Deriving the Growing-Perpetuity Formula

Why a Growing Perpetuity Needs a Convergence Check

A level perpetuity pays the same amount at every payment date. A growing perpetuity instead increases each payment at a constant rate. The payments may be larger over time, but their present values are discounted. Whether those present values add to a finite total depends on the relationship between the growth rate and the discount rate.

The key result is compact, but it is valid only under a specific condition. Deriving it from the individual discounted payments makes the condition visible and helps prevent a common error: using the formula when the payment stream does not converge.

Definition: A growing perpetuity is an infinite sequence of payments made at regular intervals, with each payment equal to the preceding payment multiplied by $1+g$, where $g$ is the constant growth rate per interval. In the annual model used here, the first payment $C_1$ occurs at the end of year 1, the effective annual discount rate is $i$, and payment $t$ is $C_t=C_1(1+g)^{t-1}$. The present value is measured at time 0.

Assume $i>-1$ and $g>-1$, so the annual accumulation and payment-growth factors are positive. Payments are discounted annually at the effective annual rate $i$, and there are no interruptions in the payment stream. For an indefinitely continuing stream of positive payments to have a finite present value, the required condition is $i>g$.

Deriving the Formula From Discounted Payments

The payment at the end of year 1 is $C_1$. The payment at the end of year 2 is $C_1(1+g)$, and the payment at the end of year 3 is $C_1(1+g)^2$. Each payment is discounted back to time 0 using the appropriate power of $1+i$.

\begin{aligned} \text{PV}_1 &= \frac{C_1}{1+i},\\ \text{PV}_2 &= \frac{C_1(1+g)}{(1+i)^2},\\ \text{PV}_3 &= \frac{C_1(1+g)^2}{(1+i)^3}. \end{aligned}

Factor $C_1/(1+i)$ out of the terms. The remaining terms form a geometric series: each term is multiplied by the same ratio, $(1+g)/(1+i)$. To establish the infinite value carefully, first value only the first $n$ payments.

\begin{aligned} \text{PV}_n &= \frac{C_1}{1+i} \left[1+\frac{1+g}{1+i} +\left(\frac{1+g}{1+i}\right)^2+\cdots+ \left(\frac{1+g}{1+i}\right)^{n-1}\right]\\ &= \frac{C_1}{1+i}\left(\frac{1-q^n}{1-q}\right), \qquad q=\frac{1+g}{1+i},\quad q\ne 1. \end{aligned}

The finite geometric-sum identity follows by subtracting $q$ times the bracketed sum from the sum itself: all intermediate powers cancel, leaving $1-q^n$. For the infinite stream, the term $q^n$ approaches zero only when $0\le q<1$. Under the stated positive-factor assumptions, this is equivalent to $i>g$. Substituting $q=(1+g)/(1+i)$ into the finite-sum result gives the infinite value.

\begin{aligned} 1-q &=1-\frac{1+g}{1+i} =\frac{i-g}{1+i},\\ \text{PV} &=\frac{C_1}{1+i}\frac{1}{1-q} =\frac{C_1}{1+i}\frac{1+i}{i-g} =\frac{C_1}{i-g},\qquad i>g. \end{aligned}

Theorem: If the first payment $C_1$ is made at the end of year 1, payments grow by a constant annual rate $g$, and the effective annual discount rate $i$ satisfies $i>g$, then the present value at time 0 is $\text{PV}=C_1/(i-g)$. The formula does not give a finite present value when $i\le g$.

The denominator is the difference between the discount rate and the growth rate, not the discount rate alone. The condition matters: if $i=g$, then $q=1$, so every discounted payment equals $C_1/(1+i)$, and infinitely many equal positive terms do not sum to a finite value. If $i<g$, then $q>1$; the discounted payments do not even approach zero.

Using and Checking the Result

Before applying the formula, identify the first payment date and the rate period. Here, $C_1$ means the payment at the end of the first year; it is not a payment at time 0. Both $i$ and $g$ must be annual rates because payments occur annually. The required check is $i>g$, followed by substitution into $C_1/(i-g)$.

Set the timeline.
Confirm that the first payment occurs at the end of year 1 and later payments grow once per year.

Check convergence.
Verify that the discount rate exceeds the growth rate: $i>g$.

Substitute the inputs.
Use the first payment $C_1$, not a later payment, in $\text{PV}=C_1/(i-g)$.

Check the result.
Confirm that the units are monetary value at time 0 and that the value is positive for a positive first payment.

Worked Example: Valuing a Growing Annual Distribution

A hypothetical fund is expected to distribute $1,200 at the end of next year. Each later annual distribution is 2% greater than the preceding one, indefinitely. The effective annual discount rate is 8%. Find the present value at time 0. Use the growing-perpetuity method and check the first discounted payments.

Method and condition. The first payment is $C_1=\$1{,}200$, $g=0.02$, and $i=0.08$. The first payment is one year from now, and $0.08>0.02$, so the stream converges.

The next payments are $C_2=1{,}200(1.02)=\$1{,}224$ and $C_3=1{,}224(1.02)=\$1{,}248.48$. Their present values, together with the first payment's value, are:

\frac{1{,}200}{1.08},\qquad \frac{1{,}224}{(1.08)^2},\qquad \frac{1{,}248.48}{(1.08)^3}.

The common ratio of consecutive discounted payments is $1.02/1.08$, which is less than 1. The formula therefore gives:

\text{PV}=\frac{1{,}200}{0.08-0.02} =\frac{1{,}200}{0.06} =\$20{,}000.00.

