Question

annuities and mortgages
john borrowed $22 000 to buy a car. calculate his monthly payment if he finances the loan over 6 years at 5.25%/year, compounded monthly.

Explanation:

Step1: Define given values

Principal \(P = 22000\), annual rate \(r = 0.0525\), years \(t = 6\), monthly compounding \(n = 12\)

Step2: Calculate monthly rate

\(i = \frac{r}{n} = \frac{0.0525}{12} = 0.004375\)

Step3: Calculate total payments

\(N = n \times t = 12 \times 6 = 72\)

Step4: Apply monthly payment formula

The formula for monthly payment \(M\) is \(M = P \times \frac{i(1+i)^N}{(1+i)^N - 1}\)
First compute \((1+i)^N = (1+0.004375)^{72} \approx 1.36699\)
Then compute numerator: \(0.004375 \times 1.36699 \approx 0.00598\)
Denominator: \(1.36699 - 1 = 0.36699\)
\(M = 22000 \times \frac{0.00598}{0.36699} \approx 22000 \times 0.01629\)

Answer:

$\approx 358.38$ (rounded to the nearest cent, so \$358.38)