Revise Financial mathematics: compound interest, annuities and amortization for IB Mathematics: Applications and Interpretation (1.3)

Simple interest is calculated only on the original amount, the principal. Compound interest is calculated on the principal AND on any interest already earned, so each period's interest is a little larger than the last. That 'interest on interest' is what makes long-term saving so powerful - the balance grows exponentially, like a snowball gathering snow on snow. The only new subtlety is how often the interest is added: this is the compounding frequency, $k$.

$FV = PV\left(1 + \frac{r}{100k}\right)^{kn}$

Notice that $k$ appears in two places. It divides the annual rate - so with monthly compounding each period gets $\frac{r}{100 \times 12}$ of interest - and it multiplies the years to give the total number of periods $kn$. Change the compounding frequency and you must update both. Compounding more often gives slightly more growth: the same 6% grows a little more when added monthly than when added once a year, because interest starts earning interest sooner.

An account that grows by 4% sounds like a gain - but if the prices of the things you buy also rise, your money buys less than the headline suggests. Economists separate two ideas. The nominal (or money) value is just the number of currency units in the account. The real value adjusts for inflation: it measures purchasing power, what the money can actually buy. When savings grow at a nominal rate while prices rise at an inflation rate, the real gain is roughly the difference between the two rates.

Not everything grows. Cars, machinery and equipment lose value over time - they depreciate. The IB models this as a reducing-balance process: the asset loses a fixed percentage of its CURRENT value each year, not a fixed number of pounds. This is compound interest with a minus sign. Because the percentage applies to a shrinking balance, the loss is largest in the early years, which matches reality: a new car loses far more in year one than in year five.

Reducing-balance depreciation: $FV = PV\left(1 - \frac{r}{100}\right)^{n}$ where $r$ is the annual depreciation rate (as a percentage) and $n$ is the number of years.

Compound interest handles a single lump sum, but many real plans involve regular payments - a fixed amount saved every month, or a pension paying out a fixed amount each year. A sequence of equal payments at equal intervals is an annuity, and the GDC finance solver is built for it. The one new ingredient is a non-zero PMT: the regular payment. Everything else is the same five-label screen, and the sign convention still rules - deposits you make are outflows (negative), withdrawals you receive are inflows (positive).

Amortization is an annuity in reverse: you receive a lump sum now (the loan) and pay it back in equal instalments, each covering the interest for that period plus a slice of the outstanding principal, until the balance reaches zero. In the solver the loan is a positive PV (money you receive) with FV = 0 (fully repaid), and you solve for the payment PMT, which comes out negative because it is money you pay. Loan questions almost always have a second part - total paid or total interest - so read to the end.

On Paper 2 the marks are analytic: each is tied to a specific line of working - a method mark (M) or an accuracy mark (A) - and an accuracy mark depends on the method mark it follows. Follow-through (FT) means an earlier slip need not cost the marks that depend on it, provided the later step is done correctly on your own figures. In AI the finance solver is the expected tool, so you must state the values you enter - that IS the method. Study how each mark below is earned by a specific line.

Financial mathematics is exponential growth and decay wearing a business suit. The compound interest factor $\left(1 + \frac{r}{100k}\right)^{kn}$ is the same geometric growth you meet in sequences and series, and the reducing-balance depreciation model is its decaying twin - both are geometric sequences in disguise. The habit built here, of naming the model, fixing the signs and showing the solver inputs before quoting an answer, is exactly the discipline every Paper 2 modelling question rewards. Master the one formula and the one calculator screen, and saving, borrowing, depreciation and inflation all reduce to the same short piece of working, marked the same predictable way.