CHAPTER 12
AN ALTERNATIVE VIEW OF RISK AND
RETURN: THE ARBITRAGE PRICING
THEORY
Answers to Concept Questions
1. Systematic risk is risk that cannot be diversified away through formation of a portfolio. Generally,
systematic risk factors are those factors that affect a large number of firms in the market, however,
those factors will not necessarily affect all firms equally. Unsystematic risk is the type of risk that
2. Any return can be explained with a large enough number of systematic risk factors. However, for a
3. The market risk premium and inflation rates are probably good choices. The price of wheat, while a
risk factor for Ultra Bread, is not a market risk factor and will not likely be priced as a risk factor
4. a. Real GNP was higher than anticipated. Since returns are positively related to the level of GNP,
b. Inflation was exactly the amount anticipated. Since there was no surprise in this announcement,
c. Interest rates are lower than anticipated. Since returns are negatively related to interest rates,
d. The President’s death is bad news. Although the president was expected to retire, his retirement
would not be effective for six months. During that period he would still contribute to the firm.
e. The poor research results are also bad news. Since Lewis-Striden must continue to test the drug,
f. The research breakthrough is positive news for Lewis Striden. Since it was unexpected, it will
g. The competitor’s announcement is also unexpected, but it is not a welcome surprise. This
5. The main difference is that the market model assumes that only one factor, usually a stock market
6. The fact that APT does not give any guidance about the factors that influence stock returns is a
commonly-cited criticism. However, in choosing factors, we should choose factors that have an
7. Assuming the market portfolio is properly scaled, it can be shown that the one-factor model is
8. It is the weighted average of expected returns plus the weighted average of each security’s beta times
9. Choosing variables because they have been shown to be related to returns is data mining. The
relation found between some attribute and returns can be accidental, thus overstated. For example,
10. Using a benchmark composed of British stocks is wrong because the stocks included are not of the
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1. Since we have the expected return of the stock, the revised expected return can be determined using
the innovation, or surprise, in the risk factors. So, the revised expected return is:
2. a. If m is the systematic risk portion of return, then:
b. The unsystematic return is the return that occurs because of a firm specific factor such as the
bad news about the company. So, the unsystematic return of the stock is –1.1 percent. The total
3. a. If m is the systematic risk portion of return, then:
b. The unsystematic return is the return that occurs because of a firm specific factor such as the
increase in market share. If is the unsystematic risk portion of the return, then:
c. The total return is the expected return, plus the two components of unexpected return: the
systematic risk portion of return and the unsystematic portion. So, the total return of the stock
is:
4. The beta for a particular risk factor in a portfolio is the weighted average of the betas of the assets.
This is true whether the betas are from a single factor model or a multi-factor model. So, the betas of
the portfolio are:
So, the expression for the return of the portfolio is:
Which means the return of the portfolio is:
5. We can express the multifactor model for each portfolio as:
where F1 and F2 are the respective risk premiums for each factor. Expressing the return equation for
each portfolio, we get:
We can solve the system of two equations with two unknowns. Multiplying each equation by the
respective F2 factor for the other equation, we get:
Summing the equations and solving F1 for gives us:
And now, using the equation for Portfolio A, we can solve for F2, which is:
6. a. The market model is specified by:
b. Since we don’t have the actual market return or unsystematic risk, we will get a formula with
those values as unknowns:
RP = .30RA + .45RB + .25RC
c. Using the market model, if the return on the market is 15 percent and the systematic risk is
zero, the return for each individual stock is:
RA = 10.5% + 1.20(15% – 14.2%)
RA = 11.46%
Alternatively, to find the portfolio return, we can use the return of each asset and its portfolio
weight, or:
7. a. Since the five stocks have the same expected returns and the same betas, the portfolio also has
the same expected return and beta. However, the unsystematic risks might be different, so the
expected return of the portfolio is:
P
R
= 11% + .84F1 + 1.69F2 + (1/5)(1 + 2 + 3 + 4 + 5)
b. Consider the expected return equation of a portfolio of five assets we calculated in part a. Since
we now have a very large number of stocks in the portfolio, as:
