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Essentials of Investments (BKM 7th Ed.) Answers to Suggested Problems Lecture 1
Chapter 2:
8.
a. At t = 0, the value of the index is: (90 + 50 + 100)/3 = 80 At t = 1, the value of the index is: (95 + 45 + 110)/3 = 83.3333 The rate of return is: (83.3333/80) 1 = 4.167%
b. In the absence of a split, stock C would sell for 110, and the value of the index would be: (95 + 45 + 110)/3 = 83.3333 After the split, stock C sells at 55. Therefore, we need to set the divisor (D) such that:
83.3333 = (95 + 45 + 55)/D or D = 2.340
c. The rate of return is zero. The index remains unchanged, as it should, since the return on each stock equals zero.
9. a. Total market value at t = 0 is: (9,000 + 10,000 + 20,000) = 39,000
Total market value at t = 1 is: (9,500 + 9,000 + 22,000) = 40,500 The return on the value-weighted index equals: (40,500/39,000) 1 = 3.85%
b. The return on each stock is as follows: Ra = (95/90) 1 = 0.0556 Rb = (45/50) 1 = 0.10 Rc = (110/100) 1 = 0.10 The return on the equally-weighted index equals: [0.0556 + (-0.10) + 0.10]/3 = 0.0185 = 1.85%
10. The after-tax yield on the corporate bonds is: [0.09 x (1 0.30)] = 0.0630 = 6.30% Therefore, municipal bonds must offer a yield of at least 6.30%.
12. The equivalent taxable yield (r) is: r = rm/(1 t)
a. 4.00% b. 4.44% c. 5.00% d. 5.71%
Chapter 3:
1. a. The underwriter charges a gross spread of 7% (and there are no other fees listed).
The total fees are therefore 1,000,000 x $50 x 7% = $3.5 million.
b. The firm raised $50 per share less the gross spread, or: 1,000,000 x $50 x (1-0.07) = $46.5 million.
c. The underpricing is $3 per share, or (53-50)/50 = 6.0%.
d. The firm paid $3.5 million in explicit fees to the underwriter. The implicit cost associated with underpricing was $3 per share. The total explicit and implicit costs were therefore $3.5 million + (1,000,000 x $3) = $6.5 million.
7.
a. You buy 200 shares of Telecom ($10,000/$50 per share). These shares increase in value by 10%, or $1,000. You pay interest of: 0.08 x 5,000 = $400 The rate of return will be:
000,5$
400$000,1$ = 0.12 = 12%
b. The value of the 200 shares is 200P. The equity in the account is (200P $5,000). You will receive a margin call when:
P200
000,5$P200 = 0.30 when P = $35.71 or lower
c. The value of the 200 shares is 200P. After one year, the equity in the account is (200P $5,000(1.08)). You will receive a margin call when:
P
P200
)08.1(000,5$200 = 0.30 when P = $38.57 or lower
8. a. Initial margin is 50% of $30,000 or $15,000.
b. Total assets are $45,000 ($30,000 from the sale of the stock and $15,000 put up for
margin). Liabilities are 500P because you must pay back the borrowed shares. The equity in the account is ($45,000 500P). A margin call will be issued when:
P
P500
500000,45$ = 0.30 when P = $69.23 or higher
18. The proceeds from the short sale (net of commission) were: ($14 x 100) ($0.50 x 100) = $1,350 You must repay the dividend to the original owner of the borrowed shares. As a result, the dividend payment of $200 is withdrawn from your account. Covering the short sale at $9 per share cost you (including commission): ($9 x 100) + ($0.50 x 100) = $950 Therefore, the value of your account is equal to the net profit on the transaction: $1350 $200 $950 = $200 Note that your profit ($200) equals (100 shares x profit per share of $2). Your net proceeds per share were:
$14 selling price of stock $ 9 repurchase price of stock $ 2 dividend per share
$ 1 2 trades x $0.50 commission per share $ 2
19. d. If the stock never traded at or below $55, your stop loss order would never be executed. In this case, your stop loss order would trigger a market sell order as soon as a transaction occurs at or below $55. After this transaction occurs, your broker will sell your shares at the best available market price. This price should be close to $55, but could be slightly above or below this price.
