Problem Set B – Simple & Compound Interest,
Payment Streams & Present Value
On all problem sets, please compute all answers in dollars to the
nearest $1 unless specified otherwise, and all answers that are rates to the
nearest .0001 (.10%). Include units
with all answers.
1. CleanEarth Inc. has entered into a
contract to build wind-powered turbines for Smog City’s municipal electricity
company. The project will require CleanEarth to invest $50,000 one year from
now, $100,000 two years from now, and $200,000 three years from now. Four years from now, CleanEarth will receive
full payment from Smog City of $2,700,000.
If the market rate of interest is 5.8%, what is the present value of the
project?
2. On June 15, 1999 Big Tobacco Co. and
the Attorney General of the State of Underdevelopment announced an agreement to
settle the state’s lawsuit against Big Tobacco for expenses incurred by the
state as a result of tobacco use by its citizens. The Underdevelopment Gazette headline reads “Big Tobacco agrees
to $3.5 billion payout”. The citizens, beguiled by the headlines, rejoice. The
governor congratulations the Attorney General for his political acuity and Big
Tobacco’s CEO receives a large bonus. The market rate of interest is 8%, and the
payments will be made at the end of each of the following:
Year
1: $10m Year
6: $60m
Year
2: $20m Year
7: $70m
Year
3: $30m Year
8: $80m
Year
4: $40m Year
9: $90m
Year
5: $50m Year
10: $3,050m
What
is the present value of the payout?
3. Larry Lawstudent is
saddened to hear that his Aunt Deepockets died suddenly. However, Larry’s grief is lessened when he
learns that his aunt remembered him in her will by leaving him $5,000 per year
for ten years, with the first payment to be made in one year. At a market rate
of interest of 7%, what is the present value of Larry’s inheritance?
4. A perpetuity that pays
$16,501 dollars per year, with the first payment to be made one year from
today, is selling for $146,936. Assume
that this is a fair price (the net present value of an investment in this
perpetuity is zero.) What is the market
rate of interest?
5. What is the present cost
of the repayment obligation for a mortgage loan that requires you to make
annual payments of $18,000 per year for 30 years, beginning one year from
today, if the market rate of interest is 9%?
Use an annuity formula.
6. Same facts as the
previous problem, except that the first payment is to be made today, and the
final annual payment will be made in 29 years.
What is the present cost of the repayment obligation?
7. A growing annuity pays
$1000, starting 1 year from today, and payouts continue in perpetuity. The
payouts grow by 4% each year. The
market interest rate is 8%.
a. What is the present value of the growing
perpetuity?
b. The company that
offers this perpetuity, MIS.Forecast, is offering the growing perpetuity for
$20,000? What is MIS Forecast’s estimate of the market rate of interest?
c. Assume now that
the payouts end after the 25th payout. What is the new present value of the growing annuity? Use the
growing annuity formula.
8. Jane Smith just
celebrated her twentieth birthday this afternoon. She wishes to retire at age 65.
She wants to purchase a retirement annuity that will pay her a lump sum
of $50,000 on her 65th birthday, and on each birthday thereafter
through and including age 85 (a total of 21 annual payments). Assume that the market rate of interest is
9% and will remain constant over the period of interest (a heroic assumption to
be sure). How much does Jane need to
pay today for this annuity, assuming that it is fairly priced?
Hint: This is
a difficult, multistep problem.
Remember that the annuity formula assumes that the annuity payments
begin one year after the date when the present value calculation is made. You may find it useful to determine the
then-present value of the annuity as of Jane’s 64th birthday (one
year before the annuity payments begin).
9. Same facts as the
previous problem, except that Jane wants her retirement payouts to keep pace
with inflation. Jane’s first payment
(on her 65th birthday) will remain at $50,000, but each payment
thereafter will grow at 3%. How much
will Jane need to pay today for this retirement annuity?
10. Quick-Buck Co. plans to
issue bonds with a maturity of 4 years from today, which pay interest semiannually, at a nominal annual
interest rate of 10% per year. What is
the annual yield on these bonds?
11. Quick-Buck also plans to
issue bonds with a maturity of 10 years from today, which pay interest semiannually, and provide an annual
yield of 11%. What is the semiannual
rate of interest on these bonds – the rate paid each six months?
12. Quick-Buck Co. also plans
to issue bonds with a maturity of 4 years from today, which pay interest semiannually, at a bond-equivalent
interest rate of 10% per year. What is
the annual yield on these bonds?
13. Bank A is offering a 4% nominal annual rate of interest, compounded
monthly. Bank B is offering a 4.05% nominal annual rate of interest, compounded
semiannually. Which bank is offering
the higher annual yield? Why?
14. What is the annual yield
on an investment that pays a nominal rate of interest of 6.5% per year,
compounded quarterly?