Deferred Geometric-Increasing Annuities

Why Deferral and Payment Growth Must Be Separated

A deferred geometric-increasing annuity has two different processes operating at once. The payment amount changes by a fixed percentage from one payment to the next, while interest discounts each payment back to a valuation date. Keeping these processes separate is essential: payment growth does not change the payment dates, and deferral does not change the growth pattern.

The first task is to define the timing precisely. In this tutorial, $m$ is the number of complete periods between time 0 and the point immediately before the first payment period. Thus, if the first payment is at the end of year 5, $m=4$: the first payment is one period after time 4. Let $n$ be the number of payments, and let $C$ be the amount of the first payment.

Definition: A deferred geometric-increasing annuity with $n$ payments has payments $C,\ C(1+g),\ C(1+g)^2,\ldots,C(1+g)^{n-1}$, made at the ends of $m+1,m+2,\ldots,m+n$ periods. Here $g$ is the effective growth rate per payment period. The present value is measured at time 0.

Unless stated otherwise, assume a constant effective interest rate $i$ per period, a constant growth rate $g$ per period, and payments at period ends. These formulas do not directly apply if the payment interval, interest rate, or growth rate changes during the stream; such cash flows must instead be valued period by period or split into appropriate sections.

Deriving the Present-Value Formula

Discount each payment separately. The first payment is discounted $m+1$ periods, the second $m+2$ periods, and so on. With $v=1/(1+i)$, the present value is

PV_0=Cv^{m+1}+C(1+g)v^{m+2}+\cdots+C(1+g)^{n-1}v^{m+n}.

Factor out $Cv^{m+1}$. Each successive term in the remaining sum is multiplied by $(1+g)v=(1+g)/(1+i)$. The terms therefore form a finite geometric series.

PV_0=Cv^{m+1}\sum_{k=0}^{n-1}\left(\frac{1+g}{1+i}\right)^k.

When $i\ne g$, the common ratio is not 1, so use the finite geometric-series identity. Since $1-(1+g)v=(i-g)/(1+i)$, this gives the two equivalent forms below.

PV_0=Cv^{m+1} \frac{1-\left(\frac{1+g}{1+i}\right)^n} {1-\frac{1+g}{1+i}} = Cv^m \frac{1-\left(\frac{1+g}{1+i}\right)^n}{i-g}, \qquad i\ne g.

If $i=g$, the ratio $(1+g)/(1+i)$ equals 1, so the unequal-rate expression has a zero denominator and must not be used. In that case, every term in the sum after factoring out $Cv^{m+1}$ equals 1.

Theorem: For a deferred geometric-increasing annuity with $n$ payments, first payment $C$ at time $m+1$, effective interest rate $i$, and payment growth rate $g$, its present value is $$ PV_0= \begin{cases} Cv^m\dfrac{1-\left(\frac{1+g}{1+i}\right)^n}{i-g},&i\ne g,\\[6pt] nCv^{m+1},&i=g. \end{cases} $$

The equal-rate result also has a direct cash-flow interpretation. If $i=g$, each payment grows at exactly the rate needed to offset the extra period of discounting. Therefore, each payment has the same present value, $Cv^{m+1}$, and there are $n$ such payments.

Applying the Formula with Unequal Rates

Worked Example: Four Payments Deferred for Two Years

A hypothetical fund makes four annual payments. The first payment is $1,000 at the end of year 3, and each later payment is 5% larger than its predecessor. The effective annual interest rate is 10%. Find the present value at time 0. Use the unequal-rate formula, and state how the deferral is counted.

Method and timing. The first payment is at the end of year 3, so there are two complete periods before the first payment period: $m=2$. There are $n=4$ payments, $C=\$1{,}000$, $g=0.05$, and $i=0.10$. Since $i\ne g$, use the unequal-rate formula.

The payment amounts at times 3 through 6 are $1,000, $1,050, $1,102.50, and $1,157.625. The ratio of successive discounted payment terms is

\frac{1+g}{1+i}=\frac{1.05}{1.10}=\frac{21}{22}.

First find the value at time 2, immediately before the first payment period. The geometric-series numerator is

1-\left(\frac{21}{22}\right)^4 =1-\frac{194481}{234256} =\frac{39775}{234256} \approx0.169792878.

Thus the value at time 2 is

PV_2 =1000\frac{1-(21/22)^4}{0.10-0.05} =1000\frac{0.169792878}{0.05} \approx \$3{,}395.86.

Discount that value two periods to time 0:

PV_0=\frac{PV_2}{(1.10)^2} =\frac{3395.85756\ldots}{1.21} \approx \$2{,}806.49.

Check. The first payment is at time 3, so the first discounted term is $1000/(1.10)^3$, not $1000/(1.10)^2$. The formula’s $v^m$ factor combined with its rate denominator accounts for that first payment period. The present value is $2,806.49, rounded to the nearest cent. The method avoids treating the time-2 value as if it were already the time-0 value.

When the Growth Rate Equals the Interest Rate

The equal-rate case is not a numerical approximation to the unequal-rate formula. It is a separate expression obtained from the geometric sum when its ratio is exactly 1. It is also a useful check: if the rates are equal, all payments should have the same present value, regardless of how much their nominal amounts increase.

Worked Example: Matching Growth and Interest Rates

A hypothetical scholarship fund will make four annual payments, with the first payment of $600 at the end of year 4. Each payment is 5% larger than the preceding payment. The effective annual interest rate is also 5%. Find the present value at time 0.

