Valuing an Annuity at an Interior Focal Date

Why Value an Annuity at an Interior Date?

An annuity’s value is often needed at a date that is neither the beginning nor the end of its payment stream. For example, a fund may need to report the value of a contract partway through its term. The key is to place every payment on the same date: accumulate payments already received to that date, and discount payments still due back to it.

The value at the focal date depends on which cash flows are being valued. The full-stream value includes past receipts carried forward as well as future payments discounted back. The remaining-stream value includes only payments still due; it does not include past receipts, even if those receipts have earned interest.

Definition: Consider an annual level annuity-immediate paying $R$ at times $1,2,\ldots,n$, with effective annual interest rate $i$. Its value at an interior focal date is the sum of the values at that date of the payments in the stream. A payment made before the focal date is accumulated to it; a payment after the focal date is discounted to it. If a payment occurs exactly at the focal date, include it once, at its payment amount.

Assume $i$ applies consistently over each year. For a focal date between annual payment dates, we will also assume compound accumulation over fractional years: a balance grows by a factor of $(1+i)^t$ over $t$ years. This specifies how to value a payment over, for example, half a year; an effective annual rate alone does not prescribe a fractional-year convention in every possible interest model.

Split the Payments at the Focal Date

Suppose first that the focal date is integer time $k$, where $1\leq k<n$. The first $k$ payments have been made by time $k$, and the remaining $n-k$ payments are still due. The past payments form an accumulated annuity-immediate value $R s_{\overline{k}|}$. The future payments form a present annuity-immediate value $R a_{\overline{n-k}|}$ at time $k$.

V_k=R s_{\overline{k}|}+R a_{\overline{n-k}|}

Here $V_k$ is the value of the full stream at time $k$. The term $R s_{\overline{k}|}$ includes the payment at time $k$, if there is one. The term $R a_{\overline{n-k}|}$ values only the payments at times $k+1,\ldots,n$. At an interior date strictly between payments, use the same split, but accumulate and discount each payment for its actual time distance from the focal date.

Locate the focal date.
Write down the payment times and identify which payments have occurred and which are still due.

Move past payments forward.
Accumulate each receipt from its payment date to the focal date.

Move future payments back.
Discount each payment from its due date to the focal date.

Add values at the common date.
Check that no payment is omitted or counted twice, especially one due exactly at the focal date.

Why the Split Formula Holds

The annuity factors are sums of the relevant payment values. At time $k$, the payment made at time $j\leq k$ has grown for $k-j$ years, while the payment due at time $j>k$ is discounted for $j-k$ years. Thus, under the stated compound-interest assumption,

V_k =R\sum_{j=1}^{k}(1+i)^{k-j} +R\sum_{j=k+1}^{n}(1+i)^{-(j-k)} =R s_{\overline{k}|}+R a_{\overline{n-k}|}.

This is a cash-flow argument: each payment is converted to an equivalent amount at time $k$, then the equivalent amounts are added. It also explains why a payment at time $k$ belongs in the first sum, where its accumulation factor is $(1+i)^0=1$, and not in both sums.

For a focal date $\tau$ between payment dates, let $k$ be the number of payments already made, so $k<\tau<k+1$. Under the fractional-year assumption, the direct cash-flow expression is

V_\tau =R\sum_{j=1}^{k}(1+i)^{\tau-j} +R\sum_{j=k+1}^{n}(1+i)^{-(j-\tau)}.

The exponents are the actual time intervals in years: for example, a receipt at time $j$ is accumulated for $\tau-j$ years if $j\leq k$, and a payment due at $j$ is discounted for $j-\tau$ years if $j>k$. When the focal date is an integer payment time, the earlier factor formula is usually quicker.

Worked Example: Value at a Payment Date

A hypothetical annuity-immediate pays $600 at the end of each year for five years. The effective annual interest rate is 5%. Find the full-stream value at time 3, immediately after the third payment. Also find the value of payments still due immediately after that payment.

Method. For the full-stream value, accumulate the first three payments to time 3 and discount the last two payments to time 3. For the remaining-stream value, include only payments at times 4 and 5. The common mistake to avoid is subtracting the time-3 payment from a value that already includes the accumulated earlier payments.

Full-stream value. The payments at times 1, 2, and 3 are carried to time 3. Their accumulation factors are $1.05^2$, $1.05$, and $1$:

600(1.05^2+1.05+1) =600(1.1025+1.05+1) =600(3.1525) =\$1{,}891.50.

The payments at times 4 and 5 are discounted by one and two years:

600\left(\frac{1}{1.05}+\frac{1}{1.05^2}\right) =600(0.95238095+0.90702948) =600(1.85941043) =\$1{,}115.65.

Adding the two components gives the full-stream value:

V_3=\$1{,}891.50+\$1{,}115.65=\$3{,}007.15.

The remaining-stream value immediately after the third payment is just the second component, $\$1{,}115.65$. The first three receipts are no longer payments still due; they are past cash flows. As a check, the payment at time 3 is included once in the accumulated component, at its amount of $600.

Focal Dates Between Payment Dates

A focal date need not coincide with an annual payment. In that case, do not round the date to the nearest payment date. Instead, use the actual time from each cash flow to the focal date. The fractional-year assumption matters here: using $(1+i)^t$ for a fraction $t$ of a year is an explicit model choice.

