Present Value of Geometric-Increasing Annuities

What Is a Geometric-Increasing Annuity?

A level annuity has the same payment in every period. In a geometric-increasing annuity, each payment is instead a fixed percentage larger than the previous one. This pattern is useful for representing cash flows that rise at a constant rate, such as a hypothetical annual benefit with a contractual percentage increase.

This tutorial considers an annuity-immediate with $n$ annual payments. The first payment, $C$, is made at the end of year 1; each later payment is multiplied by $1+g$, where $g$ is the annual effective growth rate. The annual effective interest rate is $i$. Thus, the first payment is not made at time 0.

Definition: An $n$-payment geometric-increasing annuity-immediate has payments $C, C(1+g), \ldots, C(1+g)^{n-1}$ at times $1,2,\ldots,n$, respectively. Here $C$ is the first payment, $g$ is the effective growth rate per payment period, and $i$ is the effective interest rate per payment period.

The payment index is important: payment $k$, made at time $k$, is $C(1+g)^{k-1}$. For annual payments, use an effective annual $i$ and an annual $g$. If an interest rate is quoted as nominal, first convert it to the effective rate for the payment period. The formula below does not apply with mismatched rate units.

Present Value Formula

Discount each payment to time 0 at the effective interest rate $i$. When $i \ne g$, the present value is:

Theorem: For an $n$-payment geometric-increasing annuity-immediate with first payment $C$, effective interest rate $i$, and effective growth rate $g$, where $i \ne g$,

PV = \frac{C}{i-g}\left[1-\left(\frac{1+g}{1+i}\right)^n\right].

This form assumes $i>-1$ and $g>-1$, so the annual accumulation and payment growth factors are positive. It also assumes that the payments occur at the stated regular period-ends and that the interest and growth rates remain constant over those periods. For the usual setting of positive interest and nonnegative growth, these conditions hold. The formula is not a formula for payments at the beginning of each year.

When $i=g$, the displayed formula has a zero denominator and must not be used as written. In that case, each payment discounted to time 0 has the same value, $C/(1+i)$, so the present value is $nC/(1+i)$. This is a boundary case of the same payment stream, not a reason to substitute into a zero denominator.

Why the Formula Holds

The formula follows by adding the discounted payments. The $k$th payment is $C(1+g)^{k-1}$, and its present value is that amount multiplied by $(1+i)^{-k}$. Factoring out the common first term leaves a finite geometric series.

\begin{aligned} PV &= \sum_{k=1}^{n}\frac{C(1+g)^{k-1}}{(1+i)^k}\\ &= \frac{C}{1+i}\sum_{k=0}^{n-1} \left(\frac{1+g}{1+i}\right)^k. \end{aligned}

For $r \ne 1$, the finite geometric sum is $\sum_{k=0}^{n-1}r^k=(1-r^n)/(1-r)$. Set $r=(1+g)/(1+i)$. Then $1-r=(i-g)/(1+i)$, so the factor $1+i$ cancels with the $C/(1+i)$ in front of the sum. The result is the stated present-value formula. If $r=1$, then $i=g$, every discounted term equals $C/(1+i)$, and summing the $n$ terms gives $nC/(1+i)$.

Worked Valuations

Worked Example: Valuing Four Growing Payments

A hypothetical contract pays $1,000 at the end of year 1. Each annual payment is 8% greater than the preceding payment. If the effective annual interest rate is 20%, find the present value of the four payments. Round the final answer to the nearest cent.

Method. The first payment is $C=\$1{,}000$, the annual growth rate is $g=0.08$, the effective annual interest rate is $i=0.20$, and $n=4$. The payments occur at times 1 through 4, so the exponent on the first payment is zero.

First check that $i \ne g$: $0.20 \ne 0.08$. The payment amounts are $\$1{,}000$, $\$1{,}080$, $\$1{,}166.40$, and $\$1{,}259.712$. The discount-and-growth ratio is $\frac{1+g}{1+i}=\frac{1.08}{1.20}=0.9$.

Substitute into the formula: \[ PV=\frac{1{,}000}{0.20-0.08}(1-0.9^4) =\frac{1{,}000}{0.12}(1-0.6561) =8{,}333.333333\ldots(0.3439) =2{,}865.833333\ldots \]

As a check, discount the four payments separately: $1{,}000/1.2=\$833.333333\ldots$, $1{,}080/1.2^2=\$750$, $1{,}166.40/1.2^3=\$675$, and $1{,}259.712/1.2^4=\$607.50$. Their sum is $\$2{,}865.833333\ldots$, as given by the formula. Rounded to the nearest cent, the present value is $2,865.83. The common mistake avoided here is to treat the first payment as occurring at time 0 rather than at the end of year 1.

