1.4 Financial applications of geometric sequences and seriesIB Maths: Applications and Interpretation HL: Revision notes

With compound interest, interest is added to the balance and then earns interest itself, so the balance forms a geometric sequence. The future value is $FV=PV\times\left(1+\frac{r}{100k}\right)^{kn},$ where $PV$ is the amount invested, $r\%$ is the nominal annual interest rate, $k$ is the number of compounding periods per year and $n$ is the number of years. Example: 5000 EUR at 3% compounded annually for 8 years: $5000\times1.03^{8}=6333.85$ EUR. Compare simple interest, which adds the same amount each year: $5000+8\times150=6200.00$ EUR. Simple interest is arithmetic; compound interest is geometric. You are not asked to derive the formula.

Interest can be compounded yearly ($k=1$), half-yearly ($k=2$), quarterly ($k=4$) or monthly ($k=12$). More frequent compounding gives a slightly larger balance. For 5000 EUR at a nominal 3% for 8 years: annually $6333.85$ EUR, quarterly $5000\times1.0075^{32}=6350.56$ EUR. The number of periods is $kn$ and the rate per period is $\frac{r}{k}\%$. Check that your exponent is $kn$, not $n$.

Depreciation is a fall in value. For an annual depreciation of $r\%$, the value after $n$ years is $V=V_0\left(1-\frac{r}{100}\right)^{n}.$ Example: a car costing 24 000 EUR that depreciates 12% a year is worth $24\,000\times0.88^{5}=12\,665.57$ EUR after 5 years. The total fall is $1-0.88^{5}=47.2\%$ of the original price. The fall is 12% of the current value each year, so the loss in euros gets smaller each year. It is not a straight-line fall.

Inflation is the rate at which prices rise. The real value of an amount tells you what it is worth in today's money: $\text{real value}=\frac{\text{future value}}{\left(1+\frac{i}{100}\right)^{n}},$ where $i\%$ is the annual inflation rate. Example: 8000 AED grows to 9237.74 AED in 6 years. With 1.8% inflation, the real value is $\frac{9237.74}{1.018^{6}}=8300.02$ AED. This is more than 8000 AED, so buying power has increased. If the real value is less than the amount deposited, the account has not kept up with inflation.

Your GDC finance solver has entries $N$ (years), $I\%$ (nominal annual rate), $PV$, $PMT$ (use 0), $FV$, $P/Y$ and $C/Y$ (both equal to $k$). Enter the values you know and solve for the unknown. Remember the cash-flow signs: money you pay in is negative (for example $PV=-8000$) and money you receive is positive. You can find $FV$, $PV$, the number of years $N$ or the rate $I\%$. Write down the values you entered as your working, and give money to 2 decimal places. Compound growth is an exponential model, which links to exponential functions in topic 2.