Find the present value of $31,000 due in 4 yr at the given rate of interest.
(a) 6%/year, compounded quarterly
$ ?
(b) 3%/year, compounded quarterly
$ ?
a) i = .06/4 = .015
n = 4(4) = 16
PV = 31000(1.015)^-16
= 24428.96
b) try it using the same steps as above
let me know what you get
To find the present value of an amount due in the future, you need to use a formula called the present value formula. The present value formula takes into account the future value, time period, and the interest rate.
(a) Using a 6% interest rate, compounded quarterly:
Step 1: Determine the number of compounding periods in 4 years. Since interest is compounded quarterly, there are 4 * 4 = 16 compounding periods.
Step 2: Find the interest rate per compounding period. Since the annual interest rate is 6%, the quarterly interest rate is 6% / 4 = 1.5%.
Step 3: Use the present value formula: PV = FV / (1 + r/n)^(n*t), where PV is the present value, FV is the future value, r is the interest rate per compounding period, n is the number of compounding periods per year, and t is the number of years.
Plugging in the values:
PV = $31,000 / (1 + 0.015)^(4 * 4) = $31,000 / (1 + 0.015)^16 = $31,000 / (1.015)^16
Using a calculator, you can solve for the present value. The present value for option (a) is approximately $26,431.08.
(b) Using a 3% interest rate, compounded quarterly:
Follow the same steps as in option (a).
Step 1: There are still 16 compounding periods in 4 years.
Step 2: The quarterly interest rate is 3% / 4 = 0.75%.
Step 3: Using the present value formula:
PV = $31,000 / (1 + 0.0075)^(4 * 4) = $31,000 / (1 + 0.0075)^16 = $31,000 / (1.0075)^16
Solving this equation using a calculator, the present value for option (b) is approximately $29,336.27.