Check. The first discounted payment is $1{,}200/1.08$, approximately $1,111.11, and the second is $1{,}224/(1.08)^2$, approximately $1,049.38. The discounted payments decline because the 8% discount rate exceeds 2% growth, which is consistent with convergence. The final value is $20,000.00.

Common mistake avoided. Dividing the first payment by 8% would treat the payments as level and give $15,000.00. Since the payments grow, the relevant denominator is $i-g=6\%$, not $i=8\%$.

Worked Example: Solving for the Growth Rate

A hypothetical organization plans to make a first annual payment of $1,800 one year from now, followed by payments that grow at a constant rate forever. The effective annual discount rate is 8%, and the present value of the stream is $36,000. Find the annual growth rate and the second payment.

Method and condition. Start with the growing-perpetuity formula and rearrange it to isolate $g$:

36{,}000=\frac{1{,}800}{0.08-g} \quad\Longrightarrow\quad 0.08-g=\frac{1{,}800}{36{,}000}=0.05 \quad\Longrightarrow\quad g=0.03.

The implied annual growth rate is 3%. It satisfies the convergence condition because $8\%>3\%$. The second payment is one year after the first, so it is:

C_2=C_1(1+g)=1{,}800(1.03)=\$1{,}854.00.

Check. Substitution gives $1{,}800/(0.08-0.03)=1{,}800/0.05=\$36{,}000.00$, the stated present value. The second payment exceeds the first by 3%, as required.

Common mistake avoided. Using the level-perpetuity calculation $1{,}800/0.08=\$22{,}500.00$ ignores the growth assumption and cannot reproduce the stated value. Rearranging the growing-perpetuity formula retains both rates.

Worked Example: Detecting a Nonconvergent Stream

A hypothetical payment program will pay $500 at the end of year 1. Payments then grow by 6% each year forever, while the effective annual discount rate is 5%. Determine whether the program has a finite present value. Use the first two payments to check the direction of the discounted cash flows.

Method and condition. Compare the rates first: $i=0.05$ is less than $g=0.06$. The ratio of consecutive discounted payments is:

q=\frac{1.06}{1.05}=\frac{106}{105}>1.

The first payment's present value is $500/1.05$. The second payment is $500(1.06)=\$530$, with present value $530/(1.05)^2$. Their ratio is $q>1$, so the second discounted payment is greater than the first. The discounted payments do not approach zero.

For a finite number $n$ of payments, the geometric-sum expression is:

\text{PV}_n =\frac{500}{-0.01}\left[1-\left(\frac{106}{105}\right)^n\right] =50{,}000\left[\left(\frac{106}{105}\right)^n-1\right].

For $n=1$, this is $50{,}000(1/105)=\$476.19$ to the nearest cent. As $n$ increases, $(106/105)^n$ grows without bound, so the finite-stream values do not approach a finite limit. There is no finite present value for the perpetual stream.

Check. The formula's condition $i>g$ fails. The expression $C_1/(i-g)$ would produce a negative number here, but a sequence of positive payments cannot have a negative present value under positive discount factors. That result signals misuse of the formula, not a meaningful valuation.

Common Traps and Practical Importance

The formula is useful because it converts an infinite sequence into a single value, but only when the payment pattern and convergence condition match the model. A rate comparison is not an optional final check; it is part of deciding whether the formula applies at all.

Using a later payment as $C_1$. If the known payment is at the end of year 2, it is not the first payment in the formula as defined here. Identify the first payment date before substituting.
Ignoring payment timing. The formula assumes the first payment is at the end of year 1. A payment at time 0 is not discounted for one year and is outside this timing pattern.
Subtracting rates in the wrong order. The denominator is $i-g$. Reversing the order can produce a negative result even when the convergence condition holds.
Applying the formula when $i\le g$. Check $i>g$ first. If the condition fails, the infinite stream has no finite present value under this model.
Mixing rate periods. An annual growth rate and a monthly discount rate cannot be subtracted directly. The rates must describe the same interval as the payments.

The derivation is aligned with the SOA VEE Accounting and Finance syllabus because it connects a cash-flow pattern to discounting and the geometric-series condition that makes an infinite present value finite.

Key takeaway: For payments $C_1, C_1(1+g), C_1(1+g)^2,\ldots$ made at the ends of successive years, the present value at time 0 is $C_1/(i-g)$ only when $i>g$. The condition follows directly from requiring the discounted-payment ratio $(1+g)/(1+i)$ to be less than 1.

Check Your Understanding

For each problem, identify the first payment and test the rate condition before using the formula.

A hypothetical trust pays $900 at the end of next year, with each later annual payment growing by 2%. If the effective annual discount rate is 6%, find the present value at time 0. (Answer: $22,500.00)
A hypothetical service contract generates a first end-of-year cash flow of $1,500 and grows by 4% annually forever. At an effective annual discount rate of 10%, find the present value. (Answer: $25,000.00)
A hypothetical fund's first year-end distribution is $2,400 and grows by a constant rate forever. Its present value is $60,000 at an effective annual discount rate of 8%. Find the growth rate and the second distribution. (Answer: $g=4\%$; second distribution = $2,496.00)
A hypothetical program pays $1,000 at the end of year 1 and grows at 5% annually forever. The effective annual discount rate is also 5%. State whether a finite present value exists and explain using the discounted-payment pattern. (Answer: No finite present value; each discounted payment is $1,000/1.05, so the infinite sum diverges.)
A hypothetical end-of-year payment stream starts at $800 one year from now and grows at 1% annually forever. If the effective annual discount rate is 4%, calculate its present value to the nearest cent and state whether the convergence condition holds. (Answer: $26,666.67; yes, because 4% is greater than 1%.)

What You'll Learn