Chapter 4:
13. Start of year NAV = $20
Dividends per share = $0.20
End of year NAV is based on the 8% price gain, less the 1% 12b-1 fee:
End of year NAV = $20 1.08 (1 0.01) = $21.384 Rate of return =
20$20.0$20$384.21$ + = 0.0792 = 7.92%
19.
a. After two years, each dollar invested in a fund with a 4% load and a portfolio return equal to r will grow to:
$0.96 (1 + r 0.005)2 Each dollar invested in the bank CD will grow to:
$1 (1.06)2 If the mutual fund is to be the better investment, then the portfolio return, r, must satisfy:
0.96 (1 + r 0.005)2 > (1.06)2 0.96 (1 + r 0.005)2 > 1.1236 (1 + r 0.005)2 > 1.1704 (1 + r 0.005) > 1.0819
1 + r > 1.0869
Therefore, r > 0.0869 = 8.69%
b. If you invest for six years, then the portfolio return must satisfy:
0.96 (1 + r 0.005)6 > (1.06)6 = 1.4185 (1 + r 0.005)6 > 1.4776
1 + r 0.005 > 1.0672
1 + r > 1.0722
r > 7.22%
The cutoff rate of return is lower for the six year investment because the "fixed cost" (i.e., the one-time front-end load) is spread out over a greater number of years.
c. With a 12b-1 fee instead of a front-end load, the portfolio must earn a rate of return
(r) that satisfies: (1 + r 0.005 0.0075) > 1.06
In this case, r must exceed 7.25% regardless of the investment horizon.
20. The turnover rate is 50%. This means that, on average, 50% of the portfolio is sold and replaced with other securities each year. Trading costs on the sell orders are 0.4%; and the buy orders to replace those securities entail another 0.4% in trading costs. Total trading costs will reduce portfolio returns by: 2 0.4% 0.50 = 0.4%
Essentials of Investments (BKM 7th Ed.) Answers to Suggested Problems Lecture 2
Chapter 5: 6. #3 For each portfolio: Utility = E(r) ( 4 2 ).
Investment E(r) U
1 0.12 0.30 -0.0600 2 0.15 0.50 -0.3500 3 0.21 0.16 0.1588 4 0.24 0.21 0.1518
The portfolio with the highest utility value is #3.
7. #4 When an investor is risk neutral, A = 0, so that the portfolio with the highest utility is the portfolio with the highest expected return. This is investment #4, with 24%.
9. E(RX) = [0.2 (20%)] + [0.5 18%] + [0.3 50%)] = 20%
E(RY) = [0.2 (15%)] + [0.5 20%] + [0.3 10%)] = 10% 10. X2 = [0.2 (.20 .20)2] + [0.5 (.18 .20)2] + [0.3 (.50 .20)2] = .0592 X = 24.33% Y = [0.2 (.15 .10)2] + [0.5 (.20 .10)2] + [0.3 (.10 .10)2] = .0175 Y = 13.23% 15.
a. E(RP) Rf = AP2 = 4 (0.20)2 = 0.08 = 8.0% b. 0.09 = AP2 = A (0.20)2 A = 0.09/( 0.04) = 4.5 c. Increased risk tolerance means decreased risk aversion (A), which results in a decline
in risk premiums.
18. a) Expected cash flow:
0.5($50,000) + 0.5($150,000) = $100,000 HPR = (P1 - P0)/P0 0.15 = (100,000 - P0)/P0 P0 = $86,956.52 b) HPR = (100,000 - 86956.52)/86956.52 = 15%
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
18.00%
0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00%
Standard Deviation
Expe
cted
Ret
urn
Risk-Free
Risky Portfolio
w=0.7
c) A risk-premium of 15%, leads to an expected return of 15%+5%=20%. 0.20 = (100,000 - P0)/P0 P0 = $83,333.00 d) There is an inverse relationship: price decreases as the risk premium increases. In order to earn a higher risk-premium (assuming the cash flows stay the same), you must be able to buy the security at a lower price. The investor requiring the 15% risk premium (20% HPR) is requiring a larger discount as compensation for risk.
