Practice Problem Set #2: Time value of Money
Theoretical and conceptual questions:
(see notes or textbook for solutions)
1. Describe three special case annuities.
2. Describe how to calculate the future and present values of a series of uneven cash flows.
3. What is our assumption about the timing of cash flows within a period, unless we are told
otherwise?
4. What compounding period would provide an EAR closest to continuous compounding?
Practice Problems:
1. (a) If $2,000 is placed in a savings account at the end of each year for 5 years, what is the
value of this account at the end of the fifth year, given that money paid into the account
earns 10 percent simple interest?
(b) What is the value of the account at the end of 5 years if all balances held in the account
earn interest of 10 percent compounded annually?
(c) Recompute your answer under (b) assuming that $2,000 is paid into the account at the
beginning of each year. The first of these 5 payments is made immediately.
2. (a) What is the present value of $7,000 to be received at the end of each year for 6 years if
the APR is 10 percent compounded annually?
(b) What is the present value of $3,500 to be received at the beginning of every 6-month
period for 6 years if the APR is 10 percent compounded semi-annually?
3. (a) Assume that a pension plan offers to pay a lump sum of $200,000 on a person's 65th
birthday, or an annuity of $x for the remainder of the person's life. A person’s life
expectancy has been determined statistically to be 80 years. The first annuity payment
would be on the person’s 65th birthday and it is expected that the last payment would be
on the person’s 79th birthday. Interest rates are 10 percent per year. What is the value
of x (the amount of the annuity) that would make the two alternatives equivalent on an
expected present value basis?
(b) A person joins the pension plan at age 30. How much will this person have to pay into
the pension fund at the end of each year in order to accumulate a balance of $150,000
in the fund at age 65? The first contribution will be on the person’s 31st birthday and the
final contribution will be on the person’s 65th birthday.
4. (a) Assume that Luke Smith wants to retire in 20 years. He expects to live for another 15
years after his retirement, and during retirement he wants to withdraw $12,000 at the
beginning of each year from his savings account. How much will he have to deposit in
his account at the end of each year for the next 20 years, given that the interest rate is
10 percent per year compounded annually?
(b) How would the amount computed under (a) be altered if Mr. Smith expects his wife to
live for 5 years after his death and if, in addition to the amounts provided for under (a),
he wants her to be able to withdraw $6,000 at the beginning of each year during this
additional period?
5. How much is a perpetuity of $10 per year starting today worth if the interest rate is 18% per
year?
6. After getting a $500 bonus at the nuclear power plant, Homer is deciding whether to save it
or buy a lifetime membership in the Duff Beer Club. If he joins the club, he will save $1 per
week on his weekly Duff beer purchases. If Homer earns 0.10% per week on his bank
account, how long in years does he have to live before the membership is worth
purchasing? Assume his first purchase is one week from now.
7. On their statements, MasterCard is currently quoting a daily interest rate of 0.04315% and a
yearly rate of 15.75%. Is the yearly rate effective or APR? Justify your answer.
8. Canadian mortgages are almost always compounded semi-annually, but the frequency of
payments can vary. Suppose a mortgage is for $100,000, the APR is 6% compounded
semi-annually, and the mortgage is to be paid monthly over 25 years.
(a) Find the interest rate that you would use to calculate the payment.
(b) Calculate the payment.
9. How much will you have at the end of 20 years if you invest $100 today at 15% per year,
compounded annually? How much will you have if you invest at 15% per year, compounded
continuously?
10. For the following, assume the APR is 6%.
(a) What is the effective annual interest rate if interest is compounded semiannually?
(b) What is the effective annual interest rate if interest is compounded daily?
(c) Suppose you have $100 in the bank earning interest at 6% per year compounded
monthly. How much can you withdraw at the end of each month so that your balance
will be zero after 12 equal monthly withdrawals?
11. You wish to buy a $1,250 appliance. The friendly folks at Appliance Warehouse will let you
pay for it in 36 monthly instalments at an APR of 18% per year, compounded monthly. What
will your payments be?
