Payment Indexing for Varying Annuities

Why Payment Indexing Matters

A varying annuity is a finite sequence of payments whose amounts change over time. Before calculating its value, the sequence must be described unambiguously. The first payment might occur immediately or after a delay; payments might increase by a fixed dollar amount or by a fixed percentage; and the requested value might be measured at a date that is not the beginning of the stream. Each choice affects which cash flows are included and how far each one is discounted.

The central discipline is to separate payment number from payment date. Use an index to identify the first, second, and later payments, and use a separate timeline to state when each payment occurs. Then state the valuation date and the interest-rate convention. With those details fixed, the value is a sum of individual discounted cash flows; no special varying-annuity factor is needed to define the stream.

Definition: An $n$-payment stream is specified by payment amounts $C_1,C_2,\ldots,C_n$ and payment dates $t_1,t_2,\ldots,t_n$, where $t_k$ is the time of payment number $k$, measured from a stated time origin. The first payment is $C_1$, and its payment date is $t_1$. For an arithmetic pattern, state the fixed increment $g$; for a geometric pattern, state the per-payment ratio $r$. Also state the valuation date.

For example, the words “five payments increasing by 20” do not completely define a stream. One must know the amount of the first payment, whether the increase is $20 or 20%, when the first payment occurs, and the date at which the value is required. A timeline makes these conditions visible:

\begin{array}{c|ccccc} \text{Payment number }k&1&2&3&\cdots&n\\ \text{Payment date }t_k&t_1&t_2&t_3&\cdots&t_n\\ \text{Amount }C_k&C_1&C_2&C_3&\cdots&C_n \end{array}

Indexing Arithmetic and Geometric Payments

For an arithmetic pattern, each payment after the first is larger or smaller than the preceding payment by the same dollar amount. If the increment is $g$, the $k$th payment is $C_1+(k-1)g$. The subtraction of 1 is essential: when $k=1$, the formula must return the stated first payment $C_1$.

C_k=C_1+(k-1)g,\qquad k=1,2,\ldots,n

A positive $g$ describes an increasing sequence; a negative $g$ describes a decreasing one. The amount $g$ is added once between consecutive payments. It does not, by itself, say whether those payments are annual, monthly, or otherwise spaced.

For a geometric pattern, each payment is obtained by multiplying the preceding payment by a fixed ratio $r$. The ratio is applied once per payment interval, whatever the length of that interval may be. A ratio of $1.08$ means an 8% increase from one payment to the next; a ratio of $0.95$ means a 5% decrease.

C_k=C_1r^{k-1},\qquad k=1,2,\ldots,n

The amount $C_1$ and date $t_1$ are independent pieces of information. For instance, $C_1$ can be $500 even if the first payment is not due until time 3. In that case, the payment at time 3 is still payment number 1; the deferral does not change the index.

Indexing check: Substitute $k=1$ into the payment rule. An arithmetic rule must give $C_1+(1-1)g=C_1$, and a geometric rule must give $C_1r^{1-1}=C_1$. If it does not, the index has been shifted.

Valuation Date and Discounting

Suppose payments are certain, and interest is measured by an effective annual rate $i$. A payment at time $t_k$ has value at an earlier date $s$ equal to its amount multiplied by the discount factor for the elapsed time $t_k-s$. Under annual effective compounding, for times measured in years, that factor is $(1+i)^{-(t_k-s)}$. Thus the value at $s$ of the payments occurring after $s$ is obtained by discounting each included payment separately.

V_s=\sum_{k:\,t_k>s} C_k(1+i)^{-(t_k-s)}

This expression treats $s$ as immediately after any payment made at time $s$, so a payment at $s$ is excluded. If the value is required immediately before a payment at $s$, that payment is included at its undiscounted amount, and the sum must include the term $C_k$ for which $t_k=s$. State which convention applies whenever a valuation date coincides with a payment date.

