Reading: Present Values

In the module section Compounding Frequency and Effective Annual Rate, we have learned how to deal with situations where the compounding frequency does not correspond to the time units we use to measure the investment horizon. For example, we have seen how to compute the future value of an investment that receives quarterly interest payments and runs over 5 years.

The same considerations apply for the computation of present values. In what follows, we show how to handle different compounding frequencies in the context of discrete compounding as well as continuous compounding.

Discrete Compounding

Remember from the referenced discussion that we have generalized the computation of future values as follows in a world with discrete compounding:

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

with C_t = cash flow at time t (t= 0 if cash flow is today); R = rate of return; m = number of interest payments per investment period (compounding frequency); and (T-t) = investment horizon (number of periods). 

We have also introduced the concept of the Effective Annual Rate (EAR), which is: 

\(EAR = (1+\frac{R}{m})^m-1\)

so that the computation of Future Values from above simplifies to:

\( FV_T = C_t \times (1+EAR)^{(T-t)} \)

We can proceed accordingly when computing present values. Let us illustrate this with a simple example. The relevant equations to compute the present value of a cash flow that occurs at time t (C_t) are:

\( \bf{PV_0 = \frac{C_t}{(1+EAR)^t}} \)

and:

\( \bf{PV_0 = \frac{C_t}{(1+\frac{R}{m})^{mt}}} \)

Example 4

An investment promises a cash flow of 30'000 in 4 years. An alternative investment with identical risk generates an annual return of 12% with quarterly compounding. What's the present value of the investment proposal?

Using the above notation, we can determine the effective annual rate of the alternative investment:

\(EAR = (1+\frac{R}{m})^m-1 = (1+ \frac{0.12}{4})^4-1 = 0.1255 = 12.55\%\)

Put differently, an annual return of 12% with quarterly compounding is equivalent to an annual return of 12.55% with annual compounding. With this information, the computation of the investment proposal's present value is straightforward:

\( \bf{PV_0 = \frac{C_t}{(1+EAR)^t}} = \frac{30'000}{1.1255^4} = 18'695 \)

The present value of the investment proposal is 18'695. We obtain the same result when we directly adjust the formula for present values:

\( PV_0 = \frac{C_t}{(1+\frac{R}{m})^{mt}} = \frac{30'000}{(1+\frac{0.12}{4})^{4 \times 4}} = 18'695 \)

Continuous Compounding

With continuous compounding, we have seen that the relevant expressions to compute the effective annual rate and the future value of a specific cash flow are:

 \( EAR_{R, \ continuous}=e^R-1 \)

and:

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

Again, we can reshuffle these expressions to compute the present value of C_t in an environment with continuous compounding. In particular:

\( \bf{PV_0 = C_t \times e^{-Rt}} \)

Example 5

An investment promises a cash flow of 30'000 in 3 years. The relevant cost of capital is 8% with continuous compounding. 

Based on this information, the present value of the investment proposal is 23'598.84:

\( PV_0 = C_t \times e^{-Rt} =30'000 \times e^{-0.08 \times 3} = 23'598.84 \)