acc 471 practice problem set #1 fall 2002 suggested...
TRANSCRIPT
ACC 471 Practice Problem Set #1 Fall 2002
Suggested Solutions
1. Text Problems:
1-6 a. No. Diversification calls for investing your savings in assets that do well when Ford is doing poorly.
b. No. Although Toyota is a competitor of Ford, both are subject to fluctuations in the automobile market.
1-9 The REIT manager pools the resources of many investors and uses these resources to buy a portfolio ofreal estate assets. Each investor in the REIT owns a fraction of the total portfolio according to his or herinvestment. The REIT gives the investor the ability to hold a diversified portfolio and to trade shares init more easily than the underlying real estate. Investors will be willing to pay the manager of the REIT areasonable fee for this service, which motivates qualified firms to organize and sell REITs.
1-12 In 19th century America, with a largely agrarian economy, uncertainty in crop yields and prices was amajor source of economy-wide risk. Therefore, there was a great incentive to create devices that wouldallow both producers and purchasers of agricultural commodities to hedge this risk. In contrast, the riskof paper or pencil prices was far smaller, and the need to hedge against such risk was minimal. Therewould have been very little demand for trading in securities that would allow investors to transfer risk inthe prices of these goods.
2-2 a. We have:
rBEY� 10 � 000 � P
P� 365
n� 10 � 000 � 9 � 6009 � 600
� 365180��� 0845 �
b. One reason is that the discount yield is computed by dividing the dollar discount from par by the parvalue ($10,000), rather than the bill’s price ($9,600). A second reason is that the discount yield isannualized by a 360 day year, rather than a 365 day year.
2-3 Since
P � 1 � 000�1 � rBDY
� n360 �� Pask
� 1 � 000 1 � � 0681 � 60360 � $988 � 65
Pbid� 1 � 000 1 � � 0690 � 60
360 � $988 � 50 �2-4 We have
rBEY� 1 � 000 � P
P� 365
n� 1 � 000 � 988 � 65988 � 65
� 36560��� 0698 �
which exceeds the discount yield of 6.81%. The effective annual yield is � 1 � 000 988 � 65 � 365 � 60 � 1 �7 � 19%.
2-6 a. 1 � 000 ��� 1 � � 03 � 90 360 ��� � $992 � 50.
b. The 90 day holding period return is 1 � 000 992 � 50 � 1 � 0 � 7557%.
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c. rBEY��� 007557 � � 365 90 � � 3 � 065%.
d. The effective annual yield is � 1 � 007557 � 365 � 90 � 1 � 3 � 10%.
2-9 a. At t � 0, the index is � 90 � 50 � 100 � 3 � 80. At t � 1, it is � 95 � 45 � 110 � 3 � 83 � 33, so the rateof return is 83 � 33 80 � 4 � 167%.
b. In the absence of a split, stock C would sell for $110, and the index would be 83.33. After the split,stock C sells for $55. Therefore, we need to set the divisor d such that 83 � 33 � � 95 � 45 � 55 � d, sod � 2 � 34.
c. The return is zero. The index remains unchanged, as it should, since the return on each stock sepa-rately equals zero.
2-10 a. The total market value at t � 0 is 90 � 100 ��� 50 � 200 ��� 100 � 200 � � 39 � 000. At t � 1 it is 95 � 100 ���45 � 200 ��� 110 � 200 � � 40 � 500, so the rate of return is 40 � 500 39 � 000 � 1 � 3 � 85%.
b. The return on each stock is:
rA� 95
90� 1 ��� 0556
rB� 45
50� 1 � � � 10
rC� 110
100� 1 ��� 10 �
so the equally weighted average return is � � 0556 � � 10 � � 10 � 3 � 1 � 85%.
3-5 The stop-loss order will be executed as soon as the stock price hits the limit price. If the stock price laterrebounds, the investor does not participate in the gains because the stock has been sold. In contrast, the putoption need not be exercised when the stock price falls below the exercise price. An investor who owns ashare of stock and a put option can hold on to both securities. If the stock price never rebounds, the putcan be exercised eventually, and the stock sold for the exercise price. This provides the same downsideprotection as the stop-loss order. If the price does rebound, however, the investor benefits because the stockis still held. This advantage of the put over the stop-loss order justifies the cost of the put.
3-6 Calls are options to purchase a stock at any time prior to expiration. Stop-buys require purchase as soonas the stock price hits the limit. The advantage of the call option over the stop-buy is that the investor neednot commit to buying until expiration. If the stock price later falls, the holder of the call can choose not topurchase.
3-9 a. The buy order will be filled at the best limit-sell order, $50.25.
b. At the next best price, $51.50.
c. You should increase your position. There is considerable buy pressure at prices just below $50,meaning that the downside risk is limited. In contrast, sell pressure is sparse, meaning that a moderatebuy order could result in a substantial price increase.
3-14 a. $55.50.
b. $55.25.
c. The trade will not be executed since the price on the limit sell order is higher than the bid price.
d. The trade will not be executed since the price on the limit buy order is lower than the asked price.
3-15 a. There can be price improvement for the two market orders. Brokers for each of the market orders (i.e.the buy and the sell orders) can agree to a trade inside the quoted spread. For example, they can tradeat $55.375, thus improving the price for both customers relative to the quoted bid and asked prices.The buyer gets the stock for $0.125 less than the asked price, and the seller receives $0.125 more thanthe bid price.
b. Whereas the limit buy order at $55.40 would not be executed in a dealer market (since the asked priceis $55.50), it could be executed in an exchange market. A broker for another customer with an orderto sell at market would view the limit buy order as the best bid price; the two brokers could agree tothe trade and bring it to the specialist, who would execute the trade.
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3-16 a. You buy 200 shares of BCE. These shares increase in value by 10%, or $1,000. You pay interest of� 08 � $5 � 000 � $400. The rate of return will be � 1 � 000 � 400 � � 5 � 000 � � 12%.b. You will receive a margin call if
200P � 5 � 000200P
��� 30� P � $35 � 71 �3-18 a. Initial margin is 50% of $5,000 or $2,500.
b. A margin call will be issued when
7 � 500 � 100P100P
��� 30� P � $75 � 69 �3-19 The proceeds from the short sale (net of commissions) were $14 � 100 � $0 � 50 � 100 � $1 � 350. A dividend
payment of $200 was withdrawn from the account. Coverage at $9 per share cost you (net of commissions)$900 � $50 � $950, leaving you with a profit of $1 � 350 � $200 � $950 � $200.
