Timeline Representation of Level Annuities

Why Draw an Annuity Timeline?

Annuity calculations depend on when each payment occurs and when the value is being measured. A timeline makes both dates visible before any interest calculation begins. It also helps distinguish two separate tasks: moving earlier payments forward to a valuation date, and moving later payments backward to that date.

Here, one period is one year, and the stated effective annual interest rate applies between each pair of adjacent dates. The payments are certain and level, meaning they have the same amount at every scheduled date. The timeline method works with other period lengths as well, provided the rate and dates are expressed in consistent units.

Definition: A finite level annuity-immediate has a fixed number of equal payments at the ends of consecutive periods. With $n$ payments of amount $R$, its payment dates are $1,2,\ldots,n$. A finite level annuity-due has a fixed number of equal payments at the beginnings of consecutive periods, on dates $0,1,\ldots,n-1$. A timeline records these actual dates; it does not move payments merely because a value is being calculated at another date.

Reading a Timeline

Start by marking time zero and the period boundaries. Then mark every payment at its scheduled date and label the date at which the value is required. For an annuity-immediate with three annual payments, the dates are $1,2,3$; for an annuity-due with three payments, they are $0,1,2$. The payment count is three in both cases, but the schedules differ by one period.

\begin{array}{c|cccc} \text{Time }t & 0 & 1 & 2 & 3\\ \text{Immediate cash flow} & 0 & R & R & R\\ \text{Due cash flow} & R & R & R & 0 \end{array}

The timeline does not, by itself, assign a value to the payments. It provides the dates needed to do so. If the valuation date is time zero, a later payment must be discounted. If the valuation date is after a payment, that payment must be accumulated. If the valuation date lies between scheduled payments, some payments may be in the past and others in the future. Each is moved to the valuation date separately.

Key rule: Keep the payment dates fixed. To value a cash flow at date $k$, discount it if its payment date is after $k$, accumulate it if its payment date is before $k$, and leave it unchanged if it occurs at $k$.

Moving Payments to a Valuation Date

Suppose the effective interest rate per period is $i$, and a payment $R_j$ occurs at integer time $t_j$. Under a constant effective rate, its value at time $k$ is $R_j(1+i)^{k-t_j}$. A positive exponent accumulates a payment from an earlier date; a negative exponent discounts a payment from a later date. At the payment date itself, the exponent is zero, so the payment remains $R_j$.

For several payments, add the separately moved values. This is the principle of equivalence applied at the selected focal date: every cash flow is expressed at the same date before the values are combined.

V_k=\sum_j R_j(1+i)^{k-t_j}

This expression assumes the same effective rate applies for each period between payment dates and the valuation date. If rates vary by period, each cash flow must instead be moved using the relevant sequence of period-specific accumulation or discount factors. If payment intervals are not equal, the actual elapsed time and matching rate units must be used; counting payments is not a substitute for measuring time.

Why the Timeline Formula Works

For one period, a value $C$ grows to $C(1+i)$, while a value at the end of the period is worth $C/(1+i)$ at the beginning. Repeating this one-period rule gives the factor for any whole-number gap between dates. For example, moving a payment from time $t$ to time $k$ applies the one-period rule $k-t$ times. When $k<t$, this means applying the inverse factor, which is discounting.

Mark the dates.
Write each scheduled payment date and the valuation date on the same timeline.

Determine the direction.
A payment before the valuation date moves forward; a payment after it moves backward.

Move and add.
Apply the appropriate interest factor to each payment, then add the values at the common date.

Worked Timeline Examples

Worked Example: Present and Accumulated Values of One Schedule

A hypothetical account receives four annual deposits of $500 at the ends of years 1 through 4. The effective annual interest rate is 5%. Draw the payment schedule, then find its value at time 0 and at time 4. Round money to the nearest cent.

Method: mark the four payment dates first. The deposits form an annuity-immediate: there is no payment at time 0, and the four payments occur at times 1, 2, 3, and 4.

\begin{array}{c|ccccc} \text{Time }t & 0 & 1 & 2 & 3 & 4\\ \text{Cash flow} & 0 & 500 & 500 & 500 & 500 \end{array}

Value at time 0. Each deposit is in the future, so discount each one to time 0 at 5% per year:

\begin{aligned} V_0 &=\frac{500}{1.05}+\frac{500}{1.05^2}+\frac{500}{1.05^3}+\frac{500}{1.05^4}\\ &=476.190476\ldots+453.514739\ldots+431.918799\ldots+411.351237\ldots\\ &=1772.975252\ldots \end{aligned}

Rounding the value of the exact terms at the end gives $V_0=\$1,772.98$. In particular, the exact-term total rounds to $1,772.9753 at four decimal places; adding individually rounded terms is not the method for obtaining that intermediate total.

