sample midterm 2 solutions

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FINC-UB.0002 F OUNDATIONS OF F INANCE Fall 2022 Sample Midterm 2 Solutions Multiple choice questions 1. What is the IRR on a project that requires an initial investment of $100 million (at t = 0) and will generate a single expected cash flow of $161.05 million in 5 years (at t = 5)? a. 10.0% b. 12.2% c. 61.0% d. 110.0% % 0 . 10 1 100 161 5 / 1          IRR 2. For a loan with a given (fixed) APR, what happens to the EAR as the frequency of the payments increases (e.g., from quarterly to monthly)? a. The EAR increases. b. The EAR does not change. c. The EAR decreases. d. Can’t tell from the information given. As the compounding frequency increases, the EAR increases due to the effects of compounding. For problems 3-6 assume that the expected return on the market portfolio is 10%, the standard deviation of the return on the market portfolio is 20%, and the risk-free rate is 5% (all on an annual basis). 3. Assuming the CAPM holds, what is the expected return on an asset with a beta of 1.5? a. 12.5% b. 15.0% c. 20.0% d. None of the above. Using the SML: % 5 . 12 %) 5 % 10 ( 5 . 1 % 5 ] [     r E 4. Assuming the CAPM holds, what is the expected return on an efficient portfolio with a standard deviation of 50%? a. 12.5% b. 15.0% c. 17.5% d. None of the above. Using the CML: % 5 . 17 % 50 % 20 % 5 % 10 % 5 ] [           r E

2 5. Assuming the CAPM holds, which one of the portfolios with the following characteristics cannot exist in equilibrium? a. Expected return 7.5%, standard deviation 10% b. Expected return 7.5%, standard deviation 20% c. Expected return 15.0%, standard deviation 40% d. All of the above portfolios can exist in equilibrium. All 3 portfolios have Sharpe ratios less than or equal to that of the market. 6. What is the alpha, relative to the CAPM, of a portfolio with a beta of 0.5 and an expected return of 6%? a. 1.5% b. 1.0% c. -1.5% d. None of the above. Using the SML: % 5 . 1 % 5 . 7 % 6 % 5 . 7 %) 5 % 10 ( 5 . 0 % 5 ] [          r E 7. Consider a 10-year, coupon paying bond with a coupon rate of 4% and a yield-to-maturity of 5%. What is the annual HPR over the life of the bond if all the coupon payments are reinvested at a rate of 4.5%? a. The annual HPR is greater than the 5%. b. The annual HPR is greater than 4.5%, but less than 5%. c. The annual HPR is greater than 4%, but less than 4.5%. d. Can’t tell from the information given. Over the 10-year life of the bond, some money is invested at 5%, some at 4.5%. The annual HPR will be a weighted average of these investment rates. 8. Consider a stock with a beta of 1.1. What is the standard deviation of the return on this stock? a. Less than the standard deviation of the market portfolio. b. Equal to the standard deviation of the market portfolio. c. Greater than the standard deviation of the market portfolio. d. Can’t tell from the information given. Even if the stock has no idiosyncratic risk, its variance due to systematic risk will exceed that of the market because its beta is greater than 1. 9. You invest $1,000 in a portfolio for 2 years that has an arithmetic average annual return of 0%. What is the ending value of the portfolio? a. $1,000 regardless of the pattern of returns in the 2 years. b. Greater than $1,000 as long as the returns in the two years are not the same. c. Less than $1,000 as long as the returns in the two years are not the same. d. None of the above. Unless the returns are equal, the geometric average (and the holding period return) will be negative.

3 For problems 10-12 consider a world in which there are only two possible states (both with positive, but not necessarily equal, probabilities) over the next year — recession or expansion. The returns on 2 stocks in these two states are as follows Recession Expansion Stock 1 10% 20% Stock 2 5% 25% 10. What are the relative expected returns on the two stocks? a. The expected return on stock 1 is higher than on stock 2. b. The expected return on stock 1 is lower than on stock 2. c. The expected return on stock 1 is the same as on stock 2. d. Can’t tel l from the information given. If the probability of a recession is close to 1, stock 1 has a higher expected return, if it is close to 0, stock 2 has a higher expected return. 11. What are the relative standard deviations of the returns on the two stocks? a. The standard deviation of stock 1 is higher than of stock 2. b. The standard deviation of stock 1 is lower than of stock 2. c. The standard deviation of stock 1 is the same as of stock 2. d. Can’t tell from the information given. Regardless of the probabilities, stock 2 exhibits more variation. 12. What is the correlation between the returns on the two stocks? a. The correlation is negative. b. The correlation is positive but less than 1. c. The correlation is equal to 1. d. Can’t tell from the information given. The stocks are perfectly correlated because there are only 2 possible states of the world and they move in the same direction. In a 2 state world, all risky assets are either perfectly correlated or perfectly negatively correlated. (This is probably the most difficult question on the exam.) 13. The fact that stock prices have an abnormal (after adjusting for risk) upwards drift for firms that subsequently report unexpectedly good earnings is evidence: a. Against the weak form of market efficiency. b. Against the semi-strong form of market efficiency. c. Against the strong form of market efficiency. d. None of the above. This is the example I talked about in class. A natural interpretation of the evidence is that insiders are trading in advance of the earnings report.

