2. 1 SS.02.4 - Present Value of An Annuity MCR3U - Santowski

3. 2 (A) Concept of Present Value of an Annuity The present value of an annuity is the amount of money you would invest NOW in order to provide the required periodic payments. Consider this example: Mr. S. wants to establish a yearly scholarship of $500. I want to put money in now at 5% compounded annually rather than paying $500 every year. How much should I put in now in order to have enough money to meet this obligation say for the next 5 years?

4. 3 (B) Present Value Recall that the formula for present value is: PV = A(1 + i) -n which we will use in our example So our scenario is that I will donate: End of year 1 => $500 End of year 2 => $500 End of year 3 => $500 End of year 4 => $500 End of year 5 => $500 How much do I need to deposit today to meet this obligation?

5. 4 (C) Present Value of an Annuity The $500 I need for the end of year 1has the present value of $500(1.05) -1 = $476.19 The $500 I need for the end of the second year has a PV of $500(1.05) -2 = $453.51 Likewise, the other PV’s are: $500(1.05) -3 = $431.92, $500(1.05) -4 = $411.35, $500(1.05) -5 = $391.76. So the total amount I require to meet my obligation is $476.19 + $453.51 + $431.92 + $411.35 + $391.76 = $2164.73

6. 5 (D) Present Value – Time Lines As before, we can visualize the annuity with a time line:

7. 6 (E) Present Value – Formulas The values $476.19 + $453.51 + $431.92 + $411.35 + $391.76 represent a geometric series We will rearrange the series as $391.76 + $411.35 + $431.92 + $453.51 + $476.19 So S n = a(r n – 1) / r – 1 = 391.76(1.05 5 – 1)/.05 = $2164.72 Our formula can be rearranged to give us the following equation: PV=R[(1-(1+i) -n )/i] where R is the regular deposit, i is interest rate per conversion period and n is the number of conversion periods.

8. 7 (F) Examples ex. 1. Find the present value of an annuity of 10 semi-annual payments of $750 (with the first payment due in 6 months) if money is worth 7%/a compounded semi-annually. ex. 2. Mr S. has an investment policy of a value of $50,000, redeemable at the age of 60. The money is invested at 6%/a compounded semi-annually and paid in 30 equal semi-annually payments. The first payment is to be made on his 60 th birthday. What will my monthly income be from this policy?

9. 8 (G) Homework AW text on page 162 Q2,4,9,15-18 Nelson text, p164, Q6,8,9, 11-15, 18,19