Question

A 10-year loan is to be repaid by quarter-end repayments of 8,000 starting in 3 months at an interest rate of 3.8% p.a. compounded quarterly. Or, it can be repaid by year-end repayments of $X staring in one year. Calculate the yearly repayments $X. Correct your answer to the nearest cent without any units. (Do not use "$" or "," in your answer. e.g. 12345.67)

Answer 1

First, let's find the loan amount

Quarterly rate, r = 3.8%/4 = 0.0095

Number of payments = 10 years * 4 quarters a year = 40 payments

$PV = \frac{PMT}{ r } * \left [ 1 - \frac{1}{(1+ r )^{ n }} \right ]$

8000 PV = 0.0095 * 1 (1+0.0095)40

PV = 842, 105.263157895 * 0.3149108826

PV = $265, 188.111663158

This is the loan amount. Now we need to find the annual payment

n = 10

Effective annual rate = (1 + 0.0095)^4 - 1 = 0.03854493765

$PV = \frac{PMT}{ r } * \left [ 1 - \frac{1}{(1+ r )^{ n }} \right ]$

265, 188.111663158 = PMT 0.03854493765 * (1 + 0.03854493765) 10

PMT 265, 188.111663158 = — 0.03854493765 - * 0.3149108827

265, 188.111663158 = PMT *8.1699673654

PMT = 265, 188.111663158 8.1699673654

PMT = $32, 458.894852657

Year-end payments = 32,458.89

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