TheIBAxolotl
Why it is geometric
When a bank adds interest to the balance and the next interest is calculated on the new balance, the money is multiplied by the same factor each period. That is exactly a geometric sequence. If the annual rate is \(r\%\) and interest is added \(k\) times a year, each period multiplies by \(1 + \frac{r}{100k}\), and over \(n\) years there are \(kn\) periods:
\[FV = PV\left(1 + \frac{r}{100k}\right)^{kn}.\]
Here \(PV\) is the present (starting) value and \(FV\) the future value. More frequent compounding gives slightly more money, because interest starts earning interest sooner.
Worked example: compounding
USD 5000 is invested at 4% per year, compounded monthly, for 6 years.
\[FV = 5000\left(1 + \frac{4}{1200}\right)^{72} \approx 6353.71.\]
So the investment is worth about USD 6353.71. With yearly compounding it would be \(5000 \times 1.04^6 \approx\) USD 6326.60, a little less.
Depreciation
Many assets lose a fixed percentage of their value each year. A car bought for USD 24 000 that depreciates by 15% a year is worth
\[24\,000 \times 0.85^5 \approx 10\,648.93\]
after 5 years. The multiplier is \(1 - \frac{15}{100}\). Depreciation is the same formula with a minus sign.
The finance solver
On Paper 2 the GDC finance (TVM) app solves for any one unknown: \(N\) (number of periods), \(I\%\) (annual rate), \(PV\), \(PMT\) (regular payment, 0 if none), \(FV\), and \(P/Y\), \(C/Y\) (payments and compounding periods per year). The key convention is signs: money you pay out is negative, money you receive is positive. So investing USD 5000 means \(PV = -5000\) and the answer \(FV\) appears positive.
To find how long until an investment doubles, enter \(PV = -1\), \(FV = 2\) and solve for \(N\). Remember \(N\) counts periods, so divide by \(C/Y\) to get years, and round up if the question asks when the target is first reached.
Inflation and real value
Inflation means the same amount of money buys less over time. To judge real growth, compare the investment with prices. A simple approach used in exams: real rate \(\approx\) nominal rate \(-\) inflation rate. For example, 5% interest with 2% inflation gives roughly 3% real growth per year. You can then use this real rate in the compound interest formula to express a future amount in today's money.
Comparing options
To compare two investments, compute each future value over the same time (or each time to reach the same target). Always present both numbers and state the conclusion in words.
Common mistakes
- Using the annual rate per month: 4% monthly is not \(\frac{4}{12}\%\) per year.
- Forgetting to multiply years by \(k\) in the exponent.
- Wrong signs in the finance solver, which gives an error or a negative answer.
- Rounding money too early; keep full accuracy and round to 2 decimal places at the end.
- Rounding \(N\) down when the question asks for the first time a value is exceeded.
Exam tips
- Write down every TVM entry you used; that earns method marks if the final value is wrong.
- Read whether the question wants the value, the interest earned (\(FV - PV\)) or the total paid.
- Give money to the nearest cent unless told otherwise.