Deriving Geometric Annuity Factors

Payment Pattern and Timing

A geometric-increasing annuity has payments that change by the same percentage from one payment to the next. Its present value depends on both that growth and the interest earned between payment dates. The formula is easiest to use, and easiest to misapply, when the payment index and payment timing are made explicit first.

This tutorial treats a finite annuity-immediate: there are $n$ payments, the first is made at the end of year 1, and the present value is measured at time 0. Let $C$ be the first payment, $g$ the effective annual rate of payment growth, and $i$ the effective annual interest rate.

Definition: A geometric-increasing annuity-immediate with first payment $C$, growth rate $g$, and $n$ payments pays $C(1+g)^{k-1}$ at the end of year $k$, for $k=1,2,\ldots,n$. Thus the payments are $C,\ C(1+g),\ C(1+g)^2,\ldots,C(1+g)^{n-1}$.

The exponent is $k-1$, not $k$: the first payment is the base amount $C$, with no growth applied yet. The interest rate $i$ discounts each payment according to how many years it is from time 0. Under a constant effective annual rate, a payment at the end of year $k$ is discounted by $v^k$, where $v=1/(1+i)$.

Model assumptions: There are exactly $n$ payments at the ends of years $1$ through $n$; the first payment is $C$; each subsequent payment is multiplied by $1+g$; and the effective annual interest rate $i$ is constant over the valuation period. The closed-form factor below requires $i>-1$ and $i\ne g$. If payments are to be nonnegative, also require $C\ge 0$ and $g\ge -1$. A different payment schedule or changing interest rates requires a different valuation.

Deriving the Factor

Discount each payment separately. The payment at time $k$ is $C(1+g)^{k-1}$, and its discount factor is $v^k$. The present value is therefore a sum of discounted cash flows:

\begin{aligned} PV &=\sum_{k=1}^{n} C(1+g)^{k-1}v^k\\ &=Cv\sum_{j=0}^{n-1}\big((1+g)v\big)^j. \end{aligned}

The change of index $j=k-1$ makes the first term of the series correspond to the first payment. Each term in the sum is the preceding term multiplied by the same ratio $q=(1+g)v$. For $q\ne1$, the finite geometric-series identity follows by subtracting one shifted copy of the sum:

\begin{aligned} S&=1+q+q^2+\cdots+q^{n-1},\\ qS&=q+q^2+\cdots+q^n,\\ S-qS&=1-q^n,\\ S&=\frac{1-q^n}{1-q}. \end{aligned}

Here $q\ne1$ is equivalent to $i\ne g$, because $q=(1+g)/(1+i)$. Substituting the series result gives the factor. The second form follows by simplifying the denominator:

\begin{aligned} PV &=Cv\frac{1-\big((1+g)v\big)^n}{1-(1+g)v}\\ &=\frac{C}{i-g}\left[1-\left(\frac{1+g}{1+i}\right)^n\right]. \end{aligned}

Theorem: For a finite geometric-increasing annuity-immediate with the assumptions stated above and $i\ne g$, its present value at time 0 is $ \displaystyle \frac{C}{i-g}\left[1-\left(\frac{1+g}{1+i}\right)^n\right] $. Equivalently, it is $Cv$ times the sum of the first $n$ powers of $(1+g)v$, beginning with power zero.

This result is a finite sum, not a perpetuity formula: $n$ must be the actual number of payments. The derivation also explains why the ratio combines growth and discounting. Each payment grows by $1+g$ from its predecessor, but its discount factor adds a factor of $v$; the relative size of successive discounted terms is therefore $(1+g)v$.

Using and Checking the Formula

Before substituting values, establish the first payment date, the number of payments, and whether the given growth rate is effective per payment period. For annual payments and annual effective rates, those periods agree. Do not insert a nominal rate or a rate for a different period without first converting it to the effective rate for one year.

Worked Example: A Three-Payment Contract

A hypothetical contract pays \$1,000 at the end of year 1. Each later annual payment is 10% larger than the preceding payment, and the effective annual interest rate is 25%. Find the present value at time 0. Use the finite-series method and check the result by discounting the payments individually.

Method and setup. This is an annuity-immediate with $C=1000$, $g=0.10$, $i=0.25$, and $n=3$. The rates are effective annual rates, and $i\ne g$, so the closed form applies.

The payment amounts are $1000$, $1000(1.10)=1100$, and $1100(1.10)=1210$. The common ratio of successive discounted terms is $q=(1.10)/(1.25)=0.88$. Hence $1+q+q^2=1+0.88+0.7744=2.6544$, and the first discounted term is $Cv=1000/1.25=800$.

PV=800(2.6544)=\$2{,}123.52.

Check by individual discounting. The three present values are $1000/1.25=\$800$, $1100/1.25^2=1100/1.5625=\$704$, and $1210/1.25^3=1210/1.953125=\$619.52$. Their sum is $800+704+619.52=\$2{,}123.52$, agreeing with the series calculation.

Common mistake avoided: applying growth to the first payment would incorrectly change the payment at year 1 to \$1,100. Here the first payment is $C$; growth begins only between the first and second payments.

Worked Example: Four Payments with a Simple Discounted Ratio

A hypothetical equipment lease pays \$600 at the end of year 1, with each subsequent annual payment 8% greater than the previous one. The effective annual interest rate is 20%. Find the present value and verify it using the separate payment values.

