Deferred Geometric-Gradient Annuities

Why Deferral Requires Two Valuation Steps

A geometric annuity has payments that change by the same percentage from one payment to the next. When the first payment is deferred, it is easy to count the payments correctly but discount the annuity value by the wrong number of periods. The reliable approach is to value the payment stream one period before its first payment, then discount that value back to the focal date.

This tutorial uses annual payments and an effective annual interest rate. The focal date is time 0. If there are $m$ full years before the value date of the annuity, the value date is time $m$, and the first payment is at time $m+1$. This timing convention makes the deferred stream an annuity-immediate valued at time $m$.

Definition: A deferred geometric annuity-immediate has $n$ payments at times $m+1,m+2,\ldots,m+n$. If its first payment is $C$ and each payment changes by the factor $1+g$, payment $k$ is $C(1+g)^{k-1}$, for $k=1,\ldots,n$. Here $m$ is a nonnegative integer, $n$ is a positive integer, $C>0$, $i>-1$, and $g>-1$. The rate $i$ is the effective annual interest rate and $g$ is the annual geometric payment growth rate; a negative $g$ represents a decline.

Value at the Date One Period Before the First Payment

At time $m$, the first payment is one year away. The next payment is two years away, and so on. Discounting each payment to time $m$ gives

{}_{m}V = \sum_{k=1}^{n} C(1+g)^{k-1}v^k, \qquad v=\frac{1}{1+i}.

The $k$th term has two parts: the payment $C(1+g)^{k-1}$, and the discount factor $v^k$, because that payment occurs $k$ years after time $m$. Factoring out one $v$ leaves a finite geometric series. Its common ratio is $(1+g)v$.

Result: If $i\ne g$, the value at time $m$, one period before the first payment, is

{}_{m}V = C\frac{1-\left(\frac{1+g}{1+i}\right)^n}{i-g}.

The value at time 0 is then

V_0=v^m\,{}_{m}V = \frac{C}{(1+i)^m} \frac{1-\left(\frac{1+g}{1+i}\right)^n}{i-g}.

If $i=g$, use the equal-rate result instead:

{}_{m}V=\frac{nC}{1+i}, \qquad V_0=\frac{nC}{(1+i)^{m+1}}.

Why the Formula Holds

For $i\ne g$, let $r=(1+g)/(1+i)$. The value at time $m$ can be written as

{}_{m}V = \frac{C}{1+i}\left(1+r+r^2+\cdots+r^{n-1}\right).

Multiplying the expression in parentheses by $1-r$ cancels the interior terms:

(1-r)(1+r+\cdots+r^{n-1})=1-r^n.

Since $1-r=(i-g)/(1+i)$, substitution gives

{}_{m}V = \frac{C}{1+i}\frac{1-r^n}{1-r} = C\frac{1-r^n}{i-g}.

Finally, time $m$ is $m$ years after time 0, so its value is discounted by $v^m=(1+i)^{-m}$ to obtain $V_0$. If $i=g$, each payment grows at exactly the rate at which it is discounted: $C(1+i)^{k-1}v^k=C/(1+i)$. Thus each of the $n$ payments has the same value at time $m$, giving $nC/(1+i)$. This equal-rate case should be evaluated directly, not by dividing by $i-g=0$.

Worked Examples

Worked Example: Growing Payments After a Two-Year Deferral

A hypothetical grant pays \$1,000 at the end of year 3, followed by one payment at the end of each of the next three years. Each payment is 10% larger than the preceding payment. The effective annual interest rate is 20%. Find the present value at time 0.

Method. There are $n=4$ payments, $m=2$ full deferral years, $C=\$1{,}000$, $g=0.10$, and $i=0.20$. The first payment is at time $m+1=3$, as required by the timeline.

First value the payments at time 2, one year before the first payment. The first four payments are \$1,000, \$1,100, \$1,210, and \$1,331. The ratio of payment growth to annual accumulation is $1.10/1.20=11/12$. Therefore,

{}_{2}V = 1000\frac{1-(11/12)^4}{0.20-0.10} = 1000\frac{1-14641/20736}{0.10} = 1000\frac{6095}{20736(0.10)} = \$2{,}941.07.

To check the intermediate value by discounting the four payments to time 2:

\frac{1000}{1.20} +\frac{1100}{1.20^2} +\frac{1210}{1.20^3} +\frac{1331}{1.20^4} = 833.3333+763.8889+700.2315+642.4082 = \$2{,}939.86.

The formula calculation above contains an arithmetic error if it does not agree with this direct sum, so use the direct sum to verify the factor carefully: $1-(11/12)^4=6095/20736$, and division by $0.10$ gives $2{,}941.07$, while the payment sum gives $2{,}939.86$. Recomputing the last payment term, $1331/1.20^4=1331/2.0736=\$642.12$, so the direct sum is $833.3333+763.8889+700.2315+642.1196=\$2{,}939.57$. The discrepancy signals that the geometric ratio was misapplied: the discounted terms have ratio $1.10/1.20=11/12$, and the first term is $1000/1.20$. Thus the correct geometric sum is

{}_{2}V = \frac{1000}{1.20} \frac{1-(11/12)^4}{1-11/12} = 833.3333\left(\frac{6095}{20736}\right)(12) = \$2{,}939.57.

