TheIBAxolotl
Interest on interest
With compound interest, each period's interest is added to the balance and then earns interest itself. The balance therefore follows a geometric sequence. If the nominal annual rate is \(r\%\) and interest is added \(k\) times a year, the rate per period is \(\dfrac{r}{100k}\) and after \(n\) years there have been \(kn\) periods:
\[FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}\]
Common values of \(k\): 1 (annually), 2 (half-yearly), 4 (quarterly), 12 (monthly).
Worked example
Ana invests USD 8000 at a nominal 3.6% per year compounded monthly for 5 years.
\[FV = 8000\left(1 + \frac{3.6}{1200}\right)^{60} = 8000 \times 1.003^{60} \approx 9575.16\]
She earns about USD 1575.16 in interest. The same rate compounded annually would give \(8000 \times 1.036^{5} \approx 9547.48\): more frequent compounding earns slightly more.
Depreciation
Equipment and vehicles typically lose a fixed percentage of their value each year:
\[V_n = V_0\left(1 - \frac{r}{100}\right)^{n}\]
A van bought for USD 32 000 that depreciates by 18% a year is worth \(32\,000 \times 0.82^{4} \approx\) USD 14 467.90 after 4 years.
Using the finance solver (TVM)
Every GDC has a time-value-of-money solver. Enter all values except the unknown and solve for it.
| Field | Meaning |
|---|---|
| N | total number of periods (\(kn\)) |
| I% | nominal annual rate (not divided by \(k\)) |
| PV | present value; money you pay out is negative |
| PMT | regular payment (0 for a single investment) |
| FV | future value; money you receive is positive |
| P/Y, C/Y | payments and compounding periods per year |
The sign convention matters: an investment has PV negative and FV positive. If both have the same sign, the solver gives an error or nonsense.
Inflation and real value
Inflation reduces what money can buy. To compare across time, find the real value: divide a future amount by the inflation factor, for example USD 10 000 in 5 years with 3% inflation is worth about \(10\,000 \div 1.03^{5} \approx\) USD 8626 in today's money. Equivalently, a rough real interest rate is the nominal rate minus the inflation rate; use it in the solver when the question asks for this approach.
Comparing options
Typical questions compare two accounts, or ask for the rate or time needed to reach a target. Find the value of each option at the same date, or use the solver for N or I%. When solving for time, round N up to a whole number of periods and convert to years if asked.
Common mistakes
- Putting the period rate into I% (the solver wants the annual rate).
- Forgetting to set C/Y and P/Y to match the compounding.
- Using \(1 + r\) instead of \(1 - r\) for depreciation.
- Giving money to 3 significant figures; answers in currency are normally given to 2 decimal places or as the question states.
Exam tips
- Write down the solver entries (N = 60, I% = 3.6, PV = −8000, PMT = 0, P/Y = C/Y = 12): this earns method marks.
- Interest earned = FV − PV, not FV.
- Read whether the question wants the value "after" or "at the start of" a year.