AI Math
Open App

Present Value of an Annuity Calculator

PV annuity factors, annuity-due adjustments, perpetuities and growing streams, step by step
Present value of $2,500 a year for 12 years at 7%
Same payments made at the start of each year instead
Present value annuity factor for 12 periods at 7%
Present value of $4,000 a year forever at 5%, and growing 2% a year

The Present Value Annuity Factor

The present value of nn level payments CC, each discounted at rate rr per period, is

PV=C⋅1−(1+r)−nrPV = C \cdot \frac{1 - (1+r)^{-n}}{r}

  • CC — the payment per period
  • rr — the discount rate per period, as a decimal
  • nn — the number of payments

The fraction is the present value annuity factor, written an‾∣ra_{\overline{n}|r}. It is exactly the sum ∑t=1n(1+r)−t\sum_{t=1}^{n}(1+r)^{-t} collapsed by the geometric-series formula, which is why the factor equals the PV of a \1$ annuity.

That is all a PV annuity chart is. Every cell of a present-value-of-annuity table is (1−(1+r)−n)/r(1 - (1+r)^{-n})/r evaluated at the row's nn and the column's rr; you read off the factor and multiply by your own CC. A table of (1+r)−n(1+r)^{-n} alone is a present-value-of-1 table, used for single sums rather than streams.

Annuity Due, Perpetuity and Growing Streams

Annuity due. Payments at the start of each period are discounted one period less each, so

PVdue=C⋅1−(1+r)−nr×(1+r)PV_{\text{due}} = C \cdot \frac{1 - (1+r)^{-n}}{r} \times (1+r)

Perpetuity. Let n→∞n \to \infty and (1+r)−n→0(1+r)^{-n} \to 0, leaving

PV=CrPV = \frac{C}{r}

Growing perpetuity, with payments rising by g<rg < r forever:

PV=Cr−gPV = \frac{C}{r - g}

Growing annuity, over a finite nn with r≠gr \neq g:

PV=Cr−g[1−(1+g1+r)n]PV = \frac{C}{r-g}\left[1 - \left(\frac{1+g}{1+r}\right)^{n}\right]

Inverting the ordinary factor gives the payment a lump sum can support, C=PV⋅r/(1−(1+r)−n)C = PV \cdot r / (1 - (1+r)^{-n}) - the loan payment formula. The rate rr cannot be solved for algebraically; it needs iteration.

Common Mistakes to Avoid

  • Mixing an annual rate with monthly payments: use r=rannual/12r = r_{\text{annual}}/12 and count nn in months.
  • Reading a future-value table as a present-value table: FV factors exceed 1, PV factors are below nn and approach 1/r1/r.
  • Forgetting the (1+r)(1+r) for an annuity due: it undervalues the stream by one period of discounting.
  • Using the perpetuity formula when g≥rg \geq r: the series diverges and C/(r−g)C/(r-g) is meaningless.
  • Discounting a growing stream with the level-payment factor: unequal payments need the growing-annuity form.
  • Rounding the factor too early: keep six decimals; rounding to two can move a large PV by thousands.

Examples

Step 1: (1.07)12≈2.2521916(1.07)^{12} \approx 2.2521916, so (1.07)−12≈0.4440120(1.07)^{-12} \approx 0.4440120
Step 2: 1−0.4440120=0.55598801 - 0.4440120 = 0.5559880
Step 3: Factor: 0.5559880/0.07≈7.94268630.5559880 / 0.07 \approx 7.9426863
Step 4: PV≈2500×7.9426863≈19,856.72PV \approx 2500 \times 7.9426863 \approx 19{,}856.72
Step 5: Undiscounted the payments total 12×2500=30,00012 \times 2500 = 30{,}000
Answer: PV \approx \19{,}856.72(factor(factor7.942686$)

Step 1: Multiply the ordinary factor by (1+r)(1+r): 7.9426863×1.07≈8.49867437.9426863 \times 1.07 \approx 8.4986743
Step 2: PVdue≈2500×8.4986743PV_{\text{due}} \approx 2500 \times 8.4986743
Step 3: ≈21,246.69\approx 21{,}246.69
Step 4: The gap of \approx \1{,}389.97$ is exactly 7% of the ordinary present value
Answer: PV_{\text{due}} \approx \21{,}246.69$

Step 1: Level perpetuity: PV=C/r=4,000/0.05=80,000PV = C/r = 4{,}000 / 0.05 = 80{,}000
Step 2: Growing perpetuity: PV=C/(r−g)PV = C/(r-g) with r−g=0.05−0.02=0.03r - g = 0.05 - 0.02 = 0.03
Step 3: PV=4,000/0.03≈133,333.33PV = 4{,}000 / 0.03 \approx 133{,}333.33
Step 4: Growth of 2% raises the value by two thirds, because it shrinks the denominator from 0.05 to 0.03
Answer: \80{,}000level,orlevel, or\approx $133{,}333.33$ if the payments grow 2% a year

Frequently Asked Questions

The factor is (1 − (1+r)^(−n))/r, the present value of a $1 payment made at the end of each of n periods. Multiply it by your payment C to get the annuity's present value. Every cell of a PV annuity table is this expression evaluated at that row's n and column's r.

Find the row for the number of payments and the column for the rate per period, read the factor, and multiply by the payment. A 12-year, 7% factor is 7.942686, so $2,500 a year is worth 2,500 × 7.942686 ≈ $19,856.72 today.

Multiply the ordinary annuity present value by (1+r). Each payment arrives one period earlier, so it is discounted one period less. Some tables print due factors directly; check the heading before using one.

As n grows, (1+r)^(−n) approaches zero and the factor converges to 1/r, giving PV = C/r. A perpetuity has a finite value because distant payments are discounted so heavily they contribute almost nothing. If payments grow at g < r, the value is C/(r − g).

Related Solvers

Compound Interest CalculatorAmortization Schedule CalculatorGeometric Series Calculator
Try AI-Math for Free

Get step-by-step solutions to any math problem. Upload a photo or type your question.

Start Solving