Present Value Of Growing Annuity Calculator

Payments that do not stay still

Most annuity maths assumes every payment is identical. Ninety thousand this year, ninety thousand next year, ninety thousand for as long as the arrangement runs.

Very little in life behaves that way.

Rent goes up at renewal. Salaries rise. Lease payments escalate by contract. A pension indexed to inflation grows every year by definition. Dividends from a growing company climb. In all of those the payments are regular and predictable, but they are not equal, and a flat annuity formula will value them wrongly.

A growing annuity handles that case. Same regular intervals, same fixed number of payments, but each one is a set percentage larger than the last.

Four inputs: the first payment, the discount rate, the growth rate, and how many payments there are.

Two rates racing each other

PV = [ PMT / (r - g) ] × [ 1 - ((1 + g) / (1 + r))ⁿ ]

The structure is worth reading rather than just using. Everything hinges on the ratio (1 + g) / (1 + r), which is the two rates competing.

If growth is slower than the discount rate, that ratio is below 1, raising it to higher powers makes it smaller, and later payments contribute less and less. The stream converges.

If growth is faster, the ratio is above 1, and later payments are worth more in present value terms than earlier ones. That still gives a sensible answer for a fixed number of payments, and there is a section on it below because it surprises people.

Set g to zero and the whole thing collapses back to the ordinary annuity formula, which is a decent way to check you have entered things correctly.

A stream that climbs three percent a year

Five payments, the first of 90,000, growing at 3 percent a period, discounted at 7 percent.

The actual payments are:

The calculator returns 390,270.62.

Set against the flat version, five payments of 90,000 at the same 7 percent, which is worth 369,017.77, the escalation is worth 21,252.85.

Which is about 5.8 percent more value, for a payment stream that ends 12.6 percent higher than it started. The gap between those two percentages is the discounting doing its work: the largest payments arrive last, and last is where discounting bites hardest.

That is the general lesson about escalation clauses. Growth at the far end of a long stream is worth much less than it looks on the schedule, and the longer the stream and the higher the discount rate, the more that is true.

The first payment box is doing more than it looks

The field says first payment, and that word is load bearing.

Enter the payment for period one, before any growth has been applied. The formula grows it from there. If you enter the current payment when the first payment under the arrangement will already have grown once, every figure comes out low by a factor of (1 + g).

This trips people up most often with rent and salaries, where there is a payment happening now and a different one starting next period. The rule: whatever the first payment in the stream you are valuing will actually be, that is what goes in the box.

Two other things the four inputs assume, worth saying because neither is stated on the form.

Payments land at the end of each period. This is the ordinary annuity convention. If your payments arrive at the start, as rent and lease payments usually do, multiply the answer by one plus the discount rate. That is the same adjustment our annuity due calculator applies to flat streams, and it works identically here.

Growth is constant. Every period grows by the same percentage. Real escalations often step at intervals or track an index that moves unevenly. For a rough valuation a constant rate is fine. For anything precise, discount each payment separately with our discounted cash flow calculator.

And as always, the rate and the growth rate must be per period, matching whatever the periods are. Monthly payments need monthly rates.

When growth outruns the discount rate

Here is where this formula behaves differently from its infinite cousin, and it is worth understanding because the two get confused.

Look at the denominator, (r - g). If growth equals the discount rate exactly, that is zero, and the expression breaks. The calculator will return an error rather than an answer.

But the underlying value is not undefined at all. When r equals g, every payment discounts back to the same present value, so the answer is simply the number of payments multiplied by the first payment discounted one period:

PV = n × PMT / (1 + r)

On our example at 7 percent growth and 7 percent discount, that is 5 times 90,000 divided by 1.07, which is 420,560.75. Perfectly finite and perfectly sensible. If you hit the error, that is the calculation to do by hand.

And if growth exceeds the discount rate, the formula still works. Both the numerator and the denominator go negative and the negatives cancel, giving a positive and correct answer. A fifteen year lease escalating at 8 percent discounted at 6 percent is a real thing to value and the arithmetic handles it.

That is the crucial difference from a growing perpetuity, where growth above the discount rate genuinely does break: an infinite stream growing faster than it is being discounted has no finite value. Our perpetuity calculator refuses that case for exactly that reason, and it is right to.

So the rule to carry: for a fixed number of payments, any growth rate is fine. For a stream that never ends, growth must be below the discount rate. The number of periods is what makes the difference.

Questions people ask

Which payment goes in the first payment box?

The first one in the stream you are valuing, before any growth is applied. Not the average, and not the current payment if the stream starts with a higher one.

What if the growth rate equals the discount rate?

The formula divides by zero and errors. The correct answer in that case is the number of payments multiplied by the first payment divided by one plus the rate.

Can growth be higher than the discount rate?

For a fixed number of payments, yes, and the answer is valid. For a perpetuity it cannot be, since an infinite stream growing faster than it is discounted has no finite value.

My payments arrive at the start of each period.

Multiply the answer by one plus the discount rate. That is the same adjustment that turns an ordinary annuity into an annuity due.

Should I use inflation as the growth rate?

Only if the payments are genuinely indexed to it. If a lease escalates at a fixed contractual rate, use that rate. Using inflation for a fixed escalation gives the wrong answer in either direction.

My payments grow unevenly.

Then this formula does not fit, since it assumes one constant growth rate. Discount each payment individually with the discounted cash flow calculator.

What if I set growth to zero?

You get the ordinary annuity value, which is a useful way to check your inputs against our PVIFA calculator.

References

A note on the sources. Valuing a stream of future payments is not only a modelling exercise: United States Treasury regulations require present value factors of exactly this kind when annuities and term interests are valued for tax, and they permit exact computation by software in place of rounded published tables. The rate used in those valuations is published monthly by the Internal Revenue Service, which is a rare example of a discount rate that is prescribed rather than chosen. The compounding relationship underlying both the growth term and the discount term is defined by the Securities and Exchange Commission's investor education office.

1. Cornell Law School, Legal Information Institute, 26 CFR § 1.7520-1, Valuation of annuities, unitrust interests, interests for life or terms of years, and remainder or reversionary interests, on actuarial factors and exact computational methods. https://www.law.cornell.edu/cfr/text/26/1.7520-1
2. Internal Revenue Service, Section 7520 interest rates, the monthly prescribed rate for valuing annuities and future interests. https://www.irs.gov/businesses/small-businesses-self-employed/section-7520-interest-rates
3. U.S. Securities and Exchange Commission, Office of Investor Education and Advocacy, Compound Interest, Investor.gov glossary. https://www.investor.gov/introduction-investing/investing-basics/glossary/compound-interest
4. Brealey, R.A., Myers, S.C., and Allen, F., Principles of Corporate Finance, McGraw-Hill Education, chapters on growing annuities and growing perpetuities.