Model and solve problems involving compound interest, depreciation, annuities, loans and investments using recursion and the financial solver.

What this dot point is asking

You must set up first-order recurrence relations for financial situations, convert annual rates to per-period rates, and find balances, repayments, interest paid or time taken, often with the finance solver on your calculator.

Compound interest as a recurrence

Compound interest adds interest to the balance each period, so the next balance is the current balance multiplied by a growth factor. If the interest rate is $r\%$ per period, the multiplier is

and the recurrence is $A_{n+1} = R \times A_n$, with $A_0$ the initial principal. The closed form is $A_n = A_0 R^{\,n}$.

Depreciation

Reducing-balance (declining) depreciation uses a multiplier less than one. If an asset loses $r\%$ of its value each year, then $R = 1 - \tfrac{r}{100}$ and $A_n = A_0 R^{\,n}$. Flat-rate (straight-line) depreciation instead subtracts a fixed dollar amount each period, giving the recurrence $A_{n+1} = A_n - d$.

Loans and annuities

Reducing-balance loans and annuities combine a growth factor with a fixed payment. Each period interest is added and a payment is made:

Here $d$ is the regular repayment (for a loan) and $A_0$ is the amount borrowed. For an annuity (drawing down savings) the same relation applies with $d$ the regular withdrawal. When the loan is repaid, $A_n = 0$.

Using the finance solver

The finance solver (TVM solver) links five quantities: $N$ (number of payments), $I\%$ (annual rate), $PV$ (present value), $PMT$ (payment) and $FV$ (future value). Enter any four and solve for the fifth.

The total interest paid on a loan is the sum of all repayments minus the amount borrowed: $\text{interest} = N \times d - A_0$.