AUMGT 310 – Corporate Finance Annuities Example
Jethro is planning for retirement. He is currently 25 and wants to retire at age 60. He expects to live until he is 85. When he dies, he wants to leave $150,000 to his estate (to pay for his wake and funeral). He also wants to have a monthly income, during his retirement, of $5,000 per month. When he retires he will purchase an annuity that pays 5.5% compounded semi-annually. He also plans to put a lump sum of money aside that will earn 6% compounded annually. While he is working, he will put money into a self-directed RRSP that earns 10% compounded quarterly. How much money does he have to put aside each month while working in order to achieve his retirement goals? To solve this, we need to first figure out how much money he needs to have when he retires and then we can work back and find the monthly contributions while working. The money needed at retirement is for two things – to save for the funeral and to buy an annuity to live on. First we will find the money needed to fund his funeral and wake. This is a simple future value problem. Also, we don’t need to do anything special with the interest rate since it is annual compounding. $150, 000 = PV funeral 1.06 25 (1) Now, rearrange equation (1) to get $150, 000 (2) PV funeral = = $34,949.79 1.06 25 Therefore, to have $150,000 available when he turns 85 (to pay for the wake and funeral) he needs to set aside $39,949.79 when he retires. Now we can turn to finding the size of the annuity that he needs to buy. However, in this problem we need to find the monthly interest rate to use since we have semi-annual compounding with monthly payments. We do this with the following formula
C Y Y
i=
APR C Y 100
P
+1
1
(3)
where C Y is the number of compounding periods per year and P Y is the number of payments per year. For the annuity, the monthly interest rate to use is
5.5 i= +1 2 100
2 12
1 = 0.004531682
(4)
Now we can make use of the present value formula to derive the present value of the annuity that will provide $5,000 per month in income.
PVannuity = Pmt
(1 + i )t 1 t i (1 + i )
(5)
Putting the values that we have previously determined to find the size of the annuity needed 1.00453168212 25 1 = $819,146.42 PVannuity = $5, 000 0.004531682 1.00453168212 25 Thus, he needs to purchase an annuity of $519,464.78 to be able to have a monthly income of $5,000 during retirement.
(6)
Upon his retirement he needs to have a total of $854,096.22. Now we can turn to the problem of finding out how much he needs to save each month. First, we need to find the right interest rate to use. Begin with equation (3) and plug in the appropriate values – there are 12 payments per year and 4 compounding periods.
10 i= +1 4 100
4 12
1 = 0.00826484
(7)
The future value formula is
FV = Pmt
(1 + i )t
i
1
(8)
We can rearrange equation (8) to get a formula that will tell us the monthly payments. FV (9) Pmt = (1 + i )t 1 i Putting in the values that we have found above into equation (9) we get $854,096.22 (10) Pmt = = $229.78 1.0082648412 35 1 0.00826484 Therefore, Jethro needs to be putting $229.78 per month into his RRSP in order to reach his retirement goals. Now, let’s solve this using a financial calculator. The calculation for the amount of money needed so that he can have a good wake (well, he won’t be there to enjoy it but his friends and family will have a good time) are the same as before. From our earlier calculations we know that he needs to have $34,949.79 at his retirement to put aside to leave $150,000 to his estate. Now, to find the amount of money needed for his annuity. First, clear the memory in your calculator by pressing 2nd and then CA. Now, set the payment periods and compounding periods. Press 2nd and then P Y and then 12 and then ENT (this sets the number of payments per year). Now, press the down arrow and you will see that the
display shows C Y , put in 2 and then hit ENT. Now press On/C (this gets us out of the parameter setting mode). We can now start putting in the variables into the problem. We know the APR (5.5%), we know the monthly payments needed ($5,000), and we know the FV (0) and what we want to know is the PV. Enter 10 and then hit I Y ; now hit 5000 and then hit Pmt; hit 0 and then FV. Finally, press Comp and then hit PV. The calculator will spit out an answer. Add this to the $34,949.79 (the amount needed to save for the wake and funeral). The result is the amount of money that Jethro wants to achieve. Now we can find the monthly RRSP contributions. First, we need to change the compounding periods to 4 so hit 2nd and then P Y and then 12 and Ent. Now press the down arrow and hit 4 and then Ent. The payments per year and compounding periods per year are now set so hit On/C. Now, enter the interest rate (10%) and the PV (0). Enter the future value desired (the result from above) and hit FV. Now, press Comp Pmt and the calculator will give you the monthly payment required. Now, let’s look at a variation. Suppose that rather than putting money aside to pay for the funeral, he wants to put it all into the annuity and structure the annuity that hit has a future value of $150,000 when he turns 85. You do this when you enter the parameters in the annuity calculation. What happens to the size of his monthly payments? His monthly contributions to his RRSP becomes $230.77. See if you can get this result on your own.