Worked Example: Comparing Growing and Level Payments

A hypothetical four-year payment arrangement pays $1,000 at the end of year 1, with each later payment 8% larger. A second arrangement pays $1,000 at each year-end for four years. At an effective annual interest rate of 20%, how much greater is the present value of the growing arrangement? Round the final answer to the nearest cent.

Method. Value each arrangement at time 0 using the same interest rate and payment dates, then subtract the level-payment value from the growing-payment value. For the growing arrangement, the values from the preceding calculation are $C=\$1{,}000$, $g=0.08$, $i=0.20$, and $n=4$, giving a present value of $\$2{,}865.833333\ldots$.

For the level arrangement, discount each of the four $1,000 payments: \[ PV_{\mathrm{level}} =\frac{1{,}000}{1.2}+\frac{1{,}000}{1.2^2} +\frac{1{,}000}{1.2^3}+\frac{1{,}000}{1.2^4} =833.333333\ldots+694.444444\ldots+578.703704\ldots+482.253086\ldots =2{,}588.734568\ldots \]

The difference is \[ 2{,}865.833333\ldots-2{,}588.734568\ldots =277.098765\ldots \] Therefore, the growing arrangement has a present value $277.10 greater, rounded to the nearest cent. As a check, the growing payments are never smaller than the level payments, and at least one is strictly larger, so a positive difference is expected. The common mistake avoided is comparing the payment totals without discounting them at their different payment dates.

Worked Example: Keeping the First Payment at the First Period-End

A hypothetical three-payment benefit pays $500 at the end of year 1, with each subsequent payment 10% above the preceding one. The effective annual interest rate is 25%. Find the present value, rounded to the nearest cent.

Method. Use $C=\$500$, $g=0.10$, $i=0.25$, and $n=3$. The payments are $\$500$ at time 1, $\$550$ at time 2, and $\$605$ at time 3. Since $i \ne g$, apply the formula: \[ PV=\frac{500}{0.25-0.10} \left[1-\left(\frac{1.10}{1.25}\right)^3\right] =\frac{500}{0.15}(1-0.88^3) =3{,}333.333333\ldots(1-0.681472) =1{,}061.76. \]

Check by discounting each payment: $500/1.25=\$400$, $550/1.25^2=\$352$, and $605/1.25^3=\$309.76$. The sum is $\$1{,}061.76$. The common indexing error is to use $500(1.10)$ for the year-1 payment. That incorrectly shifts the growth forward by one period and gives $1.10(\$1{,}061.76)=\$1{,}167.936$. The first payment is $C$, so the correct present value is $1,061.76.

Common Checks and Exam Traps

A reliable valuation begins by writing the payment amount alongside its time. For this annuity-immediate, the time-1 payment is $C$, the time-2 payment is $C(1+g)$, and the time-$n$ payment is $C(1+g)^{n-1}$. This makes the payment count and exponents visible before any formula is used.

Using $n$ growth periods instead of $n-1$. There are $n$ payments but only $n-1$ increases between the first and last payments.
Discounting as though the first payment is immediate. The first payment is at time 1 and is discounted once. An immediate annuity would have different payment dates.
Mixing rate periods. An annual growth rate must be paired with an effective annual interest rate for annual payments. Convert a nominal quote before using the formula.
Ignoring the equality case. If $i=g$, do not evaluate the quotient with denominator $i-g=0$; use $nC/(1+i)$.
Expecting the formula to require $i>g$. The formula is valid for $i \ne g$ under the stated rate conditions. If $g>i$, its two bracket-and-denominator signs are both negative, leaving a positive present value.

These checks matter because the present value depends on both the payment amounts and their dates. The same growing sequence has a different value if its first payment is at time 0, and a nominal rate cannot be substituted for an effective rate unless it represents the same payment-period accumulation. The formula is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume annual payments at year-end and a constant effective annual interest and growth rate. Round money answers to the nearest cent.

A fund pays $900 at the end of year 1, increasing by 4% each year for three payments. The effective annual interest rate is 10%. Find the present value. (Answer: $2,323.10)
A four-payment annuity-immediate has first payment $1,200, annual growth of 2%, and effective annual interest of 12%. Find its present value. (Answer: $3,745.14)
A five-payment annuity-immediate starts with a $2,500 payment and grows by 6% per year. At an effective annual interest rate of 10%, find the present value. (Answer: $10,566.70)
The second payment of a four-payment annuity-immediate is $1,080. Payments grow by 8% annually, and the effective annual interest rate is 20%. Find the present value. (Answer: $2,865.83)
An annuity-immediate pays $700 annually for six years, with each payment 5% greater than the preceding payment. The effective annual interest rate is also 5%. Find the present value. (Answer: $4,000.00)

What You'll Learn