Future Value

Every future value calculation reduces to two inputs multiplied against a starting sum: the rate earned per compounding period and the count of those periods. When a rate is quoted as a stated annual figure with compounding more than once a year, divide it by the number of compounding periods per year to get the periodic rate, and multiply the number of years by that same figure to get the exponent. A stated 8% compounded quarterly over three years therefore means 2% per quarter for twelve quarters, not 8% for three years.

The growth factor itself, one plus the periodic rate raised to the number of periods, is a multiplicative chain. Each period's ending balance becomes the next period's beginning balance, so interest earned in period one earns interest in every period thereafter. Under continuous compounding the chain becomes the exponential function: the growth factor is e raised to the stated annual rate multiplied by the number of years.

Single sums are the base case; most applications involve a series. For an ordinary annuity, where equal payments arrive at the end of each period, future value equals the payment multiplied by the quantity one plus the periodic rate raised to the number of periods, minus one, all divided by the periodic rate. That bracketed term is the annuity factor, and it counts how many payment-equivalents the stream has grown into by the final date. An annuity due, with payments at the beginning of each period, gives every payment one extra period of compounding, so its future value is the ordinary annuity result multiplied by one plus the periodic rate.

Uneven cash flows have no shortcut. Each amount is compounded forward individually for the number of periods remaining between its date and the valuation date, then summed. Cash flows arriving on the terminal date are compounded zero periods and enter at face amount.

Future value logic underpins any instrument whose payoff is fixed at a known date. Discount instruments, deposit maturities and guaranteed investment contracts all quote a terminal amount that is the compounded product of price and yield. Effective annual rate comparisons are future value comparisons: the EAR is simply the one-year growth factor minus one, which is why two loans with identical stated rates but different compounding frequencies are not economically identical.

On the liability side, defined benefit pension projections compound current salaries forward at assumed growth rates before discounting the resulting benefits back. In performance measurement, terminal wealth divided by beginning wealth is the holding period return, and the geometric mean return is the constant periodic rate that would have produced the same ending value. Forward exchange rates and forward interest rates are also future value relationships enforced by arbitrage.

Candidates should be able to rearrange the relationship in all directions. Solving for the periodic rate means taking the ratio of future to present value, raising it to the reciprocal of the number of periods, and subtracting one. Solving for the number of periods requires logarithms: divide the natural log of the future-to-present ratio by the natural log of one plus the periodic rate. Both appear regularly in growth-rate and doubling-time questions, where the rule of 72 provides a fast sanity estimate.

Two discipline points prevent most errors. First, the periodic rate and the period count must refer to the same interval, always. Second, on a financial calculator, present value and future value carry opposite signs because one is an outflow and the other an inflow; failing to observe that convention returns an error rather than a wrong answer, which is a useful warning.