Timelines and the ordinary-annuity present value

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Level payments at the end of each period — draw the timeline, value the stream with =PV(rate,nper,pmt) and with the annuity factor, and check that the factor is always less than n.

The timeline, and where each flow sits

An annuity is a series of equal payments at equal intervals. In an ordinary annuity each payment falls at the end of its period: the first at $t = 1$ , the last at $t = n$ . Nothing happens at $t = 0$ — and a cash flow at $t = 0$ is neither discounted nor compounded; it is simply its own value. Draw the line before you compute: the tick marks are the ends of periods, and a payment sits on a tick.

The present value is the sum of the discounted payments,

$P V = t = 1 \sum n \frac{P M T}{( 1 + r ) ^{t}} = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r}$

The bracketed quantity is the present-value annuity factor (PVAF): the PV of $1 a period for $n$ periods. Excel's =PV(rate, nper, pmt) does the whole thing; with pmt entered positive (money you receive) the result is negative (what you would pay for it), so wrap it: =-PV(…).

Worked example — Prairie Grid's lease

Prairie Grid Energy receives $1,200 at the end of each year for 10 years under a land lease. At 6 %, what is the stream worth today?

$P V A F = \frac{1 - 1.0 6 ^{- 10}}{0.06} = \frac{1 - 0.558395}{0.06} = 7.3601$

$P V = 1, 200 \times 7.3601 = 8, 832.10$

=-PV(0.06,10,1200) → 8,832.10; by the formula, =1200*(1-(1+0.06)^-10)/0.06. Ten payments of $1, 200 t o t a l$ 12,000; their value today is $8,832.10 because each one is discounted, the last by ten years.

Bigel's check: PVAF < n

The factor 7.3601 is less than 10, and it always will be: $n$ dollars received one per period cannot be worth more than $n$ dollars today when the rate is positive. A PV cell larger than $n \times P M T$ — here, more than 12,000 — is a sign or period error, not a valuation.

A monthly case

The same function prices a car loan. $350 a month for 4 years at 6 % compounded monthly: rate 0.06/12 = 0.005, nper 48. =-PV(0.005,48,350) = 14,903.11 — the loan those payments repay. Rate ÷ 12 and years × 12 together, and the payment is the monthly one, exactly as Lesson 6.3 warned. Forty-eight payments total 16,800; the loan is 14,903.11; the difference is interest.

Recompute at 8 %

The lease at 8 %: =-PV(0.08,10,1200) = 8,052.10 — a higher rate, a smaller PV, as for a lump sum. The car loan at 8 % monthly: =-PV(0.08/12,48,350) = 14,336.67. Same payments buy less loan when money costs more.

Ordinary annuity: the first payment is one period away, the last at t = n, and the PV lands at t = 0 — one period before the first payment.