Question

Find the present value of $700 due in the future under each of these conditions:15% nominal rate, semiannual compounding, discounted back 4 years. Do not round intermediate calculations. Round your answer to the nearest cent.

$
15% nominal rate, quarterly compounding, discounted back 4 years. Do not round intermediate calculations. Round your answer to the nearest cent.
$
15% nominal rate, monthly compounding, discounted back 1 year. Do not round intermediate calculations. Round your answer to the nearest cent.
$
Why do the difference in the PV's Occur?

Answer 1

The present value of $700 due in the future is calculated with semiannual, quarterly, and monthly compounding at a 15% nominal interest rate. The PV decreases with more frequent compounding because the effective interest rate increases. Compounding frequency significantly influences the discounting process and the present value.

Step-by-step explanation:

To find the present value (PV) of $700 due in the future under different compounding conditions at a 15% nominal rate, we use the formula PV = FV / (1 + r/n)^(nt), where FV is the future value, r is the nominal interest rate, n is the number of compounding periods per year, and t is the number of years.

For semiannual compounding, with n = 2 for 4 years, the PV is calculated as follows:

PV = $700 / (1 + 0.15/2)^(2*4) = $700 / (1 + 0.075)^(8) = $700 / (1.075)^8 = $700 / 1.8731 = $373.68

For quarterly compounding, with n = 4 for 4 years, the calculation is:

PV = $700 / (1 + 0.15/4)^(4*4) = $700 / (1.0375)^16 = $700 / 2.0837 = $335.99

For monthly compounding, with n = 12 for 1 year, the calculation is:

PV = $700 / (1 + 0.15/12)^(12*1) = $700 / (1.0125)^12 = $700 / 1.1608 = $602.76

The difference in the PVs occur because more frequent compounding results in a higher effective interest rate, which in turn decreases the present value. This illustrates how the compounding frequency can impact the discounting process and the overall present value of future payments.