Midterm 2 Fall 2017

Problem 1 An investment portfolio has annual returns of 6%, -20%, 30%, 14%, -40% and 19% over each of the last six years. What is the geometric average return over this six-year period? Answer: (1.06)(0.8)(1.3)(1.14)(0.6)(1.19) = 0.897310 6 0.897310 =0.982103 0.982103 – 1 = −0.17897

Problem 2 Zane purchased a zero-coupon bond earlier today for $500. The bond will mature in five years, and will pay out $1,200. What is the yield to maturity (as an effective annual rate) of this bond? Answer: 500 (1+𝑟) 5 =1200 (1+𝑟) 5 =2.4 1+𝑟=1.191358 𝑟=0.191358

Problem 3 Yani sampled a stock over a three-year time period. He found that the yearly returns in this sample were 8%, 15% and -5%. What is the standard deviation of this sample? Answer: Average = 1 3 .08+.15−.05 =.06 Variance = 1 3−1 (.08−.06) 2 + (.15−.06) 2 + (−.05−.06) 2 =.0103 Standard Deviation = .0103 =10.1489%

Problem 4 You know the following information about a bond: It has a face value of $1,600, the effective annual discount rate for the bond is 7%; there will be 12 equal annual coupon payments, starting one year from today; the bond matures 12 years from today; the present value of the bond is $1,400. How much will each coupon payment be? Answer: 1400= 𝐶 .07 1− 1 1.07 12 + 1600 1.07 12 1400=7.9427𝐶+710.42 689.58=7.9247𝐶 𝐶=$87.02

Problem 5 At 8:30 am today, company QWERTTY, Inc. planned to pay out an $8 dividend every year starting one year from today. At 1:30 pm, the company will revise its dividend to $5 per year, starting today. What is the change in this company’s planned dividend stream of payments? Assume an 8% effective annual discount rate. Answer: At 8:30: 𝑃𝑉= 8 .08 =$100 At 1:30: 𝑃𝑉=5+ 5 .08 =$67.50 𝐶ℎ𝑎𝑛𝑔𝑒=67.50−100=−$32.50

Problem 6 There are 400 stocks available to invest in the East Dakota stock market, and risk is measured in standard deviation. Gertrude is trying to figure out what the minimum standard deviation could be for a stock portfolio in this market. Systematic risk is currently at 18%. Unsystematic risk is assumed to be (50/Z)%, with Z denoting the number of stocks in Gertrude’s portfolio. What is the minimum standard deviation possible if Gertrude can only invest in the East Dakota stock market? Answer: We minimize the unsystematic risk by maximizing Z: 50 400 %=0.125% 𝑇𝑜𝑡𝑎𝑙 𝑟𝑖𝑠𝑘=𝑠𝑦𝑠𝑡𝑒𝑚𝑎𝑡𝑖𝑐+𝑢𝑛𝑠𝑦𝑠𝑡𝑒𝑚𝑎𝑡𝑖𝑐=18%+ .125%=18.125%

Problem 7 A stock will pay a $1 dividend later today. Over the next 3 years, the annual dividend (paid every 12 months) will go up by 5% each year. After that, the dividend will remain constant forever. What is the present value of this stock if the effective annual discount rate is 12%? Answer: 𝑃𝑉=1+ 1.05 1.12 + 1.05 2 1.12 2 + 1 1.12 2 × 1.05 3 .12 𝑃𝑉=$10.5068

Upper bound of 95.44% C.I. is given by: Problem 8 95.44% of the probability distribution is within 2 standard deviations of the mean of a normal distribution. Assume the historical equity risk premium is 13.5% and the standard deviation of the equity risk premium is 22.0%. 121 years of data were used to make these estimates. Find the UPPER BOUND of the 95.44% confidence interval of the historical equity risk premium. Answer: Upper bound of 95.44% C.I. is given by: 13.5%+2× 22.0% 121 =17.5%

Problem 9 Wilhelmina invested $3 in a company 40 years ago. This investment is worth $500 today. What is the arithmetic average annual rate of return of this investment? Answer: We need more information on the yearly returns in order to answer this question.

Problem 10 Walk Taller Shoes will pay its next dividend of $5 today. Every 12 months, the company will pay a dividend 4% higher than the previous dividend payment. The appropriate discount rate for this company’s stock is 10%. What will the value of the stock be in 2 years, immediately after that day’s dividend payment is made? Answer: In 2 years, the next dividend (a year from then) will be $5 (1.04) 3 , and dividends will grow at 4% each year. Using the growing perpetuity formula: 𝐹𝑉= $5 (1.04) 3 0.10−0.04 =$93.74

Problem 11 Annie’s Microphones, Inc. has the following characteristics: The beta for this company is 2.5; the next annual dividend of $8 will be paid today; the annual dividend will go up by 2% each year. You may also find the following information useful: Dividends for this stock will be paid forever; the rate of return for risk-free assets is 8%; the rate of return to the market is 14%. What is the present value of a share of Annie’s Microphones stock? Answer: First, we need to find the discount rate using the CAPM equation: 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚=14%−8%=6% 𝑟=8%+2.5 6% =23% 𝑃𝑉=$8+ $8(1.02) .23−.02 =$46.86

Problem 12 Stumpel Hotels, Inc. is currently a cash cow. Without any re-investment of their earnings, they will earn $23 per share every year forever. The effective annual discount rate for owning this stock is 13%. Assume that the next dividend payment will be made in 6 months. Subsequent dividend payments are made every 12 months after the next payment. Suppose that Stumpel Hotels could retain all of its earnings 5.5 years from today, and earn 15% on those earnings the following year. (In other words, no dividend would be paid 5.5 years from today if the company retains all of its earnings, and would continue to act as a cash cow in later years.)

Problem 12 (cont’d) (a) What is the present value of this stock if it continues to act as a cash cow? Answer: 𝑃𝑉= 23 .18 × 1.13 1 2 =$188.07 Remember that the first payment is in 6 months, so adjust the perpetuity formula accordingly.

NPVGO is greater than zero, so the company should retain its earnings. Problem 12 (cont’d) (b) Should the company retain its earnings 5.5 years from today? Why/why not? Answer: Yes, since the return on the retained earnings is higher than the discount rate. Alternatively, use part (c) to justify this. (c) How much does the stock’s present value change if the company retains its earnings 5.5 years from today? Answer: 𝑁𝑃𝑉𝐺𝑂= −$23 1.13 5.5 + $23(1.15) 1.13 6.5 =$0.207849 NPVGO is greater than zero, so the company should retain its earnings.