4-2 The offering price includes a 6% front-end load, so that every dollar paid results in only 94 cents goingtoward purchase of shares. Therefore:
Offering price � NAV1 � load
� $10 � 701 � � 06
� $11 � 38 �4-4 We have:
Stock ValueA $7,000,000B $12,000,000C $8,000,000D $15,000,000
Total $42,000,000
Therefore:
NAV � $42 � 000 � 000 � $30 � 0004 � 000 � 000
� $10 � 49 �4-5 The value of stocks sold and replaced is $15,000,000, so the turnover rate is 15 � 000 � 000 42 � 000 � 000 �
35 � 7%.
4-8 a. The start of year price P0� $12 � 00 � 1 � 02 � $12 � 24. The end of year price P1
� $12 � 10 � 0 � 93 �$11 � 25. Although the NAV increased by 10 cents, the price of the fund dropped by 99 cents. Then:
Rate of return � Distributions � P1 � P0
P0
� $1 � 50 � $0 � 99$12 � 24
� 4 � 2% �b. An investor holding the same portfolio as the fund manager would have earned a rate of return based
on the increase in the NAV of the portfolio:
Rate of return � Distributions � NAV1 � NAV0
NAV0
� $1 � 50 � $0 � 10$12 � 24
� 13 � 3% �4-9 a. Closed-end funds: diversification resulting from large scale investing, lower transaction costs associ-
ated with large scale trading, professional management that may be able to take advantage of buy orsell opportunities as they arise, record-keeping services, shares traded on organized exchanges.
b. Open-end funds: diversification resulting from large scale investing, lower transaction costs associatedwith large scale trading, professional management that may be able to take advantage of buy or sellopportunities as they arise, record-keeping services, daily redemption feature.
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c. Individual stocks and bonds: No management fee, realization of capital gains and losses can becoordinated with your personal tax situation, portfolio can be designed to your own specific riskpreferences.
4-13 We have:
NAV0� $200 � 000 � 000
10 � 000 � 000� $20
Dividends per share � $2 � 000 � 00010 � 000 � 000
� $0 � 20
NAV1� $20 � 1 � 08 � � 1 � � 01 � � $21 � 384
Rate of return � $21 � 384 � $0 � 20 � $20$20
� 7 � 92% �4-16 To purchase the shares, you would have had to spend $20 � 000 � 1 � � 04 � � $20 � 833. The shares rise in value
from $20,000 to $20 � 000 � � 1 � � 12 � � 012 � � $22 � 160. The rate of return is 22 � 160 20 � 833 � 1 � 6 � 37%.
4-17 Suppose you have $1,000 to invest. Class A funds will leave you with an initial investment of $940 net ofthe front-end load. After 4 years your investment will be worth:
$940 � 1 � 14 � $1 � 376 � 25 �Class B shares allow you to invest the full $1,000, but your investment performance net of other fees willbe only 9.5%, and you will pay an exit fee of 1% if you sell after 4 years. At that time your investment canbe sold for:
$1 � 000 � 1 � 0954 � 0 � 99 � $1 � 423 � 28 �so Class B is better if your horizon is 4 years. With a 15 year horizon, the Class A investment will be worth$940 � 1 � 115 � � $3 � 926 � 61, while the Class B investment can be sold for (note that there is no longer an exitfee) $1 � 000 � 1 � 09515 � � $3 � 901 � 32. At this longer horizon, Class A is now the better choice. The effect ofClass B’s 0.5% other charges accumulates over time and eventually exceeds the 6% load for Class A.
4-18 Suppose that finishing in the top half of all managers is purely luck and that the probability of doing soin any year is exactly 50%. The the probability that a particular manager would finish in the top halfof the sample 5 years in a row is � 1 2 � 5 � 1 32. We would then expect to find that 350 � 1 32 � 11managers finish in the top half for 5 consecutive years. This is precisely what we found. Thus, we shouldnot conclude that the consistent performance after 5 years is proof of skill: we would expect to find 11managers exhibiting precisely this level of consistent performance even if performance is due solely toluck.
4-19 a. After 2 years, each dollar invested in a fund with a 4% load, a management expense ratio of 0.5%,and a portfolio return of r will grow to 0 � 96 � � 1 � r � � 005 � 2. Each dollar invested in the GIC willgrow to 1 � 00 � � 1 � 06 � 2. If the mutual fund is to be the better investment, then:
0 � 96 � � 1 � r � � 005 � 2 � 1 � 00 � � 1 � 06 � 2� 1 � r � � 005 � 2 � 1 � 17041667� 1 � r � � 005 � � 1 � 08185797
r � � 08685797 �b. If you invest for 6 years:
0 � 96 � � 1 � r � � 005 � 6 � 1 � 00 � � 1 � 06 � 6� 1 � r � � 005 � 6 � 1 � 47762408� 1 � r � � 005 � � 1 � 06723648
r � � 07223648 �The cutoff return is lower because the fixed cost, i.e. the one time front-end load, is spread out over agreater number of years.
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c. With another charge instead of the front-end load, the portfolio must earn a rate of return r such that:
1 � r � � 005 � � 0075 � 1 � 06 �In this case, r must exceed 7.25% regardless of the investment horizon.
5-3 a. The inflation-plus GIC is safer because it guarantees the purchasing power of the investment. Usingthe approximation that the real rate equals the nominal rate minus the inflation rate, the GIC providesa real rate of 3.5% regardless of the inflation rate.
b. Th expected return depends on the expected rate of inflation over the next year. If the expected rate ofinflation is less than 3.5%, then the conventional GIC will offer a higher expected real return than theinflation-plus GIC; if the expected inflation rate is more than 3.5%, the opposite will be true.
c. If you expect the rate of inflation over the next year to be 4%, then the conventional GIC offers anexpected real return of 3%, which is 0.5% lower than the real rate on the inflation-protected GIC. Butunless you know that inflation will be 4% with certainty, the conventional GIC is also riskier. Since theinflation-plus GIC offers a higher expected real return and has lower risk, it is the better investment.
d. No, we cannot assume that the entire difference between the nominal risk free rate on conventionalGICs of 7% and the real risk free rate on inflation-protected GICs of 3.5% is the expected rate ofinflation. Part of the difference is probably a risk premium associated with the uncertainty surroundingthe real rate of return on the conventional GICs. This implies that the expected rate of inflation is lessthan 3.5% per year.