Method and timing. Here $C=\$600$, $n=4$, $i=g=0.05$, and $m=3$, because the first payment is at the end of year 4. Use the equal-rate formula; substituting $i=g$ into the unequal-rate formula would produce division by zero.

The four payments are $600, $630, $661.50, and $694.575, at times 4, 5, 6, and 7. Because the payment growth rate equals the interest rate, each payment discounts to the same value:

\frac{600}{(1.05)^4} =\frac{630}{(1.05)^5} =\frac{661.50}{(1.05)^6} =\frac{694.575}{(1.05)^7}.

Therefore,

PV_0=4(600)(1.05)^{-4} =\frac{2400}{1.21550625} \approx \$1{,}974.49.

Check. The formula gives $nCv^{m+1}=4(600)(1.05)^{-4}$, since $m+1=4$. The same answer follows by discounting each payment separately. The present value is $1,974.49, rounded to the nearest cent.

Separating Growth from Discounting in a Cash-Flow Check

A formula is safest when its result can be reconciled with the payment timeline. The first payment determines the initial discount period; the growth rate determines the later payment amounts; and the interest rate discounts them. This check is particularly valuable when the rates are close but unequal, because the geometric-series ratio can be close to 1.

Worked Example: A Four-Payment Stream Starting in Year 5

A hypothetical endowment makes four annual payments. The first is $900 at the end of year 5, and each later payment is 3% larger than the previous one. The effective annual interest rate is 7%. Find the present value at time 0. Use the formula and check the result against the payment schedule.

Method and timing. The first payment at the end of year 5 means $m=4$, and there are $n=4$ payments. Set $C=\$900$, $g=0.03$, and $i=0.07$. The ratio is $1.03/1.07=103/107$, and $i-g=0.04$.

The payment schedule is $900 at time 5, $927 at time 6, $954.81 at time 7, and $983.4543 at time 8. For the formula, calculate

\left(\frac{103}{107}\right)^4 =\frac{112550881}{131079601}, \qquad 1-\left(\frac{103}{107}\right)^4 =\frac{18528720}{131079601} \approx0.141354718.

Now apply the unequal-rate formula, discounting the value back four periods from time 4:

\begin{aligned} PV_0 &=(1.07)^{-4}(900) \frac{1-(103/107)^4}{0.07-0.03}\\ &=(1.07)^{-4}(900)\frac{0.141354718\ldots}{0.04}\\ &\approx \$2{,}426.37. \end{aligned}

The direct check uses the actual payment dates:

PV_0= \frac{900}{(1.07)^5} +\frac{927}{(1.07)^6} +\frac{954.81}{(1.07)^7} +\frac{983.4543}{(1.07)^8} \approx \$2{,}426.37.

Check. The formula and the dated cash flows agree. The present value is $2,426.37, rounded to the nearest cent. The method avoids both starting the first discount at time 4 and applying the growth rate as though it were an interest rate.

Common Timing and Rate Errors

A recurring trap is to confuse the point immediately before the first payment period with the date of the first payment. If the first payment is at the end of year 3, $m=2$, not 3. In the unequal-rate formula, the factor $v^m$ is already paired with the denominator $i-g$; adding another period of discounting incorrectly discounts the entire annuity one period too far.

For example, in the first worked example, the incorrect calculation $v^3C[1-(21/22)^4]/0.05$ gives about $2,551.36. The correct calculation uses $v^2$, giving about $2,806.49. The timeline resolves the issue: the first cash flow is at the end of year 3 and must be discounted three periods, which is exactly what the original cash-flow sum does.

Do not use the growth rate as the discount rate. Payments grow at $g$, but are discounted at $i$.
Do not use the unequal-rate formula when $i=g$. Its denominator is zero; use $nCv^{m+1}$.
Do not count the first payment date as the number of deferred periods. A first payment at the end of period 5 means $m=4$.
Do not round payments or intermediate ratios prematurely. Retain full precision and round the final monetary value to the nearest cent.

These results assume a fixed periodic growth rate and a constant effective interest rate for all periods. When either rate changes, the single geometric ratio no longer represents the whole stream. A payment-by-payment valuation or a carefully divided timeline is then needed.

Key takeaway: First establish $m$, $n$, and the first payment $C$. Use $i$ to discount and $g$ to generate payment amounts. For $i\ne g$, apply the finite geometric-series formula; for $i=g$, each discounted payment is equal and the value is $nCv^{m+1}$.

Check Your Understanding

For each problem, identify the first payment period before choosing a formula. Answers are rounded to the nearest cent when expressed as money.

A deferred stream has five annual payments. The first payment is $200 at the end of year 3, and payments grow by 4% each year. The effective annual interest rate is also 4%. Find the present value. (Answer: $889.00)
A fund makes two payments, $100 at the end of year 3 and $105 at the end of year 4. If the effective annual interest rate is 10%, find the present value. (Answer: $146.85)
A deferred stream makes three annual payments. The first payment is $300 at the end of year 5; each later payment grows by 10%, and the effective annual interest rate is 10%. Find the present value. (Answer: $558.21)
A stream makes three annual payments, beginning with $1,000 at the end of year 2. Payments grow by 4% annually, and the effective annual interest rate is 8%. Write and evaluate the present-value formula. (Answer: $1000(1.08)^{-1}[1-(26/27)^3]/0.04\approx \$2{,}478.00$)
A stream has four annual payments, the first at the end of year 3. The effective annual interest rate is 5%, the payment growth rate is 2%, and the present value is $2,000. Find the first payment $C$. (Answer: $C=2000(0.03)(1.05)^2/[1-(1.02/1.05)^4]\approx \$1{,}063.56$)

What You'll Learn