19. a) E(RP) = 0.3(7%) + 0.7(17%) = 14% P = 0.7(27%) = 18.9% b) T-Bills = 30.0% Stock A = 0.7(27%) = 18.9% (The total weight in the portfolio is 70%, and the Stock B = 0.7(33%) = 23.1% portfolio consists of 27% A, 33% B, and 40% C) Stock C = 0.7(40%) = 28.0% Total Portfolio = 100% c) Your Reward-to-variability = (RP - RF)/P = (17% - 7%)/27% = 0.3704
Clients Reward-to-variability = (14% - 7%)/18.9% = 0.3704
d) The slope of the capital allocation line equals the reward-to-variability ratio (0.3704). Note that this is the same at any point you choose on the CAL.
0 2 4 6 8
101214161820
0 10 20 30 (%)
CAL (slope=.3704)
CML (slope=.24)
20. a. Rule 1: E(RC) = RF + y(RP-RF)
0.15 = 0.07 + y(0.17 - 0.07) y = 0.80 b. T-Bills = 20.0% Stock A = 21.6% (0.8*27%) Stock B = 26.4% (0.8*33%) Stock C = 32.0% (0.8*40%) c. Rule 2: C = 0.80(0.27) = 0.216 = 21.6%
21. a. Portfolio standard deviation = y 27%. If the client wants a standard deviation of 20%, then y = 20/27 = .7407 = 74.07% in
the risky portfolio. b. Rule 2: E(R) = 7 + 10y = 7 + .7407 10 = 7 + 7.407 = 14.407%
22. a. Slope of the CML = 13 7
25 = .24 The diagram is shown below.
b. My fund has a higher reward-to-variability (or a steeper CAL). This allows an investor in my fund to achieve a higher expected return for any given standard deviation than they would earn on the passive S&P fund. In other words, my fund provides a higher return at any given level of risk.
15.0%
13.8%
12.6%
11.4%
10.2%
9.0%
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00%
CAL
Optimal Risky Portfolio(70.75% Stocks, 29.25% Bonds)
Minimum Variance Portfolio(31.42% Stocks, 68.58% Bonds)
Chapter 6:
6. The parameters of the opportunity set are:
E(RS) = 15%, E(RB) = 9%, S = 32%, B = 23%, = 0.15, Rf = 5.5% This gives Cov(RS, RB) = SB = .01104
Using Rule 1* and Rule 2*, you can calculate the return and standard deviation on each of the six portfolio combinations (in 20% increments):
Weight S Weight B E(R) Std.Dev. 100% 0% 15.0% 32.00% 80 20 13.8% 26.68 70.75 29.25 13.25 24.57 60 40 12.6% 22.50 40 60 11.4% 20.18 31.42 68.58 10.89 19.94 20 80 10.2% 20.37 0 100 9.0% 23.00
CAL
Of these six possible portfolios, the minimum variance portfolio is the combination with 40% in stocks and 60% in bonds. Note: The true minimum variance portfolio is actually between the 40% stock and 20% stock choices - the exact minimum variance portfolio has a weight of 31.42% in stocks and 68.58% in bonds. The result can be found by minimizing the variance formula, Rule 2*, as we discussed in class. The minimum-variance portfolio proportions are:
)r,r(Cov2
)r,r(Cov)S(w
BS2B
2S
BS2B
Min += 3142.0
)01104.02(0529.01024.001104.00529.0 =+
=
w Min (B) = 1 0.3142 = 0.6858 Plugging these weights into Rule 1* and Rule 2* gives the expected return and standard deviation shown in the table above.
