Young and Company owes bond holders $5,500 interest at the end of each quarter for the next five years. How much they deposit now at 8% interest compounded quarterly to yield an annuity payment of $5,500?
P(1.02)^20 = 5500
P = 5500/1.02^20 = 3701.34
To determine how much Young and Company needs to deposit now in order to yield an annuity payment of $5,500, we can use the concept of present value and the formula for calculating the present value of an annuity.
The formula for the present value of an annuity, also known as the present value of a series of cash flows, is given by:
PV = PMT × [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present value of the annuity
PMT = Payment amount per period ($5,500 in this case)
r = Interest rate per period (8% compounded quarterly = 8% / 4 = 2% = 0.02)
n = Number of periods (5 years = 5 years × 4 quarters/year = 20 quarters)
Using the given values, we can now calculate the present value of the annuity:
PV = $5,500 × [(1 - (1 + 0.02)^(-20)) / 0.02]
Now let's solve this equation step by step:
Step 1: Calculate the exponent
(1 + 0.02)^(-20) = 0.67261
Step 2: Calculate the subtraction
1 - 0.67261 = 0.32739
Step 3: Calculate the division
(0.32739 / 0.02) ≈ 16.37
Step 4: Calculate the multiplication
$5,500 × 16.37 ≈ $89,553.50
Therefore, Young and Company needs to deposit approximately $89,553.50 now, with an 8% interest rate compounded quarterly, to yield an annuity payment of $5,500 at the end of each quarter for the next five years.