When Cody's son was born, he put $4,500 in an investment that earns 7% compounded semi-annually. This investment will mature when his son turns 18 and will go straight into an annuity at 4.75% compounded and paying out quarterly at the end of the period. The investment was to help pay for his 4-years of college. Find the size of these quarterly payments received by Cody's son during his college stay.
The first part seems easy enough, but we're not at all sure about the second part. The investment is earning 7% compounded semi-annually until he turns 18, so that means the original $4,500 will have become ((1.07)^36) x $4,500 = $51,407.74 on his 18th birthday. So far, so good.
Now, we're assuming (and we could easily be wrong about this) that this "4.75% compounded and paying out quarterly" means that the annuity is accumulating interest at this rate annually, BUT paying out 16 equal payments over the four years that he's at college - at the end of which time there will be nothing left. We therefore have to find out what each of those payments will be, bearing in mind that the interest will be accumulating on a different amount every quarter. Tricky....
Suppose the quarterly payment is Q, that the annuity is purchased for A (which we already know is $51,407.74, but A is easier to write), and P is the quarterly interest rate on the annuity (i.e. whatever rate will give us 4.75% per annum). We're also going to assume that he withdraws his quarterly payment at the END of each quarter, so during the first quarter he's earning interest on the full amount. So the amount left at the end of each quarter should be....
A(1+P) - Q
(A(1+P) - Q)(1+P) - Q
((A(1+P) - Q)(1+P) - Q)(1+P) - Q
(((A(1+P) - Q)(1+P) - Q)(1+P) - Q)(1+P) - Q
P is easy enough to find: it is whatever value will give us (1+P)^4 = 1.0475, i.e. P = 1.167%. With that value, the above series of recursive relationships with any given value of Q can be fed into 16 successive cells in a column of Excel easily enough - and we then need to find Q such that the final amount in the 16th cell is zero. Having done just that, we reckon that the quarterly payment to him should be $3,540.91, which would leave him with just one cent in his account at the end of the fourth year. But arriving at that algebraically is beyond us - and obviously you'll need to show your working to get the marks, so the best we can do is show you how the problem might be tackled, and offer a possible answer to check yours against if you can do it. Sorry!
To find the size of the quarterly payments received by Cody's son during his college stay, we need to calculate the future value of the initial investment and then calculate the amount of each quarterly payment.
First, let's calculate the future value of the initial investment of $4,500 over the 18-year period at an annual interest rate of 7% compounded semi-annually. We can use the formula for compound interest:
Future Value (FV) = Principal (P) x (1 + (r/n))^(nt)
Where:
P = $4,500 (principal)
r = 7% (annual interest rate expressed as a decimal)
n = 2 (compounded semi-annually, so 2 times per year)
t = 18 (number of years)
Let's calculate the future value:
FV = $4,500 x (1 + (0.07/2))^(2 x 18)
FV = $4,500 x (1 + 0.035)^(36)
FV = $4,500 x (1.035)^(36)
FV ≈ $13,019.32
So, the future value of Cody's initial investment is approximately $13,019.32 when his son turns 18.
Now, we need to calculate the size of the quarterly payments received by Cody's son during his college stay. We can use the formula for present value of an annuity:
Present Value (PV) = Payment (PMT) x ((1 - (1 + r)^(-n))/r)
Where:
PMT = Unknown (size of quarterly payment)
r = 4.75% (annual interest rate for the annuity expressed as a decimal)
n = 4 (quarterly payments per year for 4 years)
We'll substitute the known values into the formula and solve for PMT:
$13,019.32 = PMT x ((1 - (1 + 0.0475/4)^(-4*4))/0.0475)
Let's solve for PMT:
PMT x (1 - (1.011875)^(-16))/0.0475 = $13,019.32
PMT x (1 - 0.70791723)/0.0475 = $13,019.32
PMT x 0.29208277/0.0475 = $13,019.32
PMT x 6.14245915 = $13,019.32
PMT = $13,019.32 / 6.14245915
PMT ≈ $2,119.16
Therefore, the size of the quarterly payments received by Cody's son during his college stay is approximately $2,119.16.