Growing Annuity Present Value Calculator

The growing annuity present value calculator shows what a stream of payouts that rise by a fixed percentage each period is worth to you today. Enter the first payment, how quickly each later amount grows, the interest rate and the number of periods, and you get the total value today of the whole income stream in one step. It is a quick way to price a rising payout, plan retirement cash flow, or compare two offers on equal terms.

How the growing annuity present value calculator works

A growing annuity pays you one amount and then a slightly larger amount every period after that, so the tool has to discount periodic payments of different sizes back to the present. Four numbers describe the whole stream, and one more setting decides when each payment lands:

• Initial amount (P): the payment you receive at the end of period 1, which every later payment builds on.
• Interest rate (r): the per-period discount, meaning the return you could earn elsewhere on the same money.
• Growth rate (g): the constant percentage by which each payment rises over the one before it.
• Number of periods (n): how many payments you receive in total.
• Payment timing: ordinary annuity for the end of each period, or annuity due for the start.
• Payment frequency and compounding frequency: match the interest rate and growth rate to the payment period, so a monthly stream uses monthly figures.

The calculator applies the standard growing annuity formula to those inputs and returns the present value, meaning the single amount that would fund every payment exactly. Because it works from the exact math rather than a rule of thumb, a plain annuity calculator that ignores growth will always understate the worth of a rising stream.

Depending on which unknown you solve for, a result can appear in one of these forms:

• Present value: the amount you would need today to fund every payment at your chosen interest.
• Future value: what the same stream grows to by the end of the last period if you reinvest each amount as it arrives.
• Final balance: the future value expressed as an account total after the last deposit.
• Withdrawal or income stream: the opening payout that a starting balance can support while it grows each period.

Present value of growing annuity formula and the time value of money

The math rests on the time value of money: a dollar received today can be invested, so it is worth more than a dollar received later. Discounting each payment converts a future amount into its present day value, and the sum of future cash flows, each discounted, gives the total.

For a growing annuity that starts at \(P\), is discounted at \(r\) and grows at \(g\) for \(n\) periods, the present value of a growing annuity is:

$$PV = \frac{P}{r - g}\left[1 - \left(\frac{1+g}{1+r}\right)^{n}\right]$$

This expression works whenever \(r \neq g\). In words, it divides \(P\) by the gap between discount and growth, then scales the result by how much of the growth-adjusted stream is still ahead after \(n\) periods. Online growing annuity calculators all rest on it, and textbooks often abbreviate the whole thing as the PV of growing annuity formula.

Present value of a growing annuity step by step

Each payment is one growth step larger than the last and one discounting step smaller, so the payment received at time \(k\) is worth \(\frac{P(1+g)^{k-1}}{(1+r)^{k}}\) today. These terms form a geometric series with a common ratio of \(\frac{1+g}{1+r}\), and adding them with the geometric series formula produces the expression above. Working through the present value of a growing annuity term by term is worth doing once with a small example, and the table further down does exactly that.

The special case where r equals g

The expression divides by \(r - g\), so it breaks when the two are equal. In that case each payment grows exactly as fast as it is discounted, every annuity payment has the same present value, and the answer simplifies to:

$$PV = \frac{P \times n}{1 + r}$$

The original expression also returns a sensible number when \(g\) is larger than \(r\), because the divisor and the bracket both turn negative, but a stream that grows faster than it is discounted rarely lasts long enough for that to matter.

Growing perpetuity as a limiting case

Let the stream run forever and the term \(\left(\frac{1+g}{1+r}\right)^{n}\) shrinks toward zero whenever \(r > g\). What remains is the value of a growing perpetuity, \(PV = \frac{P}{r - g}\), the same shortcut analysts use to value a dividend stock whose payout rises steadily. A perpetuity is never a literal promise, yet it gives you a ceiling: no growing annuity with the same inputs can be worth more.

Growing ordinary annuity vs growing annuity due

Timing changes the answer. A growing ordinary annuity pays at the end of each period, which finance textbooks call payment in arrears. A growing annuity due pays at the beginning of each period, or in advance, so every payment arrives one period sooner and is discounted one period less.

Because of that shift, you multiply the ordinary result by \((1 + r)\) to get the due version:

$$PV_{due} = PV \times (1 + r)$$

Rent and lease bills are the classic due case, since you pay for the month before you use it, while dividends, coupons and most withdrawal plans arrive at the end of the period. A lease whose annual amount steps up by 3% is a growing annuity due, and discounted at 8% it is worth exactly 8% more than the ordinary version.

Other annuity types follow the same logic. An immediate annuity starts paying right away, a deferred annuity waits, a fixed annuity guarantees its amounts and a variable annuity moves with the markets. The growing version, also called an increasing annuity, is the one that steps up by a set percentage on purpose.

Growing annuity calculator example with a $1,000 periodic payment

Suppose a contract pays you $1,000 at the end of the first year and then 3% more each year for ten years, and you discount the stream at 8%. Enter those figures in any growing annuity calculator and the answer is $7,550.13. Any financial calculator that supports growth should agree to the cent, so this is also a handy test case for checking a spreadsheet.

Each step follows the formula. The spread \(r - g\) is 0.05, the ratio \(\frac{1.03}{1.08}\) is about 0.9537, raising it to the tenth power gives 0.6225, and the result is \(\frac{1000}{0.05} \times 0.3775 = 7{,}550.13\). A level $1,000 payment for the same ten years is worth only $6,710.08, so the 3% annual growth adds about $840 of present value.

