Paying off a loan

Finding the balance owing on a reducible interest loan, the instalment needed to repay it in a given time, the total interest paid, and the number of instalments needed.

The balance owing

Each period the balance earns interest and then an instalment $M$ is paid. With $P$ borrowed and ratio $r = 1 + R$:

Balance after n instalments

$$A_n = Pr^n - M(1 + r + r^2 + \dots + r^{n-1}) = Pr^n - M \times \frac{r^n - 1}{r - 1}$$

The loan is repaid when $A_n = 0$.

Finding the instalment

Example 1: a home loan●●●

$\$300\,000$ is borrowed at $6\%$ p.a. compounded monthly and repaid monthly over 25 years. Find the monthly instalment and the total interest.

$r = 1.005$ and $n = 300$.

Rate and periods

$300\,000 \times 1.005^{300} = M \times \dfrac{1.005^{300} - 1}{0.005}$

Set A₃₀₀ = 0

$M = \dfrac{300\,000 \times 0.005 \times 1.005^{300}}{1.005^{300} - 1} \approx \$1932.90$

Solve

Total repaid $\approx 300 \times 1932.90 = \$579\,870$, so the interest is about $\$279\,870$.

Interest

Finding the length of a loan

Example 2●●●

$\$20\,000$ is borrowed at $1\%$ per month and repaid at $\$500$ per month. How many payments are needed?

$20\,000 \times 1.01^n = 500 \times \dfrac{1.01^n - 1}{0.01} = 50\,000(1.01^n - 1)$

$30\,000 \times 1.01^n = 50\,000$, so $1.01^n = \tfrac53$.

$n = \dfrac{\ln \frac53}{\ln 1.01} \approx 51.3$. It takes $52$ payments, with the last one smaller.

Flashcards

Balance after $n$ instalments

$A_n = Pr^n - M\dfrac{r^n - 1}{r - 1}$

Loan repaid

$A_n = 0$

Total interest

Total of the instalments minus the amount borrowed.

Practice

$\$300\,000$ at $6\%$ p.a. compounded monthly, repaid monthly over 25 years. Find the instalment, to the nearest cent.
$\$20\,000$ at $1\%$ per month, repaid at $\$500$ per month. How many payments are needed?
The balance owing after one instalment is:
The balance after two instalments is:
$\$1000$ is borrowed at $10\%$ p.a. and repaid in one instalment after a year. How much is the instalment?
$\$1000$ is borrowed at $10\%$ p.a. and repaid in two equal yearly instalments. Find the instalment to the nearest cent.
In Example 1, about how much interest is paid, to the nearest dollar? Use $M = \$1932.90$.
A loan is repaid when:
For a monthly loan at $9\%$ p.a., the ratio $r$ is: