Growing Perpetuities With Changing Payments

When Payments Grow Without a Stated End

A level perpetuity pays the same amount each period. Some long-term payment streams instead increase or decrease by a constant percentage each period. A growing perpetuity is a model for such a stream when payments continue indefinitely. Its value depends not only on the next payment, but also on how quickly payments change compared with the rate used to discount them.

This tutorial uses annual payments made at the end of each year and an effective annual interest rate $i$. The model is aligned with the SOA VEE Accounting and Finance syllabus. The payment timing matters: the value is measured at time 0, before the first payment at time 1.

Definition: A growing perpetuity has payments $C_1,C_2,\ldots$, with the first payment $C_1$ due one period from the valuation date and each later payment changing by a constant rate $g$. Thus $C_t=C_1(1+g)^{t-1}$ for $t=1,2,\ldots$. Under annual end-of-year payments and a constant effective annual discount rate $i$, its time-0 present value is $P_0=C_1/(i-g)$, provided $i>g$.

Here $C_1$ means the payment due next period. If the payment just made at time 0 is $C_0$, then the next payment is $C_1=C_0(1+g)$. Confusing these two amounts is one of the most common sources of error. The formula uses $C_1$, not $C_0$.

Why the Formula Works

Discount each payment to time 0. The first payment contributes $C_1/(1+i)$, the second contributes $C_1(1+g)/(1+i)^2$, and each following term adds another factor of $(1+g)/(1+i)$. The resulting series is geometric.

P_0=\frac{C_1}{1+i}+\frac{C_1(1+g)}{(1+i)^2}+\frac{C_1(1+g)^2}{(1+i)^3}+\cdots

Let $q=(1+g)/(1+i)$. Each term after the first is the preceding term multiplied by $q$. For a geometric series with first term $a$ and common ratio $q$, the sum is $a/(1-q)$ when $|q|<1$. Here the first term is $C_1/(1+i)$. Assuming the usual positive discount factor and nonnegative payments, convergence requires $q<1$, equivalently $g<i$.

P_0=\frac{C_1/(1+i)}{1-(1+g)/(1+i)} =\frac{C_1/(1+i)}{(i-g)/(1+i)} =\frac{C_1}{i-g}

The denominator $i-g$ is the gap between the discount rate and the growth rate. As $g$ approaches $i$ from below, the denominator approaches zero and the present value grows without bound. If $g\geq i$, the infinite series does not converge to a finite value under this model, so the formula cannot be used to assign a finite present value.

Model conditions: Use $P_0=C_1/(i-g)$ when payments occur once per period at period-end, the first payment is one period after the valuation date, the payment grows at a constant per-period rate $g$, and the discount rate is a constant effective rate $i$ for the same period. The series must converge; in the usual positive-rate setting this requires $i>g$. A finite payment stream, a different first-payment date, or rates expressed for mismatched periods needs a different valuation setup.

Using the Formula and Its Rearrangements

The base formula can be rearranged to find any one of $P_0$, $C_1$, $g$, or $i$, provided the other quantities are known and the model conditions hold. Each rate must be expressed per payment period. For annual payments, use annual rates; do not combine an annual growth rate with a monthly discount rate.

P_0=\frac{C_1}{i-g} \qquad C_1=P_0(i-g) \qquad g=i-\frac{C_1}{P_0} \qquad i=g+\frac{C_1}{P_0}

The last two rearrangements are especially useful when a price is observed and one rate is unknown. They do not eliminate the model condition: after solving, check that the resulting rates and cash flows are consistent with the assumed growing-perpetuity stream.

Set the valuation date and period.
Identify when value is measured and whether payments and rates are annual, monthly, or another frequency.

Identify the next payment.
Use the amount due one period after the valuation date. If the given amount was just paid, grow it once to obtain the next payment.

Select the rearrangement.
Use the form that isolates the unknown, keeping the units and payment timing unchanged.

Check admissibility and interpretation.
Confirm $i>g$ for a finite value and check that the computed result corresponds to the requested input.

Worked Example: Pricing a Stream From Its Next Payment

A hypothetical fund expects to distribute $1,200 one year from today. Distributions are projected to grow by 3% each year indefinitely. An investor requires an effective annual return of 8%. Find the value today.

Method. The first payment is already stated as the next payment $C_1=\$1{,}200$. The annual growth rate is $g=0.03$, and the annual required return is $i=0.08$. Both rates use the same annual period.

First check convergence: $i-g=0.08-0.03=0.05>0$. The growing-perpetuity formula therefore applies.

P_0=\frac{C_1}{i-g} =\frac{\$1{,}200}{0.08-0.03} =\frac{\$1{,}200}{0.05} =\$24{,}000

The value today is $24,000. A check is that the first discounted payment is $\$1{,}200/1.08$, and later payments are discounted by a convergent series with ratio $1.03/1.08<1$. The result is finite and consistent with the assumed growth and return.

