UNIVERSITY OF TORONTO Joseph L. Rotman School of Management. RSM332 FINAL EXAMINATION Geoffrey/Wang SOLUTIONS. (1 + r m ) r m

Transcription

1 UNIVERSITY OF TORONTO Joseph L. Rotman School of Management Dec. 9, 206 Burke/Corhay/Kan RSM332 FINAL EXAMINATION Geoffrey/Wang SOLUTIONS. (a) We first figure out the effective monthly interest rate, r m. Since ( + r m ) 2 =.08, we have r m = (.08) 2 = Instead of computing the present value of the monthly payments, we convert them into annual payments with the same present value. Let C be the annual payment at the end of year such that it has the same present value as that of the monthly payments from month 0 to, we have C.08 = $2000A2 r m ( + r m ) C =.08 $2000( + r m) C = $ r m ( + r m ) 2 Similarly, let C t be the equivalent annual payment at the end of year t that has the same present value of the monthly payments for year t. We have C t = C ( + g) t, where g = 0.. Therefore, C to C 0 is a growing annuity and its present value is given by PV = C r g ( ) 0 + g = $ ( ) 0. = $ r (b) The annual payment is C = 900m./30 = 30m. Its present value is given by PV = C ( + r) r ( + r) 30 = 30m. (.037) (.037) 30 = 558.m.

2 Note that we need an extra term of + r in the above expression because this is an annuity due. (c) Since 28% of your $30m. annual cheque is withheld by IRA, you only receive $30m. ( 0.28) = $2.6m. at the beginning of every year. In addition, you have to pay an additional tax of $30m. 0.6 = $3.48m. at the end of every year. Therefore, the present value of the after-tax winning is PV = 2.6m (.037) 30 = 40.83m m. = m. (.037) 3.48m Note that a quicker way to obtain this present value is PV = 558.m. ( 0.28) 558.m (.037) 30 = m. () The first term is the present value of the 30 annual cheques (after tax withholding), and the second term is the 30 tax payments paid at the end of every year. 2. (a) We first find out the -year, 2-year and 3-year discount factors. From the observed bond prices, we have 00DF = 97.5, 2.25DF DF DF 3 = 88.53, 5.25DF DF DF 3 = Solving the three equations, we have DF = 0.975, DF 2 = 0.88 and DF 3 = Therefore, the price of the 3-year zero coupon bond with face value 00 is 00DF 3 = The price of the 2-year 5.25% coupon bond with face value 00 is 5.25DF DF 2 = The price of the 3-year 7.25% coupon bond with face value 00 is 7.25DF DF DF 3 = (b) Since the T -period discount factor, DF T, is related to the T -period spot rate, r T, by the following relation DF T = ( + r T ), T we have r T = DF T T. Therefore, the -year, 2-year, and 3-year spot rates are given by r = DF = = 2.564%, r 2 = DF 2 2 = = 6.6%, r 3 = DF 3 3 = = 6.62%. 2

