More on Annuities
7.1 Annuities Due
- You have contributed $3000 to your RRSP at the beginning of every three
months for the past 17 years. If the RRSP has earned 8.8% compounded quarterly, what is its value today?
- Carly purchased a refrigerator under a conditional sale contract that required 30 monthly payments of $70 with the first payment due on the purchase date. The interest rate on the outstanding balance was 15% compounded monthly.
- What was the purchase price of the refrigerator?
- How much interest did Carly pay during the entire contract?
- For the past 25 years, Kiko has contributed $2500 to his RRSP at the beginning of every six months. The plan earned 8% compounded annually for the first 11 years and 7% compounded semiannually for the subsequent 14 years. What is the value of his RRSP today?
- What is the current economic value of an annuity due consisting of 22 quarterly payments of $700 if money is worth 6% compounded quarterly for the first three years, and 7% compounded quarterly thereafter?
- A loan of $35,000 is paid down to $12,750 over the span of three years by making payments at the beginning of every month. If the current interest rate is 2.6% compounded annually, determine what the payment size should be.
- Sharon takes out a loan of $45,000 and agrees to make payments of $570 at the beginning of every other week. If the agreed upon rate is 2.5% compounded bi-weekly, how long will it take her to pay the loan down to $12,000?
7.2 Deferred Annuities
- Liza has been able to transfer $25,000 into an RRSP. She plans to
let the RRSP accumulate earnings at the rate of 4.5% compounded annually for 10 years, and then purchase a 15-year annuity making payments at the end of each quarter. What size of payment can she expect if the funds in the annuity earn 5.25% compounded quarterly?
- Twelve years ago, Frank rolled a $21,000 retiring allowance into an RRSP that subsequently earned 11% compounded semiannually. Three years ago he transferred the funds to an RRIF. Since then, he has been withdrawing $2500 at the end of each quarter. If the RRIF earns 7.5% compounded quarterly, how much longer can the withdrawals continue?
- The first quarterly payment of $850 in a five-year annuity will be paid 3.75 years from now. Based on a discount rate of 6.25% compounded monthly, what is present value of the payments today?
- A $45,000 loan bearing interest at 5% compounded quarterly was repaid, after a period of deferral, by quarterly payments of $2,550 over 12 years. What was the time interval between the date of the loan and the first payment?
- What amount must be invested today to provide for quarterly payments of $3500 at the end of every quarter for 15 years after a 5 year deferral period? Assume that the funds will earn 5% compounded semiannually.
- For how long (before the first withdrawal) must a $25,000 investment earning 3.5% compounded semiannually be allowed to grow before it can provide 60 quarterly withdrawals of $1200?
7.3 Perpetuities
- Determine the present value of a perpetuity that pays $350 at the beginning of each month. Assume interest is 3.5% compounded monthly.
- If a perpetuity pays $200 at the end of each week, and the current rate of interest is 2.75% compounded monthly, what is the fair market value today of the perpetuity?
- A lottery has a grand prize of end-of-month payments of $150 for the rest of time. If the current interest rate is 5% compounded monthly, what is the fair market value today of winning this grand prize?
- Okanagan College would like to open a bank account that will pay one lucky student a $1500 bursary at the beginning of every year. If the current interest rate is 2% compounded biweekly, how much would the college have to deposit today into the bank account to ensure this perpetuity?
- The common shares of EDco are forecast to pay annual dividends of $5 at the end of each of the next five years, followed by dividends of $8 per year in perpetuity. What is the fair market value of the shares if the market requires an 8% annually compounded rate of return on shares having a similar degree of risk?
7.4 Constant Growth Annuities
- Jessica intends to make RRSP contributions on January 1 of each year. She plans to contribute $2000 in the first year and increase the contribution by 3% every year thereafter. a. Rounded to the nearest dollar, how much will she have in her RRSP at the time of her 30th contribution if the
plan earns 8% compounded annually? b. What will be the amount of her last contribution?
- David is 30 years old and intends to accumulate $2.5 million in his RRSP by age 60. He expects his income and annual RRSP contributions to keep pace with inflation, which he assumes will be 2.5% per year. Rounded to the nearest dollar, what will be his initial contribution one year from now if he assumes the RRSP will earn 8% compounded annually?
7.5 Special Case Annuities
- You purchase a car worth $25,000. You make a down payment of $5000 and repay the rest with month end payments of $350 with a final “buyout” of $7500 due on the date of the final payment. How many payments do you make? Assume the rate of interest is 5.25% compounded quarterly.
- You make month end payments of $250 into a retirement account for 30 years. At the end of each year, you contribute an additional $2500 from your yearly bonus. How much will be in the account in 30 years? Assume the account earns 6% interest compounded semi-annually.
- You wish to create a scholarship fund that pays $10,000 to a recipient 3 times each year. You only have $300,000 to invest today. If the fund into which you are investing earns 8% compounded quarterly, how long must you wait before you can award the first scholarship payment?
- A $25,000 loan bearing interest at 9% compounded monthly was repaid, after a period of deferral, by monthly payments of $430 for 10 years. What was the time interval between the date of the loan and the first payment?