Reading: Annuities

Example 1

Consider an investment proposal that pays an annual cash flow of 100 at the end of each year for the next 3 years. The cost of capital is 10%. What's the present value of this investment proposal?

With the knowledge from the section on Present Values, this question is easy to answer:

\( PV_0 = \frac{100}{1.1}+\frac{100}{1.1^2}+\frac{100}{1.1^3} = 248.69 \)

While this solution is correct, it is not particularly elegant. Especially for long series of constant cash flows, the computations can get cumbersome so that it would be nice to have an elegant shortcut (imagine, for example, that the cash flow stream above runs for 20 years instead of only 3 years).

To find that shortcut, we can build on the knowledge from the previous section on perpetuities. More specifically, what we can try to do is replicate the cash flows of our annuity with a portfolio of perpetuities:

• For the first three years, the cash flows of the project are identical to those of an ordinary perpetuity that pays a cash flow of 100.
    
• Beyond year 3, however, the ordinary perpetuity keeps delivering cash flows of 100 each year, whereas our annuity does not.
    
• To neutralize these cash flows, we could add a second ordinary perpetuity that starts in year 3 (and therefore has its first cash flow in year 4) and delivers offsetting cash flows. In our case, if the second perpetuity has a cash flow of -100 each year, the two perpetuities together will cancel out each other after year 3.
    
• The following table summarizes the two perpetuities and how they replicate the original annuity:
   
  

  Year	0	1	2	3	4	5	...
  PerpetuityStart today		100	100	100	100	100	...
  −PerpetuityStart year 3					-100	-100	...
  Annuity		100	100	100	0	0	0

Now that we have built the replicating portfolio, we can value the two perpetuities separately. All the relevant knowledge has been developed in the section on perpetuities: We know how to value a perpetuity that starts today, and we know how to value a perpetuity that starts in 3 years. 

PV Annuity = PV Ordinary Perpetuity_{Start today} − PV Ordinary Perpetuity_{Start year 3}

In keeping with the common notation for annuities, we use the capital letter T to denote the duration of the annuity (in our case, T = 3). Consequently, the first payment of the second perpetuity occurs at time (T+1). In the preceding section on perpetuities, we have denoted that first payment of the perpetuity with the letter n. The following notation substitutes n with (T+1).

\( \text{PV Annuity} = \frac{C}{R} - \frac{C}{R} \times (1+R)^{1-(T+1)} \)

This can be rewritten as:

\( \bf{\text{PV Annuity} = C \times \frac{1-(1+R)^{-T}}{R}} \)

The equation shows that the present value of an annuity corresponds to the annual payment of the annuity, times a factor that incorporates the duration of the annuity (T) and the cost of capital (R). This factor is generally referred to as the Present Value Interest Factor of an Annuity, short PFIVA.

\( \bf{\text{PV Annuity} = C \times PVIFA_{R,T}} \)

\(\bf{\text{with:} \ PVIFA_{R,T}=\frac{1-(1+R)^{-T}}{R}}\)

In our original example, R = 10% and T = 3 years. Consequently:

\( PVIFA_{10\%,3}=\frac{1-(1+R)^{-T}}{R}=\frac{1-1.1^{-3}}{0.1} = 2.4869 \)

so that:

\( \text{PV Annuity} = C \times PVIFA_{R,T} = 100 \times 2.4869 = 248.69 \),

which is the same result as under the computation above.

Since PVIFA solely depends on the duration of the annuity (T) and the discount rate (T), we can easily express it in a tabular form (click to enlarge or see the worksheet "PVIFA Ordinary Annuity" of the accompanying Excel file): 

Example 2

For the next 15 years, a project pays a constant annual cash flow of 200'000. The first cash flow occurs in exactly one year and the cost of capital is 8%. 

Based on this information, we can go to the PVIFA-table and look up the relevant value for a duration of 15 years and an interest rate of 8%. The corresponding factor is 8.5595, so that the present value of the annuity is approximately 1.7 million:
 

\( PV = C \times PVIFA_{8\%,15} = 200'000 \times 8.5595 = 1'711'900 \)

The same logic also applies to annuities that offer cash flows at the beginning of the investment period, so called annuities due. This is what the next example shows.