FINAN 431 Ch.5 Homework
Suppose your expectations regarding the stock market are as follows: State of the Economy Probability HPR Boom 0.3 44% Normal growth 0.4 14 Recession 0.3 -16 Use Equations 5.6-5.8 to compute the mean and standard deviation of the HPR on stocks.
0.2324 or 23.24%
What do you think would happen to the expected return on stocks if investors perceived an increase in the volatility of stocks?
Assuming no change in tastes, that is, an unchanged risk aversion, investors perceiving higher risk will demand a higher risk premium to hold the same portfolio they held before. If we assume that the risk-free rate is unaffected, the increase in the risk premium would require a higher expected rate of return in the equity market.
XYZ stock price and dividend history are as follows: Year Beginning-of-Year Price Dividend Paid at YearEnd 2010 $100 $4 2011 $110 $4 2012 $ 90 $4 2013 $ 95 $4 An investor buys three shares of XYZ at the beginning of 2010, buys another two shares at the beginning of 2011, sells one share at the beginning of 2012, and sells all four remaining shares at the beginning of 2013. (LO 5-1) a. What are the arithmetic and geometric average time-weighted rates of return for the investor? b. What is the dollar-weighted rate of return? ( Hint: Carefully prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 2010, to January 1, 2013. If your calculator cannot calculate internal rate of return, you will have to use a spreadsheet or trial and error.)
a. 0.0233 or 2.33% b. -0.1661%
The stock of Business Adventures sells for $40 a share. Its likely dividend payout and end-of-year price depend on the state of the economy by the end of the year as follows: Dividend Stock Price Boom $2.00 $50 Normal economy 1.00 43 Recession .50 34 a. Calculate the expected holding-period return and standard deviation of the holding- period return. All three scenarios are equally likely. b. Calculate the expected return and standard deviation of a portfolio invested half in Business Adventures and half in Treasury bills. The return on bills is 4%.
a. 0.1788 or 17.88% b. 8.94%
Suppose the same client in the previous problem decides to invest in your risky portfolio a proportion ( y) of his total investment budget so that his overall portfolio will have an expected rate of return of 15%. a. What is the proportion y? b. What are your client's investment proportions in your three stocks and the T-bill fund? c. What is the standard deviation of the rate of return on your client's portfolio?
a. 0.8 Therefore, in order to achieve an expected rate of return of 15%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills. b. The investment proportions of the client's overall portfolio can be calculated by the proportion of risky asset in the whole portfolio times the proportion allocated in each stock. c. Security Investment Proportions T-Bills 20.0% Stock A 0.8 27% = 21.6% Stock B 0.8 33% = 26.4% Stock C 0.8 40% = 32.0% d. The standard deviation of the complete portfolio is the standard deviation of the risky portfolio times the fraction of the portfolio invested in the risky asset: C = y P = 0.8 0.27 = 0.216 or 21.6% per year
Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. (LO 5-3) a. What is the expected return and standard deviation of your client's portfolio? b. Suppose your risky portfolio includes the following investments in the given proportions: Stock A 27% Stock B 33% Stock C 40% What are the investment proportions of your client's overall portfolio, including the position in T-bills? c. What is the reward-to-volatility ratio ( S) of your risky portfolio and your client's overall portfolio? d. Draw the CAL of your portfolio on an expected return/standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund's CAL.
a. Allocating 70% of the capital in the risky portfolio P, and 30% in risk-free asset, the client has an expected return on the complete portfolio calculated by adding up the expected return of the risky proportion (y) and the expected return of the proportion (1 - y) of the risk-free investment: E(rC) = y E(rP) + (1 - y) rf = (0.7 0.17) + (0.3 0.07) = 0.14 or 14% per year The standard deviation of the portfolio equals the standard deviation of the risky fund times the fraction of the complete portfolio invested in the risky fund: C = y P = 0.7 0.27 = 0.189 or 18.9% per year b. The investment proportions of the client's overall portfolio can be calculated by the proportion of risky portfolio in the complete portfolio times the proportion allocated in each stock. Security Investment Proportions T-Bills 30.0% Stock A 0.7 27% = 18.9% Stock B 0.7 33% = 23.1% Stock C 0.7 40% = 28.0% c. We calculate the reward-to-variability ratio (Sharpe ratio) using Equation 5.14. For the risky portfolio: S = = = "" /" For the client's overall portfolio: S = = = 0.3704
You estimate that a passive portfolio invested to mimic the S&P 500 stock index yields an expected rate of return of 13% with a standard deviation of 25%. Draw the CML and your fund's CAL on an expected return/standard deviation diagram. (LO 5-4) a. What is the slope of the CML? b. Characterize in one short paragraph the advantage of your fund over the passive fund.
a. Slope of the CML = = = 0.24 b. Your fund allows an investor to achieve a higher expected rate of return for any given standard deviation than would a passive strategy, i.e., a higher expected return for any given level of risk.
