Math 420
Level Annuities
Problem Set 2
1. Tim makes an initial payment of 15,000 and annual payments of 1,000 at the end of each of the first
10 years, towards an investment. Starting the end of the 11th year, his investment returns five equal
annual payments of X. If the annual effective interest rat is 8%, compute X.
2. Ted and Rob have the same amount of money. Ted buys a 10-year annuity due with annual payments
of 2500. Rob invests his money in a fund earning an effective rate of 7%. Two years later, Rob buys
a 12-year annuity-immediate with annual payments of X. The present value of both annuities is
determined using an effective interest rate of 5%. Calculate X.
3. On January 1, ABC insurance has 100,000 which is due to Chris as an insurance benefit. He would
receive the benefit annually over a period of 15 years, starting immediately. The benefit is based on
an effective interest rate of 5%.
ABC insurance earns interest at an annual effective rate of 6%. Every July 1, the company pays 200
in expenses.
At the end of 10 years, ABC has Y remaining. Calculate Y.
4. Calculate t, if the present value of the following annuities is equal:
a) t-year annuity-immediate paying 1 each year, at an effective interest rate of (i – 1)%
b) 50-year annuity-immediate paying 1 each year, at an effective interest rate of i%
c) perpetuity-immediate paying 1 each year, at an effective interest rate of 6.5%
5. You are given the following information:
a) The present value of an annuity due that pays 500 every 6 months during the first 12 years
and 300 every 12 years during the second 12 years is 5000;
b) the present value of a 12-year deferred annuity-due that pays 525 every 6 months for 12 years
is 6000.
c) The present value of an annuity due that pays 200 every 6 months during the first 12 years
and 100 every 12 years during the second 12 years is X;
Calculate X.
6. Art deposits 1000 into a fund at the beginning of every year for 10 years, and an additional deposit of
Y at the end of 15 years. At the end of 20 years, he uses the accumulated balance to buy a
perpetuity-immediate with annual payments of 2000 per year for 10 years, and 1000 per year
thereafter. If the interest is credited at an annual effective rate of 5%, compute Y.
7. The present value of a series of payments of 4 every six years, forever, is equal to 5. Calculate the
effective rate of interest.
8. A sum, P, is used to buy a deferred perpetuity-due of 1 payable annually. The effective rate of
interest is I. Calculate the deferred period.
9. Tom borrowed 10,000 from Bank X at an annual effective interest rate of 8%. He agreed to repay the
bank with five level annual payments at the end of each year. At the same time, he also borrowed
Math 420
Level Annuities
Problem Set 2
15,000 from Bank Y at an annual effective rate of 7.5%. He agreed to repay the bank with five
level annual payments at the end of each year.
Jerry borrowed 25,000 from Tom at an annual effective rate of 8.5%. Jerry agreed to repay Tom
with four level annual payments at the end of each year.
Tom can only reinvest at an annual rate of 6%. Determine how much Tom has left, immediately
after repaying the loans to the banks in full.
10. Paula borrows 5,000 from Will for 5 years. Paula repays the loan by paying interest at the end of
each year at an effective interest rate of 8%, and repaying the loan amount as a lump sum at the end
of 5 years. Immediately after the third payment, Will sells his right to future payments to Chris at a
price that will yield Chris an effective rate of return of 5%. Determine Will's overall yield rate.
11. Kevin deposits 1000 per year on his birthday into a retirement fund earning an annual effective rate
of 15%. The first deposit is made on his 39th birthday, and the last deposit is made on his 60th
birthday.
Immediately after the last deposit, the accumulated value of the fund is transferred to a fund earning
an annual effective rate of j. Five years later, a 35-year monthly annuity-due paying 1000 each month
is purchased with the funds. The purchase price of the annuity was determined using a nominal rate
of interest convertible monthly at 6%. Calculate j.
12. Dave receives payments of X at the end of each year for n years. The present value of his annuity is
500.Zach receives payments of 4X at the end of each year for 2n years. The present value of his
annuity is 2500. Both present values are calculated at the same effective interest rate. Determine vn