Deferred Annuity-Due Values and Payment Timing

Introduction

A deferred annuity-due is easy to misprice if the phrase “deferred for several years” is treated as a payment date. The reliable approach is to write the payment dates first. Once the first payment date is clear, the annuity-due factor can be placed at that date and discounted to the requested valuation date.

This tutorial considers a level annuity with a finite number of annual payments, a constant effective annual interest rate, and no contingent payments. The key is to distinguish the time when the payment stream begins from the date at which its value is calculated.

Definition: An annuity-due of $n$ level payments of 1 has payments at times $0,1,\ldots,n-1$, measured from its valuation date. Its present-value factor at that date is $\ddot{a}_{\overline{n}|}$. A deferred annuity-due is the same payment pattern shifted to a later date. If its first payment is at time $m$, its $n$ payments occur at times $m,m+1,\ldots,m+n-1$.

Find the First Payment Date Before Using a Factor

Assume annual payments and a constant effective annual rate $i$. Let $v=1/(1+i)$. If the first payment is at time $m$, the annuity-due factor is located at time $m$: at that date, the first payment is immediate and the remaining payments follow one year apart. To find the value at time zero, discount that value by $m$ years.

\text{Value at time }0 = R v^m \ddot{a}_{\overline{n}|}, \qquad \ddot{a}_{\overline{n}|}=\sum_{k=0}^{n-1}v^k

Here $R$ is the amount of each payment, $n$ is the number of payments, and $m$ is the time of the first payment in years. The annuity-due factor includes a payment at its own valuation date. It does not mean that a payment is made at time zero when the annuity is deferred.

A useful convention is to state the first payment date explicitly rather than rely on a loose phrase such as “deferred for three years.” For example, “first payment at time 3” means payments at times 3, 4, and so on. If a contract instead specifies that payments begin one year after the end of a three-year waiting period, the first payment is at time 4. That extra year changes the value.

Why the Deferred Value Formula Holds

The formula follows by discounting each payment separately. If the first payment is at time $m$, the payment at time $m+k$, for $k=0,\ldots,n-1$, has present value $R v^{m+k}$. Adding those present values gives

\begin{aligned} \text{Value at time }0 &=R v^m+R v^{m+1}+\cdots+R v^{m+n-1}\\ &=R v^m(1+v+\cdots+v^{n-1})\\ &=R v^m\ddot{a}_{\overline{n}|}. \end{aligned}

The factorization works because every payment is shifted by the same $m$ years. The sum in parentheses is exactly the present-value factor for $n$ payments of 1 in an annuity-due pattern. This reasoning also identifies the conditions behind the formula: payments are level and annual, the interest rate is constant, and the first payment occurs at the stated time $m$. If payment intervals or interest rates differ, this particular factor-and-discount calculation may not apply.

Mark the valuation date.
Identify the date at which the question asks for a value, commonly time zero.

Write all payment dates.
Use the first payment date and the payment frequency to list the schedule. Count the payments to confirm that there are $n$.

Place the due factor.
For an annuity-due, place $\ddot{a}_{\overline{n}|}$ at the date of the first payment.

Move the value to the requested date.
Discount or accumulate by the number of annual periods between the factor's date and the requested valuation date.

Worked Valuations

Worked Example: Four Payments Beginning at Time 5

A hypothetical fund promises four annual payments of $1,200, with the first payment at time 5 and the remaining payments at times 6, 7, and 8. The effective annual interest rate is 5%. Find the value at time zero. Assume payments are certain and the rate remains constant.

Method: identify the first payment date, value the due stream there, and discount to time zero. Because the first payment is at time 5, the annuity-due factor is placed at time 5. There are four payments, so use $\ddot{a}_{\overline{4}|}$.

First calculate the discount factor and the annuity-due factor:

v=\frac{1}{1.05}=0.95238095, \qquad \ddot{a}_{\overline{4}|} =1+v+v^2+v^3 =1+0.95238095+0.90702948+0.86383760 =3.72324803.

The discount factor for five years is $v^5=(1.05)^{-5}=0.78352617$. Therefore,

\text{Value at time }0 =1200(0.78352617)(3.72324803) =3500.7147\ldots

Rounded to the nearest cent, the value is $3,500.71. As a check, the payments are at times 5 through 8, so their combined value is $1200(v^5+v^6+v^7+v^8)$, the same cash-flow sum represented by the factor calculation.

Common mistake avoided: discounting by six years would put the first payment at time 6, not time 5. Using $v^6\ddot{a}_{\overline{4}|}$ would give a value one extra year too late.

