Geometric Annuities When Growth Equals Interest

Why Equality of Growth and Interest Matters

A geometric-increasing annuity has payments that rise by the same percentage each period. Usually, discounting those larger later payments produces a finite geometric series with a ratio different from 1. A special case occurs when the payment growth rate equals the effective interest rate per payment period: each payment’s present value is then identical. The present value is simple to calculate, but the usual geometric-series formula appears to have a zero denominator, so this case requires careful handling.

The essential task is to compare rates measured over the same period. An annual effective interest rate cannot be compared directly with a growth rate quoted per half-year. First convert one rate so that both describe the interval between payments.

Definition: Consider $n$ payments made at the ends of periods $1,2,\ldots,n$. If the first payment is $C$ and each later payment is $1+g$ times the preceding payment, then payment $k$ is $C(1+g)^{k-1}$. Here $g$ is the effective growth rate per payment period. If the effective interest rate per payment period is $i$, the present value at time 0 is $$ PV=\sum_{k=1}^{n} C(1+g)^{k-1}(1+i)^{-k}. $$ When $g=i$, each discounted payment is $C/(1+i)$, and therefore $$ PV=\frac{nC}{1+i}. $$

This result assumes a finite number of certain payments, a constant effective interest rate per payment period, and a constant growth rate per payment period. The first payment is at the end of the first period, not at time 0. Also, the interest rate and growth rate must use the same period unit. For example, if payments are semiannual, both rates must be effective semiannual rates.

Deriving the Limiting Value

The general present-value sum makes clear why equality is a special case. Set $q=(1+g)/(1+i)$, the ratio between the present values of consecutive payments. Rewriting each term gives a finite geometric series:

PV=\frac{C}{1+i}\left(1+q+q^2+\cdots+q^{n-1}\right).

For $q\ne 1$, the finite geometric-series formula gives $PV=\frac{C}{1+i}\frac{1-q^n}{1-q}$. If $g=i$, then $q=1$, and substituting directly into this expression gives $0/0$, which is undefined. It does not mean that the annuity has zero value. The sum before applying the formula has $n$ terms, each equal to 1 when $q=1$, so its value is $n$.

Derivation: The limiting value of the ratio in the geometric-series formula is $$ \lim_{q\to 1}\frac{1-q^n}{1-q} =\lim_{q\to 1}\frac{-nq^{n-1}}{-1} =n. $$ Therefore, when $g=i$, the present value is $$ PV=\frac{C}{1+i}\cdot n=\frac{nC}{1+i}. $$ Equivalently, payment $k$ is $C(1+i)^{k-1}$, so discounting it $k$ periods gives $C(1+i)^{k-1}(1+i)^{-k}=C/(1+i)$, independent of $k$. Summing this same present value $n$ times gives the result directly.

This argument also explains the phrase limiting value: as the growth rate approaches the interest rate, the general geometric-series expression approaches $nC/(1+i)$. At equality, it is often clearest to value each discounted payment directly rather than substitute equal rates into a formula whose denominator becomes zero.

Applying the Formula

For a standard geometric-increasing annuity, write down four items before calculating: the first payment $C$, the number of payments $n$, the effective interest rate per payment period $i$, and the growth rate $g$ per payment period. Confirm that $i=g$, then use $nC/(1+i)$. If a price is given, compare it with the present value to determine which alternative is less costly at the stated interest rate.

Worked Example: Annual Payments Growing at the Annual Interest Rate

A fund will pay $1,200 at the end of year 1. Each subsequent annual payment is 5% greater than the preceding payment, and there are eight payments in total. The effective annual interest rate is 5%. Find the present value at time 0. Then determine the net present value if the fund is available for $9,000.

Method. The payments are annual, so compare the annual growth rate with the effective annual interest rate. Both are 5%, so use the equality case.

The first payment is $C=\$1{,}200$, and $n=8$. Each payment has present value $$ \frac{C}{1+i}=\frac{1200}{1.05}=\$1{,}142.857142\ldots. $$ Thus the present value of all eight payments is $$ PV=8\left(\frac{1200}{1.05}\right) =\frac{9600}{1.05} =\$9{,}142.857142\ldots. $$ Rounded to the nearest cent, the present value is $9,142.86.

The net present value at a purchase price of $9,000 is $$ NPV=9{,}142.857142\ldots-9{,}000 =\$142.857142\ldots. $$ Rounded to the nearest cent, $NPV=$ $142.86. The fund’s present value exceeds its price by $142.86 at the stated interest rate.

Check. The first payment’s present value is $1,200/1.05. The second payment is $1,200(1.05), and discounting it for two years gives $1200(1.05)/(1.05)^2=1200/1.05$. The same cancellation applies to every payment, confirming the eight equal present values. The method avoids dividing by zero in the general formula.

