returns
1
2
3
….
10
20
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diversification
Principle of diversification: Spreading an investment
across many assets will eliminate some of the risk.
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Breaking down sources of risk
Stand-alone risk = Market risk + Firm-specific risk
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Market risk – portion of a security’s stand-alone risk
that cannot be eliminated through diversification.
Measured by beta.
Firm-specific risk – portion of a security’s stand-alone
risk that can be eliminated through proper
diversification.
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In other words
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Firm-specific or diversifiable risk is caused by such
random events as lawsuits, strikes and other events
that are unique to a particular firm. Since these
events are random, their effects on a portfolio can be
eliminated by diversification-bad events in one firm
will be offset by good events in another.
Market or non-diversifiable risk stems from factors
that systematically affect most firms: war, inflation,
recessions, and high interest rates. Since most stocks
are negatively affected by these factors, market risk
cannot be eliminated by diversification.
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single index model
Models return generating process. It offers significant new insights
into the nature of systematic vs. firm-specific risk
ri - rf = i + i(rM-rf) + i
single index model
Single-index (factor) model assumes
Var(ri - rf )= Var(i + i(rM-rf) + i)= i2Var(rM-rf) + Var(i)
Total risk = market risk + firm-specific risk
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Well-diversified portfolio
Well-diversified portfolio
rp= Σwiri = Σwi(rf +i + i(rM-rf) + i)
=rf + Σwii + (Σwii)(rM-rf) + Σwii
Last term becomes zero
rp-rf = Σwii + (Σwii)(rM-rf) = p + p(rM-rf)
So well-diversified portfolio does not have firm-specific component
Var(rp-rf )= p2Var(rM-rf)
Total risk of well-diversified portfolio is proportional to total risk of
market portfolio
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For the pricing of risk what is the relevant
measure?
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If investors are primarily concerned with the riskiness
of their portfolios rather than the riskiness of the
individual securities in the portfolio, how should the
riskiness of an individual stock be measured?
CAPM states that the relevant riskiness of an
individual stock is its contribution to the riskiness of a
well-diversified portfolio.
The risk that remains after diversifying is market risk,
or the risk that is inherent in the market, and it can
be measured by the degree to which a given stock
tends to move up or down with the market.
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contribution of a stock to the riskiness of market portfolio
If there are N securities in the market, it can be shown that
M2=w1M1M+w2M2M+.. wNMNM
So covariance of a security is directly related to the total risk of the
market portfolio. The higher a securities covariance with market,
the higher is security’s contribution to total risk of market.
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contribution of a single stock to the riskiness of a
well-diversified portfolio.
It is only the market
risk of stock A that
will affect the risk of
the well-diversified
portfolio.
Well-diversified
portfolio
Stock A
Has only market risk
But no firm-specific risk
Has both market and
firm-specific risks
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Capital Asset Pricing Model (CAPM)
Model based upon concept that a stock’s required rate of return is
equal to the risk-free rate of return plus a risk premium that
reflects the riskiness of the stock after diversification.
Assumptions:
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All investors employ Markowitz portfolio theory to find portfolios
in the efficient set and then based on their individual risk
preferences invest in one of the portfolios in the efficient set
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All investors have the same planning horizon and identical
beliefs about the distributions of security returns
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No barriers to flow of capital or information. E.g. no transaction
costs, no taxes etc.
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Standard CAPM
risk-free lending and borrowing and short sales
Recall the efficient set
•The tangency portfolio and
Rf span the efficient set.
•Since investors have
homogenous expectations,
everybody comes up with
the same tangency
portfolio.
•A risk-averse investor
(investor A) will invest
positive amounts in the riskfree asset (lends at Rf) and
the tangency portfolio.
In equilibrium, the prices for all assets must adjust
so that aggregate amount of borrowing equals
aggregate amount of lending.
•A less risk-averse investor
(investor B) will short riskfree asset (borrows at Rf)
and buy the tangency
portfolio.
