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Unformatted text preview: R. 1. R. 2. R. 3. R. 4. R. 5. R. 6. CHAPTER 12 REVIEW QUESTIONS Diversifiable risk is the portion of risk that disappears as assets are
combined in a portfolio, while non—diversifiable risk is the risk of a
portfolio remaining after all possible diversifiable risk has been removed. The
potential removal of diversifiable risk serves as the key result in modern
portfolio theory. Risk averse investors accept risk only if rewarded with higher expected
return. Most risk averse investors will make investment choices based on
comparing the risk and expected returns of various combinations of assets. A
particular portfolio will be preferred over another if it has the same
expected return but lower standard deviation, or the same standard deviation
but higher expected return. The subset of preferred portfolios are said to lie
on the efficient frontier of risky portfolios. In the Markowitz framework, the particular risky portfolio preferred by an
investor is determined by the investor’s risk preferences. Highly risk averse
people will invest in relatively low risk, high quality assets, while less
risk averse people will invest in relatively high risk, lower quality assets. In other
words, the choice among the portfolios along the efficient frontier is determined
by the level of risk aversion. The CAPM framework adds the choice of the risk free asset to the Markowitz
framework. In the CAPM framework, all investors are driven to invest in the
same portfolio of risky assets. Investors adjust the risk in their portfolio,
by adjusting the percentages of their portfolio held in risky assets versus
the risk free asset. Question 4 above outlined the result of the CAPM framework; all investors are
driven to invest in the same portfolio of risky assets. This unique portfolio
is called the market portfolio. Since every investor is striving for the same
goal —— the highest return for bearing risk —— and since every investor is
competing to purchase the same assets, we can deduce that the market portfolio
is comprised of all risky assets. Beta can be defined as a measure of systematic risk, or the risk that cannot
be diversified away by forming portfolios. The beta of a stock is defined by
the covariance between the stock’s returns and those of the market divided by
the variance of the market’s return. The numerator in beta’s formula measures
the amount of risk that an individual stock brings into an already diversified
portfolio, and the denominator represents the total risk in the market
portfolio. Beta therefore measures systematic risk relative to the risk of
the overall market. A beta of zero means no systematic risk while a beta of 1
denotes the same level of systematic risk as the overall market. A stock with 86 R. 7. R. 8. R. 9. a beta of 1.25 would be expected, on average, to rise 1.25 percent for every 1
percent unexpected rise in the overall market. The situation set up in the question is false. As described in question 6
above, beta can be used to measure the expected movement in an individual
stock’s return relative to the market. A stock with a beta of 0.75 would be
expected to fall by 0.75 percent, not by 1 percent, if the market fell by 1
percent. The graph of the security market line should have the following
specifications; the vertical axis measures the expected return of the asset
and the horizontal axis measures the beta of the asset. The security market
line should intercept the left axis at a return equal to the risk free rate
(the expected return of a zero beta asset) and should rise with a slope equal
to the expected return of the market less the risk free rate. The CAPM can be used to estimate the required rate of return used to discount
cash flows in capital budgeting analysis. Used in this way, the net present
value model uses as its estimate of risk only systematic risk, or the risk
that cannot be diversified away using portfolios. Corporate managers using
the CAPM in project analysis therefore assume that the firm’s shareholders
hold diversified portfolios. They simply insert the project’s beta into the CAPM
formula and use the resulting expected return as the NPV’s discount rate. R. 10. Project beta can be estimated for one project firms using the technique outlined in Window 12.2. However, this technique is impossible for multiple
project firms since projects are not publicly traded. The standard method of
finding the beta of a project is to find a publicly traded firm whose overall
assets have a level of risk similar to the project under consideration. The
project’s beta will be equal to the beta of the similar firm’s assets. 87 CHAPTER 12 PROBLEMS 1. The beta of the risk—free asset is 0.0 (since it has no risk) and the beta of the market is
1.0 (since beta measures responsiveness to the market). 2. 1.5. The relevant formula for beta is the covariance divided by the market’s variance.
