keymodels
Time value of money
When and how should time value of money be applied?
Contents
Would you prefer to be given a dollar today, or in one year’s time?
A dollar received now is worth more than a dollar received later because today’s dollar can be invested, earns a return and remains available for use. This principle—the time value of money—allows cash flows occurring at different dates to be compared on a common basis.
When to use it
- Compare personal investments or financing choices.
- Evaluate the economic return of a project or acquisition.
- Translate amounts received or paid at different points in time into equivalent values.
Origins
The intuition is older than modern finance: lending, interest and the preference for earlier payment have existed for centuries. Accounting historian R. H. Parker reported in 1968 that surviving interest-rate tables date to 1340. Over time, commercial practice and mathematics formalised the intuition into present-value, future-value, annuity and discounted-cash-flow techniques.
What it is
An interest rate is the price paid for using money over time. With simple interest, each period’s interest is calculated only on the original principal. If the principal is $100 and the annual rate is 10 per cent, each year adds $10:
| Period | Interest calculation | Interest | Opening balance | Closing balance |
|---|---|---|---|---|
| Year 1 | 10 per cent of $100 | $10 | $100 | $110 |
| Year 2 | 10 per cent of $100 | $10 | $110 | $120 |
| Year 3 | 10 per cent of $100 | $10 | $120 | $130 |
| Year 4 | 10 per cent of $100 | $10 | $130 | $140 |
| Year 5 | 10 per cent of $100 | $10 | $140 | $150 |
With compound interest, earned interest is reinvested, so later returns are earned on both the original principal and accumulated interest. Assuming no tax, the same investment begins as follows:
| Period | Interest calculation | Interest | Opening balance | Closing balance |
|---|---|---|---|---|
| Year 1 | 10 per cent of $100.00 | $10.00 | $100.00 | $110.00 |
- Both series begin at $100 and use a 10% annual rate over 20 years.
The next compounded balance is $121.00. The following period adds $12.10 to $121.00, producing $133.10. The sequence then continues:
| Period | Interest calculation | Interest | Opening balance | Closing balance |
|---|---|---|---|---|
| Year 4 | 10 per cent of $133.10 | $13.31 | $133.10 | $146.41 |
| Year 5 | 10 per cent of $146.41 | $14.64 | $146.41 | $161.05 |
Across 20 years, simple interest produces linear growth while compound interest produces geometric growth. The chart keeps the complete 0–20-year horizon and $0–$800 scale visible, making the widening gap directly comparable.
Compound value can be written as:
Pn = P0(1 + I)n
where Pn is the ending value after n periods, P0 is the starting value and I is the interest rate per period.
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