5-8 a. The real holding period return is:
1 � nominal holding period return1 � inflation rate
� 1 � 1 � 801 � 70
� 1 � 5 � 88% �b. The approximation gives a real holding period return of 80% less 70%, or 10%, which is clearly too
high.
5-18 Using the utility function U � E � r � � � 1 2 � Aσ2, the utility from T-bills is 7%, while the utility from theportfolio is � 12 � � 5 � A � � � 182 � ��� 12 � � 0162A. In order for the portfolio to be preferred to T-bills, it mustbe the case that � 12 � � 0162A � � 07 � � 0162A � � 05 � A � 3 � 0864 �
5-19 Given a level of σ, we determine E � r � such that U ��� 05 � E � r � � � 1 2 � � 3 � σ2, i.e. E � r � ��� 05 � � 3 2 � σ2.The following table provides some illustrative data points:
σ E � r �0.00 0.050.05 0.053750.10 0.0650.15 0.083750.20 0.110.25 0.14375
The indifference curve is plotted below:
σ
E � r �
0.0 0.1 0.2 0.3
0.05
0.10
0.15 U � A � 3 �
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5-20 Following the same steps as in 5-19, we generate the following table:
σ E � r �0.00 0.040.05 0.0450.10 0.060.15 0.0850.20 0.120.25 0.165
Plotting the results for both this problem and 5-19 gives:
σ
E � r �
0.0 0.1 0.2 0.3
0.05
0.10
0.15 U � A � 3 �U � A � 4 �
The indifference curve in this problem differs from that in the previous problem in both slope and intercept.When A increases from 3 to 4, the higher risk aversion results in a greater slope for the indifference curvesince more expected return is needed to compensate for additional σ. The lower level of utility assumedfor this problem (4% rather than 5%) shifts the vertical intercept down by 1%.
5-21 The coefficient of risk aversion for a risk-neutral investor is zero. The corresponding utility is simply equalto the expected return. As a result, the indifference curve is simply a flat line, as shown in the plot below(along with the indifference curves for the previous two problems):
σ
E � r �
0.0 0.1 0.2 0.3
0.05
0.10
0.15 U � A � 3 �U � A � 4 �
U � A � 0 �
5-22 A risk lover, rather than penalizing utility to account for risk, derives greater utility as σ increases. Thisamounts to a negative coefficient of risk aversion. If we assume A � � 2, we get the following data points:
σ E � r �0.00 0.050.05 0.04750.10 0.040.15 0.02750.20 0.010.25 -0.0125
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Indifference curves for all cases of problems 5-19 through 5-22 are provided below:
σ
E � r �
0.0 0.1 0.2 0.3
0.05
0.10
0.15 U � A � 3 �U � A � 4 �
U � A � 0 �U � A ��� 2 �
5-26 The expected return on the portfolio will be:
E � rP � � wbills� � 05 � wmarket
� � 0862 �Its standard deviation will be:
σP� wmarket
� � 162 �We obtain:
wbills wmarket E � rP � σP
0.00 1.00 0.0862 0.16200.20 0.80 0.0790 0.12960.40 0.60 0.0717 0.09720.60 0.40 0.0645 0.06480.80 0.20 0.0572 0.03241.00 0.00 0.0500 0.0000
5-27 Calculating the utility from U � E � r � � � 1 2 � Aσ2 with A � 3 gives:
wbills wmarket E � rP � σP U � A � 3 �0.00 1.00 0.0862 0.1620 0.04680.20 0.80 0.0790 0.1296 0.05380.40 0.60 0.0717 0.0972 0.05750.60 0.40 0.0645 0.0648 0.05820.80 0.20 0.0572 0.0324 0.05561.00 0.00 0.0500 0.0000 0.0500
An investor with A � 3 will prefer a position of 60% in T-bills and 40% in the market as that combinationprovides the highest level of utility.
5-28 Repeating the calculations with A � 5 gives:
wbills wmarket E � rP � σP U � A � 5 �0.00 1.00 0.0862 0.1620 0.02060.20 0.80 0.0790 0.1296 0.03700.40 0.60 0.0717 0.0972 0.04810.60 0.40 0.0645 0.0648 0.05400.80 0.20 0.0572 0.0324 0.05461.00 0.00 0.0500 0.0000 0.0500
An investor with A � 5 will prefer a position of 80% in T-bills and 20% in the market as that combinationprovides the highest level of utility. Compared with 5-27, this makes sense intuitively: more risk-averseinvestors will want to place less of their money in risky assets.
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6-1 E � r � ��� 7 � � 18 � � � 3 � � 08 � ��� 15, σ ��� 7 � � 28 � ��� 196.
6-2 Investment proportions:
Security WeightingT-bills 0.300Stock A � 7 � � 27 � � 0 � 189Stock B � 7 � � 33 � � 0 � 231Stock C � 7 � � 40 � � 0 � 280Total 1.000
6-3 My reward-to-variability ratio is � � 18 � � 08 � � 28 � 0 � 3571. The reward-to-variability ratio for my client is� � 15 � � 08 � � 196 � 0 � 3571.
6-4 We have:
σ
E � r �
0 0.10 0.20 0.300
0.05
0.10
0.15
0.20
0.25
CAL (slope = 0.3571)
My portfolio
Client’s portfolio
6-5 a. Since � 16 ��� 08 � 1 � y � � � 18y, we have y ��� 2. Therefore 20% must be invested in T-bills and 80% inthe risky portfolio.
b. Investment proportions:Security WeightingT-bills 0.200Stock A � 8 � � 27 � � 0 � 216Stock B � 8 � � 33 � � 0 � 264Stock C � 8 � � 40 � � 0 � 320Total 1.000
c. σ ��� 8 � � 28 � � 0 � 224.
6-6 a. Since � 18 ��� 28y, we have y � 0 � 6429.b. E � r � ��� 6429 � � 18 � � � 3571 � � 08 � ��� 14429.
6-7 a. We have:
y� � E � rP � � r f
Aσ2P� � 18 � � 08
3 � 5 � � 282 � � 0 � 3644 �so the client’s optimal proportions are 36.44% in the risky portfolio and 63.56% in T-bills.
b. E � r � ��� 6356 � � 08 � � � 3644 � � 18 � � 11 � 644%, and σ ��� 3644 � � 28 � � 10 � 20%.