Worked Example: Comparing a Payment Stream with a Quoted Price

A service contract pays $500 at the end of year 1, with each annual payment increasing by 4% for a total of six payments. The effective annual interest rate is 4%. A buyer can purchase the contract for $2,800. Calculate the present value and determine whether the price is below or above that value.

Method. Since the payment growth and effective interest rates are equal and both are annual, discount each payment using the equality result.

Here $C=\$500$, $n=6$, and $i=g=0.04$. The present value is $$ PV=\frac{nC}{1+i} =\frac{6(500)}{1.04} =\frac{3000}{1.04} =\$2{,}884.615384\ldots. $$ Rounded to the nearest cent, $PV=$ $2,884.62.

Compare the value with the price: $$ 2{,}884.615384\ldots-2{,}800 =\$84.615384\ldots. $$ The present value is $84.62 greater than the quoted price, rounded to the nearest cent. At the stated interest rate, the contract’s payments are worth more than the purchase price.

Check. Every payment has present value $500/1.04=\$480.769230\ldots$. Six such values total $6(\$480.769230\ldots)=\$2,884.615384\ldots$, as calculated. The comparison uses the present value of the payments, not their undiscounted total.

Worked Example: Converting an Annual Rate for Semiannual Payments

A hypothetical investment makes ten payments, one at the end of each half-year. The first payment is $900, and each later payment is 4% greater than the previous one. The effective annual interest rate is 8.16%. A seller asks $8,700 for the payment stream. Find its present value and compare it with the asking price.

Method. Convert the annual effective rate to an effective half-year rate, because the payments and growth are semiannual. For two half-year periods, the accumulation factor must equal the annual accumulation factor: $$ (1+i_{\text{half-year}})^2=1.0816. $$ Since $1.04^2=1.0816$, the effective half-year interest rate is $i_{\text{half-year}}=4\%$. This equals the 4% growth rate per half-year.

The first payment is $C=\$900$, and there are $n=10$ payments. Therefore, $$ PV=\frac{nC}{1+i_{\text{half-year}}} =\frac{10(900)}{1.04} =\frac{9000}{1.04} =\$8{,}653.846153\ldots. $$ Rounded to the nearest cent, the present value is $8,653.85.

The present value less the asking price is $$ 8{,}653.846153\ldots-8{,}700 =-\$46.153846\ldots. $$ Rounded to the nearest cent, this is −$46.15. The asking price is $46.15 above the present value of the payments at the stated rate.

Check. The annual rate conversion is consistent because $(1.04)^2=1.0816$, exactly the annual accumulation factor. Each semiannual payment has present value $900/1.04=\$865.384615\ldots$; ten payments total $8,653.85 after rounding. Comparing rates without first converting the annual rate would not be valid.

Common Exam Trap: Misreading the Zero Denominator

A frequent error is to use the general finite geometric-series formula at $g=i$, observe that $1-(1+g)/(1+i)=0$, and conclude that the present value is zero. That denominator is zero because the ratio of consecutive discounted payments is 1. The numerator is also zero, so the expression is undefined in that form; it does not represent a zero-valued annuity.

Another error is to calculate $nC$ without discounting. For the eight-payment example above, this incorrect calculation gives $8(1{,}200)=\$9{,}600$. It is the sum of the nominal payments only if all eight payments were $1,200; in the actual stream, later payments grow. Neither way of summing nominal payments gives the present value. The correct present value is $8(1{,}200)/1.05=\$9{,}142.86$, because each growing payment is discounted for its own payment period.

Key takeaway: When the growth rate and effective interest rate are equal per payment period, the discounted value of every payment in a finite geometric-increasing annuity is $C/(1+i)$. Multiply by the number of payments. Do not substitute into the general geometric-series formula as though its zero denominator were an ordinary value.

Check Your Understanding

For each problem, match rates to the payment period before using the limiting present-value result.

A stream pays $700 at each year’s end initially, with each payment increasing by 3.5% annually. There are five payments, and the effective annual interest rate is 3.5%. Find the present value. (Answer: $3,381.64)
A stream pays $400 at the end of each half-year for eight half-years, with each payment increasing by 5% per half-year. The effective annual interest rate is 10.25%. Find the present value. (Answer: $3,047.62)
A seven-payment annual stream begins with $1,200 at the end of year 1 and grows by 4% annually. The effective annual interest rate is 4%, and the stream costs $8,000. Find its net present value. (Answer: $76.92)
An annual payment stream has twelve payments, the first of which is $525 at the end of year 1. Payments grow by 5% annually, and the effective annual interest rate is 5%. Find the present value. (Answer: $6,000.00)
A settlement offer of $8,800 can replace nine annual payments, beginning with $1,000 at the end of year 1 and growing by 3% each year. The effective annual interest rate is 3%. Compare the settlement amount with the present value of the payments. (Answer: The payment stream has present value $8,737.86, so it is $62.14 less than the settlement amount.)

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