Temporary Annuity-Due Valuations

What a Temporary Annuity-Due Pays

A life annuity-due makes payments at the beginning of each payment period, provided the annuitant is alive when the payment is due. If the annuity is temporary, payments are scheduled only for a fixed number of periods. The value therefore depends on two separate questions: whether a payment is scheduled at a given time, and whether the annuitant is alive to receive it.

For an $n$-year annual annuity-due issued to a person aged $x$, the possible payment times are $0,1,\ldots,n-1$. The payment at time 0 is made immediately, assuming the person is alive at issue. A payment at time $k>0$ is made only if the person survives $k$ years. Death before the end of the term prevents later payments; it does not create extra payments or extend the term.

Definition: An $n$-year temporary life annuity-due provides payments at times $0,1,\ldots,n-1$, with each payment after time 0 contingent on the annuitant being alive at its payment time. The annuity factor $\ddot{a}_{x:\overline{n}|}$ is the expected present value of these payments when each payment is one unit. For $n=0$, the factor is 0.

Assume a constant effective annual interest rate $i$, with $v=1/(1+i)$, and annual discrete mortality. Let ${}_kp_x$ be the probability that a life aged $x$ survives for $k$ years; in particular, ${}_0p_x=1$. Payments are annual and level, are made only while the annuitant is alive, and have no additional death benefit or refund. Under these assumptions, the temporary annuity-due factor is:

\ddot{a}_{x:\overline{n}|} = \sum_{k=0}^{n-1} v^k\,{}_kp_x.

For a level payment of $R$ dollars, the expected present value is $R\ddot{a}_{x:\overline{n}|}$. The term for time 0 is $1$, because the payment is immediate and the life is alive at issue. Each later term combines the probability of being alive at the payment date with the discount factor for that date.

Why the Formula Works

The present value of a payment at time $k$ is not received in every outcome. It equals $v^k$ times the payment amount if the annuitant is alive at time $k$, and zero otherwise. Taking its expectation multiplies the discounted amount by the probability of survival to that time. Adding the expected present values of the scheduled payments gives the expected present value of the whole annuity.

E[\text{present value}] = \sum_{k=0}^{n-1} v^k R\,P(\text{alive at time }k) = R\sum_{k=0}^{n-1}v^k\,{}_kp_x.

The factor also satisfies a one-year recursion. After the immediate payment, the annuitant must survive one year for any later payments to be possible. Conditional on survival, the life is aged $x+1$, and $n-1$ years of the original term remain.

Theorem: For $n\geq 1$, with $p_x={}_1p_x$, $$ \ddot{a}_{x:\overline{n}|} = 1+vp_x\ddot{a}_{x+1:\overline{n-1}|}, \qquad \ddot{a}_{x:\overline{0}|}=0. $$

To justify the recursion, split the value into the time-0 payment and all payments from time 1 onward. The time-0 payment has value 1. The remaining value is zero if death occurs during the first year. If the life survives, the remaining payments are those of a temporary annuity-due with $n-1$ years left, valued one year later. Its value at issue is discounted by $v$, and the survival probability is $p_x$. Thus the remaining expected present value is $vp_x\ddot{a}_{x+1:\overline{n-1}|}$, as stated.

Calculating the Survival Probabilities

When mortality is supplied as one-year death probabilities, calculate survival to each payment date by multiplying the relevant one-year survival probabilities. If $q_x$ is the probability of death within one year for a life aged $x$, then $p_x=1-q_x$. For example, survival for two years from age $x$ is ${}_2p_x=p_xp_{x+1}$. In the annuity formula, the probability used for a payment at time $k$ is survival for exactly $k$ years—not survival for the full term unless $k=n$.

Worked Example: A Three-Year Annuity-Due With Mortality

A hypothetical contract pays \$1,000 annually at the beginning of each year for at most three years to a life aged $x$. The annual effective interest rate is 5%. Assume annual death probabilities $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.30$. Find the expected present value.

Method. List the three scheduled payment times, find survival to each time, and discount each payment. The third death probability is not needed for the correct three-year value, but it will make the term-counting trap clear.

The scheduled payment times are 0, 1, and 2. The one-year survival probabilities are $p_x=0.90$, $p_{x+1}=0.80$, and $p_{x+2}=0.70$. Therefore, survival to times 0, 1, and 2 is respectively $1$, $0.90$, and $0.90(0.80)=0.72$. The discount factor is $v=1/1.05=20/21$.

\begin{aligned} \text{EPV} &=1000\left(1+v(0.90)+v^2(0.72)\right)\\ &=1000\left(1+\frac{20}{21}(0.90) +\left(\frac{20}{21}\right)^2(0.72)\right)\\ &=1000\left(1+\frac{6}{7}+\frac{32}{49}\right)\\ &=1000\left(\frac{123}{49}\right) =\$2{,}510.20408\ldots \end{aligned}

Rounded to the nearest cent, the expected present value is \$2,510.20. As a check, the value includes the certain immediate payment of \$1,000, plus discounted expected payments of \$857.14 at time 1 and \$653.06 at time 2.

Common mistake avoided: A three-year annuity-due has three payments, at times 0, 1, and 2—not four payments through time 3. Including a time-3 payment would incorrectly add $1000(0.90)(0.80)(0.70)v^3$, or about \$435.37. That extra payment lies outside the contract term.