Now discount this time-2 value two years to time 0:

V_0=\frac{2{,}939.57}{1.20^2} =\frac{2{,}939.57}{1.44} =\$2{,}041.37.

The method is to value the annuity at time 2 and then discount two years. The key trap is using the geometric-series formula without its initial factor $1/(1+i)$; the direct payment-by-payment calculation exposes that error.

Worked Example: Growth Equals the Interest Rate

A hypothetical research fund pays \$100 at the end of year 2, then makes two further annual payments, each 10% larger than the preceding payment. The effective annual interest rate is 10%. Find the present value at time 0.

Here $m=1$, $n=3$, $C=\$100$, and $g=i=0.10$. At time 1, each payment has the same value: the payments are \$100, \$110, and \$121, and their time-1 values are

\frac{100}{1.10}=\$90.9091,\qquad \frac{110}{1.10^2}=\$90.9091,\qquad \frac{121}{1.10^3}=\$90.9091.

Thus the value at time 1 is $3(100/1.10)=\$272.7273$. Discount it one year to time 0:

V_0=\frac{272.7273}{1.10} =\frac{300}{1.10^2} =\$247.93.

The check is direct: each of the three payments has present value $100/1.10^2=\$82.6446$, so their total is $3(\$82.6446)=\$247.93$. The method uses the equal-rate case. A common mistake is substituting $i=g$ into the formula with denominator $i-g$, which would divide by zero.

Worked Example: Declining Payments with a One-Year Deferral

A hypothetical equipment reserve pays \$1,000 at the end of year 2. Each later annual payment is 20% lower than the preceding one, and there are three payments in total. The effective annual interest rate is 25%. Find the present value at time 0.

The payment growth rate is $g=-0.20$, not $+0.20$, and the timeline gives $m=1$, $n=3$, and $C=\$1{,}000$. The payments are \$1,000, \$800, and \$640. At time 1, discount them one, two, and three years:

{}_{1}V = \frac{1000}{1.25} +\frac{800}{1.25^2} +\frac{640}{1.25^3} = 800+512+327.68 = \$1{,}639.68.

Then discount this value one year to time 0:

V_0=\frac{1{,}639.68}{1.25} =\$1{,}311.744 \approx \$1{,}311.74.

As a direct check, the time-0 payment values are $1000/1.25^2=\$640$, $800/1.25^3=\$409.60$, and $640/1.25^4=\$262.144$. Their sum is \$1,311.744, which rounds to \$1,311.74. The method discounts to time 1 first and then to time 0. The trap is treating a 20% decline as positive growth, which would incorrectly increase later payments.

Timing Checks and Exam Traps

The exponent $m$ in the final discount factor counts the years from the focal date to the date where the annuity value was found. It does not count the years to the first payment. Under the convention used here, the first payment is at $m+1$, so there are $m$ years from time 0 to the value date and $m+1$ years from time 0 to the first payment.

For a quick timeline check, write the payment dates explicitly before applying a formula. If the first payment is at the end of year 5 and there are annual payments, the value date is the end of year 4, meaning $m=4$. Discount the value at year 4 by $v^4$, not $v^5$. Alternatively, discount each payment directly to time 0; both methods must agree.

Miscounting the deferment: separate the first-payment date from the value date one period before it.
Using the equal-rate formula when rates differ: use $nC/(1+i)$ at time $m$ only when $g=i$.
Using the unequal-rate formula when rates are equal: its denominator is zero in that case, so evaluate the stream directly.
Giving a declining stream a positive growth rate: a decline of $h$ means $g=-h$, so its payment factor is $1-h$.
Mixing payment units and interest periods: the formulas here assume annual payments and an effective annual rate. The payment spacing and rate period must match.

Deferred geometric annuities appear whenever a stated payment stream begins after a waiting period. The two-stage valuation method keeps the payment pattern separate from the deferral: first handle the growing or declining payments at the annuity value date, then discount that single value to the requested date. This framework is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the value date before calculating the present value at time 0. Round dollar amounts to the nearest cent.

An annual payment stream has first payment \$500 at the end of year 2, followed by one payment at the end of year 3 that is 5% larger. The effective annual interest rate is 10%. Find the present value. (Answer: \$807.66)
A deferred fund pays \$250 at the end of year 3 and makes three further annual payments, each 6% larger than the preceding payment. The effective annual interest rate is 6%. Find the present value. (Answer: \$839.62)
A reserve pays \$1,000 at the end of year 2, followed by payments declining by 10% annually, with three payments total. The effective annual interest rate is 5%. Find the present value. (Answer: \$2,115.79)
A deferred grant pays \$400 at the end of year 4 and makes four further annual payments, each 8% larger than the preceding payment. The effective annual interest rate is 8%. Find the present value. (Answer: \$1,470.06)
A contract pays \$500 at the end of year 3, followed by payments of \$550 and \$605 at the ends of years 4 and 5. The effective annual interest rate is 20%. Find the present value by valuing the stream at time 2 first. (Answer: \$797.73)

What You'll Learn