Why Due and Immediate Annuities Differ by One Payment

Why the Payment Timing Matters

Two annual life annuities can have the same payment amount, the same insured life, and the same interest and mortality assumptions, yet different present values. The distinction is when the first payment is made. An annuity-due payment is made at the beginning of a payment period; an annuity-immediate payment is made at its end. With annual payments, that means the due annuity starts at time zero and the immediate annuity starts at time one.

The time-zero payment of a life annuity-due is made while the insured is alive at issue, so it is not discounted and is not weighted by a future survival probability. Each payment after time zero occurs at the same date, and under the same survival condition, as the corresponding payment of the annuity-immediate. This date-by-date match explains why the difference is one payment, not an interest adjustment to every payment.

Definition: For an annual life annuity with level payment amount 1, $\ddot{a}_x$ denotes the present value at age $x$ of payments at times $0,1,2,\ldots$, made only if the life is alive at the payment time. The symbol $a_x$ denotes the present value of payments at times $1,2,3,\ldots$, made only if the life is alive at the payment time. These are an annual life annuity-due and an annual life annuity-immediate, respectively.

Assume a deterministic effective annual interest rate $i$, with $v=1/(1+i)$. The probability that a life aged $x$ survives at least $k$ years is ${}_kp_x$, where ${}_0p_x=1$. A payment at time $k$ has expected present value equal to its amount, multiplied by $v^k$ and by the probability that it is payable.

Matching Payments by Their Dates

For unit annual payments, the due annuity includes a certain payment at time zero. Its payment at time $k$, for $k\geq 1$, is made only if the life survives to time $k$. The immediate annuity has exactly those same payment dates and survival conditions for $k\geq 1$. Thus their present-value sums are

\ddot{a}_x =1+\sum_{k=1}^{\infty}v^k\,{}_kp_x, \qquad a_x=\sum_{k=1}^{\infty}v^k\,{}_kp_x.

Subtracting the common terms leaves only the time-zero payment. This argument applies to a whole-life annuity when the sums converge; it also applies to finite payment schedules when both annuities have the same payments from time one onward.

Theorem: Under the annual payment timing and survival-contingency definitions above, $\ddot{a}_x=1+a_x$. For a level annual payment $R$, the corresponding present values are $R\ddot{a}_x$ and $Ra_x$, so their difference is $R$.

R\ddot{a}_x-Ra_x =R\left(1+\sum_{k=1}^{\infty}v^k\,{}_kp_x\right) -R\sum_{k=1}^{\infty}v^k\,{}_kp_x =R.

The derivation depends on comparing payments at the same dates. It is not a claim that the two annuities have identical payments after shifting one stream by a year. In particular, multiplying the immediate value by $1+i$ shifts its payment dates and changes the survival probabilities attached to the payments. That is a different operation.

Worked Calculations

Worked Example: Evaluating Both Values from Survival Probabilities

A hypothetical annual life annuity pays $\$750$ while its recipient is alive. The effective annual interest rate is $25\%$. Assume ${}_1p_x=0.80$, ${}_2p_x=0.40$, and ${}_3p_x=0$. Find the present values of the annuity-immediate and annuity-due at issue, and verify their difference.

Method. Evaluate each payment at its payment date, using the corresponding survival probability. Since ${}_3p_x=0$, the life cannot survive to receive any payment at time three or later, so no later terms contribute.

The discount factor is $v=1/1.25=0.8$. The immediate annuity has payments at times one and two with nonzero survival probabilities. Their expected present values are

\begin{aligned} \text{Time 1: }&750(0.8)(0.80)=\$480.00,\\ \text{Time 2: }&750(0.8^2)(0.40)=\$192.00. \end{aligned}

At time three, the expected present value is $750(0.8^3)(0)=\$0$, and all later terms are also zero. Therefore, the annuity-immediate value is $\$480.00+\$192.00=\$672.00$. The annuity-due also pays $\$750.00$ at time zero, so its value is $\$750.00+\$672.00=\$1{,}422.00$.

Check. The difference is $\$1{,}422.00-\$672.00=\$750.00$, exactly the time-zero payment. The common mistake avoided here is omitting the time-two payment: surviving to time two has probability $0.40$, not zero.

Worked Example: Checking the Relationship at a Different Interest Rate

A hypothetical contract pays $\$1{,}200$ annually while the insured is alive. The effective annual interest rate is $10\%$, and the survival probabilities are ${}_1p_x=0.75$, ${}_2p_x=0.50$, ${}_3p_x=0.25$, and ${}_4p_x=0$. Calculate both present values at issue. Round money to the nearest cent only after completing the sums.

Method. First value the immediate annuity by payment date. The discount factor is $v=1/1.10=10/11$. The terms at times one, two, and three are

\begin{aligned} \text{Time 1: }&1200\left(\frac{10}{11}\right)(0.75) =\frac{9000}{11}\approx \$818.181818,\\ \text{Time 2: }&1200\left(\frac{10}{11}\right)^2(0.50) =\frac{60000}{121}\approx \$495.867769,\\ \text{Time 3: }&1200\left(\frac{10}{11}\right)^3(0.25) =\frac{300000}{1331}\approx \$225.394440. \end{aligned}

The time-four term is zero because ${}_4p_x=0$, and the same is true of all later terms. Adding the three nonzero terms gives an immediate-annuity value of $\$1{,}539.444027\ldots$, or $\$1{,}539.44$ to the nearest cent. The due value adds the certain payment at time zero:

\text{Due value} =\$1{,}200+\$1{,}539.444027\ldots =\$2{,}739.44.

