Financial Economics Spring 2018
Assignment 1 of 5: Allocating Resources over Time
This homework will be due in class on March 27, 2018. Late papers will not be accepted.
Answer all questions completely in A4 size paper.
1. As winner of a breakfast cereal competition, you can choose one of the following prizes:
a. $100,000 now
b. $180,000 at the end of 5 years
c. $11,400 a year forever, starting next year
d. $16,000 for each of 10 years, with the first payment to be received today
e. $6,500 next year, and increasing thereafter by 5% a year forever
If the interest rate is 12%, which is the most valuable prize?
Solution: Compute the present value of each choice. Option b is the best choice.
a. $100,000 now is just worth $100,000
b. $18,000/1.12^5 = $102,136.83
c. $11,400/0.12 = $95,000
1−1.12−10
d. $16000 × × 1.12 = $101,252
0.12
$6,500
e. = $92,857.14
0.12−0.05
2. Suppose that you take out a 30-year mortgage loan of $200,000 at an interest rate of 10%.
a. What is your total monthly payment?
b. How much of the first month’s payment goes to reduce the size of the loan?
c. If you can afford to pay $2,000 per month, how long would it take you to pay for this
loan (still at 10% interest)?
d. If you can only pay $1,700 per month, and still want to finish paying in 30 years, what is
the highest (annual) interest rate that you could pay?
Solution:
a. Use the formula for an annuity as follows:
𝑖 −𝑛𝑚
1−(1+ )
𝑚
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 = 𝑃𝑀𝑇 × [ 𝑖 ]
𝑚
where 𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 = $200,000, 𝑖 = 10%, 𝑚 = 12, 𝑛 = 30 216
to find that 𝑃𝑀𝑇 = $1,755.14
b. The first month’s interest is $200,000*10%/12=$1,666.67. Since the monthly payment
is $1755.14, only $88.47 goes to reduce the size of the loan.
c. Again, use the formula for an annuity, where 𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 = $200,000, 𝑖 = 10%, 𝑚 = 12,
𝑃𝑀𝑇 = $2,000 to find that n=18 years
d. 9.625145%
3. As a manager of short-term projects, you are trying to decide whether or not to invest in a
short-term project that pays one cash flow of $1,000 one year from today. The total cost of the
project is $950. Your alternative investment is to deposit the money in a one-year bank
Certificate of Deposit (CD), which pays 4%, compounded annually.
a. Assuming that the cash flow of $1,000 is guaranteed (there is no risk you will not receive
it), what would be a logical discount rate to use to determine the present value of the
project?
b. What is the net present value of that investment? Should you invest in that project?
c. What would you do if the bank increases its quoted rate on one-year CDs to 5.5%?
d. At what bank one-year CD rate would you be indifferent between the two investments?
Solution:
a. Use 4%, the only alternative you have in this case
b. NPV=(1000/1.04)-950=11.54>0 so you should invest in the project
c. NPV=(1000/1.055)-950=-2.13<0 so you should not invest in the project. The CD is
better.
d. Find the IRR: (1000/(1+IRR))=950, IRR=5.26%.
4. The exchange rate between the pound sterling and the dollar is currently $1.50 per pound, the
dollar interest rate is 7% per year, and the pound interest rate is 9% per year. You have
$100,000 in a one-year account that allows you to choose between either currency, and it pays
the corresponding interest rate.
a. If you expect the dollar/pound exchange rate to be $1.40 per pound a year from now,
and are indifferent to risk, which currency should you choose?
b. What is the break-even value of the dollar/pound exchange rate one year from now?
Solution:
a. Dollar: gives you $107,000 gross return
£1 $1.40
Pound: gives you ($100000 × $1.50 × 1.09 × £1 ) = $101,733.33 gross return
If you are indifferent to risk, choose the dollar investment.
£1 $𝑋
b. Solve for X in $100000 × $1.50 × 1.09 × £1 = $107,000, 𝑋 = 1.47
5. David and Helen Zhang are saving to buy a boat at the end of five years. The boat costs $20,000
now. They can earn 10% a year on their savings. Inflation is projected to be 5% a year. In
order to be able to buy this boat at the end of 5 years, how much should they put in the bank
now?
Solution:
The real future value of the boat is still $20,000. Find the real present value investment
required by discounting $20,000 using the real interest rate
1+𝑛𝑜𝑚𝑖𝑛𝑎𝑙 1.10
Compute real interest rate: 1 + 𝑟𝑒𝑎𝑙 = 1+𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 = 1.05
Discount $20,000 to today using the real rate: $20,000/(1.10/1.05)^5=$15,849.41
Alternatively, use nominal values
Compute the nominal future value of the boat: $20,000*1.05^5=$25,525.63
Discount the nominal future value to today using nominal rate:
$25,525,63/(1.10)^5=$15,849.41
