Money & Capital Markets
Fall 2007
Assignment 5 --- Due 9/20
Answer only 5 of the questions --- Selected Answers
1. Explain the significant advance made by Louis Bachelier. What are the implications
of this advance for investment strategies?
No answer provided
2. Compare and contrast the work of Charles Dow and Alfred Cowles.
No answer provided
3. Compare and contrast the essential points in the work of Harry Markowitz and
William Sharpe.
No answer provided
4. Write an essay that clearly and fully discusses the intuition behind the equilibrium
concept in CAPM and how the adjustment process to equilibrium is supposed to take
place.
No answer provided
5. Assume the expected return on the market to be 15%, risk-free rate of 8%, expected
rate of return on XYZ security of 17%, and beta of XYZ security of 1.25. Within the
context of CAPM, is XYZ overpriced, underpriced, or fairly priced? Explain your
answer.
The expected rate of return based on CAPM would be
CAPM
rXYZ
 8%  1.25(15%  8%)  16.75%
Since the return (17%) exceeds the CAPM, then we say that it is underpriced.
People sometimes have trouble thinking about the over- or under- priced asset in
the CAPM context. If you do, try changing the story a bit. Suppose you looked at
the results of XYZ for the past year and found that it did in fact have a 17% rate of
return. You also go back and calculate what the CAPM rate of return should have
been for the past year – you calculate the 16.75%. Now, looking back, you might
think “I wish that I would have bought some of XYZ!” – in that case you are really
saying that XYZ was underpriced during the past year. Remember, other things
equal, if you begin purchasing the stock, you’ll drive the price upward and as you
do the actual rate of return will fall --- following CAPM you would do this until the
return fell to the CAPM level.
6. If CAPM is valid, then discuss which of the following situations are possible.
a.
Portfolio (or, asset)
A
B
Expected Return
20%
25%
Beta
1.4
1.2
This surely could not be correct according to CAPM. According to CAPM, the
greater the beta – hence, the risk – the higher the expected rate of return. Since A
has a higher systematic risk – beta – it should have a higher expected return.
b.
Portfolio (or, asset)
A
B
Expected Return
30%
15%
Beta
1.5
.75
Suppose the risk-free rate is 12% and the market return is 24%, then for A,
E (rA )  r f   A [ E (rM )  r f ]  12%  1.5(24  12)  30%
What about B?
E (rB )  r f   B [ E (rM )  r f ]  12%  .75(24  12)  21%
So, in fact, the returns are not correct by CAPM standards. On the other hand, if
the 30% and 15% had been “Expected Excess Returns” (i.e., expected return minus
the risk-free rate), then things would be fine according to CAPM – in fact, this is the
way I intended to have this particular option – often times it is easier to always think
in terms of excess returns which I tend to do. It’s easier because you see that you
can directly translate relative betas into relative excess returns. For example, solve
the CAPM for beta.
A 
B 
E (rA )  r f
E (rM )  r f
E (rB )  r f
E (rM )  r f
Now, we can simply state the relative betas –
E (rA )  r f
E (rA )  r f 30%
 A E (rM )  r f



2
E (rB )  r f
B
E (rB )  r f 15%
E (rM )  r f
Thus, the relative excess returns must be equal to the relative betas. It is not true
that the relative rates of return will be equal to relative betas. These are the points I
wanted to bring out in this question.
c.
Portfolio (or, asset)
risk-free
market
A
Expected Return
10%
18%
22%
Beta
1
1.5
Using the CAPM equation,
E (r )  r f   [ E (rM )  r f ]  10%  1.5(18%  10%)  22%
So, yes, this is correct according to CAPM.
7. An entrepreneur is considering purchasing a business. The entrepreneur expects the
business to have yearly profits – forever – of $1 million. The risk-free rate is currently
8% and the expected rate of return on the market is 18%. The entrepreneur believes that
the beta for this business is 1.15. How much should the entrepreneur pay for this
business?
First, we calculate the CAPM expected rate of return.
