CHAPTER 8
CAPITAL MARKET THEORY
QUESTIONS AND PROBLEMS
Questions for Discussion
1.
A security's beta measures its market-related or systematic risk. Expressed
another way, it measures how sensitive the security's returns are to market
movements. If a security's beta is greater than +1.0, then its returns fluctuate
more than those of the overall market do and they move in the same
direction. If the beta is less than +1.0, then its returns fluctuate less than
those of the overall market. For example, if the market return is 3% greater
than that of a riskless(risk-free) security, and the beta of a risky security is
+2.0, the security's expected return according to the CAPM should be about
6% greater than that of the riskless (risk free) asset. A negative beta
indicates that a security's returns tend to move in an opposite direction than
those of the market. However, because the performance of most firms is tied
to the overall state of the economy, securities that have negative betas are
rarely found. A security's beta is estimated by regressing returns of a
security against corresponding market returns for a number of past periods.
Beta, then, is the slope of the regression line.
2. The availability of riskless (or risk free) assets alters the set of attainable
efficient portfolios.
Because investors retain the full benefit of diversification in holding the market
portfolio, better combinations of risk and expected return are now possible.
Any possible investor preference for a particular risk level can be met by
adjusting the proportions of the total invested in the riskless asset and the
market portfolio. No combination of stock portfolios alone will be able to
exceed the expected return for any given level of risk.
3.
It might be worthwhile for an investor to combine them.
It really depends on the correlation coefficient between the Canadian stocks and
the Hong Kong stocks. We can think of three different possibilities:
8-1
1) if the Canadian and Hong Kong assets are perfectly positively
correlated, the correlation coefficient between the two is +1.
Then, the portfolio which mixes the two assets will follow the locus,
which is a straight line from the point corresponding to all Canadian
assets to the point corresponding to all H.K. asset.
a
b
c
Suppose that you start with point a which has 100% of the
Canadian assets, and move on toward point b and finally point c.
Compared with point a, point b is no better, and in fact is not so
good as point a. It gets worse as you are moving from 100% of
Canadian assets to some diversification with Hong Kong assets.
2) if the correlation coefficient between the Canadian and Hong Kong
assets are –1 as being perfectly negatively correlated.
a
b
c
3) In general, the correlation coefficient between the two assets should
be between –1 and 1.
On its own, a Hong Kong portfolio might not be a good investment, but if
it has a low correlation with the Canadian market returns, it could be a
worthwhile investment for diversification purposes. An investor might
invest in Hong Kong to lower his/her risk. Graphically,
8-2
4.
5.
CAPM: E[R]  R f   (E[RM ]  E[R f ])
The CAPM deals with expected returns, not returns that are actually realized.
The fact that some stocks had returns unequal to the returns “predicted” by
the CAPM is only because the returns were not as expected, not because
CAPM is wrong. In other words, the CAPM does not “predict” returns, it only
gives the expected return.
Yes, CAPM:
E[R]  R f  (E[R M ]  E[R f ])
We know that: R f  0
E[R M ]  R f
But, if
  0 , then R f  E[R] . Remember that  
Cov[Ri , RM ]
 M2
There is nothing to prevent a stock from being negatively correlated with the
market. Thus, beta can be negative and the expected return on the stock
can be less than the risk-free rate. Intuitively, a negative beta implies that
adding that stock to a diversified portfolio actually reduces the systematic risk
of the portfolio as a whole. This benefit induces investors to accept an
expected return less than the risk-free rate.
8-3
ADDITIONAL PROBLEMS
Problems 8-1
Total investment is 40 + 60 + 25 = $125 thousand
 40 
 60 
 25 
+
.95
+
1.65
125
125 = 0.866
125
 p = .25 
Expected return = .08 + 0.866(.14 - .08) = 13.20%
or:
Expected return (security 1) = .08 + .25(.14 - .08) = .095
Expected return (security 2) = .08 + .95(.14 - .08) = .137
Expected return (security 3) = .08 + 1.65(.14 - .08) = .179
Expected return (portfolio)
 40 
 60 
 25 
+
.137
+
.179
= .095 
125
125 
125
= 13.20%
8-4
Problem 8-2(a)
The investor uses $4000 of his/her own money, so
Weight on index fund = 10000/4000 = 2.5
Weight on risk-free security = -6000/4000 = -1.5
We know,  M  1 and  RF  0
Therefore,  P  2.5(1)  ( 1.5)(0),   P  2.5
8-5
Problem 8-2(b)
No, the answer under (a) does not depend on the assumption that
one borrows at the riskless rate. If the borrowing rate were higher,
the SML would shift upwards but the ß would not change.
For more advanced classes, instructors may go beyond this answer
and have students deal with the issue of whether or not risky debt
can have a beta equal to zero.