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6-8 a. The slope of the CML is � � 13 � � 08 � � 25 ��� 20. The plot is:
σ
E � r �
0 0.10 0.20 0.300
0.05
0.10
0.15
0.20
0.25
CAL (slope = 0.3571)
CML (slope = 0.20)
b. My fund allows an investor to achieve a higher expected return for any given level of risk (i.e. standarddeviation) than would a passive strategy. This is true regardless of the investor’s risk preferences:anyone (no matter how risk-averse) would be better off combining my fund with a risk-free investmentthan combining the passive portfolio with a risk-free investment.
6-9 a. If the client were to switch to 70% in the passive portfolio and 30% in T-bills, then E � r � � � 3 � � 08 ���� 7 � � 13 � � 0 � 115 and σ ��� 7 � � 25 � � 0 � 175. These values can be compared to the current situation withan expected return of 15% and a standard deviation of 19.6%. Note that both the expected return andthe standard deviation would fall if the client were to switch to the passive strategy, so it is not yetclear that the client is worse off to do so. However, consider the following argument. If the clientreally desires a portfolio expected return of 11.5%, this could be achieved as follows. Since:� 115 ��� 08 � 1 � y ��� � 18 � y � � y � 0 � 35 �the client can attain this return by putting 35% in my fund and 65% in T-bills. This would have astandard deviation of � 35 � � 28 � � 9 � 8%, which is lower than that achievable using the passive portfolio.
b. The fee would reduce the reward-to-variability ratio, i.e. the slope of the CAL. Clients would beindifferent between my fund and the passive portfolio if the slopes of the after-fee CAL and the CMLwere equal. Letting f denote the fee, we have:� 18 � � 08 � f� 28
� � 20
f � � 18 � � 08 � � 056 ��� 044 �so the fee could be as high as 4.4% per year.
6-10 a. We have:
y� � E � rM � � r f
Aσ2M� � 13 � � 08
3 � 5 � � 252 � � 0 � 2286 �so the client’s optimal proportions are 22.86% in the risky passive portfolio and 77.14% in T-bills.
b. The answer here is the same as in 6-9. The fee that you can charge a client is the same regardlessof the asset allocation mix of your client’s portfolio. You can charge a fee that will equalize thereward-to-variability ratio of your portfolio with that of your competition.
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6-11 If the risk-free lending rate is 5% and the risk-free borrowing rate is 9%, then the CML and indifferencecurves are as follows:
σ
E � r �
ME � rM � ��� 13
σM ��� 25
rBf ��� 09
rLf ��� 05
lending
borrowing
6-12 For y � 1 (so that the investor is a lender, risk aversion must be large enough that:
y � E � rM � � r f
Aσ2M
� 1� y � � 13 � � 05A � � 252 � � 1� A �� 13 � � 05� 252
� 1 � 28 �while for y � 1 (so that the investor is a borrower), we must have:
y � E � rM � � r f
Aσ2M
� 1� y � � 13 � � 09A � � 252 � � 1� A �� 13 � � 09� 252
� 0 � 64 �Therefore, for values of risk-aversion between 0.64 and 1.28, the investor neither lends nor borrows, butinstead holds a complete portfolio comprised only of the risky portfolio M.
6-14 The maximum feasible fee, denoted by f , depends on the reward-to-variability ratio. For y � 1, the lendingrate of 5% is the relevant risk free rate, and we have:� 11 � � 05 � f� 15
� � 13 � � 05� 25� f � 1 � 2% �
For y � 1, the borrowing rate of 9% is the relevant risk free rate. Then we notice that even without a fee,the active fund is inferior to the passive fund because� 11 � � 09� 15
��� 1333 �� 13 � � 09� 25
��� 16 �More risk tolerant investors (who are more inclined to borrow) will therefore not be clients of the fundeven without a fee. If you solve for the fee that would make investors who borrow indifferent between the
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active and passive portfolios, you will find that f is negative, i.e. you would need to pay them to chooseyour active fund. The reason is that these investors desire higher risk-higher return portfolios and thus arein the borrowing range of the relevant CAL. In the range the reward-to-variability raiot of the passive fundis better than that of the active fund.
6-21 Using the formula provided on p. 205 of the text, we have
wS� σ2
B � σBS
σ2S � σ2
B � 2σBS� � 152 � � � 15 � � � 30 � � � 10 �� 302 � � 152 � 2 � � 15 � � � 30 � � � 10 ���� 1739
wB��� 8261 �
Then:
E � rP � ��� 1739 � � 20 ��� � 8261 � � 12 � ��� 1339
σP� � � � 17392 � � � 302 ��� � � 82612 � � � 152 ��� 2 � � 1739 � � � 8261 � � � 15 � � � 30 � � � 10 ��� 1 � 2 ��� 1392 �
6-22 We have:
wS wB E � rP � σP
0% 100% 12.0% 15.00%20% 80% 13.6% 13.94%40% 60% 15.2% 15.70%60% 40% 16.8% 19.53%80% 20% 18.4% 24.48%
100% 0% 20.0% 30.00%
6-23 We have:
0 5 10 15 20 25 30 35 400
5
10
15
20
25
30
σP (%)
E � rP � (%)
CML
Minimum variance portfolio
Tangency portfolio
The graph approximates the points:
E � rP � σP
Minimum variance portfolio 13.4% 13.9%Tangency portfolio 15.6% 16.5%
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The tangency portfolio is the optimal portfolio of risky assets. Investors can move anywhere along theCML by combining the tangency portfolio with the risk free asset.
