Midterm 2 Spring 2017

Problem 1 In a sample of three stocks, the annual rate of returns last year were 7%, 18%, and 35%. The standard deviation of this sample is__? Answer: 𝐴𝑣𝑒𝑟𝑎𝑔𝑒= .07+.18+.35 3 =.2 𝑉𝑎𝑟= 1 3−1 .07−.2 2 + .18−.2 2 + .35−.2 2 𝑉𝑎𝑟=.0199 𝑺𝒕𝒅𝑫𝒆𝒗= .𝟎𝟏𝟗𝟗 𝟏 𝟐 =.𝟏𝟒𝟏𝟎𝟔𝟕

Problem 2 Aeber purchased a zero-coupon bond earlier today for $500. The bond has a yield to maturity of 10% and matures in two years. If the bond now has a yield to maturity of 8%, what is the current bond value? Assume all yields are effective annual rates. Answer: 𝑃𝑟𝑖𝑐𝑒=500= 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 1.1 2 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒=500 1,1 2 =605 If YTM changes to 8% - 𝑷𝒓𝒊𝒄𝒆= 𝟔𝟎𝟓 𝟏.𝟎𝟖 𝟐 =𝟓𝟏𝟖.𝟔𝟗

Problem 3 Assume that the risk free return in the market is currently 10%, and a stock with beta of 5 has an expected return of 35%. What is the expected return on the market portfolio? Answer: CAPM: 𝑟 𝑟 = 𝑟 𝑓 +𝛽 𝑟 𝑚 − 𝑟 𝑓 .35=.1+5 𝑟 𝑚 −.1 𝒓 𝒎 = .𝟑𝟓−.𝟏 𝟓 +.𝟏=.𝟎𝟓+.𝟏=.𝟏𝟓

Using the information from the last problem: Assume that the risk free return in the market is currently 10%, and a stock with beta of 5 has an expected return of 35%. What is the risk premium? Answer: Using the information from the last problem: 𝑟 𝑚 =.15 Risk Premium = 𝑟 𝑚 − 𝑟 𝑓 .𝟏𝟓−.𝟏=.𝟎𝟓

𝑠.𝑒.= 𝑠𝑑 (𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒)^(1/2) = .22 121 1 2 =.02 Problem 5 From Jan 1, 1890, to Jan 1, 2011, the historical average annual rate of return in the hypothetical county of Wachatame was 10%. The annual standard deviation of the rate of return was 22%. What is the upper bound of the 95.4% confidence interval for the annual rate of return based on this information? Hint: you need to be within two standard error of the average to find the upper and lower bounds of the 95.4% confidence interval. Answer: 𝑢𝑝𝑝𝑒𝑟 𝑏𝑜𝑢𝑛𝑑=10%+2∗𝑠.𝑒. 𝑠.𝑒.= 𝑠𝑑 (𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒)^(1/2) = .22 121 1 2 =.02 𝒖𝒑𝒑𝒆𝒓 𝒃𝒐𝒖𝒏𝒅=𝟏𝟎%+𝟐∗.𝟎𝟐=.𝟏𝟒

Problem 6 A stock will pay its next dividend of $2 per share later today. The dividend over each of the next 5 years will increase by 20% per year. After that, the dividend will not change. Assume that dividends will be paid forever. What is the present value? EAR is 25% Answer: 𝑃𝑉=2 1 .25−.2 − 1 .25−.2 1.2 1.25 6 ∗ 1.25 + 2 1.2 5 .25 ∗ 1 1.25 5 =17.3851 Or 𝑃𝑉=2+ 2 1.2 1.25 + 2 1.2 2 1.25 2 + 2 1.2 3 1.25 3 + 2 1.2 4 1.25 4 + 2 1.2 5 .25 ∗ 1 1.25 4 =17.3851

If $100 is invested in each stock, Problem 7 Suppose the Stock F and Stock G have a correlation value of rho=01. Stock F has an expected return of 20% and a standard deviation of 30%. Stock G has an expected return of 20% and a standard deviation of 30%. Today each stock is valued at $100 per share. Over the next year, Stock F will go up by $10. How much will Stock G go up by next year? Answer: If $100 is invested in each stock, 𝑥 𝐹 = 𝑥 𝐺 =.5 𝜎 𝐹,𝐺 = 𝜌 𝐹,𝐺 𝜎 𝐹 𝜎 𝐺 =−1 .3 .3 =−.09 𝜎 𝑃𝑜𝑟𝑡 2 = .5 2 ∗. 3 2 + .5 2 ∗. 3 2 +2∗.5∗.5∗ −.09 =0 Since variance is zero, rate of return on portfolio will always be 200*.2=40 𝑟 𝐹 + 𝑟 𝑔 =40 𝟏𝟎+ 𝒓 𝑮 =𝟒𝟎⇒ ⇒ 𝒓 𝑮 =𝟑𝟎