Value at time 4. The payment at time 4 stays at its date. Accumulate the other payments forward to time 4:

\begin{aligned} V_4 &=500(1.05)^3+500(1.05)^2+500(1.05)+500\\ &=578.8125+551.25+525+500\\ &=2155.0625. \end{aligned}

Thus the accumulated value is $2,155.06. The timeline check is that all four scheduled deposits appear once in each calculation, and the time-4 deposit is neither discounted nor accumulated.

Worked Example: A Due Schedule Valued Between Payments

A hypothetical grant makes four equal payments of $800 at times 0, 1, 2, and 3. The effective annual interest rate is 4%. Find the value at time 2, rounding to the nearest cent.

Method: locate the focal date before moving any cash flows. This is an annuity-due schedule because the first payment occurs at time 0. At the time-2 valuation date, payments at times 0 and 1 are in the past, the time-2 payment is at the focal date, and the time-3 payment is in the future.

\begin{array}{c|cccc} \text{Time }t & 0 & 1 & 2 & 3\\ \text{Cash flow} & 800 & 800 & 800 & 800\\ \text{Relation to }t=2 & \text{past} & \text{past} & \text{focal date} & \text{future} \end{array}

Accumulate the first two payments, leave the payment at time 2 unchanged, and discount the payment at time 3:

\begin{aligned} V_2 &=800(1.04)^2+800(1.04)+800+\frac{800}{1.04}\\ &=865.28+832+800+769.230769\ldots\\ &=3266.510769\ldots \end{aligned}

The value is $3,266.51 to the nearest cent. The common mistake is to discount the entire schedule to time 0, or to move every payment in the same direction. The timeline shows why neither is appropriate: the focal date is time 2, so both past and future payments contribute.

Worked Example: Counting Every Payment at an Intermediate Date

A hypothetical savings plan pays $1,000 at the end of each year for five years, at times 1 through 5. The effective annual interest rate is 4%. Find the value at time 3, rounding to the nearest cent.

Method: list all five dates and assign the exponent from each date to time 3. Payments at times 1 and 2 are accumulated; the payment at time 3 is unchanged; payments at times 4 and 5 are discounted.

\begin{array}{c|cccccc} \text{Time }t & 0 & 1 & 2 & 3 & 4 & 5\\ \text{Cash flow} & 0 & 1000 & 1000 & 1000 & 1000 & 1000\\ \text{Exponent to }t=3 & - & 2 & 1 & 0 & -1 & -2 \end{array}

\begin{aligned} V_3 &=1000(1.04)^2+1000(1.04)+1000+\frac{1000}{1.04}+\frac{1000}{1.04^2}\\ &=1081.60+1040+1000+961.538461\ldots+924.556213\ldots\\ &=5007.694674\ldots \end{aligned}

The value is $5,007.69. The check is the payment count: the expression has five terms, one for each of the five scheduled payments. Omitting the payment at time 3 would be wrong even though its interest factor is 1.

Common Timeline Traps

A timeline is useful not only for setting up a calculation but also for checking whether the result answers the question asked. Before evaluating any expression, confirm the payment dates, number of payments, valuation date, and rate period.

Shifting the schedule to match the valuation date. A value at time 2 does not move the payment dates. Move the values, not the scheduled payments.
Confusing an immediate annuity with a due annuity. An immediate schedule starts at time 1; a due schedule starts at time 0. The payment amount and count alone do not determine the dates.
Counting intervals instead of payments. Four payments occupy four dates, even though the gap from the first to the last may be three periods.
Using one direction for every payment. At an intermediate focal date, payments on opposite sides of that date require opposite directions of movement.
Mixing units. If the rate is effective per year, an exponent of 3 represents three years. A monthly date gap must not be treated as three years.

These checks matter whenever a schedule is valued, compared, or used in an equation of value. A correctly applied interest formula cannot repair a timeline that assigns a cash flow to the wrong date. This date-first discipline is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, draw a timeline before calculating. Assume the stated effective rate applies per year, and round dollar answers to the nearest cent.

A fund receives three deposits of $250 at times 0, 1, and 2. At 4% interest, find the value at time 0. (Answer: $721.52)
A contract pays $120 at the end of each of five years, at times 1 through 5. At 5% interest, find the accumulated value at time 5. (Answer: $663.08)
Five payments of $90 occur at times 0, 1, 2, 3, and 4. At 3% interest, find their value at time 2. (Answer: $450.39)
A scholarship makes four payments of $600 at times 0, 1, 2, and 3. At 6% interest, find the value at time 3. (Answer: $2,624.77)
A plan makes six payments of $400 at times 1 through 6. At 4% interest, find the value at time 2, identifying which payments are accumulated and which are discounted. (Answer: $2,267.96)

What You'll Learn