6-24 Applying the formula from p. 211 of the text, we have:
wS� � E � rS � � r f � σ2
B � �E � rB � � r f � σBS� E � rS � � r f � σ2B �
�E � rB � � r f � σ2S � �E � rS � � r f � E � rB � � r f � σBS� � � 20 � � 08 � � � 152 � � � � 12 � � 08 � � � 30 � � � 15 � � � 10 �� � 20 � � 08 � � � 152 � � � � 12 � � 08 � � � 302 � � � � 20 � � 08 � � 12 � � 08� � � 30 � � � 15 � � � 10 �� 0 � 4516 �
wB� 0 � 5484 �
Therefore, the expected return and standard deviation of the optimal risky portfolio are:
E � rP � � � � 4516 � � � 30 ��� � 5484 � � 12 �� 0 � 1561 �σP� � � � 45162 � � � 302 ��� � � 54842 � � � 152 ��� 2 � � 4516 � � � 5484 � � � 30 � � � 15 � � � 10 � � 1 � 2� 0 � 1654 �
6-25 The reward-to-variability ratio of the optimal CAL is:
E � rP � � r f
σP
� 0 � 1561 � 0 � 080 � 1654
� 0 � 4601 �6-26 a. The standard deviation can be found from the equation of the optimal CAL. For any combination C
of the risk free asset and the optimal portfolio of risky assets P, we have:
E � rC � � r f �E � rP � � r f
σPσC� 0 � 14 � 0 � 08 � 0 � 4601σC� σC
� 0 � 1304 �b. Since:
E � rC � � � 1 � y � r f � yE � rP �� 0 � 14 � � 1 � y � � � 08 � � y � � 1561 �� y � 0 � 7884 � 1 � y � 0 � 2116 �Therefore, 21.16% is invested in T-bills, 0 � 7884 � 0 � 4516 � � 35 � 60% is invested in S, and0 � 7884 � 0 � 5484 � � 43 � 24% is invested in B.
6-27 If we use only B and S to obtain an expected return of 14%, we have:
0 � 14 � wS � � 20 ��� � 1 � wS � � � 12 �� wS� 0 � 25 � wB
� 0 � 75 �Then:
σP� � � � 252 � � � 302 ��� � � 752 � � � 152 � � 2 � � 25 � � � 75 � � � 30 � � � 15 � � � 10 � � 1 � 2 � 0 � 1413 �
which is larger than the standard deviation of 13.04% which can be attained by using T-bills and theoptimal portfolio.
6-28 If you cannot borrow, then you can only use B and S to form your portfolio. In this case:
0 � 24 � wS � � 20 ��� � 1 � wS � � � 12 �� wS� 1 � 50 � wB
� � 0 � 50 �12
This means that you should sell short B and invest fully in S. The standard deviation of this portfolio wouldbe:
σP� � � 1 � 502 � � � 302 ��� � � � 502 � � � 152 � � 2 � 1 � 50 � � � � 50 � � � 30 � � � 15 � � � 10 � � 1 � 2 � 0 � 4487 �
If you were allowed to borrow at the risk free rate, then you should do so and invest in the optimal portfolioof risky assets. In this case:
E � rC � � r f �E � rP � � r f
σPσC� 0 � 24 � 0 � 08 � 0 � 4601σC� σC
� 0 � 3478 �which is considerably lower than the standard deviation of 44.87% for the situation where you cannotborrow at the risk free rate. The portfolio proportions can be found as follows:
E � rC � � � 1 � y � r f � yE � rP �� 0 � 24 � � 1 � y � � 08 � y � � 1561 �y � 2 � 1025 � 1 � y � � 1 � 1025 �
7-5 a. Call the aggressive stock A and the defensive stock D. We calculate each stock’s β by taking thedifference in its return across the two scenarios divided by the difference in the market return:
βA� � 32 � � 02� 20 � � 05
� 2 � 00
βB� � 14 � � 035� 20 � � 05
� 0 � 70 �b. We have:
E � rA � ��� 5 � � 32 � � � 5 � � 02 � � 0 � 17
E � rB � ��� 5 � � 14 � � � 5 � � 035 � � 0 � 875 �c. The SML is determined by the market expected return of � 5 � � 20 � � � 5 � � 05 � � 12 � 5%, with a β of 1,
and the T-bill rate of 8%, with a β of 0. The plot is as follows.
0 0 � 5 1 � 0 1 � 5 2 � 00
0 � 05
0 � 10
0 � 15
0 � 20
β
E � r �SML
d. Based on its risk, A has a required expected return of � 08 � 2 � � 125 � � 08 � � 17%, and the analyst’sforecast of expected return is also 17%. Therefore it has αA
� 0. Similarly, the required returnon D is � 08 � � 7 � � 125 � � 08 � � 11 � 15%, but the analyst’s forecast is only 8.75%. Therefore αD
�� 0875 � � 1115 � � 2 � 4%. The plot is:
13
0 0 � 5 1 � 0 1 � 5 2 � 00
0 � 05
0 � 10
0 � 15
0 � 20
β
E � r �SML
A
M
D
αD
e. The hurdle rate is determined by the project β of 0.7, not by the aggressive firm’s β. The correctdiscount rate is 11.15%, the fair rate of return on stock D (given its systematic risk).
7-16 a. To determine which investor was a better selector of individual stocks, we examine abnormal returns,which is the ex post α, i.e. the difference between the actual return and that predicted by the SML.Without information about the market parameters (the risk free rate and the market rate of return), wecannot tell which investor was more accurate.
b. If r f� 6% and rM
� 14%, then:
α1��� 19 � � � 06 � 1 � 5 � � 14 � � 06 ��� ��� 01
α2��� 16 � � � 06 � 1 � 0 � � 14 � � 06 ��� ��� 02 �
Here, the second investor has the larger abnormal return and thus appears to be the superior stock se-lector. By making better predictions the second investor appears to have purchased more underpricedstocks.
c. If r f� 3% and rM
� 15%, then:
α1��� 19 � � � 03 � 1 � 5 � � 15 � � 03 ��� � � � 02
α2��� 16 � � � 03 � 1 � 0 � � 15 � � 03 ��� ��� 01 �
Here, not only does the second investor appear to be the superior stock selector, but the first investor’spredictions appear valueless (or worse).
7-17 a. Since the market portfolio by definition has a β of 1, its expected rate of return is 12%.
b. A β of zero means no systematic risk, so the stock’s expected rate of return is the risk free rate of 5%.
c. Using the SML, the fair expected rate of return for a stock with β � � 0 � 5 is � 05 � � 5 � � 12 � � 05 � �1 � 5%. The actually expected rate of return, using the expected price and dividend for next year is� 41 � 3 � 40 � 40 � 10%. Because the actual return exceeds the fair return, the stock is underpriced.
7-18 a. The risky portfolio selected by all defensive investors is at the tangency point between the minimumvariance frontier and the line originating at r f , depicted as the point R on the graph below. Point Qrepresents the risky portfolio selected by all aggressive investors. It is the tangency point between theminimum variance frontier and the line starting at rB
f .