7. If you consider only the six portfolios you created at 20% increments, the optimal risky portfolio is the portfolio that includes 60% stocks and 40% bonds. This portfolio has an expected return of 12.6% and a standard deviation of 22.5%. You can use this portfolio to answer questions 8 through 10. (Note: The true optimal risky portfolio (P) is somewhere between the 60% stock and 80% stock choices. The portfolio with 70% stocks and 30% bonds is close to the optimal risky portfolio. This portfolio has an expected return of 13.2% and a standard deviation of approximately 24.41%. Although I will not require you to do this calculation, you can solve for the exact weights in the optimal portfolio using the formula in footnote 3 on page 185 of the text: the weights are 70.75% stock and 29.25% bonds. This portfolio has an expected return of 13.25% and a standard deviation of 24.57%.)
8. The reward-to-variability ratio of the optimal CAL (using the 60/40 portfolio) is:
3156.05.22
5.56.12)( ==p
fp rrE
9. a) The equation for the CAL (using the 60/40 portfolio) is:
CCp
fpfC
rrErrE 3156.0%5.5
)()( +=+=
Setting E(rC) equal to 12% yields a standard deviation of 20.6%.
b. The mean of the complete portfolio as a function of the proportion invested in the risky portfolio (y) is:
E(rC) = (l y)rf + yE(rP) = rf + y[E(rP) rf] = 5.5 + y(12.6 5.5) Setting E(rC) = 12% y = 0.9155 (91.55% in the risky portfolio) 1 y = 0.0845 (8.45% in T-bills) From the composition of the optimal risky portfolio:
Proportion of stocks in complete portfolio = 0.9155 0.60 = 0.5493 Proportion of bonds in complete portfolio = 0.9155 0.40 = 0.4507
10. Using only the stock and bond funds to achieve a mean of 12% we solve:
12 = 15wS + 9(1 wS ) = 9 + 6wS wS = 0.5 Investing 50% in stocks and 50% in bonds yields a mean of 12% and standard deviation of: P = [(0.502 1024) + (0.502 529) + (2 0.50 0.50 110.4)] 1/2 = 21.06% The efficient portfolio with a mean of 12% has a standard deviation of only 20.6%. Using
the CAL reduces the SD by 46 basis points (0.46%).
12. If = -1, a zero-risk portfolio can be created:
To find the answer, substitute = -1 into Rule 2* and then set the equation equal to zero. This gives the minimum variance portfolio when = -1. We discussed the solution to this equation in class. Here, the weight in stock A that gives a perfect hedge (zero risk) equals B/(A+B)=0.6/(0.4+0.6) = 60%. In other words, a portfolio with wA = 0.60 and wB = 0.40 would have = 0. This portfolio has a return of: E(R) = 0.6(.08) + 0.4(0.13) = 10% Since this portfolio is riskless, the risk-free rate must be exactly equal to the expected return on this portfolio or 10%.
14. All investors will have the same optimal risky portfolio, since the optimal risky portfolio is independent of investor preferences. However, investors will adjust portfolios to obtain the appropriate levels of risk by combining the risky portfolio with a risk-free asset such as T-Bills. (Note: This answer assumes that there is a risk-free security in the economy and investors can borrow and lend at this risk-free rate. If there is no risk-free security, then the optimal risky portfolio may depend on the risk-aversion level of the individual).
15. No. It isnt possible to get such a diagram. With a correlation between 1.0 and +1.0, the graph should be a smooth curve passing through points A and B and bowing toward the Y-axis. (Even if the correlation between A and B were exactly +1.0, the frontier would be a straight line connecting A and B.)
16. The two measures are equal if =+1. Otherwise, the portfolio standard deviation will be
less than the weighted average of the standard deviations of the component assets. In other words, for any
Essentials of Investments (BKM 7th Ed.) Answers to Selected Problems Lecture 3
Note: The solutions to Example 6.4 and the concept checks are provided in the text.
Chapter 6:
19. a. The risk of the diversified portfolio consists primarily of systematic risk. Beta measures
systematic risk, which is the slope of the security characteristic line (SCL). The two figures depict the stocks' SCLs. Stock B's SCL is steeper, and hence Stock B's systematic risk is greater. The slope of the SCL, and hence the systematic risk, of Stock A is lower. Thus, for this investor, stock B is the riskiest.
b. The undiversified investor is exposed to both systematic and firm-specific risk. Stock A has higher total risk because the total variation of the observations around the SCL is larger for Stock A than for Stock B. Stock A is therefore riskiest to this investor.