The tricky part of this problem is that the market’s standard deviation is given rather
than the variance. Since the variance is simply the standard deviation squared, we can
compute the market variance by squaring the standard deviation: 0.2*0.2=0.04 Now the beta can be found as the covariance (0.06) divided by the variance of the market
beta = 0.06 / 0.04 = 1.5 3. a. Drucker mean = 10%, Market mean = 5%
b. Variance of market = 0.015
c. Covariance = 0.025
d. Beta = Covariance / Market Variance = 0025/0015 = 1.667
Returns Deviations Mkt Dev Cross
Drucker Market Drucker Market Squared Product
—— 15.00% — 10.00% — 0.2500 — 0.1500 0.0225 0.0375
10.00% 5.00% 0.0000 0.0000 0.0000 0.0000
35.00% 20.00% 0.2500 0.1500 0.0225 0.0375 10.00% 5.00% (means) 0.0150 0.0250
4. 11% E(R) = 6% + 0.5 (16% — 6%) = 6% + 5% = 11%
5. a. 0.00 b. 0.25 c. 0.50 d. 0.75 e. 1.00 The beta of each portfolio is the weighted average of the betas of the assets in
the portfolio. As discussed in Problem #1, the beta of the risk—free asset is
0.0 and the beta of the market is 1.0. a. beta = (1.00 * 0.00) + (0.00 * 1.00) = 0.00
b. beta = (0.75 * 0.00) + (0.25 * 1.00) = 0.25
c. beta = (0.50 * 0.00) + (0.50 * 1.00) = 0.50
(:1. beta = (0.25 * 0.00) + (0.75 * 1.00) = 0.75
e. beta = (0.00 * 0.00) + (1.00 * 1.00) = 1.00 6. As shown in detail in Window 12.3, any of the variables in the CAPM can be
computed given the other three variables. a. 18% b. 13% c. 23% d. 17.2% e. —0.2%
f. 0.5 g. 0.25 h. 1.5 i. —1.0 j. 15% k. 15% 1.19% III. 19.5% n. 13.2%
0.6% p. 10% q. 5% r. 6.4% 88 7. a. Arnold = 25%; Ziffel = 40%; Douglas = 35%
The portfolio weights are found as the proportion of the entire portfolio’s
market value attributable to each of the individual securities: Total Portfolio Value = (100*025) + (100*s40) + (100*s35)
Total Portfolio Value = $2,500 + $4,000 + $3,500 = $10,000 Arnold weight = $2,500/$10,000 = 0.25 or 25% '
Ziffel weight = $4,000 / $10,000 = 0.40 or 40%
Douglas weight = $3,500/ $10,000 = 0.35 or 35% b. 12.2% The expected return of the portfolio is the weighted average of the expected
returns of the securities in the portfolio: Portfolio E(R) = (0.25*10%) + (0.40*12%) + (0.35*14%) = 12.2%
c. 0.75, 1.00 & 1.25. Using Formula 12.2: Arnold beta = Covariance / Mkt. Variance = 0.03 / 0.04 = 0.75
Ziffel beta = Covariance / Mkt. Variance = 0.04 / 0.04 = 1.00
Douglas beta = Covariance / Mkt. Variance = 0.05 / 0.04 = 1.25 d. 1.025. From formula 12.3:
Portflio beta = (0.25*0.75) + (0.40*1.00) + (0.35*1.25) = 1.025 e. 0.875. In the CAPM all securities must trade on the security market line
and all portfolios with equal expected returns must have equal betas. Note
that a portfolio with 50% of Arnold Corp. and 50% of Ziffel Corp. would have
to have an expected return of 11% (found as a weighted average of the expected
returns of the two securities). Note further that the beta of the portfolio with
50% Arnold Corp. and 50% Ziffel Corp. must be 0.875 since the beta of the total
portfolio is a weighted average of the betas of the securities within the portolfio.
Finally, we can deduce that the beta of any portfolio with an expected return
of 11% (discussed in the problem) must have a beta of 0.875 since we know that it must share the same beta as the above portfolio which also has an expected
return of 11%. 8. a. Homer Co. mean = 28.18%; b. Market mean = 12.02% b. Homer Co. beta = Covariance / Mkt Variance = 1.25
Homer Co. Market Homer Co. Market Cross Mkt. Dev.
Return Return Deviation Deviation Product Squared 34.09% 14.63% 0.0591 0.0261 0.0015 0.0007
1.00% 2.03% —0.2718 —0.0999 0.0271 0.0100
17.05% 12.41% —0.1113 0.0039 —0.0004 0.0000 89 73.15% 27.26% 0.4497 0.1524 0.0685 0.0232
20.40% —6.56% —0.0778 — 0.1858 0.0145 0.0345
51.84% 26.31% 0.2366 0.1429 0.0338 0.0204
30.50% 4.46% 0.0232 — 0.0756 —0.0018 0.0057 2.22% 7.06% — 0.2596 0.0496 0.0129 0.0025
11.43% —1.54% —0.1675 —0.1356 0.0227 0.0184
40.16% 34.11% 0.1198 0.2209 0.0265 0.0488 Covariance Mkt. Var.