14
σ
E � r �
R
Q
rBf
r f
b. Investors who do not wish to borrow or lend will each have a unique portfolio of risky assets at thetangency of their own individual indifference curves with the minimum variance frontier in the sectionbetween R and Q.
c. The market portfolio is defined as the portfolio of all risky securities, with weights in proportion totheir market values. Thus, by design, the average investor holds the market portfolio. The averageinvestor, in turn, neither borrows nor lends. Hence, the market portfolio is on the efficient frontierbetween R and Q.
d. Yes, the zero β CAPM will be valid. The market portfolio M will lie between R and Q, as noted above.The zero β portfolio corresponding to M, denoted by Z, will lie on the inefficient part of the frontier,and E � rZ � will be where the tangency line from M intersects the expected return axis. We can thenuse the property of the efficient set which states that the expected return on any individual securitycan be expressed as a function of any two frontier portfolios, in particular Z and M, in order to obtainthe zero β CAPM. The graph below illustrates.
σ
E � r �
R
Q
rBf
r f
M
ZE � rZ �
7-19 To simplify the discussion, assume that stocks pay no dividends and thus the rate of return on stocksis essentially tax-free (since investors do not have to realize their capital gains or losses). Thus, bothtaxed and untaxed investors will compute identical efficient frontiers. The situation is analogous to thatwith different borrowing and lending rates as analyzed in Problem 7-18. Taxed investors are analogous tolenders with a lending rate of r f � 1 � t � . Their relevant CML is drawn from r f � 1 � t � to the efficient frontierwith tangency at point R on the graph. Untaxed investors are analogous to borrowers who must use the(now higher) rate of r f to get a tangency at Q. Between them, both classes of investors hold the market
15
portfolio which is a weighted average of R and Q, with weights proportional to the aggregate wealth of theinvestors in each class.Since any combination of two efficient frontier portfolios is also efficient, the average (market) portfolio Mwill also be efficient. Moreover, the zero β model must now apply, because M is efficient and all investorschoose risky portfolios that lie on the efficient frontier. As a result, the line from the expected return onthe efficient portfolio with zero correlation with M (and hence zero β) will be tangent at M. This can onlyhappen if r f � 1 � t � � E � rZ � � r f .More generally, consider the case of any number of classes of investors with individual risk free borrowingand lending rates. As long as the same efficient frontier of risky assets applies to all of them, the zero βmodel will apply, and the equilibrium zero β rate will be a weighted average of each individual’s risk freeborrowing and lending rates.
8-2 a. Using the formula:
σi� �
β2i σ2
M � σ2 � ei � � 1 � 2 �we obtain
σA� � � � 82 � � � 222 ��� � 32 � 1 � 2 � 0 � 3478
σB� � � 1 � 22 � � � 222 � � � 42 � 1 � 2 � 0 � 4793 �
b. For portfolio expected return:
E � rP � � wAE � rA ��� wBE � rB ��� w f r f� � � 30 � � � 13 ��� � � 45 � � � 18 � � � � 25 � � � 08 � � 0 � 14 �For portfolio β:
βP� wAβA � wBβB � w f β f� � � 30 � � � 8 ��� � � 45 � � 1 � 2 ��� � � 25 � � 0 � � 0 � 78 �
The variance of the portfolio is:σ2
P� β2
Pσ2M � σ2 � eP � �
where β2Pσ2
M is the systematic component and σ2 � eP � is the nonsystematic component. Since theresiduals ei are uncorrelated, the nonsystematic variance is:
σ2 � eP � � w2Aσ2 � eA ��� w2
Bσ2 � eB ��� w2f σ2 � e f �� � � 302 � � � 302 ��� � � 452 � � � 402 � � � � 252 � � 0 � � 0 � 0405 �
The nonsystematic standard deviation of the portfolio is therefore�
0 � 0405 � 20 � 12%. The totalvariance of the portfolio is:
σ2P� � � 782 � � � 222 ��� � 0405 � 0 � 06995 �
and the standard deviation is 26.45%.
8-3 a. The two figures depict the stocks’ security characteristic lines (SCLs). Stock A has a higher firm-specific risk because the deviations of the observations from the SCL are larger for A than for B.Deviations are measured by the vertical distance of each observation from the SCL.
b. The slope of the SCL, β, is the measure of systematic risk. Stock B’s SCL is steeper, hence stock B’ssystematic risk is greater.
c. The R2 of the SCL is the ratio of the explained variance of the stock’s return to its total variance:
R2 � β2i σ2
M
β2i σ2
M � σ2 � ei � �Since stock B’s explained variance is higher (β2
Bσ2M
� β2Aσ2
M because βB� βA), and its residual vari-
ance σ2 � eB � is smaller, its R2 is higher than that of stock A.
16
d. α is the intercept of the SCL with the expected return axis. Stock A has a small positive α whereasstock B has a negative α; hence stock A’s α is larger.
e. The correlation coefficient is the square root of R2, so stock B’s correlation with the market is higher.
8-4 a. Firm-specific risk is measured by the residual standard deviation, so A has more firm-specific risk(since 10.3% exceeds 9.1%).
b. Market risk is measured by β, the slope coefficient of the regression. A has larger market risk (since1 � 2 � 0 � 8).
c. R2 measures the fraction of total variance of return explained by the market return. A’s R2 is largerthan B’s ( � 576 � � 436).
d. The average rate of return in excess of that predicted by CAPM is α, the intercept of the SCL. SinceαA
� 0 (while αB � 0), A had an average return higher than predicted by the CAPM.e. Rewriting the SCL equation in terms of total return r rather than excess return:
rA � r f� α � β � rM � r f �� rA� α � r f � 1 � β ��� βrM
so the intercept is now equal to
α � r f � 1 � β � ��� 01 � � 06 � 1 � 1 � 2 � � � 0 � 002 �8-5 Since:
R2i� β2
i σ2M
σ2i
�σ2
i� β2
i σ2M
R2i
�� σA
� � � 72 � � � 22 �� 2 1 � 2 � 0 � 3130
σB� � 1 � 22 � � � 22 �� 12 1 � 2 � 0 � 6928 �
8-6 The systematic risk for A is:β2
Aσ2M� � � 72 � � � 22 � ��� 0196 �
and its firm-specific risk (residual variance) is the difference between its total risk and its systematic risk,� 3132 � � 0196 � 0 � 0783. Similarly, for B:
β2Bσ2
M� � 1 � 22 � � � 22 � ��� 0576 �
so its firm-specific risk is � 69282 � � 0576 � 0 � 4224.
