Econ 134 A Fall 2012 Test 2 solution

Econ 134 A Fall 2012 Test 2 solution sketches Average: 41. 68 points What counts as 100%: 54. 55 points (2 students with 55 points; this also counts as 100%)

Joe Izu takes out a car loan of $50, 000 today n Joe Izu takes out a car loan of $50, 000 today. He makes 72 monthly payments of $1, 000 each, starting one month from today. He also makes 2 additional payments in order to fully pay off the loan. One of these payments will be 2 months from today and one will be 84 months from today. The payment 84 months from today will be twice the amount of the payment 2 months from today. How much will the final payment be if the stated annual discount rate is 18%, compounded monthly?

Joe Izu takes out a car loan of $50, 000 today n n Monthly rate is 0. 18/12 = 0. 015 = 1. 5% PV of $1, 000 payments n n PV of other 2 payments n n (1, 000/0. 015)(1 – 1/1. 01572) = $43, 844. 67 $50, 000 - $43, 844. 67 = $6, 155. 33 Let Y be the payment 84 months from now n n 6, 155. 33 = (Y/2)/1. 0152 + Y/1. 01584 Solve for Y to get $7, 976. 83

Solve each of the following n (a) An investment portfolio has annual returns of 10%, – 60%, 45%, 14%, 9%, 2%, and 30% over each of seven years. What is the geometric average return over this seven-year period? n n Take the seventh root of (1. 1)(0. 4)(1. 45)(1. 14)(1. 09)(1. 02)(1. 3) to get 1. 00716208 The geometric average is 1. 00716208 – 1, or 0. 00716208

Solve each of the following n (b) Stella will receive 4 payments. Each payment will be $1, 000 every six months, starting six months from today. The effective annual discount rate is 13%, and interest is compounded continuously. What is the total present value of the 4 payments?

Solve each of the following n n (b) Stella will receive 4 payments. Each payment will be $1, 000 every six months, starting six months from today. The n effective annual discount rate is 13%, and interest is compounded continuously. n What is the total present value of the 4 payments? Two ways to find the rate every 6 months n Since the effective rate is 13% annually, we can just take the square root of 1. 13 – 1, or 6. 30146% n Find the stated rate, which is ln(1. 13), or 12. 2218%; then take exp(0. 5*. 0122218) – 1, or 6. 30146% PV = 1000/(1. 0630146) + 1000/(1. 0630146)2 + 1000/(1. 0630146)3 + 1000/(1. 0630146)4 = $3, 441. 33 OR PV = 1000/(1. 13)1/2 + 1000/1. 13 + 1000/(1. 13)3/2 + 1000/(1. 13)2 = $3, 441. 33

Solve each of the following n (c) A perpetuity pays $5, 000 every three years, starting one year from today. What is the present value of this perpetuity if the effective annual discount rate is 16%? n n Effective rate every 3 years is (1. 16)3 – 1 = 56. 0896% PV = (5, 000/0. 560896) * (1. 16)2 = $11, 995. 09

Solve each of the following n (d) There are three states of the world, each with one-third probability of occurring: High, Medium, and Low. When times are High, Stock X has a rate of return of 35%, and stock Y has a rate of return of 3%. When times are Medium, Stock X has a rate of return of 24% and stock Y has a rate of return of 12%. When times are Low, Stock X has a rate of return of 7% and stock Y has a rate of return of 11%. What is the correlation of Stock X and Stock Y?

Solve each of the following n (d) There are three states of the world, each with one-third probability of occurring: High, Medium, and Low. When times are High, Stock X has a rate of return of 35%, and stock Y has a rate of return of 3%. When times are Medium, Stock X has a rate of return of 24% and stock Y has a rate of return of 12%. When times are Low, Stock X has a rate of return of 7% and stock Y has a rate of return of 11%. What is the correlation of Stock X and Stock Y? n n Find the arithmetic average of each stock: 22% for Stock X and 8. 67% for Stock Y Find the covariance of X and Y n n (1/3) [(. 35 -. 22)(. 03 -. 0867) + (. 24 -. 22)(. 12 -. 0867) + (. 07 -. 22)(. 11 -. 0867), or – 0. 0034 σX 2 = (1/3)[(. 35 -. 22)2 + (. 24 -. 22)2 + (. 07 2 -. 22) ] σX = 0. 11518 σY 2 = (1/3)[(. 03 -. 0867)2 + (. 12 -. 0867)2 + (. 11 -. 0867)2] σY 2 = 0. 040277 Corr(X, Y) = – 0. 0034/σXσY = – 0. 7329