Present value schedule year by year

The amortization table below treats the $7,550.13 as an opening amount that earns 8% each year while the growing amounts are drawn out of it. The last row closes at zero, which confirms the result.

Each row adds the periodic interest for that year to the opening amount, subtracts the periodic payment and carries the rest forward. Every payment exceeds that year's interest, so each row shows a principal reduction from the starting principal of $7,550.13, and the reductions grow as the amounts rise. Run the same stream forward instead and you get an accumulation schedule that ends in the future value.

A real-use walkthrough: lump sum or rising payout

A former employer offers to settle a consulting contract in one of two ways: $118,500 today, or $9,600 at the end of this year followed by 14 more annual payments, each 2.5% larger than the last. The stream adds up to $172,146.50 on paper, which makes the lump sum look small until you discount it.

You enter 9,600 as the opening amount, 2.5% growth, 15 periods and a 6.2% discount rate, the yield your adviser quoted on a 15-year investment-grade bond ladder. The result is $107,033.22, which is $11,466.78 below the cash offer, so at 6.2% the lump sum wins. The final payout of $13,564.55 looks generous, yet it arrives in year 15, exactly where discounting shrinks a dollar the most, and that back-loading is why the paper total overstates the offer.

The offer letter leaves timing loose, so you test both readings:

The contract says payments land on January 2, which makes it an annuity due, and switching the timing lifts the value by exactly 6.2% to $113,669.28. That is still $4,830.72 short. Rerunning with the discount lowered until the two values match shows a break-even near 4.76%: the rising payout only wins if you expect to earn less than that, and the 6.2% quote clears the hurdle by 1.44 points.

So you accept the $118,500 and ask for it as a single transfer before the offer expires on the 30th, keeping the 15-year payout schedule only as a fallback if the bond yield ever drops below 4.76%.

How growth rate and interest rate change the present value of a growing annuity

Growth and discounting pull in opposite directions, so the gap between them drives the result. Holding the $1,000 first payment, ten periods and an 8% discount constant, the table shows how the present value climbs as growth speeds up.

The relationship is not a straight line. As \(g\) approaches \(r\), the denominator \(r - g\) shrinks and each extra point of growth adds more value than the last, which is why a stream growing at 6% is worth far more than one growing at 3%. Raise the discount instead and the effect reverses, because distant, larger amounts are shrunk the most. A level annuity is simply the 0% row, and the standard present value of annuity formula gives that row exactly.

The chart makes the same point payment by payment. The year-one payment is worth $926 either way, but by year ten the growing payment still carries $604 of present value against $463 for the level one. Anyone comparing the PV of growing annuity cash flows with a flat alternative should look at that widening gap rather than only at the total.

Compounding frequency and continuous compounding

Interest that compounds monthly or daily grows faster than interest that compounds once a year, so a monthly stream needs monthly discounting and monthly growth. Under continuous compounding the discount factor becomes \(e^{-rt}\), and \(1 + r\) is replaced by \(e^{r}\) throughout. Most finance courses treat it as a theoretical benchmark, and a stated rate quoted as an annual figure should be converted before it goes into the model.

Two related inputs are easy to mix up. The annual growth rate is the yearly percentage increase, while the periodic growth rate is that increase applied per payment period. A 12% annual growth rate paid monthly works out to roughly 0.949% per month, and the discount needs the same conversion. A nominal interest rate quoted as 8% compounded monthly, for example, becomes about 0.667% per month, whereas an annual interest rate of 8% compounded yearly stays at 8%.

Where the PV of a growing annuity matters: retirement, dividends and insurance

Anywhere a payment is expected to rise on a schedule, the PV of a growing annuity gives a single number to compare against a lump sum or another offer. The most common cases are these:

• Retirement income: a retiree who wants withdrawals that keep pace with inflation can price the whole plan today and see whether savings cover it.
• Stock valuation: an investor who expects a company's payout to rise 3% a year can weigh the payout stream against today's share price.
• Insurance and settlements: a contract that pays a rising amount, such as a structured settlement with yearly increases, can be compared with a lump sum offered instead.
• Pension and salary planning: an employer can value a promise that steps up with wages.
• Financial and tax planning: an adviser can compare a rising payout with a fixed one on an equal footing.

Businesses use the same idea for capital budgeting: a project whose cash flow grows steadily can be valued like a growing annuity. A savings plan that adds a bigger deposit every year is the mirror image, and the future value formula then gives the matching final balance. Subtract everything you deposited from that total and what is left is your total return.

Choosing the interest rate and number of periods for an increasing annuity

Two inputs deserve the most thought. The discount should reflect the rate of return available on an alternative investment of similar risk, such as a bond ladder or a low-cost index fund. A figure near 8% suits an equity-heavy portfolio, whereas a safe pension or insurance payout may deserve 4%.

The number of periods is the number of payments you expect to receive, so a monthly payout for 25 years is 300 periods, not 25. When the term is uncertain, run the calculation at a short, a middle and a long horizon and compare the results, because a longer term raises the value only until the discounted amounts become negligible. If you would rather see every payment laid out, list the payment schedule in a spreadsheet and confirm that the discounted amounts add up to the same total.

Inflation, tax and other real-world adjustments

The model assumes payments increase by the same percentage every period, which is rarely exactly true. If you set \(g\) equal to the expected price rise, the result shows the present value in today's purchasing power, while a higher \(g\) bakes in real raises. Fees and surrender charges are not part of the equation, so treat the output as one of many estimates rather than a quote, and rely on a constant rate of growth only for as long as you can defend it.