8-7 Under the assumptions of the index model, the residuals for A and B are uncorrelated, so the covariancebetween returns of A and B is:
Cov � rA � rB � � βAβBσ2M� � � 7 � � 1 � 2 � � � 22 � � 0 � 0336 �
The correlation coefficient is:
ρAB� Cov � rA � rB �
σAσB
� � 0336� � 313 � � � 6928 � � 0 � 1549 �8-8 Note that the correlation coefficient is the square root of R2: ρ � �
R2. Then:
Cov � rA � rM � � ρAMσAσM� � � 2 � � 313 � � � 2 � � 0 � 0280
Cov � rB � rM � � ρBMσBσM� � � 12 � � 6928 � � � 2 � � 0 � 0480 �
17
8-9 The non-zero αs from the regressions are inconsistent with the CAPM. The question is whether the αestimates reflect sampling errors or real mispricing. To test the hypothesis of whether the intercepts (3%for A, -2% for B) are significantly different from zero, we would need to calculate t-statistics for eachintercept.
8-10 For portfolio P we can calculate:
σP� � � � 62 � � � 3132 ��� � � 42 � � � 69282 � � 2 � � 6 � � � 4 � � � 0336 � � 1 � 2 � 0 � 3580
βP� � � 6 � � � 7 ��� � � 4 � � 1 � 2 � � 0 � 90
σ2 � eP � � σ2P � β2
Pσ2M��� 35802 � � � 92 � � � 22 � � 0 � 0958
Cov � rP � rM � � βPσ2M� � � 9 � � � 22 � � 0 � 036 �
This same result can also be obtained using the covariances of the individual stocks with the market:
Cov � rP � rM � � Cov � � 6rA � � 4rB � rM ���� 6Cov � rA � rM ��� � 4Cov � rB � rM ���� 6 � � 0280 ��� � 4 � � 0480 � � 0 � 036 �8-11 Note that the variance of T-bills and their covariance with any asset are zero. Therefore:
σQ� �
w2Pσ2
P � w2Mσ2
M � 2wPwMσPM � 1 � 2� � � � 52 � � � 3582 ��� � � 32 � � � 22 ��� 2 � � 5 � � � 3 � � � 036 � � 1 � 2 � 0 � 2155
βQ� � � 5 � � � 9 ��� � � 3 � � 1 � 0 ��� � � 2 � � 0 � � 0 � 75
σ2 � eQ � � σ2Q � β2
Qσ2M��� 21552 � � � 752 � � � 22 � � 0 � 0239
Cov � rQ � rM � � βQσ2M� � � 75 � � � 22 � � 0 � 03 �
8-15 We have:E � rP � � r f � βP1
�E � r1 � � r f � � βP2� E � r2 � � r f � �
We need to find the risk premium for each of the factors (the terms in square brackets in the above equa-tions). To do so, we solve the following system of two equations in two unknowns:
0 � 31 ��� 06 � 1 � 5 �E � r1 � � � 06 � � 2 � 0 �E � r2 � � � 06 �0 � 27 ��� 06 � 2 � 2 �E � r1 � � � 06 � � 0 � 2 �E � r2 � � � 06 �� E � r1 � � 0 � 16 � E � r2 � � 0 � 11 �
Therefore, the expected return-β relationship is:
E � rP � ��� 06 � βP1 � � 10 ��� βP2 � � 05 � �8-16 Note that since βF
� 0, E � rF � � r f��� 06. An implication of absence of arbitrage is that
E � rA � � r f
βA
� E � rB � � r f
βB� � 12 � � 061 � 2 ��� 05
�� � 08 � � 060 � 6 ��� 033 �
so there is an arbitrage opportunity. This could be exploited by creating a portfolio G with βG� βB
� 0 � 6by combining portfolios A and F with equal weights. Notice that βG
� � 5 � 1 � 2 � � � 5 � 0 � � 0 � 6, and E � rG � �� 5 � � 12 � � � 5 � � 06 � � � 09. Comparing G to B, G has the same β and higher return. Therefore, the arbitragestrategy would be to buy portfolio G and sell the same amount of portfolio B. If you do so, your profit willbe:
rG � rB� � � 09 � � 6F � � � � 08 � � 6F � ��� 01 �
or 1% of the funds (long or short) in each portfolio.
18
8-25 The APT required rate of return on the stock based on r f and the factor β is � 06 � 1 � 0 � � 06 � � � 5 � � 02 � �� 75 � � 04 � ��� 16. According to the equation given for the return on the stock, the actually expected return is15% (because the expected surprises on all factors are by definition zero). Because the actually expectedreturn is less than the required return, we conclude that the stock is overpriced.
9-1 Zero. If not, one could use returns from one period to predict returns in later periods and make abnormalprofits.
9-2 c. This is a predictable pattern in returns which should not occur if markets are weak-form efficient.
9-3 c. This is a classic filter rule which should not work in an efficient market.
9-4 b. This is basically the definition of an efficient market.
9-5 c. The P/E ratio is public information and should not be predictive of abnormal security returns.
9-13 The question for market efficiency is whether investors can earn abnormal risk-adjusted profits. If the stockprice run-up occurs when only insiders are aware of the coming dividend increase, then it is a violation ofstrong form, but not semi-strong form efficiency. If the public already knows of the increase, then it is aviolation of semi-strong form efficiency.
9-14 While positive β stocks will respond well to favourable new information about the economy’s progressthroughout the business cycle, they should not show abnormal returns around already anticipated events.If a recovery, for example, already is anticipated, the actual recovery is not news. Stock prices shouldalready reflect the coming recovery.
9-18 a. Based on broad market trends, the CAPM suggests that AmbChaser stock should have increased by� 01 � 2 � � 015 � � 01 � � 2%. Its firm-specific return due to the lawsuit is $1 million per $100 million ofinitial equity, or 1%. Therefore, the total return should be 3%. (It is assumed here that the outcome ofthe lawsuit had an expected value of zero.)
b. If the settlement was expected to be $2 million, then the actual settlement was a $1 million “disap-pointment”, and so the firm-specific return would be -1%, for a total return of � 02 � � 01 � 1%.
9-19 Given market performance, predicted returns on the two stocks would be:
Apex: 0 � 002 � 1 � 4 � � 03 � ��� 044
Bpex: � 0 � 001 � 0 � 6 � � 03 � ��� 017 �Apex underperformed this prediction; Bpex outperformed. We conclude that Bpex won the suit.