25. In the regression of the excess return of Stock ABC on the market, the square of the correlation
coefficient is 0.296 (this is the R2 of the regression). This indicates that 29.6% of the variance of the excess return of ABC is explained by the market (systematic risk).
Chapter 7: 3. E(Rp) = Rf + p[E(RM) - Rf]
0.20 = 0.05 + (0.15 - 0.05) = 0.15/0.10 = 1.5
6. a) False: =0 implies E(R)=Rf, not zero. b) False: Investors of a diversified portfolio require a risk premium for systematic risk. Only the systematic portion of total risk is compensated. c) False: 75% of the portfolio should be in the market and 25% in T-bills. p=(0.75 * 1) + (0.25 * 0) = 0.75
8. Not possible. Portfolio A has a higher beta than B, but a lower expected return.
9. Possible. If the CAPM is valid, the expected rate of return compensates only for market risk (beta), rather than for nonsystematic risk. Part of As risk may be nonsystematic.
10. Not possible. If the CAPM is valid, the market portfolio is the most efficient and a higher reward-to-variability ratio than any other security. In other words, the CML must be better than the CAL for any other security. Here, the slope of the CAL for A is 0.5 while the slope of the CML is 0.33.
11. Not possible. Portfolio A clearly dominates the market portfolio with a lower standard deviation and a higher expected return. The CML must be better than the CAL for security A.
12. Not possible. Security A has an expected return of 22% based on CAPM and an actual return of 16%. Security A is below the SML and is therefore overpriced. It is also clear that security A has a
higher beta than the market, but a lower return which is not consistent with CAPM.
13. Not possible. Security A has an expected return of 17.2% and an actual return of 16%. Security A is below the SML and is therefore overpriced.
14. Possible. Portfolio A has a lower expected return and lower standard deviation than the market and thus plots below the CML.
18. Using the SML: 6 = 8 + (18 8)
= 2/10 = .2
23. The expected return of portfolio F equals the risk-free rate since its beta equals 0. Portfolio As ratio of risk premium to beta is: (10-4)/1 = 6.0%. You can think of this as the slope of the pricing line for Security A. Portfolio Es ratio of risk premium to beta is: (9-4)/(2/3) = 7.5%, suggesting that Portfolio E is not on the same pricing line as security A. In other words, there is an arbitrage opportunity here. For example, if you created a new portfolio by investing 1/3 in the risk-free security and 2/3 in security A, you would have a portfolio with a beta of 2/3 and an expected return equal to (1/3)*4% + (2/3)*10% = 8%. Since this new portfolio has the same beta as security E (2/3) but a lower expected return (8% vs. 9%) there is clearly an arbitrage opportunity.
27. The APT factors must correlate with major sources of uncertainty in the economy. These factors would correlate with unexpected changes in consumption and investment opportunities. DNP, the rate of inflation, and interest rates are candidates for factors that can be expected to determine risk premia. Industrial production varies with the business cycle, and thus is a candidate for a factor that is correlated with uncertainties related to investment opportunities in the economy.
28. A revised estimate of the rate of return on this stock would be the old estimate plus the sum of the expected changes in the factors multiplied by the sensitivity coefficients to each factor: revised Ri = 14% + 1.0(1%) + 0.4(1%) = 15.4%
29. E(RP) = rf + P1[E(R1) - Rf] + P2[E(R2) - Rf] Use each securitys sensitivity to the factors to solve for the risk premia on the factors: Portfolio A: 40% = 7% + 1.81 + 2.12 Portfolio B: 10% = 7% + 2.01 + (-0.5)2 Solving these two equations simultaneously gives 1 = 4.47 and 2 = 11.88. This gives the following expected return beta relationship for the economy: E(RP) = 0.07 + 4.47P1 + 11.88P2
33. d. The expected return on the market.
34. d. You need to know the risk-free rate. For example, if we assume a risk-free rate of 4%, then the alpha of security R is 2.0% and it lies above the SML. If we assume a risk-free rate of 8%, then the alpha of security R is zero and it lies on the SML.
