Find the present value of an ordinary annuity with deposits of $16,435 quarterly for 4 years at 11.2% compounded quarterly. Question content area bottom Part 1 What is the present value?
The correct answer and explanation is:
To find the present value of an ordinary annuity, use the following formula: PV=P×(1−(1+r)−ntr)PV = P \times \left( \frac{1 – (1 + r)^{-nt}}{r} \right)
Where:
- PVPV is the present value
- PP is the periodic payment = $16,435
- rr is the interest rate per period = 11.2% annually ÷ 4 = 0.112 ÷ 4 = 0.028
- nn is the number of periods per year = 4
- tt is the number of years = 4
Now substitute the values: PV=16435×(1−(1+0.028)−160.028)PV = 16435 \times \left( \frac{1 – (1 + 0.028)^{-16}}{0.028} \right) PV=16435×(1−(1.028)−160.028)PV = 16435 \times \left( \frac{1 – (1.028)^{-16}}{0.028} \right)
First calculate (1.028)−16(1.028)^{-16}: (1.028)−16≈0.641661(1.028)^{-16} \approx 0.641661
Now compute the full expression: PV=16435×(1−0.6416610.028)PV = 16435 \times \left( \frac{1 – 0.641661}{0.028} \right) PV=16435×(0.3583390.028)PV = 16435 \times \left( \frac{0.358339}{0.028} \right) PV=16435×12.7971≈210,254.79PV = 16435 \times 12.7971 \approx 210,254.79
Correct Answer:
The present value of the ordinary annuity is $210,254.79
Explanation:
The present value of an annuity represents the amount of money needed today to fund a series of regular payments made in the future, given a specific interest rate. This concept is useful in retirement planning, loan evaluation, or any financial scenario involving regular payments.
In this question, the annuity makes quarterly deposits of $16,435 for 4 years, meaning there are 16 total payments. Since the interest is compounded quarterly, the effective interest rate per period is 2.8 percent.
By applying the present value formula for ordinary annuities, which assumes payments are made at the end of each period, the calculation discounts each future payment back to its equivalent value today using the given interest rate. This shows how much an investor would need to invest now to be able to make those future payments.