9-28 The negative abnormal returns (downward drift in CAR) just prior to stock purchases suggests that insidersdeferred their purchases until after bad news was released to the public. This is evidence of valuableinside information. The positive abnormal returns after purchases by insiders suggests insider purchasesin anticipation of good news. The analysis is symmetric for insider sales.
9-30 a. Some empirical evidence that supports the efficient markets hypothesis is that (i) professional moneymanagers do not typically earn higher returns than comparable risk, passive index strategies; (ii) eventstudies typically show that stocks respond immediately to the public release of relevant news; and (iii)most tests of technical analysis find that it is difficult to identify price trends that can be exploited toearn superior risk-adjusted investment returns.
b. Some evidence that is difficult to reconcile with the efficient markets hypothesis is that some sim-ple portfolio strategies would have apparently provided high risk-adjusted returns in the past. Someexamples of portfolios with attractive historical returns include (i) low P/E stocks; (ii) high book-to-market ratio stocks; (iii) small firms in January; and (iv) firms with very poor stock price performancein the last few months. Other evidence concerns post-earnings announcement stock price drift andintermediate-term price momentum.
c. An investor may choose not to index even if markets are efficient because he or she may want to tailora portfolio to specific tax considerations (e.g. capital gains instead of dividends) or to specific riskmanagement issues (e.g. the need to hedge risk exposure to a particular source of risk).
19
2. Doug’s rate of return was:1750 � � 010 � � 1400 � � 008 �
1400 � � 008 � � 0 � 5625 �while the rate of return for a Japanese domestic investor was:
1750 � 14001400
� 0 � 25 �3. This can be argued either way. If it can be shown that the public did not know about the magnitude of the costs
involved, it is more a demonstration of efficiency as the market reacted quickly to negative news. If investorswere already aware of the size of the liability, then it would be evidence of inefficiency because this is simplyan accounting change with no economic impact.
4. (a) We have:
wX� � � 20 � � 08 � � � 22 � � � � 12 � � 08 � � � 3 � � � 2 � � � 25 �� � 20 � � 08 � � � 22 ��� � � 12 � � 08 � � � 32 � � � � 20 � � 08 � � 12 � � 08 � � � 3 � � � 2 � � � 25 �� 0 � 7
wY� 0 � 3� E � rP � ��� 7 � � 2 ��� � 3 � � 12 � � 0 � 176
σP� � � � 72 � � � 32 ��� � � 32 � � � 22 ��� 2 � � 7 � � � 3 � � � 3 � � � 2 � � � 25 � � 1 � 2� 0 � 2324 �
Letting x be the percentage invested in the optimal portfolio of risky assets:
x � � 176 ��� � 1 � x � � � 08 � ��� 14� x � 0 � 625
σ � 0 � 625 � 0 � 2324 � � 0 � 1452 �(b) We have:
y� � � 176 � � 08
5 � � 23242 � � 0 � 3555 �so the investor would invest 35.55% of his funds in the risky portfolio P from part (a) and the remainder(64.45% of his funds) in the risk free asset. Then:
E � r � ��� 3555 � � 176 ��� � 6445 � � 08 � � 0 � 1141 �σ ��� 3555 � � 2324 � � 0 � 0826 �
(c) We have: � � 20 � R f � � � 22 � � � � 12 � R f � � � 3 � � � 2 � � � 25 �� � 20 � R f � � � 22 � � � � 12 � R f � � � 32 � � � � 20 � R f � � 12 � R f � � � 3 � � � 2 � � � 25 � � � 0062 � � 025R f� 014 � � 10R f
� 1� R f��� 104
y� � � 20 � � 104
5 � � 32 � � 0 � 2133 �so the investor puts 21.33% of his money in asset X and 78.67% in the risk free asset. Then:
E � r � ��� 2133 � � 2 � � � 7867 � � 104 � � 0 � 1240 �σ ��� 2133 � � 3 � � 0 � 064 �
This investor is better off since expected return is now higher and standard deviation is lower. An investorwith a very low level of risk aversion would be made worse off. Such an individual would want to borrowand would end up paying a higher interest rate after the increase in R f .
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5. (a) For investor I:
E � rM � ��� 75 � � 30 ��� � 25 � � 20 � � 0 � 275 �βA� σAM
σ2M
� � 75 � � 092 �� � 752 � � � 092 ��� � � 252 � � � 052 � � 1 � 2891 �For investor J:
E � rM � ��� 5 � � 30 ��� � 5 � � 20 � � 0 � 25 �βA� σAM
σ2M
� � 5 � � 092 �� � 52 � � � 092 ��� � � 52 � � � 052 � � 1 � 5283 �(b) Note that:
Cov � RZ � RM � � xAw � � 092 ��� xB � 1 � w � � � 052 ���� 0081xAw � � 0025xB � 1 � w � �where w is the percentage invested in A in the zero-β portfolio. We need this covariance to be zero in orderto have a zero-β portfolio. For investor I:
Cov � RZ � RM � ��� 00545w � � 000625 � 0� w � � 0 � 1147 �Therefore, investor I’s zero-β portfolio has -11.47% invested in A and 111.47% invested in B. The expectedreturn on this portfolio is � � 1147 � � 3 ��� 1 � 1147 � � 2 � � 0 � 1885. For investor J:
Cov � RZ � RM � ��� 0028w � � 00125 � 0� w � � 0 � 4464 �Thus investor J’s zero-β portfolio has -44.64% invested in A and 144.64% invested in B. This portfoliohas an expected return of � � 4464 � � 3 � � 1 � 4464 � � 2 � � 0 � 1554.
(c) For investor I:
E � Ri � � 0 � 1885 � � 0 � 275 � 0 � 1885 � βi
E � RA � � 0 � 1885 � � 0 � 275 � 0 � 1885 � � 1 � 2891 � � 0 � 3 �and for investor J:
E � Ri � � 0 � 1554 � � 0 � 250 � 0 � 1554 � βi
E � RA � � 0 � 1554 � � 0 � 250 � 0 � 1554 � � 1 � 5283 � � 0 � 3 �(d) In this problem with only two assets, every combination of A and B is on the efficient frontier. Roll noted
that there is an exact linear relationship between any two efficient portfolios and the expected return on anyasset. We see this here—the “market portfolios” for each investor are different, yet they each will calculatean exact linear relationship involving β and the expected returns on the market portfolio and its associatedzero-β portfolio and end up computing the same expected return for each individual security.
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