NPV of Perpetuity Calculator & Growing Perpetuity Guide

What is the Net Present Value of a Perpetuity?

In corporate valuation, a Perpetuity is a continuous, unending stream of cash flows with no terminal date. Calculating the Net Present Value involves finding the mathematical limit of an infinite geometric series discounted to the present day, and deducting any initial purchase price or investment outlay.

Preferred Stock & Consols

Fixed dividend streams paid without expiration, valued as constant perpetuities ($PV = Div / r$).

Commercial Ground Leases

99-year or infinite land leases with CPI annual rent escalators modeled as growing perpetuities.

DCF Terminal Value (TV)

Capturing 60%–80% of total enterprise value beyond the 5-year discrete forecast horizon.

1. Constant Perpetuity Formula

When the periodic cash flow ($C$) remains identical across all infinite periods:

PV = C / r | NPV = (C / r) − Initial_CostMathematical Proof: The infinite sum ∑_t=1^∞ [ C / (1+r)^t ] is an infinite geometric series with initial term a = C / (1+r) and common ratio k = 1 / (1+r). Using the infinite sum formula S = a / (1 − k), we obtain: [C / (1+r)] / [1 − 1/(1+r)] = C / r.

2. Growing Perpetuity Formula (Gordon Growth Model)

When cash flows grow at a constant annual percentage rate ($g$) forever:

PV = C₁ / (r − g) | NPV = [ C₁ / (r − g) ] − Initial_CostCrucial Convergence Constraint: The discount rate ($r$) must strictly exceed the perpetual growth rate ($g$). If $g \ge r$, the present value is mathematically infinite and cannot be discounted.

Application: Terminal Value in DCF Modeling

In professional Discounted Cash Flow (DCF) models, it is impossible to forecast line-item revenues 50 years into the future. Instead, analysts project explicit cash flows for years 1 through 5, and then apply the Growing Perpetuity formula to establish the Terminal Value (TV) representing all subsequent cash flows into perpetuity.

For finite multi-year project cash flows, use our Annual NPV Calculator or explore our guide on the NPV Formula & Manual Calculation Steps.

Frequently Asked Questions

Q1What is a Perpetuity in finance?

A Perpetuity is an infinite stream of identical or growing cash flows that continues indefinitely with no maturity date. Common examples include perpetual bonds (consols), preferred stock dividend payments, real estate ground leases, and endowment funds.

Q2What is the formula for the Present Value and NPV of a Perpetuity?

For a constant perpetuity:
PV = C / r and NPV = (C / r) − Initial_Outlay
where C is the constant periodic cash payment and r is the discount rate.

Q3What is a Growing Perpetuity (Gordon Growth Model)?

A growing perpetuity has cash flows that increase at a constant annual growth rate (g) forever:
PV = C_1 / (r − g)
where C_1 is the cash flow received in Year 1, r is the discount rate, and g is the constant growth rate. Note that r must be strictly greater than g (r > g) for the sum to converge.

Q4How is perpetuity math used to calculate Terminal Value in DCF models?

In a multi-stage Discounted Cash Flow (DCF) model, analysts forecast explicit cash flows for 5 to 10 years, and then calculate the Terminal Value at the end of the forecast period using the Gordon Growing Perpetuity formula: Terminal_Value = [ FCF_n × (1 + g) ] / (WACC − g).