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4 14. Consider two risk-averse investors in a CAPM world. The investor who is more risk averse a. Will hold a greater percentage of her wealth in the market portfolio. b. Will definitely hold a negative position in the market portfolio. c. Will definitely hold a positive position in the risk-free asset. d. None of the above. The more risk averse investor could still be sufficiently risk tolerant to want to borrow. The optimal portfolio depends on the level on the specific utility function and the properties of the market portfolio. 15. Consider two portfolios, one with a weight of 68% in Google stock, the other with a weight of -34% in Google stock, with the residual in both cases invested in the risk-free asset. What is the correlation between the returns on these portfolios? a. 1.0 b. -1.0 c. -0.5 d. C an’t tell from the information given. These 2 portfolios are on the CAL (long position in Google) and the reflection of the CAL (short position in Google). 16. If 2 risky assets have the same expected return and the same standard deviation, i.e, they plot on top of each other in expected return/standard deviation space, what does the investment opportunity set look like? a. The IOS is a horizontal line as long as the assets are not perfectly correlated. b. The IOS is a single point regardless of the correlation. c. The IOS is a vertical line as long as the assets are not perfectly correlated. d. None of the above. As long as the correlation is not perfect, there is still a diversification benefit. This benefit takes the form of a lower standard deviation. However, expected returns in all portfolios of these 2 assets are the same. Thus, the IOS is a horizontal line.

5 Numerical problems 17. If you buy a 7-year, risk-free, zero coupon bond with a yield-to-maturity of 4%, then sell it 3 years later, when the yield-to-maturity on 4-year, risk-free, zero coupon bonds is 5%, what is your annual HPR? (Assume annual compounding.) The annual HPR doesn’t depend on the face amount of the bond, it is return per dollar invested (see below). However, for simplicity, assume a face amount of $100. The purchase price is 99 . 75 %) 4 1 ( 100 7 0    V The sale price is 27 . 82 %) 5 1 ( 100 4 3    V Therefore the annual HPR is % 68 . 2 1 99 . 75 27 . 82 1 3 / 1 3 / 1 0 3                    V V HPR Or equivalently (to see that the face amount cancels in this calculation) % 68 . 2 1 7599 . 0 8227 . 0 1 %) 4 1 /( %) 5 1 /( 3 / 1 3 / 1 7 4                      F F HPR 18. Consider the following 2 stocks and the market portfolio (all quantities are annual): Expected Return Standard Deviation Beta Google 12% 60% 2.0 Yahoo 10% 50% 1.5 Market portfolio 8% 20% 1.0 The correlation between the returns on Google and Yahoo is 0.5, and the annual risk-free rate is 4%. What fraction of the variance of a portfolio that is invested 50% in Google, 50% in Yahoo is systematic? (Assume the CAPM holds.) First, find the beta of the portfolio. It may (or may not) be obvious that the beta of a portfolio is just the weighted average of the betas of the securities in the portfolio 75 . 1 ) 5 . 1 ( 5 . 0 ) 2 ( 5 . 0      Y Y G G P w w    Alternatively, calculate the expected return on the portfolio % 11 %) 10 ( 5 . 0 %) 12 ( 5 . 0 ] [ ] [ ] [      Y Y G G P r E w r E w r E Then use the SML to find the beta 75 . 1 % 4 % 8 % 4 % 11 ] [ ] [        f M f P P r r E r r E  Second, find the variance of the portfolio 2275 . 0 %) 50 %)( 60 )( 5 . 0 )( 5 . 0 )( 5 . 0 ( 2 %) 50 ( 5 . 0 %) 60 ( 5 . 0 2 2 2 2 2 2 1 12 2 1 2 2 2 2 2 1 2 1 2              w w w w P

6 Finally, calculate the fraction that is systematic % 85 . 53 2275 . 0 %) 20 ( 75 . 1 2 2 2 2 2   P M    19. The current price of Walmart stock is $40. You estimate that the return on the stock has a standard deviation of 50% and a correlation of 0.2 with the market portfolio. The market portfolio has an expected return of 10% and a standard deviation of 20%, and the risk-free rate is 5%. If you expect Walmart to pay a dividend of $2 in one year and that the price after this dividend payment will be $44, what is your expected alpha on the stock (relative to the CAPM) over the next year? First, find the beta of Walmart 5 . 0 % 20 % 50 2 . 0    M i iM     Second, find the expected return under the CAPM % 5 . 7 %) 5 % 10 ( 5 . 0 % 5 ) ] [ ( ] [        f M f r r E r r E  Third, find the expected return under the price and dividend assumptions % 15 1 40 2 44 ] [ *     r E Finally, the alpha is just the difference between these expectations % 5 . 7 % 5 . 7 % 15 ] [ ] [ *      r E r E 

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