BA 504 - Chapter 11 HW

Top hedge fund manager Sally Buffit believes that a stock with the same market risk as the S&P 500 will sell at year-end at a price of $50. The stock will pay a dividend at year-end of $2. Assume that risk-free Treasury securities currently offer an interest rate of 2%. Average rates of return on Treasury bills, government bonds, and common stocks, 1900-2015 (figures in percent per year) are as follows. Portfolio: Ave Annual RoR | Ave Premium (Extra return vs Treasury bills) Treasury bills: 3.8% Treasury bonds: 5.3% | 1.5% Common stocks 11.4% | 7.6% a) What is the discount rate on the stock? b) What price should she be willing to pay for the stock today?

Based on the historical risk premium of the S&P 500 (7.6%) and a risk-free rate of 2.0%, one would predict an expected rate of return of 9.60%. If the stock has the same systematic risk, it also should provide this expected return. Therefore, the stock price equals the present value of cash flows for a 1-year horizon. P0 =[$2 + $50]/1.096 = $47.45

The common stock of Leaning Tower of Pita Inc., a restaurant chain, will generate payoffs to investors next year, which depend on the state of the economy, as follows: Dividend | Stock Price Boom: $8 | $240 Normal economy: $4 | $90 Recession: $0 | $0 The company goes out of business if a recession hits. Assume for simplicity that the three possible states of the economy are equally likely. The stock is selling today for $80. a-1. Calculate the rate of return to Leaning Tower of Pita shareholders for each economic state. a-2. Calculate the expected rate of return and standard deviation of return to Leaning Tower of Pita shareholders.

Boom:($8 + ($240 − 80))/$80=210% Normal: ($4 + ($90 − 80))/$80 =17.5% Recession:($0 + ($0 − 80))/$80=−100% r =[210 + 17.5 + (−100)]/3 = 42.5% Variance = [1/3 × (2.10 − 0.425)^2 + [1/3× (0.175 − 0.425)^2] + [1/3× (−1.00 − 0.425)^2] = 1.6327 Standard deviation = √1.6327 = 1.2778, or 127.78%

The common stock of Escapist Films sells for $25 a share and offers the following payoffs next year: Dividend | Stock Price Boom: $0 | $18 Normal economy: $1 | $26 Recession: $3 | $34 The common stock of Leaning Tower of Pita Inc. is selling for $80 and offers these payoffs next year: Dividend | Stock Price Boom: $8 | $240 Normal economy: $4 | $90 Recession: $0 | $0 a-1. Calculate the rate of return of Escapist Films for each economic state. a-2. Calculate the expected return and standard deviation of Escapist if all three economic scenarios are equally likely to occur. b-1. Calculate the rate of return of a portfolio half invested in Escapist and half in Leaning Tower of Pita for each economic state. b-2. Calculate the expected rate of return and standard deviation of a portfolio half invested in Escapist and half in Leaning Tower of Pita. All three economic scenarios are equally likely to occur.

Escapist Films: a-1. Boom: [$0 + ($18 − $25)]/ $80 =−28% Normal: [$1 + ($26 − $25)]/$80 = 8% Recession: [$3 + ($34 − $25)]/$25=48% a-2. Expected return: =[(−28) + 8 + 48]/3=9.33 Variance =[1/3× (−0.28 − 0.0933)^2] + [1/3 × (0.08 − 0.0933)^2] + [1/3× (0.48 − 0.0933)^2] = 0.096356 Standard deviation = √0.096356 = 31.04% Portfolio rate of return: b-1.Boom: (-28 + 210) / 2 = 91% Normal: = (8 + 18) / 2 = 13% Recession: = (48 - 100) / 2 = -26% b-2. Expected return = 26% Variance = [1/3 × (0.91 - 0.26)^2] + [1/3 × (0.13 - 0.26)^2] + [1/3 × (-0.26 - 0.26)^2] = 0.236585 Standard deviation = √0.236585 = 0.4864, or 48.64%

A stock will provide a rate of return of either −18% or 26%. If both possibilities are equally likely, calculate the stock's expected return and standard deviation.