3 (c) The two 3-year bonds do not have the same yield-to-maturity because they the term structure of interest rates is not flat. Instead, we face an increasing term structure of interest rates, i.e., cashflows at year 3 will be discounted using a higher interest rates than cashflows in year and 2. Since the bond with a low coupon rate (2.25%) has most of its cashflows coming in year 3, its yield-to-maturity is higher than the bond with a high coupon rate (5.25%). However, a bond with higher yield-to-maturity does not necessarily imply it is underpriced. Even when all the bonds are priced properly, they can have different yield-to-maturity simply because the term structure of interest rates is not flat. 3. (a) The expected return of portfolio p is ER p = 0.6ER A + 0.2ER B + 0.2ER C = = The expected return of portfolio q is ER q = 0.2ER A + 0.2ER B + 0.6ER C = = The standard deviation of portfolio p is σ p = (0.6) 2 σ 2 A + (0.2) 2 σ 2 B + (0.2) 2 σ 2 C + 2(0.6)(0.2)σ AB + 2(0.6)(0.2)σ AC + 2(0.2)(0.2)σ BC 2 = (0.6) 2 (0.05) 2 + (0.2) 2 (0.) 2 + (0.2) 2 (0.2) 2 + 2(0.6)(0.2)(0.05)(0.)(0.5) + 2(0.6)(0.2)(0.05)(0.2)(0) + 2(0.2)(0.2)(0.)(0.2)(0.5) 2 = The standard deviation of portfolio q is σ q = (0.2) 2 σ 2 A + (0.2) 2 σ 2 B + (0.6) 2 σ 2 C + 2(0.2)(0.2)σ AB + 2(0.2)(0.6)σ AC + 2(0.2)(0.6)σ BC 2 = (0.2) 2 (0.05) 2 + (0.2) 2 (0.) 2 + (0.6) 2 (0.2) 2 + 2(0.2)(0.2)(0.05)(0.)(0.5) + 2(0.2)(0.6)(0.05)(0.2)(0) + 2(0.2)(0.6)(0.)(0.2)(0.5) 2 = (b) The covariance between the two portfolios is σ pq = (0.6)(0.2)σ 2 A + (0.6)(0.2)σ AB + (0.6)(0.6)σ AC + (0.2)(0.2)σ AB + (0.2)(0.2)σ 2 B + (0.2)(0.6)σ BC + (0.2)(0.2)σ AC + (0.2)(0.2)σ BC + (0.2)(0.6)σ 2 C = (0.6)(0.2)(0.05) 2 + (0.6)(0.2)(0.05)(0.)(0.5) + (0.6)(0.6)(0.05)(0.2)(0) + (0.2)(0.2)(0.05)(0.)(0.5) + (0.2)(0.2)(0.) 2 + (0.2)(0.6)(0.)(0.2)(0.5) + (0.2)(0.2)(0.05)(0.2)(0) + (0.2)(0.2)(0.)(0.2)(0.5) + (0.2)(0.6)(0.2) 2 =

4 It follows that the correlation between the two portfolios is ρ pq = σ pq σ p σ q = = (c) The Sharpe ratios of the two portfolios are given by SR p = µ p r σ p = SR q = µ q r σ q = = , = Since the tangency portfolio has the highest Sharpe ratio, portfolio p cannot be the tangency portfolio. So if one of the two portfolios were the tangency portfolio, then portfolio q would be the tangency portfolio. (d) Let w p be the weight on portfolio p, we have 0. = w p µ p + ( w p )r 0. = w p (0.062) + ( w p )(0.02) w p = Therefore, you need to put 24.3% of your money in portfolio p by borrowing 4.3% of your money at the risk-free rate. The standard deviation of such a portfolio is 2.43σ p = (e) Let w q be the weight on portfolio q, we have 0.5 = w q σ q 0.5 = w q ( ) w q =.753. Therefore, you need to put 75.3% of your money in portfolio q by borrowing 75.3% of your money at the risk-free rate. The expected return of this portfolio is.753µ q 0.753r = = (a) Having α A = 0 tells us that advisor A has matched the performance of the market on a risk adjusted basis over the past five years. Since β A =.0 over the past five years, we know that the level of market risk in advisor A s investments equaled that of the market portfolio. Therefore, the return of the market over the past five years is same as the return advisor A produced over the same period, i.e., 6.0%, assuming the CAPM holds. Note that w q =.753 is also a solution but since µ q > r, such a portfolio would have negative expected return, so we do not choose this solution. 4

5 (b) The historical beta for the past five years for advisor B is β B = σ BM σ 2 M = ρ BMσ B σ M = =.76. (c) To utilize the CAPM for this question we first need all components of the CAPM. We need a measure for the risk free rate for the ten year period which we can solve for. r = µ M (µ M r) = = Advisor A has generated an average return of 2% over the past 0 years. Given the level of market risk advisor A has taken, they should have generated an average return of (0.09) = 0.2. Advisor A s performance on a risk adjusted basis is α A = = Advisor B has generated an average return of 5%. Given the level of market risk advisor B has taken, they should have generated an average return of (0.09) = Advisor B s performance on a risk adjusted basis is α B = = Advisor B had been outperforming advisor A on the average return. However, when exposure to market risk is taken into account advisor B underperformed advisor A. On a risk adjusted basis advisor A has outperformed advisor B by ( 0.034) = This suggests that advisor B s absolute outperformance has been achieved by taking on high amounts of market risk. (d) Possible reasons include: The CAPM, or specifically beta, only reflects market risk. If you care about other risks, the CAPM may not be appropriate to base your financial decision making criterion on. Since your parents are near retirement age there are many other things they should be concerned about including liquidity risk, shortfall risk, tail risk. The analysis performed in this question was done so using historical returns and historical risk metrics. Perhaps the analysis of past data periods is not appropriate for the near future leading up to your parents retirement. As always when using historical data, they are subject to sampling errors and may not provide an accurate estimates of the expected returns and betas. The CAPM is obtained by making a number of assumptions including that investors are price-takers, no taxes and transaction costs, investors only have a single period investment horizon and have identical expectations about asset returns, etc. These questionable assumptions may lead the CAPM to have little support from empirical research. If you use it to make projections, there is a possibility that the resulting recommendation will be wrong. 5