Method and setup. Set $C=600$, $g=0.08$, $i=0.20$, and $n=4$. The effective annual rates match the annual payment interval, and $i\ne g$. The common ratio of the discounted terms is $q=1.08/1.20=0.9$.

The finite series is $1+0.9+0.9^2+0.9^3=1+0.9+0.81+0.729=3.439$. Also, $Cv=600/1.20=500$. Therefore the present value is $500(3.439)=\$1{,}719.50$.

Check by individual discounting. The payments are \$600, \$648, \$699.84, and \$755.8272. Discounting gives $600/1.2=\$500$, $648/1.2^2=648/1.44=\$450$, $699.84/1.2^3=699.84/1.728=\$405$, and $755.8272/1.2^4=755.8272/2.0736=\$364.50$. Their sum is $\$1{,}719.50$.

Common mistake avoided: using the payment growth rate itself as the discounted-series ratio. The ratio is $0.9$, not $1.08$, because each later term is also discounted one additional year.

Worked Example: Applying the Closed Form with a Nontrivial Power

A hypothetical grant pays \$900 at the end of year 1. Payments then increase by 5% each year for a total of five payments. If the effective annual interest rate is 12%, calculate the present value to the nearest cent.

Method and setup. Here $C=900$, $g=0.05$, $i=0.12$, and $n=5$. The payment interval and rate period are both one year, and $i-g=0.07\ne0$.

The discounted ratio is $(1.05)/(1.12)=105/112=15/16$. Its fifth power is $15^5/16^5=759{,}375/1{,}048{,}576$. Thus the bracket in the closed form is $1-759{,}375/1{,}048{,}576=289{,}201/1{,}048{,}576$.

\begin{aligned} PV &=\frac{900}{0.12-0.05} \left(1-\left(\frac{15}{16}\right)^5\right)\\ &=\frac{90{,}000}{7}\cdot\frac{289{,}201}{1{,}048{,}576}\\ &=3{,}546.0458\ldots\\ &=\$3{,}546.05. \end{aligned}

Check. The equivalent series factor is $1+15/16+(15/16)^2+(15/16)^3+(15/16)^4=4.4128570557\ldots$, and $Cv=900/1.12=803.5714286\ldots$. Their product is $3{,}546.0458\ldots$, which rounds to the same amount. Rounding is to two decimal places, at the final step only.

Common mistake avoided: using $n-1=4$ as the exponent in the closed-form bracket. There are five discounted terms, so the last power in the series is four, but the geometric-series formula uses $q^n$, which here is $q^5$.

Common Indexing and Formula Traps

A reliable check is to write out the first few cash flows with their dates before using a factor. The payment at time $k$ has growth exponent $k-1$ and discount exponent $k$. These exponents differ because growth starts after the first payment, while discounting starts as soon as a payment is one year away.

For example, in the first worked example, an incorrect time-0 payment pattern of $1000,\ 1100,\ 1210$ at times $0,\ 1,\ 2$ would give $1000+1100/1.25+1210/1.25^2=\$2{,}654.40$. That is not the stated annuity-immediate: it moves every payment one year earlier. The correct dates are years 1, 2, and 3, producing \$2,123.52.

Do not omit the first discount. The first payment is at time 1, so it is discounted by $v$, not by 1.
Do not grow the first payment. The payment at time 1 is $C$; the payment at time $k$ is $C(1+g)^{k-1}$.
Do not confuse $g$ with the series ratio. The ratio is $(1+g)/(1+i)$, incorporating both growth and discount.
Check the conditions before dividing by $i-g$. The displayed closed form requires $i\ne g$; if that condition fails, the quotient form is not applicable.
Match rate periods. The derivation uses one interest period and one growth step per payment interval. Convert rates or rebuild the cash-flow sum if the periods differ.

Key takeaway: For an annuity-immediate, write the $k$th payment as $C(1+g)^{k-1}$ at time $k$, discount it by $v^k$, and sum the resulting finite geometric series. This makes both the formula and its timing conditions transparent.

Check Your Understanding

For each problem, identify the first payment date and use the stated effective annual rates. Round monetary answers to the nearest cent.

A fund pays \$400 at the end of year 1 and increases each later annual payment by 4%. The effective annual interest rate is 10%, and there are three payments. Find the present value. (Answer: \$1,032.49)
A hypothetical service contract pays \$1,200 at the end of year 1, with annual payments growing by 2%. The effective annual interest rate is 6%, and there are four payments. Find the present value. (Answer: \$4,278.37)
A scholarship pays \$900 at the end of year 1 and grows by 5% annually for five total payments. At an effective annual interest rate of 12%, find the present value. (Answer: \$3,546.05)
For an annuity-immediate with first payment $C$, $n$ annual payments, effective annual interest rate $i$, and annual growth rate $g$, state the common ratio of the discounted terms and the condition under which the closed-form quotient applies. (Answer: The ratio is $(1+g)/(1+i)$; the quotient requires $i\ne g$, as well as $i>-1$ for the discount factor.)
A hypothetical asset pays \$500 at the end of year 1, with each later payment 5% greater than the previous payment. There are four payments and the effective annual interest rate is 15%. Find the present value. (Answer: \$1,525.15)

What You'll Learn