Essentials of Investments (BKM 7th Ed.) Answers to Selected Problems Lecture 4
Note: The solution to the concept check is provided in the text.
Chapter 5: 14b. Time Cash flow Explanation 0 300 Purchase of three shares at $100 each. 1 208 Purchase of two shares at $110 less dividend income on three shares held. 2 110 Dividends on five shares plus sale of one share at price of $90 each. 3 396 Dividends on four shares plus sale of four shares at price of $95 each. +110 +396 | | | | Date: 1/1/96 1/1/97 1/1/98 1/1/99 | | | | -300 -208 The Dollar-weighted return can be determined by doing an internal rate of return (IRR) calculation. In other words, set the present value of the outflows equal to the present value of the inflows (or the net present value to zero):
%1661.0001661.0)1(
396)1(
110)1(
208300 321 ==+++=++ RRRR Chapter 18: 5. We need to distinguish between timing and selection abilities. The intercept of the scatter
diagram is a measure of stock selection ability. If the manager tends to have a positive excess return even when the markets performance is merely neutral (i.e., has zero excess return), then we conclude that the manager has on average made good stock picks stock selection must be the source of the positive excess returns. Timing ability is indicated by curvature in the plotted line. Lines that become steeper as you move to the right of the graph show good timing ability. An upward curved relationship indicates that the portfolio was more sensitive to market moves when the market was doing well and less sensitive to market moves when the market was doing poorly -- this indicates good market timing skill. A downward curvature would indicate poor market timing skill.
We can therefore classify performance ability for the four managers as follows: Selection Ability Timing Ability a. Bad Good b. Good Good c. Good Bad d. Bad Bad
9. The managers alpha is: 10 - [6 + 0.5(14-6)] = 0
10. a) (A) = 24 - [12 + 1.0(21-12)] = 3.0% (B) = 30 - [12 + 1.5(21-12)] = 4.5% T(A) = (24 - 12)/1 = 12 T(B) = (30-12)/1.5 = 12 As an addition to a passive diversified portfolio, both A and B are candidates because they both have positive alphas. b) (i) The funds may have been trying to time the market. In that case, the SCL of the funds may be non-linear (curved). (ii) One years worth of data is too small a sample to make clear conclusions. (iii) The funds may have significantly different levels of diversification. If both have the same risk-adjusted return, the fund with the less diversified portfolio has a higher exposure to risk because of its higher firm-specific risk. Since the above measure adjusts only for systematic risk, it does not tell the entire story.
Essentials of Investments (BKM 7th Ed.) Answers to Selected Problems Lecture 5
Chapter 8: 1. Zero. If not, one could use returns from one period to predict returns in later periods and
make abnormal profits.
2. c. The January Effect implies that one can predict January prices based on past January prices. This is a predictable pattern in returns which should not occur if weak-form EMH is valid.
3. c. This is a classic filter rule which should not be profitable in an efficient market.
5. c. The P/E ratio is public information and should not predict abnormal security returns. 13. a) The grandson is referring to the small-firm effect (which can also be described as the
January effect). b) 1 - Building a portfolio of only small firms results in increased risk, as the portfolio is less diversified. 2 - Because the anomaly has existed in the past is not a predictor that the anomaly will exist in the future. 3 - After the results of these studies became publicly known, investors may bid up the prices of these securities to reflect the now-known opportunity.
14. No, this is not a violation of the EMH. Microsofts continuing large profits do not imply
that stock market investors who purchased Microsoft shares after its success already was evident would have earned a high return on their investments.
15. No, this is not a violation of the EMH. This empirical tendency does not provide investors
with a tool that will enable them to earn abnormal returns; in other words, it does not suggest that investors are failing to use all available information. An investor could not use this phenomenon to choose undervalued stocks today. The phenomenon instead reflects the fact that dividends occur as a response to good performance. After the fact, the stocks that happen to have performed the best will pay higher dividends, but this does not imply that you can identify the best performers early enough to earn abnormal returns.
