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

Joe Izu takes out a car loan of $50,000 today 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 Monthly rate is 0.18/12 = = 1.5% PV of $1,000 payments (1,000/0.015)(1 – 1/ ) = $43, PV of other 2 payments $50,000 - $43, = $6, Let Y be the payment 84 months from now 6, = (Y/2)/ Y/ Solve for Y to get $7,976.83

Solve each of the following (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? Take the seventh root of (1.1)(0.4)(1.45)(1.14)(1.09)(1.02)(1.3) to get The geometric average is – 1, or

Solve each of the following (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 (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? Two ways to find the rate every 6 months Since the effective rate is 13% annually, we can just take the square root of 1.13 – 1, or % Find the stated rate, which is ln(1.13), or %; then take exp(0.5* ) – 1, or % PV = 1000/( ) /( ) /( ) /( ) 4 = $3, OR PV = 1000/(1.13) 1/ / /(1.13) 3/ /(1.13) 2 = $3,441.33

Solve each of the following (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%? Effective rate every 3 years is (1.16) 3 – 1 = % PV = (5,000/ ) * (1.16) 2 = $11,995.09

Solve each of the following (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 (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? Find the arithmetic average of each stock: 22% for Stock X and 8.67% for Stock Y Find the covariance of X and Y (1/3) [( )( ) + ( )( ) + ( )( ), or – σ X 2 = (1/3)[( ) 2 + ( ) 2 + ( ) 2 ]  σ X = σ Y 2 = (1/3)[( ) 2 + ( ) 2 + ( ) 2 ]  σ Y 2 = Corr(X,Y) = –0.0034/σ X σ Y = –

Solve each of the following (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? Two possibilities: (up, down) or (down, up) P(up, down) =.4 *.6 =.24 P(down, up) =.6 *.4 =.24 Total probability is , or.48

Solve each of the following (f) The ZipDoodle 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 (f) The ZipDoodle 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.) Total cost = /1.21 3/ / /1.21 9/2 = $6, EAC  = (C/.21)[1 – 1/ ]  C = $2,070.48

Leo’s Batons, Inc. 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?

Leo’s Batons, Inc. 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 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 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 X Q = X K = 0.5 σ QK = Corr(Q,K) * s.d.(Q) * s.d.(K) = –1 * 0.1 * 0.1 = –0.01 Variance of a portfolio with $150 invested in each stock is X Q 2 σ Q 2 + 2X Q X K σ QK + X K 2 σ K 2 =.5 2 * *.5 *.5 * (–.01) *.1 2 = 0 Since the variance of the portfolio is 0, then investing in one share of each stock guarantees a return of 5% 5% of $300 is $15 If Stock K has a return of $5, then the return of Stock Q must be $10

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