Worked Example: A Payment at the End of a Two-Year Deferral

A hypothetical benefit pays $900 annually for six years. The first payment is due at time 3, exactly three years from the valuation date. The effective annual interest rate is 4%. Find the value at time zero, assuming all payments are certain.

Method: translate the stated payment date directly into the discount exponent. The payment dates are 3, 4, 5, 6, 7, and 8. Since the first payment is at time 3, the six-payment annuity-due factor is located at time 3.

At 4%, $v=1/1.04=0.96153846$. The due factor is

\begin{aligned} \ddot{a}_{\overline{6}|} &=1+v+v^2+v^3+v^4+v^5\\ &=1+0.96153846+0.92455621+0.88899636+0.85480419+0.82192711\\ &=5.45182233. \end{aligned}

The three-year discount factor is $v^3=0.88899636$. Thus

\text{Value at time }0 =900(5.45182233)(0.88899636) =4361.985\ldots

The value is $4,361.99, rounded to the nearest cent. The timing check is that the earliest cash flow is at time 3; the formula discounts the due-factor value at time 3 back to time zero.

Common mistake avoided: “three years from now” means the first payment is at time 3, not time 4. Adding an extra year of deferral would understate the value by a factor of $v$.

Worked Example: Distinguishing Two Possible Start Dates

A hypothetical contract pays $1,500 annually for five years at an effective annual interest rate of 5%. Compare its value at time zero if the first payment is at time 4 with its value if the first payment is at time 5. Assume certain payments and no change in the rate.

Method: keep the payment count and annuity-due factor fixed, and change only the first payment date. At 5%,

\ddot{a}_{\overline{5}|} =1+0.95238095+0.90702948+0.86383760+0.82270247 =4.54595050.

If the first payment is at time 4, discount the factor value by four years:

1500v^4\ddot{a}_{\overline{5}|} =1500(0.82270247)(4.54595050) =5609.95\text{ (to the nearest cent).}

If instead the first payment is at time 5, discount by five years:

1500v^5\ddot{a}_{\overline{5}|} =1500(0.78352617)(4.54595050) =5342.81\text{ (to the nearest cent).}

The second value is lower because every payment is one year later. In fact, the second value is the first value multiplied by $v=1/1.05$. This is a useful check: shifting an otherwise identical stream one year later should reduce its present value when $i>0$.

Common mistake avoided: using the same discount exponent for both start dates hides the timing difference. A one-year change in the first payment date changes the discounting of the entire stream.

Common Timing Traps

The annuity-due factor is determined by the payment pattern at its own valuation date. It includes a payment immediately at that date. The deferral discount is then determined by how far that date is from the requested valuation date. Keeping these two jobs separate prevents most timing errors.

Counting the waiting period instead of locating the payment. Write the first payment time explicitly. A first payment at time $m$ leads to $v^m$, not to an exponent chosen from a vague description.
Using the annuity-due factor at time zero. For a deferred stream, the factor is placed at the first payment date, not automatically at the valuation date.
Using the number of payments as the deferral period. The payment count $n$ determines the factor; the first payment date $m$ determines the deferral discount.
Changing the payment count when shifting the start date. If the contract still has $n$ payments, shifting the schedule later does not add or remove payments.

The factor form is appropriate when the annual payment interval matches the period used for $i$, and the effective annual rate is constant throughout the deferral and payment periods. If the rate varies by year, each cash flow must be discounted using the applicable rates rather than a single $v^m\ddot{a}_{\overline{n}|}$ expression. Careful timing remains essential in either approach.

Key takeaway: For a level annual annuity-due whose first payment is at time $m$, place $\ddot{a}_{\overline{n}|}$ at time $m$, then move its value to the requested date. At time zero, the value is $R v^m\ddot{a}_{\overline{n}|}$. The formula is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the first payment date before calculating. Assume annual payments, certain cash flows, and a constant effective annual interest rate.

A fund pays $800 annually for three years, with the first payment at time 4. The effective annual rate is 5%. Find the value at time zero. (Answer: $1,881.96)
A hypothetical benefit pays $1,000 annually for four years, with the first payment at time 3. The effective annual rate is 6%. Find the value at time zero. (Answer: $3,083.93)
A contract pays $250 annually for five years, with the first payment at time 6. The effective annual rate is 4%. Find the value at time zero. (Answer: $914.77)
A hypothetical payment stream has three annual payments of $1,000 at times 2, 3, and 4. The effective annual rate is 10%. Write it as a deferred annuity-due and find its value at time zero. (Answer: $1000v^2\ddot{a}_{\overline{3}|}=$$2,260.77)

What You'll Learn