Question

A demand loan of $8000.00 is repaid by payments of $4000.00 after two years, $4000.00 after four years, and a final payment after six years. Interest is 5% compounded quarterly for the first two years, 6% compounded monthly for the next two years, and 6% compounded semi-annually thereafter. What is the size of the final payment? The final payment is $ 1. (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

Answer 1

Loan Amount = PV = $8000

Let Payment in Year n be Pn

Given, P2 = $4000
P4 = $4000
Let Final Payment = P6 = X

The Present Value of the future payments should be equal to the Loan Amount

Interest Rate for Year 1 and 2 = r1 = 5% compounded Quarterly
Interest Rate for Year 3 and 4 = r2 = 6% compounded monthly
Interest Rate for Year 5 and 6 = r3 = 6% compounded semi annually

Number of quarters in an year = 4
Number of months in an year = 12
Number of semiannual periods in an year = 2

Hence, PV = P2/(1+r1/4)^4*2 + P4/(1+r1/4)^4*2(1+r2/12)^12*2 + P6/(1+r1/4)^4*2(1+r2/12)^12*2(1+r3/2)^2*2

=> 8000 = 4000/(1+0.05/4)⁸ + 4000/(1+0.05/4)⁸(1+0.06/12)²⁴ + X/(1+0.05/4)⁸(1+0.06/12)²⁴(1+0.06/2)⁴

=> X = [ 8000 - 4000/(1+0.05/4)⁸ - 4000/(1+0.05/4)⁸(1+0.06/12)²⁴ ] (1+0.05/4)⁸(1+0.06/12)²⁴(1+0.06/2)⁴

= $1632.91