11.21: Fractional Time Periods
So far, we have dealt only with whole periods and, therefore, whole exponents. True, you may argue that we have also broken whole periods into halves (or other pieces), such as (whole period) half years, so that, for example, an annual rate of 10% compounded / discounted semi-annually became 5% per period; exponents were still whole numbers. But let’s not stop here!
Occasionally, cash flows may occur before the end of a period. Suppose you earn 10% per year, compounded daily. What would be the FV after ¾ of a year?
If we assume a 360-day year 13 , ¾ × 360 = 270 days. (In short-term financial analysis, we often deal with, or otherwise assume, twelve 30-day months to a year.) Further, 10% ÷ 360 = .0002777. Therefore, after 270 days, you will have earned:
$1 (1 + .10/360) 270 =
$1 (1.0002777) 270 = 1.077850
By comparison, if the compounding period is three months or a quarter of a year, then:
1 (1 + .10/4) 3 = 1.0769
If, in fact, the compounding period is a whole year, then we may employ a fractional exponent:
1 (1.10) 3/4 = 1.0741
All of the foregoing choices are arithmetically correct. The compounding convention you use matters. In practice, you will have to know the proper convention commonly used in each situation.
Here are some more examples. Notice that if we calculate the FV of $1 at 10% annually, after half a year we would have arrived at an FV of $1.05 ($1 (1 + .10/ 2) 1 ) – as we may have assumed until now. However, if we use fractional exponents, we get a slightly different result:
$1 (1.10) 1/2 = $1.048809
$1 (1 + .10/2) 1 P ≠ $1 (1.10) 1/2
It is therefore very important to understand the context of the problem and the use of the appropriate compounding/discounting convention.
The Take-away : Compounding assumptions matter. Be sure you use the correct assumptions.