Chapter 10, Arbitrage Pricing Theory and Multifactor Models of Risk and Return Video Solutions, Investments

Problem 1

Suppose that two factors have been identified for the U.S. economy: the growth rate of industrial production, IP, and the inflation rate, IR. IP is expected to be $3 \%$, and IR $5 \%$. A stock with a beta of 1 on IP and .5 on IR currently is expected to provide a rate of return of $12 \%$. If industrial production actually grows by $5 \%$, while the inflation rate turns out to be $8 \%$, what is your revised estimate of the expected rate of return on the stock?

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Problem 2

The APT itself does not provide guidance concerning the factors that one might expect to determine risk premiums. How should researchers decide which factors to investigate? Why, for example, is industrial production a reasonable factor to test for a risk premium?

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Problem 3

If the APT is to be a useful theory, the number of systematic factors in the economy must be small. Why?

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Problem 4

Suppose that there are two independent economic factors, $F_1$ and $F_2$. The risk-free rate is $6 \%$, and all stocks have independent firm-specific components with a standard deviation of $45 \%$. Portfolios $A$ and $B$ are both well-diversified with the following properties:
$$
\begin{array}{cccc}
\text { Portfolio } & \text { Beta on } F_1 & \text { Beta on } F_2 & \text { Expected Retum } \\
\hline \text { A } & 1.5 & 2.0 & 31 \% \\
B & 2.2 & -0.2 & 27 \%
\end{array}
$$
What is the expected return-beta relationship in this economy?

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Problem 5

Consider the following data for a one-factor economy. Both portfolios are well diversified.
$$
\begin{array}{crr}
\text { Portiolio } & \text { E(r) } & \text { Beta } \\
\hline \text { A } & 12 \% & 1.2 \\
\text { F } & 6 \% & 0.0
\end{array}
$$
Suppose that another portfolio, porffolio $E$, is well diversified with a beta of .6 and expected return of $8 \%$. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy?

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Problem 6

Assume that portfolios $A$ and $B$ are both well diversified and that $E\left(r_A\right)=12 \%$ and $E\left(r_B\right)=9 \%$. If the economy has only one factor, and $\beta_A=1.2$, whereas $\beta_B=.8$, what must be the risk-free rate?

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Problem 7

Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard deviation of $30 \%$.
Suppose that an analyst studies 20 stocks and finds that one-half of them have an alpha of $+2 \%$, and the other half have an alpha of $-2 \%$. Suppose the analyst invests $$\$1$$ million in an equally weighted portfolio of the positive alpha stocks, and shorts $$\$ 1$$ million of an equally weighted portfolio of the negative alpha stocks.
a. What are the expected profit (in dollars) and standard deviation of the analyst's profit?
b. How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks?

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Problem 8

Assume that security returns are generated by the single-index model,
$$
R_i=\alpha_i+\beta_i R_M+c_i
$$
where $R_i$ is the excess return for security $i$ and $R_M$ is the market's excess return. The risk-free rate is $2 \%$, Suppose also that there are three securities, $A, B$, and $C$, characterized by the following dala:
$$
\begin{array}{cccc}
\text { Security } & \beta & E(R) & \text { ole. } \\
\hline \text { A } & 0.8 & 10 \% & 25 \% \\
B & 1.0 & 12 & 10 \\
C & 1.2 & 14 & 20
\end{array}
$$
a. If $\sigma_M=20 \%$, calculate the variance of returns of securities $A, B$, and $C$.
b. Now assume that there are an infinite number of assets with return characteristics identical to those of $A, B$, and $C$, respectively. If one forms a well-diversified portfolio of type $A$ securities, what will be the mean and variance of the portfolio's excess returns? What about portfolios composed only of type $B$ or $C$ stocks?
c. Is there an arbitrage opportunity in this market? What is it? Analyze the opportunity graphically.