Common mistake avoided: treating the growth rate as an amount to add to the payment rather than using the difference $i-g$ in the denominator. The relevant annual rates are 8% and 3%, so the denominator is 5%, not 11%.

Worked Example: Solving for the Next Payment

A hypothetical endowment is valued at $36,000 immediately after a payment date. Its annual distributions are expected to grow by 3% indefinitely, and the appropriate effective annual return is 8%. Find the distribution due one year from the valuation date.

Method. The known value is $P_0=\$36{,}000$, $i=0.08$, and $g=0.03$. Since the question asks for the next payment, solve $C_1=P_0(i-g)$.

i-g=0.08-0.03=0.05 $$ $$ C_1=\$36{,}000(0.05)=\$1{,}800

The payment due one year from now is $1,800. Substituting it back verifies the answer:

\frac{\$1{,}800}{0.08-0.03} =\frac{\$1{,}800}{0.05} =\$36{,}000

Common mistake avoided: reporting $P_0$ as the payment. The $36,000 is the present value of the entire future stream; it is not a cash flow received in one year.

Worked Example: Inferring the Growth Rate

A hypothetical asset is priced at $50 per share. Its next annual payment is $3 per share, and the required effective annual return is 9%. Assuming the payment stream grows at a constant annual rate forever, find that rate.

Method. The price is $P_0=\$50$, the next payment is $C_1=\$3$, and $i=0.09$. Rearrange the formula to $g=i-C_1/P_0$.

\frac{C_1}{P_0}=\frac{\$3}{\$50}=0.06 $$ $$ g=0.09-0.06=0.03

The implied annual growth rate is 3%. Check that $i>g$: $9\%>3\%$. Substitution also recovers the stated price:

\frac{\$3}{0.09-0.03} =\frac{\$3}{0.06} =\$50

This inferred rate is meaningful only if the constant-growth model is appropriate. It is not a guarantee that actual future payments will grow at that rate.

Worked Example: Finding the Required Return When the Given Payment Was Just Made

A hypothetical income-producing asset is priced at $40. The payment just made was $1.80 per unit, and future payments are expected to grow by 4% per year indefinitely. Find the required effective annual return implied by the price.

Method. The given $1.80 is $C_0$, not $C_1$. Grow it one year to find the payment due next:

C_1=C_0(1+g) =\$1.80(1.04) =\$1.872

Now use $i=g+C_1/P_0$, with $g=0.04$ and $P_0=\$40$:

\frac{C_1}{P_0} =\frac{\$1.872}{\$40} =0.0468 $$ $$ i=0.04+0.0468=0.0868=8.68\%

The implied required return is 8.68%. The convergence check holds because $8.68\%>4\%$. Substitution verifies the price: $\$1.872/(0.0868-0.04)=\$1.872/0.0468=\$40$.

Common mistake and correction: using the just-paid amount directly gives $i=0.04+\$1.80/\$40=8.50\%$. That is wrong because the formula requires the next payment. Growing $1.80 by 4% gives $1.872, which produces the correct 8.68%.

Timing, Units, and Interpretation

A reliable calculation begins by naming the payment that belongs in the numerator. At a valuation date just after a payment, the amount already received is $C_0$, and the model requires $C_1=C_0(1+g)$. If the valuation date is one period before the first payment, that first payment is already $C_1$; it must not be grown again.

Keep rate periods consistent. If payments occur annually, $i$ and $g$ must both be annual effective rates. If the cash flows are monthly, both the discount and growth assumptions must be stated for monthly periods before using the formula. A quoted annual rate cannot automatically be treated as the rate for every month.

Also distinguish an assumption from a forecast. The formula represents a stream with a constant growth rate forever. It does not establish that a real payment stream will actually grow at a constant rate, nor does it apply unchanged to a stream that ends after a fixed number of payments. Its value is a model-based estimate conditional on the stated assumptions.

Key takeaway: For a growing perpetuity valued one period before its first payment, use the next payment $C_1$ and the same-period rates $i$ and $g$: $P_0=C_1/(i-g)$. Check that $i>g$, and convert a payment just made into the next payment before applying the formula.

Check Your Understanding

For each problem, identify whether the stated amount is the next payment or the payment just made before calculating.

A hypothetical trust will pay $900 one year from now, with payments growing at 2.5% annually forever. The effective annual required return is 7.5%. Find the present value. (Answer: $18,000)
A hypothetical contract has just paid $1.80. Payments grow at 5% annually, and the required effective annual return is 11%. Find the value immediately after the payment just made. (Answer: $31.50)
A payment stream has a present value of $64, its next annual payment is $3.20, and its required effective annual return is 9%. Find the constant annual growth rate. (Answer: 4%)
A hypothetical asset costs $28 and will pay $1.40 one year from now, with payments growing at 3% annually forever. Find the required effective annual return implied by the price. (Answer: 8%)
A stream has just made a payment of $4.00. Its next payment is 2% higher, and payments then grow at 2% annually forever. If the effective annual discount rate is 6%, find the value immediately after the payment just made. (Answer: $102.00)

What You'll Learn