The formula follows from the definition of effective interest. A dollar invested at date $s$ accumulates to $(1+i)^{t_k-s}$ dollars at date $t_k$. Therefore the amount required at $s$ to fund $C_k$ at $t_k$ is $C_k/(1+i)^{t_k-s}$. Adding the required amounts for each payment gives the total value at $s$. This reasoning works for any finite payment pattern; it does not depend on whether the amounts are level, arithmetic, or geometric.

If payments are not annual, describe both the payment dates and the compounding convention. With an effective annual rate and fractional-year dates, the same formula uses fractional exponents. If instead an effective rate is given per payment period, use that period consistently in both the dates and the discounting. Do not silently treat a monthly payment as though one full year had elapsed.

Worked Examples

Worked Example: An Arithmetic Stream With Its First Payment Deferred

A hypothetical fund will pay five certain annual amounts. The first payment is $240 at time 1, and each later payment is $35 more than the preceding payment. Find the value at time 0 using an effective annual interest rate of 5%. The payments are deterministic; no mortality or probability distribution is assumed.

Method. First index the payments, then place each amount at its stated date, and discount each amount to time 0. Since the first payment is at time 1, it is payment number 1 at $t_1=1$, not a payment at time 0.

Here $n=5$, $C_1=240$, $g=35$, and $t_k=k$ for $k=1,\ldots,5$. The payment rule is $C_k=240+35(k-1)$, giving:

\begin{array}{c|ccccc} k&1&2&3&4&5\\ t_k&1&2&3&4&5\\ C_k&240&275&310&345&380 \end{array}

With $v=1/1.05=0.9523809524$, the value is the sum of the five discounted payments:

\begin{aligned} V_0 &=240v+275v^2+310v^3+345v^4+380v^5\\ &=240(0.9523809524)+275(0.9070294785)+310(0.8638375986)\\ &\quad+345(0.8227024749)+380(0.7835261666)\\ &=228.5714286+249.4331066+267.7896556+283.8323539+297.7399433\\ &=1327.366488. \end{aligned}

Rounded to the nearest cent, the value is $1,327.37. The method avoids treating the deferred first payment as though it were due immediately: its discount exponent is 1, not 0. As a check, the amounts increase by exactly $35 at each step, and each payment is discounted for exactly its time from date 0.

Worked Example: A Geometric Stream and a Separate Payment Index

A hypothetical contract pays three certain annual amounts at times 2, 3, and 4. The first amount is $200, and each payment is 10% greater than the preceding one. Find the value at time 0 at an effective annual interest rate of 4%. There is no mortality contingency or random payment model.

Method. The growth ratio applies by payment number, not by the time measured from date 0. The first payment is $C_1=200$ at $t_1=2$, and the ratio is $r=1.10$. Thus $C_k=200(1.10)^{k-1}$.

\begin{array}{c|ccc} k&1&2&3\\ t_k&2&3&4\\ C_k&200&220&242 \end{array}

The discount exponents are the payment dates 2, 3, and 4 because the valuation date is 0. Using $v=1/1.04$:

\begin{aligned} V_0 &=200v^2+220v^3+242v^4\\ &=200(0.9245562130)+220(0.8889963587)+242(0.8548041912)\\ &=184.9112426+195.5791989+206.8626143\\ &=587.3530558. \end{aligned}

The value rounded to the nearest cent is $587.35. The amounts grow by the ratio 1.10 from one payment to the next, while the exponents depend on dates from the valuation point. These are different roles: using $r^2$ for the first payment would incorrectly make its amount $242 instead of $200.

Worked Example: Revaluing a Stream at a Later Date

A hypothetical fund makes six certain annual payments at times 1 through 6. The first payment is $80, and the amount increases by $12 each year. Find the value at time 2 immediately after the payment at time 2, using an effective annual interest rate of 6%. No mortality or probability distribution is involved.

