Worked Problems on Geometric-Gradient Annuities

Setting Up a Geometric-Gradient Stream

A geometric-gradient annuity has payments that change by the same percentage from one payment to the next. The central task is to keep three features aligned: the first payment amount, the percentage change, and the payment dates. A correct formula applied to the wrong first payment date can still produce a wrong value.

Unless stated otherwise, assume payments occur once per year, the effective annual interest rate $i$ is constant, and the first payment amount is $C$. Let $g$ be the effective annual growth rate of the payments, so each payment is $1+g$ times the preceding payment. For a decreasing stream, $g$ is negative; for example, a 10% annual decrease means $g=-0.10$. Assume $i>-1$, $g>-1$, and a positive integer number of payments $n$.

Definition: An increasing or decreasing geometric annuity-immediate with first payment $C$, growth rate $g$, and $n$ payments has payments $C,C(1+g),\ldots,C(1+g)^{n-1}$ at times $1,2,\ldots,n$. A geometric annuity-due has the same sequence of amounts at times $0,1,\ldots,n-1$. A deferred annuity-immediate with $m$ full years of deferral makes its first payment at time $m+1$.

The present value is measured at time 0. With $v=1/(1+i)$, payment $k$ of an annuity-immediate is $C(1+g)^{k-1}$ and is discounted $k$ years. This gives the finite geometric series below.

\operatorname{PV}_{\text{immediate}} = C v\sum_{j=0}^{n-1}\big((1+g)v\big)^j = C v\frac{1-\big((1+g)v\big)^n}{1-(1+g)v}, \qquad (1+g)v\ne 1.

For an annuity-due, the first payment is at time 0 rather than time 1. It therefore has one fewer discount period than the corresponding payment in the immediate stream.

\operatorname{PV}_{\text{due}} = C\sum_{j=0}^{n-1}\big((1+g)v\big)^j = C\frac{1-\big((1+g)v\big)^n}{1-(1+g)v}, \qquad (1+g)v\ne 1.

Both formulas have a special case when $(1+g)v=1$, which is equivalent to $g=i$. In that case every discounted payment has the same value, so the value is $n$ times that value. This case should be handled directly rather than by dividing by zero.

Special case: When $g=i$, the present value of an annuity-immediate is $nCv$, and the present value of an annuity-due is $nC$. The first payment still determines which value applies: the immediate stream's first $C$ is paid at time 1, while the due stream's first $C$ is paid at time 0.

Why the Geometric-Series Formula Works

For an annuity-immediate, the discounted payment at time $k$ is $C(1+g)^{k-1}v^k=Cv((1+g)v)^{k-1}$. Thus the discounted cash flows are themselves a geometric sequence, with first term $Cv$ and common ratio $r=(1+g)v$. The sum of $n$ terms with first term $a$ and ratio $r\ne1$ is $a(1-r^n)/(1-r)$. Applying that identity gives the immediate formula. If $r=1$, each term equals $a$, so the sum is $na$, which gives $nCv$. For a due annuity, each payment is one period earlier, so the first discounted term is $C$, giving $nC$ in the equal-growth-and-interest case.

This derivation also explains the main practical rule: write the discounted first payment and the ratio before substituting into a formula. That makes payment timing visible and reveals whether the special case applies.

Accumulated Value and Deferral

An accumulated value at time $n$ is the value at that date of payments made by then, assuming each payment earns interest at the same effective annual rate $i$. It can be found by accumulating the present value for $n$ years. If $g=i$, each payment in an annuity-immediate accumulates to the same amount at time $n$, because earlier payments have more time to earn interest while later payments are larger.

\operatorname{AV}_{n,\text{immediate}} =\operatorname{PV}_{\text{immediate}}(1+i)^n, \qquad g=i:\quad \operatorname{AV}_{n,\text{immediate}} =nC(1+i)^{n-1}.

For a deferred annuity-immediate with $m$ full years of deferral, the stream has the ordinary immediate-annuity pattern starting at time $m+1$. Its value at time $m$ is the value of an immediate annuity with first payment $C$. Discount that value back $m$ years to time 0. A deferred annuity-due instead has its first payment at time $m$, so its value at time $m$ is the due value.

\operatorname{PV}_{\text{deferred immediate}} =v^m\operatorname{PV}_{\text{immediate}}, \qquad \operatorname{PV}_{\text{deferred due}} =v^m\operatorname{PV}_{\text{due}}.

These deferral formulas assume the stated number of full years with no earlier payments. If the first payment date differs, the number of discount periods must be adjusted to match the timeline.

Worked Example: Growth Equals Interest

A fund is expected to pay $500$ at the end of each year for 8 years. Each payment is 6% greater than the previous one. The effective annual interest rate is 6%. Find the present value at time 0 and the accumulated value at the end of year 8. Assume annual payments and a constant effective annual rate.

Method. This is an annuity-immediate with $C=500$, $n=8$, $g=0.06$, and $i=0.06$. Because $g=i$, use the equal-growth-and-interest case.

For present value, the first payment is one year away. Each discounted payment is $500/1.06$, since the growth factor and discount factor cancel in every subsequent period. Therefore,

\operatorname{PV}=8\left(\frac{500}{1.06}\right) =8(471.6981132\ldots) =3773.5849057\ldots

For accumulated value at time 8, each payment accumulates to $500(1.06)^7$. For instance, the first payment grows for 7 years, while the eighth payment, $500(1.06)^7$, grows for no years. Hence,

\operatorname{AV}_8 =8(500)(1.06)^7 =4000(1.5036302589\ldots) =6014.5210356\ldots

Rounded to the nearest cent, the present value is \$3,773.58 and the accumulated value is \$6,014.52. As a check, $3773.5849057\ldots(1.06)^8=6014.5210356\ldots$, consistent with accumulating the present value for 8 years. A common mistake is to use $nC$ for the present value: the discounted value of each payment is $Cv$, not $C$.

