How a rising stream of payments becomes one value today
Money you receive in the future is worth less than money in hand, so to value a stream of payments you must discount each one back to today and add them up. A growing annuity makes this concrete: the first payment is PMT, and every payment after it is larger by the growth rate g, while each is discounted at the rate r. Rather than summing period by period, the geometric series collapses to a single closed-form expression, which is exactly what this calculator evaluates. The result is the present value — the lump sum today that is financially equivalent to the entire rising stream.
The growing annuity formula
PV = PMT ÷ (r − g) × (1 − ((1 + g) ÷ (1 + r))ⁿ)
where PMT = the first payment, r = the discount rate per period, g = the growth rate per period, and n = the number of periods. When g = r the formula uses the equal-rate case PV = n × PMT ÷ (1 + r), and as ntends to infinity it becomes the growing perpetuityPV = PMT ÷ (r − g).
What makes this calculator different
- It handles the r = g edge case. Instead of returning a division-by-zero error when the growth rate equals the discount rate, it switches to the correct limiting formula, PV = n × PMT ÷ (1 + r).
- It shows the first and final payment. Because the payments grow, the last one can be far larger than the first — seeing both makes the effect of the growth rate tangible rather than abstract.
- It links to the growing-perpetuity idea. A growing annuity with infinite periods is a growing perpetuity, so you can extend the same inputs to an endless horizon with theperpetuity calculator.
Frequently asked questions
What is a growing annuity?+
A growing annuity is a finite stream of periodic payments where each payment is larger than the one before it by a constant growth rate g. Its present value compresses that whole stream into a single value today, discounting each future payment at the rate r. The closed-form result is PV = PMT ÷ (r − g) × (1 − ((1 + g) ÷ (1 + r))ⁿ), where PMT is the first payment. It is the natural tool whenever cash flows are expected to escalate rather than stay flat.
How does it differ from an ordinary (level) annuity?+
An ordinary annuity pays the same fixed amount every period, so its present value uses the standard PV = PMT ÷ r × (1 − (1 + r)⁻ⁿ). A growing annuity instead lets each payment rise by the growth rate g, which raises the present value relative to a level annuity that starts at the same first payment. In effect the level annuity is just the special case of a growing annuity with g = 0. Modelling growth matters most over long horizons, where even a small g compounds into a large difference.
What happens when the growth rate equals the discount rate?+
When g equals r the main formula breaks down because the term (r − g) in the denominator becomes zero, producing a division by zero. The limit is well defined, though, and the present value collapses to a simple expression: PV = n × PMT ÷ (1 + r). Intuitively, every discounted payment has the same present value, so the total is just n copies of that single value. This calculator detects the equal-rate case and switches to this formula automatically.
How is it related to the growing perpetuity?+
A growing perpetuity is a growing annuity carried out to an infinite number of periods. As n grows without bound and provided the discount rate exceeds the growth rate (r > g), the term ((1 + g) ÷ (1 + r))ⁿ shrinks toward zero, and the present value simplifies to the famous PV = PMT ÷ (r − g). So the growing annuity sits between a level annuity and a growing perpetuity, and you can explore the infinite-horizon version with the perpetuity calculator.
What real situations does it model?+
Growing annuities show up wherever income or expenses are expected to escalate at a steady rate. Common examples include escalating commercial leases where rent steps up each year, cost-of-living-adjusted (COLA) pensions whose payouts rise with inflation, and equities valued on a stream of growing dividends. They also describe inflation-linked income generally, where a nominally fixed payment is replaced by one that keeps pace with rising prices.
Disclaimer: This calculator is foreducation and illustration only. Present-value results depend entirely on the assumptions you supply — the discount rate, the growth rate, and the number of periods — and real cash flows rarely grow at a perfectly constant rate. Nothing here is investment, tax, or financial advice.