17. a. Consistent. Half of managers should beat the market based on pure luck in any year. b. Inconsistent. This would be the basis of an "easy money" rule: simply invest with last
year's best managers. c. Consistent. Predictable volatility does not convey a means to earn abnormal returns.
d. Inconsistent. The abnormal performance ought to occur in January when earnings are announced.
e. Inconsistent. Reversals offer a means to earn easy money: just buy last week's losers. 24. You should buy the stock. In your view, the firms management is not as bad as everyone
else believes it to be. Therefore, you view the firm as undervalued by the market. You are less pessimistic about the firms prospects than the beliefs built into the stock price.
26. The market may have anticipated even greater earnings. Compared to prior expectations,
the announcement was a disappointment.
1
Essentials of Investments (BKM 7th Ed.) Answers to Suggested Problems Lecture 6
Chapter 10: 3. The bond callable at 105 should sell at a lower price because the call provision is more
valuable to the firm when the call price is lower. Because the call feature is more valuable to the firm (and more costly to investors), investors will require a higher yield to maturity, resulting in a lower price.
4. Lower. Interest rates have fallen since the bond was issued. Thus, the bond is selling at a
premium and the price will decrease (toward par value) as the bond approaches maturity.
5. True. Under the Expectations Hypothesis, there are no risk premia built into bond prices. The only reason for an upward sloping yield curve is the expectation of increased short-term rates in the future.
8. If the yield curve is upward sloping, you cannot conclude that investors expect short-term interest rates to rise because the rising slope could either be due to expectations of future increases in rates or due to a liquidity premium. In fact, with a liquidity premium, the yield curve can be upward sloping even if future short-term rates are expected to remain flat or even decrease.
9. a) The bond pays $50 every 6 months Current price = $1052.42 Assuming that market interest rates remain at 4% per half year:
the price 6 months from now = $1044.52 b) Rate of return = [1044.52 - 1052.42 + 50]/1052.42 = .04 or 4% per 6 months. Because the yield has not changed, the bond prices adjust such that the bond earns exactly the YTM.
35. a) The forward rate, f, is the rate that makes rolling over one-year bonds equally attractive
as investing in the two-year maturity bond and holding until maturity: (1.08)(1 + f) = (1.09)2 which implies that f = 0.1001 or 10.01% b) According to the expectations hypothesis, the forward rate equals the expected short rate next year, so the best guess would be 10.01%. c) According to the liquidity preference (liquidity premium) hypothesis, the forward rate exceeds the expected short-term rate for next year (by the amount of the liquidity premium), so the best guess would be less than 10.01%.
37. a. We obtain forward rates from the following table:
2
Maturity (years)
YTM
Forward rate
Price (for part c)
1 10.0% $909.09 ($1000/1.10) 2 11.0% 12.01% [(1.112/1.10) 1] $811.62 ($1000/1.112) 3 12.0% 14.03% [(1.123/1.112) 1] $711.78 ($1000/1.123)
b. We obtain next years prices and yields by discounting each zeros face value at the
forward rates derived in part (a): Maturity (years)
Price
YTM
1 $892.78 [ = 1000/1.1201] 12.01% 2 $782.93 [ = 1000/(1.1201 x 1.1403)] 13.02%
Note that this years upward sloping yield curve implies, according to the expectations hypothesis, a shift upward in next years curve.
c. Next year, the two-year zero will be a one-year zero, and it will therefore sell at: $1000/1.1201 = $892.78
Similarly, the current three-year zero will be a two-year zero, and it will sell for: $782.93
Expected total rate of return:
two-year bond: %00.101000.0162.811$78.892$ ==
three-year bond: %00.101000.0178.711$93.782$ ==
Chapter 11: Solutions to the concept checks are provided at the end of the chapter.
1. BB
D yy
= +1 -7.194 * (.005/1.10) = -.0327 or a decline of 3.27%
2. If YTM=6%, Duration=2.833 years
If YTM=10%, Duration=2.824 years
3
6. a) Bond B has a higher yield since it is selling at a discount (perhaps because it has lower credit quality). Thus, the duration of bond B is lower (it is less sensitive to interest rate changes).
b) Bond B is callable, has a higher coupon, and has a higher yield (both because it is callable and because Y=C when the bond is selling for par value). Thus, the duration of bond B is therefore lower (it is less sensitive to interest rate changes).