Method. Define the full stream first: $C_k=80+12(k-1)$ and $t_k=k$. The amounts at times 1 through 6 are $80, $92, $104, $116, $128, and $140. Because the valuation date is immediately after time 2's payment, the first two payments are past and excluded. The future payment amounts and elapsed times from time 2 are:

\begin{array}{c|cccc} \text{Payment date}&3&4&5&6\\ \text{Amount}&104&116&128&140\\ \text{Elapsed time from }2&1&2&3&4 \end{array}

Discounting by the elapsed times gives:

\begin{aligned} V_2 &=\frac{104}{1.06}+\frac{116}{(1.06)^2} +\frac{128}{(1.06)^3}+\frac{140}{(1.06)^4}\\ &=98.1132075+103.2395870+107.4712682+110.8931129\\ &=419.7171756. \end{aligned}

Rounded to the nearest cent, the value is $419.72. The common mistake is to discount from time 0 and label the result $V_2$, or to include the time-2 payment even though the value is immediately after it. The correction is to measure every discount period from the stated valuation date and apply the stated before-or-after convention. The four future payments are discounted for 1, 2, 3, and 4 years, respectively.

Common Indexing Traps

Many errors occur before any financial calculation begins. The following checks help prevent them:

Starting the payment index at zero without changing the rule. If the first payment is called $C_1$, its amount must be generated at $k=1$. The standard arithmetic expression $C_1+(k-1)g$ would be wrong if one substituted a zero-based index without rewriting the expression.
Confusing a payment date with a payment number. If the first payment is at time 3, it is still payment number 1. A geometric ratio of 1.10 makes the second payment 1.10 times the first, not the payment at time 4 times the first three times.
Applying the increment or ratio according to elapsed years from time 0. A stream can be deferred or have nonannual payment intervals. Apply a payment-to-payment change once per step in the payment sequence, as specified.
Discounting from the wrong date. At valuation date $s$, the elapsed time for a payment at $t_k$ is $t_k-s$, not $t_k$, unless $s=0$.
Leaving the payment at the valuation date ambiguous. State whether the value is immediately before or immediately after a payment due on that date. The payment is included in the former case and excluded in the latter.

A timeline is especially useful when the first payment is deferred, when the valuation date falls partway through the stream, or when the payment frequency differs from the interest-rate period. Writing the payment number, date, and amount on separate rows exposes an off-by-one error before it contaminates the valuation.

Key takeaway: Before valuing a varying annuity, state $n$, $C_1$, the payment dates $t_k$, and either the arithmetic increment $g$ or geometric ratio $r$. Then specify the valuation date, interest convention, and whether a payment at that date is included.

Check Your Understanding

For each problem, make the payment index and dates explicit before calculating or identifying the requested amount.

A five-payment arithmetic stream has its first payment of $75 at time 2, with an increase of $9 per payment. List the amounts in order and identify the amount paid at time 5. (Answer: $75, $84, $93, $102, and $111; the payment at time 5 is $102.)
A four-payment geometric stream starts with $250 at time 0. Each payment is 0.80 times the preceding payment, and payments occur annually. List the payment dates and amounts. (Answer: At times 0, 1, 2, and 3, the amounts are $250, $200, $160, and $128.)
A certain three-payment arithmetic stream pays $40, $50, and $60 at times 1, 2, and 3. Find its value at time 0 at an effective annual interest rate of 10%. (Answer: $40/1.10+50/(1.10)^2+60/(1.10)^3=$$122.76.)
A certain three-payment geometric stream pays $50 at time 3, with each following payment 1.20 times the previous one. Find its value at time 0 at an effective annual interest rate of 5%. (Answer: Payments are $50, $60, and $72 at times 3, 4, and 5; value $=50/(1.05)^3+60/(1.05)^4+72/(1.05)^5=$$148.97.)
A certain stream pays five annual amounts at times 0 through 4, starting at $30 and increasing by $7 per payment. Find the value at time 2 immediately after that date's payment, using an effective annual interest rate of 10%. (Answer: The future payments are $51 at time 3 and $58 at time 4; value $=51/1.10+58/(1.10)^2=$$94.30.)

What You'll Learn