Young and Company owes bond holders $5,500 interest at the end of each quarter for the next five years. How much must they deposit now at 8% interest compounded quarterly to yield an annuity
payment of $5,500?
You want the present value of 20 payments of 5500 at the end of each quarter for 5 years
i = .08/4 = .02
n = 4(5) = 20
PV = 5500( 1 - 1.02^-20)/.02
= $89932.88
To calculate the amount Young and Company must deposit now to yield an annuity payment of $5,500, we need to use the formula for the present value of an annuity.
The formula for the present value of an annuity is:
PV = PMT * [(1 - (1+r)^(-n)) / r]
Where:
PV is the present value (the amount Young and Company needs to deposit now)
PMT is the annuity payment ($5,500)
r is the interest rate per period (8% divided by 4 because it's compounded quarterly, so 0.08 / 4 = 0.02)
n is the number of periods (5 years multiplied by 4 because it's quarterly, so 5 * 4 = 20)
Let's calculate it:
PV = $5,500 * [(1 - (1+0.02)^(-20)) / 0.02]
First, calculate (1+0.02)^(-20):
(1+0.02)^(-20) ≈ 0.672
Now, plug it into the formula:
PV = $5,500 * [(1 - 0.672) / 0.02]
PV = $5,500 * [0.328 / 0.02]
PV ≈ $90,100
Therefore, Young and Company must deposit approximately $90,100 now at 8% interest compounded quarterly to yield an annuity payment of $5,500.