Suppose the same client as in the previous problem prefers to invest in your portfolio a proportion ( y) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio's standard deviation will not exceed 20%. (LO 5-3) a. What is the investment proportion, y? b. What is the expected rate of return on the overall portfolio?
a. Standard deviation of the complete portfolio= C = y 0.27 If the client wants the standard deviation to be equal or less than 20%, then: y = (0.20/0.27) = 0.7407 = 74.07% He should invest, at most, 74.07% in the risky fund. b. E(rC) = rf + y [E(rP) - rf] = 0.07 + 0.7407 0.10 = 0.1441 or 14.41%
Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $50,000 or $150,000, with equal probabilities of .5. The alternative riskless investment in T-bills pays 5%. a. If you require a risk premium of 10%, how much will you be willing to pay for the portfolio? b. Suppose the portfolio can be purchased for the amount you found in ( a). What will the expected rate of return on the portfolio be? c. Now suppose you require a risk premium of 15%. What is the price you will be willing to pay now? d. Comparing your answers to ( a) and ( c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell?
a. The expected cash flow is: (0.5 $50,000) + (0.5 $150,000) = $100,000 With a risk premium of 10%, the required rate of return is 15%. Therefore, if the value of the portfolio is X, then, in order to earn a 15% expected return: Solving X (1 + 0.15) = $100,000, we get X = $86,957 b. If the portfolio is purchased at $86,957, and the expected payoff is $100,000, then the expected rate of return, E(r), is: = 0.15 = 15% The portfolio price is set to equate the expected return with the required rate of return. c. If the risk premium over T-bills is now 15%, then the required return is: 5% + 15% = 20% The value of the portfolio (X) must satisfy:X (1 + 0.20) = $100, 000 X = $83,333 d. For a given expected cash flow, portfolios that command greater risk premiums must sell at lower prices. The extra discount in the purchase price from the expected value is to compensate the investor for bearing additional risk.
Your client (see previous problem) wonders whether to switch the 70% that is invested in your fund to the passive portfolio. a. Explain to your client the disadvantage of the switch. b. Show your client the maximum fee you could charge (as a percent of the investment in your fund deducted at the end of the year) that would still leave him at least as well off investing in your fund as in the passive one. ( Hint: The fee will lower the slope of your client's CAL by reducing the expected return net of the fee.)
a. With 70% of his money in your fund's portfolio, the client has an expected rate of return of 14% per year and a standard deviation of 18.9% per year. If he shifts that money to the passive portfolio (which has an expected rate of return of 13% and standard deviation of 25%), his overall expected return and standard deviation would become: E(rC) = rf + 0.7 E(rM) - rf] In this case, rf = 7% and E(rM) = 13%. Therefore: E(rC) = 0.07 + (0.7 0.06) = 0.112 or 11.2% The standard deviation of the complete portfolio using the passive portfolio would be: C = 0.7 M = 0.7 0.25 = 0.175 or 17.5% Therefore, the shift entails a decline in the mean from 14% to 11.2% and a decline in the standard deviation from 18.9% to 17.5%. Since both mean return and standard deviation fall, it is not yet clear whether the move is beneficial. The disadvantage of the shift is apparent from the fact that, if your client is willing to accept an expected return on his total portfolio of 11.2%, he can achieve that return with a lower standard deviation using your fund portfolio rather than the passive portfolio. To achieve a target mean of 11.2%, we first write the mean of the complete portfolio as a function of the proportions invested in your fund portfolio, y: E(rC) = 7% + y (17% - 7%) = 7% + 10% y Because our target is E(rC) = 11.2%, the proportion that must be invested in your fund is determined as follows: 11.2% = 7% + 10% y y = = 0.42 The standard deviation of the portfolio would be: C = y 27% = 0.42 27% = 11.34% Thus, by using your portfolio, the same 11.2% expected rate of return can be achieved with a standard deviation of only 11.34% as opposed to the standard deviation of 17.5% using the passive portfolio. b. The fee would reduce the reward-to-variability ratio, i.e., the slope of the CAL. Clients will be indifferent between your fund and the passive portfolio if the slope of the after-fee CAL and the CML are equal. Let f denote the fee: Slope of CAL with fee = = Slope of CML (which requires no fee) = = 0.24 Setting these slopes equal and solving for f: = 0.24 10% - f = 27% 0.24 = 6.48% f = 10% 6.48% = 3.52% per year
