Reading: Compounding Frequency

In our original example, we were confronted with an investment of EUR 100'000 that makes semi-annual interest payments of 4% over an investment horizon of 5 years.

Instead of computing the investment's effective annual return, we could simply determine how many interest payments the investment will make over the investment horizon. Put differently, we could express the investment horizon (T) in half-years instead of years. In our case, an investment horizon of 5 years is equivalent to an investment horizon of 10 half-years (T = 10). Consequently, the EUR 100'000 we invest today will grow at a rate of 4% over the next 10 half-years, for a future value of EUR 148'024:

\( FV_5 = 100'000 \times 1.04^{10} = EUR \ 148'024 \)

This is the same result as in the preceding section, where we have first computed the EAR and then applied that EAR to an investment horizon of 5 years.

Generally speaking, the Future Value at time T of an investment C₀ that pays an annual indicated interest rate of R with compounding frequency m is:

\( \bf{FV_T = C_t \times \bigg(1+\frac{R}{m}\bigg)^{m \times (T-t)}} \)

This is the same equation as the one that we have derived in the concluding remarks of the preceding section. Not surprisingly, when we switch to continuous compounding, we will also find the same equation as in the preceding section, namely:

\( \bf{FV_T = C_t \times e^{R \times (T-t)}} \)

To practice these equations, see the example at the end of the preceding section.