7.3 Annuities

Since, in practice, very few financial arrangements include “forever” as a reasonable timeframe, we should consider the value of cash flows over shorter periods. Any set of constant periodic payments lasting for a fixed amount of time is called an annuityConstant periodic payments over a fixed amount of time.. Paying down a credit card, mortgages, lottery payouts, and more can all be modeled as annuities. All annutities can be valued using the “long way”, by calculating the NPV of the cash flows, but we can also value them using mathematical intuition.

If we want to figure out the value of $10/year for 15 years, and the current interest rate is 5%, then we can use our perpetuity formula to determine the value of the annuity in the following way:

1. Determine the value of an equivalent perpetuity (in this case, $10/year at 5% gives a PV for the perpetuity of $200).
2. Determine the PV of the value of the perpetuity at the end of the annuity period (in this case, the PV of $200 in 15 years is $96.20).
3. Subtract #2 from #1 to get the value of the annuity ($200−$96.20 = $103.80).

$\text{PV }=\frac{PMT}{r}-\frac{\frac{PMT}{r}}{{(1+r)}^n}$ $\text{PV }=\frac{PMT}{r}\times (1-\frac{1}{{(1 + r)}^n})$

Sometimes an annuity will also include an additional one-time cash flow at the conclusion of the annuity. In this case, it is a simple matter of adding the PV of this cash flow to your calculated value to get the total value of the annuity. Continuing our example, if the annuity delivered $100 at its conclusion, then the PV would increase by the PV of $100 in 15 years at 5%, or $48.10, making the total value $103.80 + $48.10 = $151.90.

Financial calculators and spreadsheets can also easily solve annuities. On the calculator, the PMT key accepts the input for the periodic payment, and any four inputs can be used to calculate the fifth.

Using a financial calculator:

<CLR TVM> 15 <N> 5 <I> 10 <PMT> 100 <FV> <CPT> <PV> should show the proper solution of −151.90.

Spreadsheet:

=PV(rate, nper, pmt, fv) =PV(5%, 15, 10, 100) −151.90

Also, PMT can be solved for given the other four inputs. =PMT(rate, nper, pv, fv) is the corresponding spreadsheet function. It is critical that we always input our rate, number of periods, and payment based upon the same time period. For example, if payments are monthly, then rate should be a monthly rate, and number of periods will be the number of months.

With most annuities, each payment is assumed to occur at the end of each period. When payment in each period occurs at the beginning instead of the end, the annuity is called an annuity dueAn annuity with payments due at the beginning of each period instead of at the end..

While technically each payment is occurring at the beginning of its period instead of the end, in effect, the only change is that the very last payment has been moved to the very beginning! Thus, the value of an annuity due is higher by the difference between the PV of a payment at the end and the PV of a payment now (which, by definition, is equal to the total payment).

$P{V}_{due}=P{V}_{ordinary}+(\text{PMT}-\frac{PMT}{{(1+r)}^n})$

If our example were an annuity due, its value would increase by $10 minus the PV of $10 in 15 years at 5%, or ($10 – $4.81) = $5.19.

Financial calculators can be set to solve for annuity due (see appendix). Most spreadsheet functions can be set to automatically calculate an annuity due by taking a fifth argument, “type”. When type is omitted, it is assumed to be calculating an ordinary annuity, but by using an input of “1” instead, an annuity due is calculated:

=PV(rate, nper, pmt, fv, type) =PV(5%, 15, 10, 100, 1) −157.09