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7.3 Annuities

PLEASE NOTE: This book is currently in draft form; material is not final.

Learning Objectives

  1. For an annuity, calculate the fifth variable given four of: PV, FV, rate, PMT, and n.
  2. Calculate the PV of an annuity due.

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.

Figure 7.6 Annuity Timeline (General)

Figure 7.7 Annuity Example Timeline

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).

Equation 7.4 PV of an Annuity

PV of annuity = PV of equivalent perpetuity – PV of perpetuity at end of annuity
PV = PMTrPMTr(1+r)n PV = PMTr × (11(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.

Watch the Direction of Cash Flows

As we get more complex with or calculations, it becomes all the more important to keep track of whether our cash flows are inflows or outflows. For example, if a one-time payment in this example were instead a cash outflow, then our value would drop by $48.10 to a total value of $55.70.

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.

Why Is the Result Negative?

Calculators and spreadsheets always assume that a financial transaction is occurring, so they figure out the “fair value” based on the inputs. Since the NPV of the annuity’s inflows is $151.90, the calculated answer is that you should be willing to pay $151.90 (an outflow) to receive the scheduled payments in a completely “fair value” transaction.

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..

Figure 7.8 Annuity Due Timeline

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).

Equation 7.5 PV of an Annuity Due

PV of annuity due = PV of annuity + (payment – PV of payment at end)
PVdue= PVordinary + (PMTPMT(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

Key Takeaways

  • Annuities are a fixed stream of payments over a finite time. Given four of PV, FV, rate, PMT, and n, the fifth variable can be solved for.
  • Rate, PMT, and n should all be in the same time units!
  • An annuity due is a type of annuity where the payments are due at the beginning of the period.

Exercise

  1. Jill purchases an annuity that pays $1,000 per month for the next 300 months, with no value at the end of the 300 months (FV=0). If interest rates are 12% APR, what is the fair PV of this annuity? What would it be worth if it were an annuity due?