Expected return = 0.5 × (-18) + 0.5 × 26 = 4% Variance = 0.5 × (-18 - 4)2 + 0.5 × (26 - 4)2 = 484. Standard deviation = √484 = 22%

State whether the following statements are true or false? a) Investors prefer diversified portfolios because they are less risky b) If stocks were perfectly correlated, diversification would not reduce risk. c) Diversification with an indefinitely large number of securities completely eliminates risk d) Diversification works only when returns are uncorrelated e) The risk of a diversified portfolio depends on the specific risk of the individual stocks. f) The risk that you can't avoid no matter how much you diversify is known as market risk. g) For a well-diversified portfolio, only market risk matters.

a) True b) True c) False d) False e) False f) True g) True

You purchase 100 shares of stock for $40 a share. The stock pays a $2 per share dividend at year-end. a. What is the rate of return on your investment if the end-of-year stock price is (i) $38; (ii) $40; (iii) $42? b. What is your real (inflation-adjusted) rate of return if the inflation rate is 4%?

a. (i) Rate of return=Capital gain + Dividend=($38 − 40) + $2=0%Initial share price$40 (ii) Rate of return=($40 − 40) + $2=0.05, or 5%$40 (iii) Rate of return=($42 − 40) + $2=0.10, or 10%$40 b. (i) Rate of return= 1 + Nominal rate of return−1 =1 + 0=−1 = −0.0385, or −3.85% 1 + Inflation rate1 + 0.04 (ii) Rate of return= 1 + Nominal rate of return−1 =1.05=−1 = 0.0096, or 0.96% 1 + Inflation rate1.04 (iii) Rate of return= 1 + Nominal rate of return−1 =1.10=−1 = 0.0577, or 5.77% 1 + Inflation rate1.04

Consider the following scenario analysis: Scenario: Recession | Normal | Boom Probability: 0.20 | 0.60 | 0.20 ROR/Stocks: -5% | 15% | 25% ROR/Bonds: 14% | 8% | 4% Assume a portfolio with weights of .60 in stocks and .40 in bonds. a. What is the rate of return on the portfolio in each scenario? b. What are the expected rate of return and standard deviation of the portfolio?

a. Recession: (-0.05 × 0.6) + (0.14 × 0.4) = 0.026 or 2.6% Normal economy: (0.15 × 0.6) + (0.08 × 0.4) = 0.122 or 12.2% Boom: (0.25 × 0.6) + (0.04 × 0.4) = 0.166 or 16.6% b. Expected return = (0.2 × 0.026) + (0.6 × 0.122) + (0.2 × 0.166) = 0.1116 or 11.16% Variance = [0.2 × (0.026 - 0.1116)^2] + [0.6 × (0.122 - 0.1116)^2] + [0.2 × (0.166 - 0.1116)^2] = 0.002125 Standard deviation = √0.002125 = 4.61%

A stock is selling today for $40 per share. At the end of the year, it pays a dividend of $2 per share and sells for $44. a. What is the total rate of return on the stock? b. What are the dividend yield and percentage capital gain? c. Now suppose the year-end stock price after the dividend is paid is $36. What are the dividend yield and percentage capital gain in this case? d. Is there any change in the dividend yield calculated in parts (b) and (c)?

a. Total percentage return=(Capital gain + Dividend) / Initial share price =[($44 - 40) + $2] / $40 =0.15, or 15% b. Dividend yield=Dividend / Initial share price =$2 / $40 =0.05, or 5% Capital gains yield=Capital gain / Initial share price =($44 - 40) / $40 =0.10, or 10% c. Dividend yield=Dividend / Initial share price =$2 / $40 =0.05, or 15% Dividend yield=Capital gain / Initial share price =($36 - 40) / $40 =-0.10, or -10% d. The dividend yield is unaffected by the ending share price because the yield is based on the initial price.