8-6
Problem 8-3(a)
XA 
5000
 0.25
20000
XB 
5000
 0.25
20000
XC 
6000
 0.3
20000
XD 
4000
 0.2
20000
Problem 8-3(b)
E[Rp ]  0.25(9%)  0.25(10%)  0.3(11%)  0.2(12%)  10.45%
Problem 8-3(c)
p  0.25(0.8)  0.25(1)  0.3(1.2)  0.2(1.4)  1.09
8-7
Problem 8-4
First, find the beta of the stock;
40
35 
 r  0.143 or 14.3%
1  r 
0.143  .07   (.15  .07)
  0.9125
Next, test the effect of change in Covariance on beta;

Cov[RM , R])
 M2
Formula indicates that if covariance
doubles, beta will double.
0.9125 x 2 = 1.825 = new beta.
ER  .07  1.825(.15  .07)
ER  0.216 or 21.6%
Thus, the new price will be P0 
40
 $32.89
1.216
8-8
Problem 8-5(a)
ERp   9%,  p  0
You are totally invested in a risk-free asset.
Problem 8-5(b)
Wf  1
3,
WM  2
3
ERp   1 (9%)  2 (15%)
3
3
=13%
 p2  Wf2 f2  WM2 M2  2Wf WM fM
 0  WM2  M2  0
  p  WM M
 2 (0.21)
3
= 0.14 or 14%
8-9
Problem 8-5(c)
Wf   1 , WM  4
3
3
E[R P ]   1 (0.09)  4 (0.15)  17%
3
3
 P  WM M  4 (0.21)
3
 P  0.28
8 - 10
Problem 8-6
Stock A:
According to CAPM;
ERA   4%  1.2(10%  4%)
= 11.2%
If CAPM is to hold in the long run, the E[R] on A will have to decrease
from 14% to 11.2%. This decrease is accomplished through a rise in
the stock price. Therefore, you should buy Stock A now.
Stock B:
According to CAPM;
ERB   4%  2.9(6%)
=21.4%
The E[R] on B will increase, resulting in a fall in the stock price.
Therefore, you should sell B now.
8 - 11
Problem 8-7
According to CAPM,
R j = Rf +  j (Rm - Rf )
.16 = .06 +  j (.12 - .06)
 j = 1.67
8 - 12
Problem 8-8
To calculate the expected stock price at year-end, we use CAPM to
derive the expected return, and apply this return to the current share
price.
We have Rf= 0.08, βj = 1.4 and Rm = 0.14, from which we derive:
R = 0.08 + 1.4(0.14 - 0.08)
Rj= 16.4%
Using this value as the one-period total return, we obtain:
+( - )
R = D1 P1 P0
P0
0 + P1 - 10
.164 =
10
P1 = 10(.164) + 10 = $11.64
8 - 13
Problem 8-9(a)
Using CAPM, the expected return on the stock (Rj) is calculated as
follows:
R j = Rf +  j (Rm - Rf )
= .10 + 1.6(.05)
= 18%
Problem 8-9(b)
If the stock price is expected to remain unchanged yet investors
require or expect a return of 18%, then this return must come
completely in the form of dividends. Hence, investors must receive a
dividend yield of 18%; for a $10 share price, this yield implies a
dividend per share figure of $1.80.
8 - 14
Problem 8-9(c)
If the risk-free rate dropped to 6%, investors would require a return of:
Rj = 0.06 + 1.6(.05) = 14%
If the firm continues to pay out the same amount of dividends as
under (b), the yield will be higher than the market requires if P0
remains at its current level (since no capital appreciation is expected).
Hence, the share price will be bid up.
r = D1
+ (P1 - P0 )
P0
= D1 , since P1 = P0
P0
D1 1.80
P0 = =
r
.14
P0 = $12.86
Problem 8-9(d)
In this case, with the risk-free rate at 16%, investors would require a
return of:
Rj = 0.16 + 1.6(.05) = 24%
If the firm continues to pay out the same amount of dividends as
under (b), the yield will be lower than the market requires if P0
remains at its current level (since no capital appreciation is expected).
Hence, the share price will drop until a 24% return can be achieved.
D1 1.80
P0 = =
r
.24
P0 = $7.50
8 - 15
Problem 8-9(e)
Generalizing our findings under (c) and (d), we would expect stock
prices to increase as interest rates fall and to decrease as interest
rates rise, other factors remaining constant. Hence, stock prices and
interest rates are inversely related. This should be intuitively obvious,
since as interest rates fall, stocks become a more attractive
investment relative to others such as savings accounts and treasury
bills. Hence, prices of stocks will be bid up.
8 - 16
Problem 8-10
First find what the portfolio weights must be:
E[R p ]  w M (.15)  (1  w M )(.06)
.12  .06  .09w M
wM  2
3
 w RF  1
3
Next, use the weights to find out the market’s standard deviation.
 p  w M M  w RF RF  2Cov( M , RF )
0.2  2  M  1 (0)  2(0)
3
3
0. 2  2  M
 M  0.3
3
8 - 17