Check. The difference is $\$2{,}739.44-\$1{,}539.44=\$1{,}200.00$. The common mistake avoided is applying an extra discount factor to the due annuity's time-zero payment; no discounting is needed at time zero.

Worked Example: Distinguishing an Extra Initial Payment from a Shift

Consider a hypothetical life-contingent cash-flow schedule with a $\$400$ payment at time zero, followed by payments of $\$500$, $\$600$, and $\$700$ at times one, two, and three if the life is then alive. The effective annual interest rate is $5\%$, and ${}_1p_x=0.80$, ${}_2p_x=0.60$, ${}_3p_x=0.30$, and ${}_4p_x=0$. Find the present value, and compare it with the otherwise identical schedule that omits the time-zero payment.

Method. This schedule is not level, so the fixed-payment annuity identity should not be applied using one common annual amount. Instead, value each dated cash flow directly. Here $v=1/1.05=20/21$. The payments at times one through three have expected present values

\begin{aligned} \text{Time 1: }&500(0.80)\left(\frac{20}{21}\right) \approx \$380.952381,\\ \text{Time 2: }&600(0.60)\left(\frac{20}{21}\right)^2 \approx \$326.530612,\\ \text{Time 3: }&700(0.30)\left(\frac{20}{21}\right)^3 \approx \$181.405896. \end{aligned}

The schedule without the time-zero payment is worth $\$380.952381+\$326.530612+\$181.405896=\$888.888889$, or $\$888.89$. Adding the $\$400.00$ paid at time zero gives a value of $\$1{,}288.89$ for the full schedule.

Check. The difference is $\$1{,}288.89-\$888.89=\$400.00$, the actual time-zero payment. This illustrates the more general matching-dates argument: when two schedules have identical payments from time one onward, their value difference is the amount of the initial payment. It is not necessarily the later annual payment amount if the schedule is nonlevel.

Conditions and Common Exam Traps

For a level annual life annuity, the simple difference $\ddot{a}_x-a_x=1$ assumes that both values refer to the same life, payment amount, interest basis, and payment dates from time one onward. The payment at time zero is certain at issue. Payments at later integer times are survival-contingent and discounted according to their actual dates.

Counting the first payment at the wrong time. The due annuity starts at time zero; the immediate annuity starts at time one. Do not make the due annuity's first payment contingent on surviving one year.
Dropping a later payment incorrectly. A zero probability at time three eliminates the time-three and later payments; it does not eliminate a time-two payment when ${}_2p_x$ is positive.
Discounting every due payment one period less. The payment at time one is discounted by $v$ under both annuities. For those later payments, the two present-value terms match exactly.
Using the relation when schedules do not match. If a contract changes the payment amount after time zero, state the actual schedules and compare their matching dates. The difference is the unmatched time-zero cash flow, not automatically a later level payment.
Confusing annuity timing with a survival shift. The identity follows by matching the same payment dates and survival probabilities. A calculation that changes the survival condition or shifts all payments by a year is evaluating a different cash-flow stream.

A reliable calculation therefore starts with a small date table: record each payment time, its amount, the relevant survival probability, and the discount factor. Compare the two streams row by row. For a level annual annuity, the only unmatched row is time zero, making the result aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Key takeaway: An annual life annuity-due includes one certain payment at time zero. Its later payments match those of the corresponding annuity-immediate, so for level annual payments $R$, the due value exceeds the immediate value by exactly $R$.

Check Your Understanding

For each problem, use the stated annual interest and survival assumptions, and compare the values using payment dates.

A hypothetical annuity pays $\$400$ annually. The effective annual interest rate is $5\%$, with ${}_1p_x=0.50$, ${}_2p_x=0.25$, and ${}_3p_x=0$. Find the immediate and due values at issue. (Answer: immediate $\$281.18$; due $\$681.18$.)
A hypothetical level annuity pays $\$900$ each year. The effective annual interest rate is $25\%$, with ${}_1p_x=0.60$, ${}_2p_x=0.30$, and ${}_3p_x=0$. Find both values and their difference. (Answer: immediate $\$604.80$; due $\$1{,}504.80$; difference $\$900.00$.)
A hypothetical life annuity pays $\$500$ annually. The effective annual interest rate is $10\%$, with ${}_1p_x=0.80$, ${}_2p_x=0.50$, ${}_3p_x=0.20$, and ${}_4p_x=0$. Find both present values, rounded to cents. (Answer: immediate $\$645.38$; due $\$1{,}145.38$.)
For the same life and interest basis, a level annual annuity-due is worth $\$2{,}460$, while the corresponding annuity-immediate is worth $\$1{,}960$. What is the annual payment? (Answer: $\$500$, since the difference is the certain time-zero payment.)
Suppose $a_x=2.5$ and the effective annual interest rate is $20\%$. A candidate claims that $\ddot{a}_x=(1+i)a_x=3.0$. Is this correct, and what is $\ddot{a}_x$? (Answer: No. $\ddot{a}_x=1+a_x=3.5$; the due-immediate difference is one unit payment, not an interest multiplier.)

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