Worked Example: A Focal Date Halfway Between Payments

A hypothetical annuity-immediate pays $1,000 at the end of each year for four years. The effective annual interest rate is 8%. Assuming compound growth by $(1.08)^t$ over $t$ years, find the full-stream value at time 2.5.

Method. Payments at times 1 and 2 have already been received, so accumulate each for 1.5 and 0.5 years, respectively. Payments at times 3 and 4 are still due, so discount each for 0.5 and 1.5 years. The common mistake to avoid is treating time 2.5 as if it were time 2 or time 3.

V_{2.5} =1{,}000\left((1.08)^{1.5}+(1.08)^{0.5} +(1.08)^{-0.5}+(1.08)^{-1.5}\right).

The required factors are $(1.08)^{0.5}=1.03923048$, $(1.08)^{1.5}=1.12236892$, $(1.08)^{-0.5}=0.96225045$, and $(1.08)^{-1.5}=0.89097264$. Therefore, the accumulated past payments are

1{,}000(1.12236892+1.03923048)=\$2{,}161.60,

and the discounted future payments are

1{,}000(0.96225045+0.89097264)=\$1{,}853.22.

Thus $V_{2.5}=\$2{,}161.60+\$1{,}853.22=\$4{,}014.82$, rounded to the nearest cent. As a check, all four payments have been included exactly once and valued at time 2.5.

Using the Factors to Find an Unknown Payment

The split can also turn a target value at an interior date into an equation for the level payment. At integer time $k$, the coefficient of $R$ is the sum of the accumulated factor for past payments and the present-value factor for future payments. Solve for $R$ only after matching each factor to its side of the focal date.

Worked Example: Find the Level Payment from an Interior Value

A hypothetical annuity-immediate pays an unknown amount $R$ at the end of each year for six years. The effective annual interest rate is 5%, and its full-stream value at time 3 is $7,000. Find $R$, to the nearest cent.

Method. There are three payments accumulated to time 3 and three payments discounted to time 3. The common mistake to avoid is using a present-value factor for all six payments at time 3, which would value the stream at time 0 instead.

The accumulated factor for the first three payments is

s_{\overline{3}|}=1+1.05+1.05^2=1+1.05+1.1025=3.1525.

The present-value factor for the three payments at times 4, 5, and 6, valued at time 3, is

a_{\overline{3}|} =\frac{1}{1.05}+\frac{1}{1.05^2}+\frac{1}{1.05^3} =0.95238095+0.90702948+0.86383760 =2.72324803.

The value equation is therefore

7{,}000=R(3.1525+2.72324803)=R(5.87574803), \qquad R=\frac{7{,}000}{5.87574803}=\$1{,}191.34.

The answer is rounded to the nearest cent. Checking with the unrounded payment, $R(5.87574803)=\$7{,}000$ to the nearest cent, as required.

Common Timing Traps

The phrase “value after the payment” can refer either to the value of payments still due or to the value of the entire cash-flow history. These are different quantities. At time 3 in the first example, payments still due are worth $1,115.65. The full-stream value is $3,007.15 because it also carries the first three receipts to time 3.

A common incorrect calculation is to take the full-stream value immediately before the third payment, $3,007.15, and subtract $600 to claim that the remaining payments are worth $2,407.15. This is wrong: the $3,007.15 already includes the third payment, due at time 3, and also includes the earlier receipts. Subtracting $600 removes the payment at time 3 but leaves the earlier receipts in the result. For the remaining stream, calculate directly from payments at times 4 and 5:

600\left(\frac{1}{1.05}+\frac{1}{1.05^2}\right)=\$1{,}115.65.

Likewise, do not add $600 to the full-stream value when moving from immediately before to immediately after the time-3 payment. That payment was already counted as due at time 3 in the value immediately before payment; adding it again would double-count it.

Key takeaway: Choose the quantity first. For a full-stream value, carry past receipts forward and bring future payments back. For the value of payments still due, leave out all past receipts and value only the future payments. In either case, place every included cash flow at the requested focal date exactly once.

Check Your Understanding

For each problem, use the stated timing and interest assumptions, and round money to the nearest cent.

A hypothetical annuity-immediate pays $500 at the end of each year for five years. At an effective annual rate of 5%, find the full-stream value at time 2. (Answer: $2,386.62)
A hypothetical annuity-immediate pays $800 at the end of each year for seven years. At an effective annual rate of 4%, find the full-stream value at time 4. (Answer: $5,617.24)
A hypothetical annuity-immediate pays $1,200 at the end of each year for four years. At an effective annual rate of 6%, assume fractional-year growth by $(1.06)^t$. Find the full-stream value at time 1.5. (Answer: $4,537.92)
A hypothetical annuity-immediate pays $R$ at the end of each year for six years. At an effective annual rate of 5%, its full-stream value at time 3 is $7,000. Find $R$. (Answer: $1,191.34)
A hypothetical annuity-immediate pays $1,000 at the end of each year for eight years. At an effective annual rate of 5%, find the value of payments still due immediately after the third payment. (Answer: $4,329.48)

What You'll Learn