04 // Interactive Formula Compendium // FINANCEPV = PMT · [ 1 − (1 + r)⁻ⁿ ] / r
Present Value of an Annuity & Loan Amortization Formula Sheet
The Present Value of an Ordinary Annuity formula PV = PMT × [1 − (1 + r)⁻ⁿ] / r calculates the current lump-sum value of n equal end-of-period payments PMT discounted at periodic interest rate r.
Formula Reference
Main Formula
PV = PMT · [ 1 − (1 + r)⁻ⁿ ] / r
Rearranged Forms
PMT = PV · [ r(1 + r)ⁿ ] / [ (1 + r)ⁿ − 1 ] (Fixed Loan Payment)
FV = PMT · [ (1 + r)ⁿ − 1 ] / r (Future Value of Ordinary Annuity)
PV_due = PV_ordinary · (1 + r) (Annuity Due: Payments at Start of Period)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| PV | Present Value of Annuity (PV) | USD |
| PMT | Periodic Payment Amount (PMT) | USD |
| r | Periodic Discount Rate (r) | 1/period |
| n | Total Number of Periods (n) | periods |
Inputs & Parameters
[USD]
[1/period]
[periods]
Calculated Result (Present Value of Annuity (PV))
PV =166,791.6USD
Formula Notes & Background
- Write PV as the sum of discounted payments: PV = PMT/(1+r)¹ + PMT/(1+r)² + … + PMT/(1+r)ⁿ.
- Factor out PMT/(1+r) and apply the finite geometric series sum formula with common ratio q = 1/(1+r).
- Simplifying [1 − (1+r)⁻ⁿ] / [1 − 1/(1+r)] cancels the (1+r) factor and leaves PV = PMT · [1 − (1+r)⁻ⁿ] / r.
Step-by-Step Examples
#1: Find the maximum mortgage loan PV supported by a $1,000/month payment at 6% APR (r = 0.005/mo) for 30 years (n = 360).
PMT = $1,000r = 0.06 / 12 = 0.005n = 360 months
PV = 1000 × [1 − (1.005)⁻³⁶⁰] / 0.005 = 1000 × [1 − 0.16604] / 0.005
→ PV = $166,791.61
#2: Find the lump sum needed today to fund $50,000/year retirement withdrawals for 20 years at 5% annual return.
PMT = $50,000r = 0.05n = 20
PV = 50000 × [1 − (1.05)⁻²⁰] / 0.05 = 50000 × 12.46221
→ PV = $623,110.52
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Frequently Asked Questions
What is the difference between an Ordinary Annuity and an Annuity Due?+
An ordinary annuity pays at the end of each period (like mortgages and bonds), whereas an annuity due pays at the beginning of each period (like rent and leases), making its PV higher by a factor of (1 + r).
What happens to the annuity formula as n → ∞ (a perpetuity)?+
For r > 0, (1 + r)⁻ⁿ → 0 as n → ∞, so the formula simplifies to the perpetuity equation PV = PMT / r.
Why must r and n use matching time units?+
If payments are monthly, r must be the monthly rate (APR / 12) and n must be the total number of months (years × 12).