6 5. (a) For the portfolio p, the expected return is µ p = (0.30)(0.2) + (0.6)(0.) + (0.24)(0.4) + (0.2)(0.3) = The sensitivity of portfolio p to factor is β p, = (.2)(0.2) + (0.8)(0.) + (0.6)(0.4) + (0.2)(0.3) = The sensitivity of portfolio p to factor 2 is β p2 = ( 0.4)(0.2) + (.0)(0.) + (0.2)(0.4) + ( 0.2)(0.3) = 0.6. (b) Such a Factor portfolio can be found by solving.2x A + 0.8x B + 0.6x C + 0.2x D = 0.4x A x B + 0.2x C 0.2x D = 0 x A + x B + x C + x D = where x A, x B, x C, x D are the portfolio weights. There are four unknowns and three equations, so there exist multiple solutions. For example, ( 5, 5, 7 ) 5, 0 is one solution or one of such portfolios. When the APT holds, all Factor portfolios have the same expected return: given their betas, their expected returns are all equal to λ 0 + λ. That is, according to the APT, µ q = λ 0 + β q λ + β q2 λ 2 = λ 0 + λ + 0 λ 2 = λ 0 + λ. It not, then there exist an arbitrage opportunity between two Factor portfolios. (c) We can use any three stocks of the four, say, A, B, and C. By the APT, we set up three equations for three unknowns: λ 0 +.2λ 0.4λ 2 = 0.30, λ λ λ 2 = 0.6, λ λ + 0.2λ 2 = 0.24, and the solution is: λ 0 = 0., λ = 0.2 and λ 2 = 0.. (d) According to the APT, the expected return on stock E should be µ E = (0.5)(0.20) + ( 0.3)(0.0) = 0.7. If the expected return is 0.2, less than the APT value, an arbitrage opportunity exists. 6

7 We can create a portfolio E such that its betas are identical to those of stock E:.2x A + 0.8x B + 0.6x C + 0.2x D = 0.5, 0.4x A x B + 0.2x C 0.2x D = 0.3, x A + x B + x C + x D =. There are more than one solution. One solution is ( 0, 2 7, 9 28, ). 28 We short-sell stock E and buy portfolio E by the same dollar amount, say, $,000. This strategy will have an expected profit of 0.7, ,000 = $50, but it has no systematic risk since the betas of E and E offset each other. 6. (a) (i) To implement this trading strategy, one needs accounting data, which are publicly available. Because abnormal returns can be earned, this statement violates the semi-strong (and thus strong) form of market efficiency. (ii) No violations here. The returns generated were not adjusted for risk. The higher return could be explained by the fact that high-tech firms are more risky than pharmaceuticals. (iii) Because the trading strategy uses only price and volume data, this statement violates the weak (and thus also the semi-strong and strong) form of market efficiency. (iv) No violations here. The stock price just reacted to positive news. (v) Public information alone does not seem to be yielding positive abnormal returns but trading on private information does. This is a violation of the strong form of market efficiency. (b) (i) The price is obtained by using the Gordon growth formula: P A = D r g = 8 0% 3% = $4.29 (ii) The expected dividend next year is D = = 6. Thus, the new apple stock price after the announcement is P A = D r g = 6 0% 5% = $

8 The post-announcement return is: R = P A P A = = 5% (iii) It is a violation of the semi-strong form of market efficiency because prices did not adjust to all publicly available information in a timely fashion. Under the assumption that analysts are able to make correct forecasts and no other news were released that day, the price should have increased to P A at 0am rather than 2pm. 8