Question

Two payments of $15,000 and $4,500 are due in 1 year and 2 years, respectively. Calculate the two equal payments that would replace these payments, made in 6 months and in 3 years if money is worth 6.5% compounded quarterly.

Answer 1

Payment Due at Year 1 = $15,000 | Payment Due at Year 2 = $4,500 | Rate of interest = 6.5% compounded quarterly

Quarterly rate = 6.5% / 4

Effective Annual Rate = (1 + APR / 4)⁴ - 1

Effective Annual Rate = (1 + 6.5% / 4)⁴ - 1

Effective Annual Rate = 6.66%

First we will calculate the Present Value of Two payments at Year 1 and 2.

PV of Two payments = Year 1 Payment * PVIF(EAR,1) + Year 2 Payment * PVIF(EAR,2)

PV of Two payments = 15,000 / (1+6.6%) + 4,500 / (1+6.6%)²

PV of Two payments = 14,063.36 + 3,955.56

PV of Two payments = $18,018.92

Now using the PV of Two payments, we can calculate the equal payments at 6 months and 3 years.

Let the equal payment be X

Before calculating the payment, as first payment is in 6 months, we need Effective semiannual rate.

(1+effective semi-annual rate)² = (1+Effective annual rate)

Effective Semi-annual rate = (1+EAR)^1/2 - 1

Effective Semi-annual rate = (1+6.66%)^1/2 - 1

Effective Semi-annual rate or ESR = 3.28%

Now we need PV of an equal payment at 6 months and 3 years to equal PV of the two payments calculated earlier.

6 months = 1 period | 3 years = 3 * 2 = 6 periods

PV of equal payments at 6 months and 3 years = X * PVIF(ESR, 1) + X * PVIF(ESR, 6)

PV of payments = $18,018.92

=> 18,018.92 = X (PVIF(3.28%, 1) + PVIF(3.28%, 6))

=> 18,018.92 = X (1 / (1+3.28%) + 1 / (1+3.28%)⁶)

X = 18,018.92 / (1 / (1+3.28%) + 1 / (1+3.28%)⁶)

X = 18,018.92 / 1.7924

Equal Payments at 6 month and 3 years = $10,052.95

Hence, Equal payments at 6 month and 3 years is $10,052.95