Present Value of an Ordinary Annuity

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

$PV$: what the whole stream of payments is worth today
$PMT$: the equal payment received at the end of every period
$(1+r)^{-n}$: a negative exponent means divide: this shrinks 1 by n periods of discounting
$\frac{1-(1+r)^{-n}}{r}$: the present-value annuity factor: today's price of intuition: ` per period for n periods

Why it must be true

A stream of equal end-of-period payments is worth the sum of each payment's present value. Nearby payments are worth almost their face amount; distant ones are worth much less. The factor $\frac{1-(1+r)^{-n}}{r}$ collapses that whole discounted sum.

This is also the loan formula read backwards: a lender hands over $PV$ today precisely because the repayments $PMT$ discount back to it. Solving for $PMT$ prices a mortgage.

The derivation

Discount each payment individually:

PV = \frac{PMT}{(1+r)} + \frac{PMT}{(1+r)^2} + \dots + \frac{PMT}{(1+r)^n}

This is a geometric series with first term $\frac{PMT}{1+r}$ and ratio $\frac{1}{1+r}$ . Applying the geometric-sum identity and simplifying:

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

As $n \to \infty$ the factor tends to $\frac{1}{r}$ — the perpetuity formula, its natural limit.

When to reach for it

Valuing a stream of equal end-of-period payments today — pricing a settlement or pension, or (rearranged) finding a loan's installment.

Listen for

“equal payments for n periods”“amortizing loan / installments”“what lump sum replaces the stream”“coupon payments (the bond's income leg)”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Ceiling check: the annuity factor can never exceed pitfalls: [/r$ (the perpetuity). At 8%, a stream of \ pitfalls: [ payments is worth at most \ pitfalls: [2.50 — instant upper bound.
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Short streams: factor ≈ n minus a bit. 5 payments at 6% → factor ≈ 4.2 (not 5, not 3).
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Loan payments must beat interest-only: $PMT > PV \times r$ , or the balance never shrinks. Any answer below interest-only is wrong on sight.

Traps in applying it

✗The factor prices the stream one period BEFORE the first payment — if the first payment is today, it's an annuity due.
✗n is the number of payments, not years — a 5-year monthly loan has n = 60.
✗Using the coupon rate as the discount rate — the market yield discounts; the coupon only sets the payment.

Limits & criticisms

Same idealization as all annuity math: level payments, one flat rate. Long annuities are violently rate-sensitive, so a small error in r compounds into a large pricing error. The formula also prices the promise, not the promisor — the payer's credit risk has to enter through the discount rate or not at all.

Where it came from

The first scientifically priced financial product was an annuity. Johan de Witt, ruler of the Dutch Republic and a trained mathematician, published The Worth of Life Annuities in 1671, showing the state was selling annuities too cheap by discounting each future payment. Edmond Halley's 1693 life table put the method on real mortality data.

Today the same factor prices the other side of the trade: every mortgage, auto loan and structured settlement is an annuity valued this way — and a bond's coupon leg is nothing else.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Value an income stream

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

The pricing face: each payment discounted back to today, collapsed into one factor. Pensions, settlements and bonds' coupon legs are valued this way.

Drill this face →

Loan (amortization) payment

PMT = \frac{PV \cdot r}{1-(1+r)^{-n}}

The borrowing face, read in reverse: the lender hands over PV precisely because the installments discount back to it. Every mortgage payment comes from here.

Drill this face →

On the BA II Plus

Worked example: A loan of $37,410.92 is repaid with equal end-of-period payments over 14 periods at 2.5% per period. What is the payment?

1.[2ND] [CLR TVM]always clear the worksheet first
2.14 [N]
3.2.5 [I/Y]the rate per period, as a percent
4.37,410.92 [+|-] [PV]cash paid out is negative
5.0 [FV]
6.[CPT] [PMT]compute the unknown

→ $3,200.00

Where it leads

Master this and the following come almost for free:

Present Value of a Perpetuity Bond Price & Yield to Maturity

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