Worked Valuations With Multiple Survival Periods

A reliable calculation keeps the payment date, survival probability, and discount factor together. A short table can help prevent mixing up the number of elapsed years and the number of payments. At time $k$, use ${}_kp_x$ and $v^k$; the term ends after the payment at time $n-1$.

Worked Example: Four Payments With Age-Specific Survival

A hypothetical four-year temporary annuity-due pays \$500 at times 0, 1, 2, and 3, if the annuitant is alive at each payment time. The annual effective interest rate is 4%. The one-year survival probabilities at the relevant ages are $p_x=0.95$, $p_{x+1}=0.90$, and $p_{x+2}=0.80$. Find the expected present value.

Method. Build survival probabilities by multiplication, then sum the discounted expected payments. This avoids treating the successive annual survival probabilities as though they were all measured from the original age.

Survival to time 1 is $0.95$. Survival to time 2 is $0.95(0.90)=0.855$. Survival to time 3 is $0.855(0.80)=0.684$. At 4%, $v=1/1.04=25/26$, $v^2=625/676$, and $v^3=15625/17576$.

\begin{aligned} \text{EPV} &=500\left(1+0.95v+0.855v^2+0.684v^3\right)\\ &=500\left(1+0.91346154+0.79049556+0.60807351\right)\\ &=500(3.31203061)\\ &=\$1{,}656.015305 \end{aligned}

Rounded to the nearest cent, the expected present value is \$1,656.02. The four terms correspond to expected present values of approximately \$500.00, \$456.73, \$395.25, and \$304.04. Their sum checks to \$1,656.02 after rounding the final total.

Common mistake avoided: The time-3 survival probability is $0.95(0.90)(0.80)=0.684$, not just the last one-year probability, $0.80$. The last probability describes survival over one year conditional on reaching that age; it does not describe survival from issue.

Worked Example: Measuring the Effect of Death Before the Term

A hypothetical annuity-due pays \$1,200 annually for at most four years. The annual effective interest rate is 4%, and the one-year death probabilities at the relevant ages are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Find its expected present value and compare it with four certain payments at the same times.

Method. Calculate the survival-weighted value, then subtract it from the value of four certain payments. This isolates the value lost because some later payments are not made after death.

The annual survival probabilities are $0.90$, $0.80$, and $0.75$. Survival to payment times 0 through 3 is $1$, $0.90$, $0.90(0.80)=0.72$, and $0.72(0.75)=0.54$. At 4%, the discount factors are $1$, $25/26$, $625/676$, and $15625/17576$.

\begin{aligned} \ddot{a}_{x:\overline{4}|} &=1+0.90v+0.72v^2+0.54v^3\\ &=1+0.86538462+0.66568047+0.48005803\\ &=3.01112312. \end{aligned}

Multiplying by the payment amount gives $1200(3.01112312)=\$3,613.347744$, or \$3,613.35 to the nearest cent. For four certain payments, the factor is $1+v+v^2+v^3=3.77509103$. The certain-payment value is $1200(3.77509103)=\$4,530.109236$, or \$4,530.11.

The difference is about $\$4,530.109236-\$3,613.347744=\$916.761492$, or \$916.76 to the nearest cent. It can also be checked directly: the probabilities of missing the payments at times 1, 2, and 3 are $0.10$, $0.28$, and $0.46$. Thus the discounted expected value of the missing payments is $1200(0.10v+0.28v^2+0.46v^3)=\$916.76$, agreeing with the difference.

Common mistake avoided: The time-0 payment is still made under the stated assumption that the annuitant is alive at issue. Mortality reduces the later survival-contingent payments; it does not reduce the immediate payment.

Common Checks Before You Finish

A temporary annuity-due valuation is especially sensitive to the payment schedule. Before accepting a result, confirm that the number of terms is $n$, that the first term is undiscounted, and that each later term uses survival to that payment time. The following checks help catch frequent errors:

Count payment dates, not elapsed periods. An $n$-year annuity-due has payment dates 0 through $n-1$.
Keep survival conditional on reaching each age. Multiply annual survival probabilities to get survival from issue to a later payment date.
Match discounting to timing. A payment at time $k$ is discounted by $v^k$, not by $v^{k-1}$.
Check the immediate payment. If the life is alive at issue, the time-0 payment has survival probability 1 and discount factor 1.

These calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. In an exam-style problem, writing the payment times explicitly before substituting values is often the simplest way to avoid a one-payment error.

Check Your Understanding

For each question, identify the scheduled payment times before calculating.

A hypothetical two-year annuity-due pays \$800 per year. The annual effective interest rate is 5%, and $q_x=0.20$. Find the expected present value. (Answer: \$1,409.52)
A hypothetical three-year annuity-due pays \$1,000 per year at 6% effective annual interest. The one-year survival probabilities are $p_x=0.90$ and $p_{x+1}=0.80$. Find the expected present value. (Answer: \$2,489.85)
A hypothetical four-year annuity-due pays \$250 per year, with no interest. The one-year survival probabilities are $0.96$, $0.90$, and $0.75$, in order from issue. Find the expected present value. (Answer: \$868.00)
A hypothetical three-year annuity-due is valued at 5% effective annual interest. Its death probabilities are $q_x=0.10$ and $q_{x+1}=0.20$. What level annual payment gives an expected present value of \$3,000? (Answer: \$1,195.12)
A hypothetical four-year annuity-due pays \$1,200 per year at 4% effective annual interest. The one-year survival probabilities are $0.90$, $0.80$, and $0.75$, in order from issue. Find the expected present value of the payment at time 3 alone. (Answer: \$576.07)

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