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CAPM
All efficient portfolios are combinations of market with risk free security
rp=wMP rM+wrfP rf
E(rP)=wMP E(rM)+(1-wMP) rf = rf + wMP [E(rM)-rf ]
P2=wMP2M2 or P=wMPM
Efficient portfolios’ total risk depends on market risk
[E(rP)- rf ]/[E(rM)-rf ]=P/M
E(rP)= rf +P [E(rM)-rf ]/M
The last relation is called Capital Market Line.
We need a relation between risk and return that holds for all
securities/portfolios. This is call Security Market Line in CAPM.
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A measure of market risk: Beta
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The tendency of a stock to move up or down with the
market is reflected in its beta coefficient.
Indicates how risky a stock is if the stock is held in a
well-diversified portfolio.
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Calculating betas
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Run a regression of past returns of a security against
past returns on the market.
The slope of the regression line (sometimes called
the security’s characteristic line) is defined as the
beta coefficient for the security.
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Illustrating the calculation of beta
Returns of asset i and market are excess returns. Practitioners
often use total rather than excess returns. This practice is most
common when daily data is used where total and excess returns
are almost indistinguishable.
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Comments on beta
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If beta = 1.0, the security is just as risky as the average stock.
If beta > 1.0, the security is riskier than average.
If beta < 1.0, the security is less risky than average.
Most stocks have betas in the range of 0.5 to 1.5.
Can the beta of a security be negative?
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Yes, if the correlation between Stock i and the market is
negative (i.e., ρi,m < 0).
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If the correlation is negative, the regression line would slope
downward, and the beta would be negative.
 However, a negative beta is highly unlikely.
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Beta coefficients for HT, Coll, and T-Bills
Slope of the regression line is given by the following formula:
Given our payoff matrix we can calculate:
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Comparing expected return and beta coefficients
Security
HT
Market
USR
T-Bills
Coll.
Exp. Ret.
17.4%
15.0
13.8
8.0
1.7
Beta
1.30
1.00
0.89
0.00
-0.87
?
Riskier securities have higher returns, so the rank order is OK.
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The Security Market Line (SML): Calculating
required rates of return
Recall that CAPM is based upon concept that a stock’s required
rate of return is equal to the risk-free rate of return plus a risk
premium that reflects the riskiness of the stock after diversification.
SML equation states that the risk premium is the product of risk and
extra compensation per unit of risk. Risk is measured by beta, and
extra compensation by excess return on market portfolio.
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What is the market risk premium?
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Additional return over the risk-free rate needed to
compensate investors for assuming an average
amount of risk.
Its size depends on the perceived risk of the stock
market and investors’ degree of risk aversion.
Varies from year to year, but most estimates suggest
that it ranges between 4% and 8% per year.
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Calculating required rates of return
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Expected vs. Required returns
Ceteris paribus as price rises expected return falls
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Illustrating the Security Market Line
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Calculating portfolio beta
Example: Equally-weighted two-stock portfolio
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Create a portfolio with 50% invested in HT
and 50% invested in Collections.
The beta of a portfolio is the weighted
average of each of the stock’s betas.
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Calculating portfolio required returns
The required return of a portfolio is the weighted
average of each of the stock’s required returns.
kP = wHT kHT + wColl kColl
kP = 0.5 (17.1%) + 0.5 (1.9%)
kP = 9.5%
Or, using the portfolio’s beta, CAPM can be used to
solve for required return.
kP = kRF + (kM – kRF) βP
kP = 8.0% + (15.0% – 8.0%) (0.215)
kP = 9.5%
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Factors that change the SML
What if investors raise inflation expectations by 3%,
what would happen to the SML?
recall that kRF =k*+IP
kRF will increase by 3%, RPM stays constant since kM
also increases by the same amount
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Increase in risk aversion
What if investors’ risk aversion increased, causing the market risk
premium to increase by 3%, what would happen to the SML?
Investors would require higher risk premium per unit of risk
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how can we derive the formula for  assuming
that SML holds (by using single index model)
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decomposition of total (stand-alone) risk
Again you may assume that kA and kM show excess returns.
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Useful formulas
recall: cov(x,y)=E(xy) – E(x) E(y)
if cov(x,y)=0 then E(xy) = E(x) E(y)
therefore E(xy) = 0 if E(x)=0 or E(y)=0
so since cov(kM, A)=0 and E( A)=0 then E(kM  A)=0
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