Geometrically Varying Annuities

Instead of paying the same amount every period, a geometrically growing annuity pays $P, P(1+g), P(1+g)^2, \ldots$ — each payment g% bigger than the last. Discount it at an adjusted rate $j$ and the whole growing mess collapses back into an ordinary level annuity you already know how to value.

By the end you'll be able to convert a geometrically growing payment stream into an equivalent level annuity using the adjusted rate $j=(1+i)/(1+g)-1$, and value growing perpetuities like the Gordon growth (dividend discount) model.

Predict: as the growth rate g approaches the interest rate i, what happens to the present value of a long payment stream? Drag g toward i below and check.

The bars below are 20 payments starting at 1,000 and growing by g each year. Discount them all at rate i and you get the present value shown. Check the box to watch the same bars collapse to one flat height — that's the annuity valued at the adjusted rate j instead, and it lands on the exact same present value.

Growth rate g = 3.0%

Interest rate i = 8.0%

Collapse to level annuity at rate j

j = 4.854% · PV of 20 payments = 12,250.0 · growing-perpetuity limit P/(i−g) = 20,000.0

20 payments starting at 1,000, growing at g, discounted at i

growing payment stream level equivalent (rate j)

A geometrically growing annuity pays $P(1+g)^{k-1}$ at the end of period k. Discounting it directly means summing n different growth-and-discount factors — but a change of rate turns that sum right back into a plain level annuity.

Formal

The present value is $PV = \sum_{k=1}^{n} P(1+g)^{k-1}v^k$, where $v = 1/(1+i)$. Pull out $P/(1+g)$ and what's left is a geometric series in the ratio $(1+g)/(1+i)$. Define the adjusted rate j by $1+j = (1+i)/(1+g)$, so $j = (1+i)/(1+g) - 1$. The sum becomes $a_n$ — the ordinary level annuity-immediate factor — evaluated at j instead of i: $PV = \dfrac{P}{1+g}\cdot a_n\text{ at rate }j$. Let $n \to \infty$: the series converges only when $g < i$ (equivalently $j > 0$), giving the growing-perpetuity value $PV = P/(i-g)$.

Applied

This is exactly the dividend discount model (also called the Gordon growth model) used to price stocks: if next year's dividend is $D_1$ and dividends are expected to grow forever at g, with required return k, the stock's fair value is $D_1/(k-g)$ — a growing perpetuity in disguise. Salary streams with a fixed annual raise, or benefits indexed to inflation, are valued the same way for a finite n.

Worked example

Payments start at 1,000, growing 3% per year for 20 years; $i = 8\%$. $j = 1.08/1.03 - 1 \approx 4.854\%$. $a_{20}$ at rate j $= (1-1.04854^{-20})/0.04854 \approx 12.6175$. $PV = (1000/1.03)\cdot 12.6175 \approx 970.874 \cdot 12.6175 \approx$ 12,250.0.

Your turn

Same idea, different numbers: payments start at 500, growing 2% per year for 15 years; $i = 6\%$. $j = 1.06/1.02 - 1 \approx 3.922\%$. $a_{15}$ at rate j $\approx 11.1796$. Now finish it: $PV = (500/1.02)\cdot 11.1796 = 490.196 \cdot 11.1796 =$ ____

Reveal the answer

$PV = 490.196 \cdot 11.1796 \approx$ 5,480.2. Set g = 2.0% and i = 6.0% on the slider above (n there is 20, not 15, so the exact number won't match — but you can confirm j and the collapse-to-level behavior).

More info — why the adjusted rate works

Every payment in the stream is really two effects multiplied together: growth $(1+g)^{k-1}$ pushing the value up, and discounting $v^k$ pulling it down. Combining them into a single per-period ratio, $(1+g)\cdot v = (1+g)/(1+i)$, is the same as discounting at one new rate j where $1+j$ is the reciprocal of that ratio. That's why the messy growing sum is secretly the same shape as the plain level annuity from the annuity-immediate lesson — just with i swapped for j. See the Marcel Finan guide in Dive deeper for the full derivation.

Check your understanding

Question 1 of 4

A stock's next dividend is expected to be $2, growing at a constant 4% per year forever. The required return is 10%. What is the stock's value today?

Question 2 of 4

A payment stream starts at 1,000, growing 3% per year, discounted at $i = 8\%$. What is the adjusted rate j used to collapse this into a level annuity?

Question 3 of 4

Once you discount a geometrically growing annuity at the adjusted rate $j=(1+i)/(1+g)-1$, its present value becomes $P/(1+g)$ times which already-known quantity?

Question 4 of 4

For a growing perpetuity with payments increasing by g forever, discounted at rate i, what happens to the present-value formula $P/(i-g)$ if $g \geq i$?

Recap

A geometric annuity pays $P(1+g)^{k-1}$ at the end of period k; discount at i and it still has a closed-form value.
Adjusted rate: $1+j = (1+i)/(1+g)$, so $j = (1+i)/(1+g)-1$. Its PV collapses to $\dfrac{P}{1+g}\cdot a_n$ evaluated at rate j — an ordinary level annuity in disguise.
As $n \to \infty$ with $g < i$, the growing-perpetuity value is $PV = P/(i-g)$ — the Gordon growth / dividend discount model.
If $g \geq i$, the infinite sum never converges — the growing-perpetuity formula is only valid for $g < i$.

Dive deeper

Corporate Finance Institute — Gordon Growth Model Connect the growing-perpetuity limit P/(i−g) to the dividend discount model.
Marcel Finan — A Basic Course in the Theory of Interest (Exam FM study guide, PDF) Derive the geometric-annuity value via the adjusted rate j.

Sources