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Problem 9

The SML relationship states that the expected risk premium on a security in a one-factor model must be directly proportional to the security's beta. Suppose that this were not the case. For example, suppose that expected return rises more than proportionately with beta as in the figure below.
a. How could you construct an arbitrage portfolio? (Hint: Consider combinations of portfolios $A$ and $B$, and compare the resultant portfolio to $C$.)
b. Some researchers have examined the relationship between average returns on diversified portfolios and the $\beta$ and $\beta^2$ of those portfolios. What should they have discovered about the effect of $\beta^2$ on portfolio return?

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Problem 10

Consider the following multifactor (APT) model of security returns for a particular stock.
$$
\begin{array}{lcc}
\text { Factor } & \text { Factor Beta } & \text { Factor Risk Premium } \\
\hline \text { Inflation } & 1.2 & 6 \% \\
\text { Industral production } & 0.5 & 8 \\
\text { Oil prices } & 0.3 & 3
\end{array}
$$
a. If T-bills currently offer a $6 \%$ yield, find the expected rate of return on this stock if the market views the stock as falrty priced
b. Suppose that the market expects the values for the three macro factors given in column 1 below, but that the actual values turn out as given in column 2 . Calculate the revised expectations for the rate of return on the stock once the "surprises" become known.
$$
\begin{array}{lcc}
\text { Factor } & \text { Expected Value } & \text { Actual Value } \\
\hline \text { Infiation } & 5 \% & 4 \% \\
\text { Industrial production } & 3 & 6 \\
\text { Oil prices } & 2 & 0
\end{array}
$$

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Problem 11

Suppose that the market can be described by the following three sources of systematic risk with associated risk premiums.
$$
\begin{array}{lc}
\text { Factor } & \text { Risk Premium } \\
\hline \text { Industrial production } \boldsymbol{\eta} & 6 \% \\
\text { Interest rates }(f) & 2 \\
\text { Consumer confidence }(C) & 4
\end{array}
$$
The return on a particular stock is generated according to the following equation:
$$
r=15 \%+1.0 I+.5 R+.75 C+e
$$

Find the equilibrium rate of return on this stock using the APT. The T-bill rate is $6 \%$. Is the stock over- or underpriced? Explain.

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Problem 12

As a finance intern at Pork Products, Jennifer Wainwright's assignment is to come up with fresh insights concerning the firm's cost of capital. She decides that this would be a good opportunity to try out the new material on the APT that she learned last semester. She decides that three promising factors would be (a) the return on a broad-based index such as the S\&P 500; (b) the level of interest rates, as represented by the yield to maturity on 10 -year Treasury bonds; and (c) the price of hogs, which is particularly important to her firm. Her plan is to find the beta of Pork Products against each of these factors by using a multiple regression and to estimate the risk premium associated with each exposure factor. Comment on Jennifer's choice of factors. Which are most promising with respect to the likely impact on her firm's cost of capital? Can you suggest improvements to her specification?

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Problem 13

According to the APT, if the risk-free rate is $4 \%$, what should be McCracken's estimate of the expected return of Orb's High Growth Fund?

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Problem 14

With respect to McCracken's APT model estimate of Orb's Large Cap Fund and the information Kwon provides, is an arbitrage opportunity available?

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Problem 15

If the GDP Fund is constructed from the other three funds, which of the following would be its weight in the Utility Fund? (a) -2.2 ; (b) -3.2 ; or (c) 3 .

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Problem 16

With respect to the comments of Stiles and McCracken concerning for whom the GDP Fund would be appropriate:
a. McCracken is correct and Stiles is wrong.
b. Both are correct.
c. Stiles is correct and McCracken is wrong.

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Problem 17

Assume a universe of $n$ (large) securities for which the largest residual variance is not larger than $n \sigma_M^2$. Construct as many different weighting schemes as you can that generate welldiversified portfolios.

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Problem 18

Small firms generally have relatively high loadings (high betas) on the SMB (small minus big) factor,
a. Explain why this is not surprising.
b. Now suppose two unrelated small firms merge. Each will be operated as an independent unit of the merged company. Would you expect the stock market behavior of the merged firm to differ from that of a portfolio of the two previously independent firms?
c. How does the merger affect market capitalization?
d. What is the prediction of the Fama-French 3-factor model for the risk premium on the merged firm compared to the weighted average of the two component companies?
e. Do we see here a problem in applying the FF model?

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