Econ 134A Fall 2012Test 2 solution sketches

Average: 41.68 pointsWhat 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 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 = 0.015 = 1.5% PV of $1,000 payments

(1,000/0.015)(1 – 1/1.01572) = $43,844.67 PV of other 2 payments

$50,000 - $43,844.67 = $6,155.33 Let Y be the payment 84 months from

now 6,155.33 = (Y/2)/1.0152 + Y/1.01584 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 1.00716208 The geometric average is 1.00716208 –

1, or 0.00716208

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 6.30146%

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

(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 = 56.0896% PV = (5,000/0.560896) * (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) [(.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-.22)2] σX = 0.11518

σY2 = (1/3)[(.03-.0867)2 + (.12-.0867)2 +

(.11-.0867)2] σY2 = 0.040277

Corr(X,Y) = –0.0034/σXσY = – 0.7329

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 .24 + .24, 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 = 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. 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 XQ = XK = 0.5 σQK = 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

XQ2σQ

2 + 2XQXK σQK + XK2σK

2 = .52 * .12 + 2 * .5 * .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%

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