Question

Find the present value of $400 due in the future under each of these conditions:

(a) 15% nominal rate, semiannual compounding, discounted back 7 years. Do not round intermediate calculations. Round your answar to the nearest cent.
(b) 15% nominal rate, quarterly compounding, discounted back 7 years, Do not round intermediate calculations. Round your answer to the nearest cent.
(c) 15% nominal rate, monthly compounding, discounted back 1 year, Do not round intermediate calculations. Round your answer to the nearest cent.
(d) Why do the differences in the PVs occur?

Answer 1

The present value of $400 discounted at a 15% nominal rate varies with the compounding frequency: $127.99 semiannually, $123.48 quarterly, and $349.14 monthly. This demonstrates how compounding affects the present value calculation.

The present value of $400 due in the future at a nominal rate of 15% under various compounding conditions can be calculated using the present value formula: PV = FV / (1 + r/n)^(nt), where PV is the present value, FV is the future value, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

1. (a) For semiannual compounding, n equals 2: PV = $400 / (1 + 0.15/2)^(2*7).
2. (b) For quarterly compounding, n equals 4: PV = $400 / (1 + 0.15/4)^(4*7).
3. (c) For monthly compounding, n equals 12: PV = $400 / (1 + 0.15/12)^(12*1).

In each scenario, calculating the PV using the appropriate formula yields:

• (a) $127.99
• (b) $123.48
• (c) $349.14

The differences in PV occur due to the frequency of compounding, which impacts how often interest is added to the principal.