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. P = Po(1+r)^n.
P = $31,000. = Value in 4 yrs.
Po = Initial value.
r = (6%/4) / 100% = 0.015 = Quarterly %
rate expressed as a decimal.
n = 4Comp./yr * 4yrs = 16 Compounding
periods.
P = Po(1.015)^16 = $31,000.
Po*1.26899 = 31000
Po = $24,428.96 = Present value.
b. Same procedure as "a".
To find the present value of future cash flows, we can use the formula for compound interest:
Present Value = Future Value / (1 + r/n)^(n*t)
where:
- "r" is the interest rate per period,
- "n" is the number of compounding periods per year, and
- "t" is the number of years.
(a) Let's start with the first case, where the interest rate is 6% per year, compounded quarterly.
Here, the interest rate per period would be 6%/4 = 1.5%, and the number of compounding periods would be 4 years * 4 quarters per year = 16 quarters.
Using the formula, we can plug in the values:
Present Value = $31,000 / (1 + 0.015)^(16)
Simplifying the formula:
Present Value = $31,000 / (1.015)^16
Calculating Present Value:
Present Value = $31,000 / 1.2871407
Present Value ≈ $24,071.02
Therefore, the present value of $31,000 due in 4 years at an interest rate of 6% per year, compounded quarterly, is approximately $24,071.02.
(b) Now, let's find the present value when the interest rate is 3% per year, compounded quarterly.
Similarly, the interest rate per period would be 3%/4 = 0.75%, and the number of compounding periods would still be 16 quarters.
Using the formula:
Present Value = $31,000 / (1 + 0.0075)^(16)
Simplifying the formula:
Present Value = $31,000 / (1.0075)^16
Calculating Present Value:
Present Value = $31,000 / 1.1284777
Present Value ≈ $27,441.22
Therefore, the present value of $31,000 due in 4 years at an interest rate of 3% per year, compounded quarterly, is approximately $27,441.22.