Final Payment Value Compound Interest Question

Question

A demand loan of $\$4000.00$ is repaid by payments of $\$2000.00$ after two years, $\$2000.00$ after four years, and a final payment after six years.

Interest is $6\%$ compounded quarterly for the first two years, $7\%$ compounded annually for the next two years, and 7% compounded semi-annually thereafter. What is the size of the final payment?

I know how to make all these precursor calculations but I'm very lost on which formula to use to find the final amount.

1. payment. $3000(1+0.0175)^{0.5}$

2. Payment $3000(1+0.04)^{2}$

Answer 1

The final payment can be calculated in steps by figuring how much the loan has grown with interest over each time period before each of the stage payments. I'm assuming the interest for each time period is simply the annual rate divided by the number of time periods per year.

$$P_F = ((4000\cdot 1.015^8 -2000)1.07^2 - 2000)1.035^4 = \$997.30$$

Answer 2

$L=4000$, $p_2=2000$, $p_4=2000$, $i^{(4)} =6\%$, $i^{(1)} =7\%$, $i^{(2)} =7\%$. Find $p_6$ solving

$$ L=\frac{p_2}{\left(1+\frac{i^{(4)}}{4}\right)^{2\times 4}} +\frac{p_4}{\left(1+\frac{i^{(4)}}{4}\right)^{2\times 4}\times\left(1+\frac{i^{(1)}}{1}\right)^{2\times 1}}+\frac{p_6}{\left(1+\frac{i^{(4)}}{4}\right)^{2\times 4}\times\left(1+\frac{i^{(1)}}{1}\right)^{2\times 1}\times\left(1+\frac{i^{(2)}}{2}\right)^{2\times 2}} $$