Econ 134A Test 1Fall 2012

Solution sketches

Solve each of the following (a) (5 points) Yongli will receive $750 later

today. He will receive $825, or 10% more, one year from today. For each of the three years after that, he will receive $75 more than the year before. After these five payments have been made, he will receive nothing more. Find the present value of these five payments if the effective annual discount rate is 4%. PV = 750 + 825/1.04 + 900/1.042 + 975/1.043 +

1050/1.044 = $4,139.69

Solve each of the following (b) (6 points) Yusuf is quoted a price for a

bond of $500. This bond has a face value of $475 and pays a 10% coupon once per year. Two coupons will be paid, one later today and the other one year from today. If the bond matures one year from today, what is the yield on this bond (expressed as an effective annual rate)? 475 / (1+r) + 47.5 + 47.5 / (1+r) = 500 522.5 / (1+r) = 452.5 1 + r = 1.154696 r = .154696 or 15.4696%

Pepsi and $1 billion Years ago, Pepsi offered a chance to win $1 billion to many

contestants. Assume that there are 40 yearly payments as follows:

$5 million for each of the first 20 payments $10 million for each of the next 19 payments $710 million for the final payment

For this problem, assume that the first payment is made today and there is an effective annual discount rate of 7%. Make the calculations below assuming that the prize is won.

HINTS: Pay careful attention to when each payment is made. A single mistake could lead to deductions on multiple parts of this problem. Also note that each payment is made one year apart.

(a) (4 points) What is the present value of the first 20 payments?

Note that we use the annuity formula for 19 years and add in one more payment today

PV = 5 + (5/0.07)[1 – (1/1.07)19] = 5 + 51.6780 = 56.6780

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Pepsi and $1 billion

(b) (6 points) What is the present value of the next 19 payments? The 1st $10 million payment will be

made 20 years from today So we need to discount the annuity

formula by 19 years PV = [1/1.0719] * (10/0.07)[1 –

1/1.0719] PV = 28.5788

Pepsi and $1 billion

(c) (2 points) What is the present value of the final payment? This payment is made 39 years from

now PV = 710 / 1.0739

PV = 50.7331

Pepsi and $1 billion (d) (4 points) Pepsi offered a single payment of

$250 million today as an alternative to the payments mentioned at the beginning of this problem. Given the information from this problem, should someone who wins this prize accept the single payment? Justify your answer with math and/or a written justification of 40 words or less.

PV of all payments is 56.6780 + 28.5788 + 50.7331, or 135.990

This PV is less than the single payment of 250 made today

Choose the single payment, because it has a higher PV

C&L Shopping Trolleys (9 points) C&L Shopping Trolleys, Inc. has just paid out its

annual dividend of $5.70 earlier today. The annual dividend will go up by 3% each of the next 5 years. This will be followed by 5% growth in the annual dividend every year after that forever. What will the price of this stock be 4 years from today if the effective annual discount rate is 8%? (Note: Provide the price AFTER the dividend has been paid.)

We need to find the future value 4 years from now Only dividends paid 5 years or more in the future are included in

this future value Dividend in year 5: $5.70 * 1.035 = 6.60786 We can use the growing perpetuity formula for payments made 5

or more years into the future The future value (year 4) of the stock is then 6.60786/(0.08-0.05)

$220.26

Slacking Sean Slacking Sean likes to procrastinate on paying back loans.

He is currently negotiating with Patient Paula on a loan. Today, Paula is giving Sean $600 for a loan. Assume the effective annual interest rate is 15%.

(a) (5 points) If Sean pays back a constant amount of the principal each year for five years to pay off the loan, and payments are made yearly, how much will the payment 4 years from today be? Assume that the first payment will be one year from today.

Principal must be reduced by $120 per year to pay off the loan $240 three years from today

Four years from today, $120 in principal must be paid, along with interest on $240

$120 + $240*0.15 = $156

Slacking Sean (b) (6 points) If Sean has to make five equal

payments to pay off the loan, how much would each one have to be if one payment will be made two years from today, one payment will be made three years from today, one payment will be made four years from today, and two payments will be made five years from today? Let X be the amount of each payment

X/1.152 + X/1.153 + X/1.154 + 2X/1.155 = 600 2.9798X = 600 X = $201.36

A bond with maturity date 4½ years from today… (7 points) A bond with maturity date 4½ years from

today has a face value of $1500. There are five coupon payments of $100 each to be made. These payments will be made annually starting six months from today. If the effective annual discount rate is 17%, what is the present value of future payments that will be paid by this bond?

Note that we discount the first payment by 6 months Each subsequent payment is discounted by an additional year

100/1.170.5 + 100/1.171.5 + 100/1.172.5 + 100/1.173.5 + 100/1.174.5 + 1500/1.174.5

$1,086.10