28.18% 12.02% (means) 0.0205 0.0164 9. a. 50% risk—free asset and 50% market portfolio
b. 10% risk—free asset and 90% market portfolio See Appendix Formula A125 and discussion immediately following. 10. a. 12% & 16.8% b. 12% & 16.8% The answers to (a) are found using the betas, the risk—free rate, the expected
return of the market and the algebra of Window 12.3. In order to answer (b), please
note that two securities (or two portfolios) that have the same beta must have the
same expected and required rates of return. Thus the answers to (a) and (b) are the same. Further note that in a perfect market, expected returns are
equal to required rates of return. 11. Underpriced. lts price would rise until its expected return fell to lie
on the security market line. 12. a. 0.84 b. 0.60 As shown in Window 12.4, the beta of the assets is the weighted average of the betas
of the debt (0.0) and equity (1.2). a. (30%*0.0) + (70%*1.2) = 0.84
b. (50%*0.0) + (50%*1.2) = 0.60 13. a. 16.4% b. 14% As shown in Window 12.3, given all other variables, the expected (required) rate of
of return may be computed as follows: a. E(R) = 8% + 0.84*(18%—8%) = 8% + 8.4% = 16.4%
b. E(R) = 8% + O.60*(18%—8%) = 8% + 6% = 14% 90 CHAPTER 12 DISCUSSION QUESTIONS 1. The expected rate of return is a statistical measure of return. The required
rate of return is the expected return that an investor would require to be just
willing to invest $1 (more precisely to be indifferent with regard to investing). In a perfect market required rates of return for all investors must equal expected
rates of return. The reason is that if expected returns exceeded required returns
for any or all investors, then the investors would purchase additional securities,
driving prices up and expected rates of return down. Similarly, if expected returns
were below required rates of return, investors would sell until they were equal. In imperfect markets and especially in the case of assets that are not regularly
traded, assets can offer expected returns that differ from the rates of return
that investors would require. 2. Since not all investors hold the market portfolio — as predicted by the CAPM —
we know that the CAPM does not predict perfectly in the real world. In fact, the
CAPM has revealed serious problems in fully explaining risk and return behaviors.
However, the concepts of diversifiable and undiversifiable risk as well as the idea
that the risk—free rate and market portfolio provide reasonable reference points
about which to estimate required rates of return continue to appear to be useful. 3. No. His systematic risk is equal to half of the systematic risk of the market. However,
with only one stock in his portfolio, we would expect that he is also exposing
himself to a tremendous level of diversifiable risk. 4. Regardless of the products or product lines, the CAPM guides the user through the
important exercise of separating the risk of a decision into the diversifiable and
undiversifiable components. Accordingly, the financial manager can have a better
understanding of which risk will vanish through diversification and which will
remain and needs to be rewarded. 5. The important point of this exercise is to note that mosts risks facing the producer
of a particular product are diversifiable rather than systematic. A diversifiable
risk is a risk that is independent of whether the overall economy does well or
poorly. A systematic risk is a risk that is completely dependent upon the overall
performance of the economy. For example, even one of the most economy related
products such as a car has far more diversifiable risk than systematic risk since
poor sales for a particular car model depend mostly upon the attraction of the
car to buyers and the competition. 6. Lower. Investments with negative betas must offer expected returns lower than the risk—~free rate since these investments can be combined with positive beta
assets to produce zero beta portfolios. If the negative beta asset did not provide 91 a return below the risk—free rate, then the zero beta portfolio would offer an
expected return above the risk—free rate and an arbitrage opportunity would exist. There are no common stocks that offer a consistently negative beta. However,
numerous derivative securities such as options and futures contracts (to be discussed in latter chapters) can have very negative betas. Some of these
securities even offer negative expected returns. Why would an investor select securities with such low or even negative expected
returns? The answer is that these securities are like insurance contracts because
they reduce rather than add to the risk of a portfolio. Investors must pay a
price in the form of reduced portfolio expected return for the privelage of
owning one of these portfolio risk—reducing securities. While the CAPM requires that expected returns lie exactly on a straight line
when graphed against betas, the CAPM does not require that actual returns
lie on a straight line. In fact, the CAPM predicts that actual returns will usually
deviate from a straight line relationship due to unique risks. Only the actual
returns of well diversified portfolios are predicted to lie on a straight line. 92 ...
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 Spring '10
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