E (r )  r f   [ E (rM )  r f ]  8%  1.15(18%  8%)  19.5%
Next, we calculate the present value of a stream of income of $1 million annually
forever. I attempted to simplify the problem by assuming the income would be the
same forever. This makes the investment appear much like a consol. In capital
budgeting problems, such as this one, one can use the CAPM expected rate of return
as the appropriate discount factor.
$1,000,000
 $5,128,205 This is the approximate price, rounding to the nearest
.195
dollar.
P
Notice, when one first learns to use present value formulas, some sort of interest rate
is typically used as the discount factor. This interest rate could be the interest rate
on the loan or the risk-free rate (as the opportunity cost of the money). In any case,
this sort of procedure ignores the relevant risk of the investment. If we had used the
risk-free rate as the discount factor, then the price we would be willing to pay for a
$1 million annual income stream would be the following.
P
$1,000,000
 $12,500,000
.08
Hence, we would have been willing to pay much, much more. The intuitive
explanation for us be willing to pay more is that we ignored the risk of the
investment in coming up with this price.
8. Suppose equity holders’ investment in the publicly regulate utility company is $100
million and beta is .60. If the risk-free rate is 3% and the expected rate of return on the
market is 5%, then calculate the ‘fair’ – according to CAPM - rate of return for investors
and their profits.
First, we calculate the CAPM expected rate of return.
E (r )  r f   [ E (rM )  r f ]  3%  .6(5%  3%)  4.2%
Notice, the CAPM rate of return for the utility company is less than the market
return. Why? Because the beta of .60 is less than 1 tells us that the utility’s return
fluctuates less than the market – it is less risky than the market. On the other hand,
it is greater than the risk-free rate because it is still risky.
Second, the investment is $100 million, so we can easily calculate the ‘fair’ profits.
$100 million x .042 = $4,200,000
The regulatory agency will now allow the company to set a per unit price to target
the profit figure.
9. For this question, assume that the risk-free rate is 4% and the market rate of return is
14%.
a. An investor using CAPM is considering purchasing a particular stock with a beta of .5.
According to CAPM, what would be the expected rate of return on this stock?
E (r )  4%  .5(14%  4%)  9%
b. A judge uses CAPM to determine the damages in a copyright infringement case. The
judge has been advised that the a beta of 2.0 is typical for the publishing industry. What
would be the ‘fair’ – according to CAPM - rate of return to be used to determine
damages?
E (r )  4%  2(14%  4%)  24%
c. The owner of an investment firm is deciding on whether or not to pay a bonus to his
two portfolio managers. One manager (Mr. Marko) was able to obtain a 22% rate of
return on his portfolio – the portfolio had a beta of 1.5. The other manager (Mr. Witz)
was able to earn a 12% rate of return on his portfolio – the portfolio had a beta of 1.0.
Assuming the owner uses a CAPM criteria to determine bonuses, which manager - if any
– deserves the bonus?
First, calculate the expected rates of return based on CAPM.
E (rMarko )  4%  1.5(14%  4%)  19%
E (rW itz )  4%  1.0(14%  4%)  14%
Now, note that based on the risk of Marko’s portfolio he should have earned only
19%. In fact, Marko has been able to ‘beat the market’ or ‘beat CAPM’ by earning
a 22% rate of return. Graphically, you can picture Marko’s returns as lying above
the security market line (SML). Witz on the other hand should have been able to
earn a 14% rate of return on is portfolio according to CAPM. He actually only
earned a 12% rate of return – his portfolio lies below the SML (in saying this, please
do not confuse the reference to the SML here with our use of it when determining
when a particular asset is under or over priced). Mr. Marko deserves the bonus –
not because he achieved a higher return than Witz, but rather because he obtained a
higher return than expected by CAPM! Given Marko’s riskier portfolio we would
expect him (on average, over time) to earn a higher return than Witz – however, we
would not expect him to consistently beat CAPM. By doing so, especially if this was
on a long-term basis, we would say that Marko is superior at picking stocks –
something we’ll begin to check out.