Worked Example: A Decreasing Annuity-Due

A hypothetical maintenance reserve pays $1{,}000$ immediately and then makes four more annual payments, each 10% below the previous one. The effective annual interest rate is 5%. Find the present value of all five payments at time 0. Assume the payment dates are times 0 through 4.

Method. The stream is an annuity-due with $C=1{,}000$, $n=5$, $g=-0.10$, and $i=0.05$. The growth factor is $1+g=0.90$, so the ratio of successive discounted terms is $0.90v=0.90/1.05=6/7$.

\operatorname{PV} =1000\left(1+\frac67+\left(\frac67\right)^2+ \left(\frac67\right)^3+\left(\frac67\right)^4\right)

The five terms inside the parentheses are $1$, $6/7$, $36/49$, $216/343$, and $1296/2401$. Using denominator 2401, their sum is $(2401+2058+1764+1512+1296)/2401=9031/2401$. Thus,

\operatorname{PV} =1000\left(\frac{9031}{2401}\right) =3761.3494377\ldots

The present value is \$3,761.35, rounded to the nearest cent. As a cash-flow check, the payments are $1{,}000$, $900$, $810$, $729$, and $656.10$ at times 0 through 4. Discounting those payments gives $1000+900/1.05+810/(1.05)^2+729/(1.05)^3+656.10/(1.05)^4=3761.3494\ldots$. A common mistake is to discount the first payment by one year; because it is paid immediately, its discount factor is 1.

Worked Example: A Deferred Increasing Annuity-Immediate

A hypothetical grant makes 6 annual payments, beginning with $800$ at the end of year 4. Each later payment is 2% greater than the preceding payment. The effective annual interest rate is 4%. Find the present value at time 0. Assume there are no payments before year 4.

Method. There are 3 full years of deferral, followed by a 6-payment annuity-immediate. Here $C=800$, $n=6$, $g=0.02$, and $i=0.04$. At time 3, the first payment is one year away. The ratio of successive discounted terms is $(1.02)/1.04=51/52$. Discount the time-3 value back three years.

\operatorname{PV}_0 =v^3\left[800v\frac{1-(51/52)^6}{1-51/52}\right] =\frac{800}{(1.04)^4}\left[52\left(1-\left(\frac{51}{52}\right)^6\right)\right].

The geometric sum is $52[1-(51/52)^6]=5.7188290\ldots$, and $800/(1.04)^4=683.84335\ldots$. Therefore,

\operatorname{PV}_0 =(683.84335\ldots)(5.7188290\ldots) =3910.7832\ldots

The present value is \$3,910.78, rounded to the nearest cent. The first payment is at time 4, so the first $800$ is discounted four years; the next payment, $816$, is discounted five years. The formula includes that initial four-year discount through $v^4$. A common mistake is to discount by only three years, forgetting that an immediate annuity's first payment is one period after its valuation date.

Common Traps and a Reliable Workflow

Geometric-gradient problems are often difficult not because the series is complicated, but because the timeline is easy to misread. In particular, “deferred 3 years” does not by itself tell you whether the first payment is at time 3 or time 4; the wording must specify whether the stream is due or immediate. The formulas above define the convention explicitly.

A second trap is using an ordinary geometric-series formula when the ratio equals 1. If $g=i$, the numerator and denominator in the closed form are both zero. That does not mean the present value is zero or undefined; it means the discounted payment terms are equal, and their sum is $n$ times one term.

Mark the payment dates.
Identify whether the first payment is at time 0 or time 1, and note any deferral.

Write the first amount and payment ratio.
Use $C$ for the first payment and $1+g$ for the ratio of consecutive payments. A percentage decrease gives a negative $g$.

Determine the discounted ratio.
For annual payments, it is $(1+g)v=(1+g)/(1+i)$. Check whether this ratio equals 1 before using the closed form.

Apply timing adjustments and check.
For a deferred stream, discount the value at the start of its annuity period to time 0. Finally, compare the formula with the cash-flow dates and the number of payments.

These methods are aligned with the SOA Exam FM (Financial Mathematics) syllabus. The key is not to memorize separate formulas for every growth rate and payment pattern. The timeline, the first discounted amount, and the common ratio determine the valuation; the equal-growth-and-interest case follows directly when the ratio is 1.

Check Your Understanding

For each problem, assume a constant effective annual interest rate and annual payments unless otherwise stated.

An annuity-immediate pays $300$ at the end of each year for 4 years, with each payment 5% greater than the preceding payment. The effective annual interest rate is 5%. Find the present value. (Answer: $\$1,142.86$)
An annuity-due pays $600$ immediately and then makes three further annual payments, each 20% lower than the preceding one. The effective annual interest rate is 5%. Find the present value. (Answer: $\$1,670.81$)
A deferred annuity-immediate makes 5 payments, starting with $1{,}000$ at the end of year 3; each payment grows by 3% annually. The effective annual interest rate is 7%. Give the present value at time 0, rounded to the nearest cent. (Answer: $\$3,785.66$)
An annuity-immediate pays $250$ at the end of each year for 10 years. The effective annual interest rate is 4%, and payments decrease by 4% per year. Find its present value. (Answer: $10(250/1.04)=\$2,403.85$)
A deferred annuity-due has 4 payments, with the first payment of $900$ at time 3. Payments grow by 2% per year, and the effective annual interest rate is 5%. Find the present value at time 0, rounded to the nearest cent. (Answer: $900v^3[1+(1.02)v+((1.02)v)^2+((1.02)v)^3]=\$3,132.42$)

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