9. a) PV = 10,000/(1.08) + 10,000/((1.08)2) = $17,832.65 Duration = (9259.26/17832.65)*1 + (8573.39/17832.65)*2 = 1.4808 years
b) A zero-coupon bond with 1.4808 years to maturity (duration=1.4808) would immunize the obligation against interest rate risk.
c) We need a bond position with a present value of $17,832.65. Thus, the face value of the bond position must be:
$17,832.65*(1.08)1.4808 = $19,985.26 If interest rates increase to 9%, the value of the bond would be:
$19,985.26/((1.09)1.4808) = $17,590.92
The tuition obligation would be: 10,000/1.09 + 10,000/((1.09)2) = $17,591.11
or a net position change of only $0.19. If interest rates decrease to 7%, the value of the bond would be: $19,985.26/((1.07)1.4808) = $18,079.99 The tuition obligation would be: 10,000/(1.07) + 10,000((1.07)2) = $18,080.18
or a net position change of $0.19. **The slight differences result from the fact that duration is only a linear approximation of the true convex relationship between fixed-income values and interest rates.
4
11. a) The duration of the perpetuity is 1.05/.05 = 21 years. Let w be the weight of the zero-coupon bond. Then we find w by solving:
w 5 + (1 w) 21 = 10 21 16w = 10 w = 11/16 or .6875 Therefore, your portfolio would be 11/16 invested in the zero and 5/16 in the
perpetuity. b) The zero-coupon bond now will have a duration of 4 years while the perpetuity will
still have a 21-year duration. To get a portfolio duration of 9 years, which is now the duration of the obligation, we again solve for w:
w 4 + (1 w) 21 = 9 21 17w = 9 w = 12/17 or .7059 So the proportion invested in the zero has to increase to 12/17 and the proportion in the
perpetuity has to fall to 5/17. 24. a) 4
b) 4 c) 4 d) 2
29. Choose the longer-duration bond to benefit from a rate decrease. a) The Aaa-rated bond will have the lower yield to maturity and, therefore, the longer
duration. b) The lower-coupon bond will have the longer duration ( it also has more de facto call
protection, leading to a lower yield and, thus, a longer duration). c) The lower-coupon bond will have a longer duration.
1
Essentials of Investments (BKM 7th Ed.) Answers to Suggested Problems Lecture 7
Handout: Answers to the options handout are provided at the back of the handout. Supplemental Problems: I) The following options would be more valuable:
a) Call option with X=90 - with lower exercise price, the probability of being in the money and
expected value of S-X are greater. b) Put option the Put option is currently in the money, while the Call is not. c) Call option with 6 months to expiration - with longer maturity, the probability of being in the money
and expected value of S-X are greater. d) Call option on Intel - with higher volatility, the probability of being in the money and expected value
of S-X are greater.
II) The payoffs and profits on each of these positions is shown in the course notes. See the associated page numbers below: a) Page 7-3 b) Page 7-6 c) Page 7-9 d) Page 7-11 e) Page 7-12
III) Using the Google option prices provided in your notes, calculate the payoff and profit for investments in each of the following June maturity options. Assume the stock price on Google at the maturity date is $590. a) Long Call option, X=500
This option would be exercised, since S>X. The payoff is 590-500=$90 and the profit is 90-54.78=$35.22, where $54.78 is the cost of the option.
b) Long Put option, X=500 This option would not be exercised, since S>X. The payoff is zero and the profit is 0-13.70 or a loss of $13.70, the cost of the option.
c) Long Call option, X=600 This option would not be exercised, since S
2
e) Short Call option, X=600 As noted above, this option would not be exercised, since S
3
b. Vertical combination Position S < X1 X1 < S < X2 S >X2 Long call (X2) 0 0 S X2
Long put (X1) X1 S 0 0
Total X1 S 0 S X2
X1
STX1 X2
Payoff