Solve each of the following n (e) Suppose that the daily price for each share of Alominyo, Inc. , stock is a random walk with each day’s movement in price independent of the previous day’s. Every day, the stock can either go up with probability 60% or down by $1 with probability 40%. However, over the past five days, the stock has gone up by $1 every day. What is the probability that the stock will be the same price two days from today? n n Two possibilities: (up, down) or (down, up) P(up, down) =. 4 *. 6 =. 24 P(down, up) =. 6 *. 4 =. 24 Total probability is. 24 +. 24, or. 48

Solve each of the following n (f) The Zip. Doodle machine can be purchased today for $5, 500, and lasts 6 years. Maintenance costs of $700 have to be incurred three times. The first maintenance cost occurs 18 months from today, the second 3 years from today, and the third 54 months from today. If the effective annual discount rate is 21%, what is the equivalent annual cost of the machine? (Note: All costs are in real dollars. )

Solve each of the following n (f) The Zip. Doodle machine can be purchased today for $5, 500, and lasts 6 years. Maintenance costs of $700 have to be incurred three times. The first maintenance cost occurs 18 months from today, the second 3 years from today, and the third 54 months from today. If the effective annual discount rate is 21%, what is the equivalent annual cost of the machine? (Note: All costs are in real dollars. ) n n Total cost = 5500 + 700/1. 213/2 + 700/1. 213 + 700/1. 219/2 = $6, 717. 92 EAC 6717. 92 = (C/. 21)[1 – 1/1. 216] C = $2, 070. 48

Leo’s Batons, Inc. n Leo’s Batons, Inc. , has the following characteristics: The beta for the company is 1. 6; the annual dividend of $6 will be paid later today; the annual dividend will go up by 4% each year. You may also find the following information useful in solving this problem: Dividends for this stock will be paid forever; the rate of return for riskfree assets is 3%; the rate of return to the market is 8%. What is the present value of a share of Leo’s Batons stock?

Leo’s Batons, Inc. n n n Leo’s Batons, Inc. , has the following characteristics: The beta for the company is 1. 6; the annual dividend of $6 will be paid later today; the annual dividend will go up by 4% each year. You may also find the following information useful in solving this problem: Dividends for this stock will be paid forever; the rate of return for risk-free assets is 3%; the rate of return to the market is 8%. What is the present value of a share of Leo’s Batons stock? Return = risk-free rate + beta * market premium = 3% + 1. 6(8% – 3%) = 11% PV = 6 + 6(1. 04)/(0. 11 – 0. 04) = $95. 14

Stock Q and Stock K n Suppose that Stock Q and Stock K have a correlation value of ρ = – 1. Stock Q has an expected return of 5% and standard deviation 10%. Stock K also has an expected return of 5% and standard deviation 10%. Today, each stock is valued at $150 per share. Over the next year, Stock K will go up by $5. How much will Stock Q go up by next year? (Please completely justify your answer to get full credit. )

Stock Q and Stock K n n n Suppose that Stock Q and Stock K have a correlation value of ρ = – 1. Stock Q has an expected return of 5% and standard deviation 10%. Stock K also has an expected return of 5% and standard deviation 10%. Today, each stock is valued at $150 per share. Over the next year, Stock K will go up by $5. How much will Stock Q go up by next year? (Please completely justify your answer to get full credit. ) If someone invests $150 in each stock, then XQ = XK = 0. 5 σQK = Corr(Q, K) * s. d. (Q) * s. d. (K) = – 1 * 0. 1 = – 0. 01 Variance of a portfolio with $150 invested in each stock is 2 2 n XQ σQ + 2 XQXK σQK + XK σK =. 52 *. 12 + 2 *. 5 * (–. 01) +. 52 *. 12 = 0 Since the variance of the portfolio is 0, then investing in one share of each stock guarantees a return of 5% n 5% of $300 is $15 n If Stock K has a return of $5, then the return of Stock Q must be $10

Level of difficulty n Easy (34 points) n n n Joe Izu Geometric average A perpetuity pays $5 K every 3 years… Zip. Doodle Leo’s Batons n Easy-medium (5 points) n n Hard (8 points) n n n … 3 states of the world… Stella Alominyo, Inc. Very hard (8 points) n Stock Q/Stock K