Temporary Annuity-Immediate Valuations

Introduction

A temporary life annuity-immediate combines two limits on payments: payments stop after a fixed number of years, and a payment is made only if the annuitant is alive at its due date. Its value therefore depends both on interest and on survival probabilities at the payment dates. The key is to value each possible payment separately, using the probability of being alive when that payment is due.

This tutorial develops the expected present value for annual payments made at the end of each year, for at most $n$ years. The payment amount is fixed, and the valuation is at the annuity’s issue date. Payments are assumed to be made only when the annuitant has survived to the payment date.

Definition: A temporary life annuity-immediate with term $n$ pays a specified amount at the end of each year $k=1,2,\ldots,n$, provided the annuitant aged $x$ is alive at time $k$. It makes no payments after time $n$. For a unit annual payment, its expected present value is denoted $a_{x:\overline{n}|}$.

Valuation Formula

Let $i$ be the effective annual interest rate and $v=1/(1+i)$ the discount factor for one year. Let ${}_kp_x$ be the probability that a life aged $x$ survives for at least $k$ years. A payment of 1 at time $k$ has present value $v^k$ if it is made, and zero otherwise. Thus the expected present value of that payment is $v^k{}_kp_x$.

Adding the expected present values for the $n$ possible payment dates gives the temporary annuity-immediate factor. For a level annual payment $B$, multiply this factor by $B$.

a_{x:\overline{n}|}=\sum_{k=1}^{n}v^k\,{}_kp_x, \qquad \operatorname{EPV}=B\,a_{x:\overline{n}|}.

The survival probability in the term for payment $k$ is ${}_kp_x$, not ${}_{k-1}p_x$: the annuitant must survive to the end of year $k$, when that payment is due. In the special case of an annuity certain, where every payment is guaranteed, each survival probability is 1 and the formula reduces to $a_{\overline{n}|}=\sum_{k=1}^{n}v^k$.

Why the Formula Holds

The formula follows by writing the present value as a sum of payment indicators. Let $I_k$ equal 1 if the annuitant is alive at time $k$, and 0 otherwise. The present value of a unit annuity-immediate is $\sum_{k=1}^{n}v^k I_k$. By linearity of expectation, its expected present value is the sum of the expected values of these terms. Since $E[I_k]=P(\text{alive at time }k)={}_kp_x$, each term contributes $v^k{}_kp_x$. No assumption of independence between survival at different dates is needed.

E\!\left[\sum_{k=1}^{n}v^kI_k\right] =\sum_{k=1}^{n}v^kE[I_k] =\sum_{k=1}^{n}v^k{}_kp_x.

This reasoning requires that the payment amounts and dates are as specified, that the interest rate is appropriate for discounting each annual period, and that the survival probabilities match the payment dates. If payments are made at the beginning of each year, or more frequently than annually, the dates and corresponding survival probabilities must be changed.

Worked Valuations

Worked Example: Constant Annual Mortality for a Three-Year Term

A hypothetical contract pays $1,000 at the end of each year for at most three years, provided the annuitant is alive at the payment date. The effective annual interest rate is 5%. Assume a discrete annual mortality model in which the conditional probability of surviving each year is 0.90, independently of the year reached. Find the expected present value.

Method. First calculate the probability of surviving to each payment date. The annual survival probability is 0.90, so the cumulative probabilities are ${}_1p_x=0.90$, ${}_2p_x=0.90^2=0.81$, and ${}_3p_x=0.90^3=0.729$. The discount factors are $v=1/1.05$, $v^2=1/1.05^2$, and $v^3=1/1.05^3$.

The expected present value of the first payment is $1000(0.90)/1.05=\$857.142857\ldots$. For the second payment it is $1000(0.81)/1.05^2=\$734.693877\ldots$. For the third payment it is $1000(0.729)/1.05^3=\$629.737609\ldots$.

Adding the three expected present values gives $\$857.142857\ldots+\$734.693877\ldots+\$629.737609\ldots=\$2,221.574344\ldots$. The expected present value, rounded to the nearest cent, is $2,221.57.

Check and trap avoided. The result is less than the present value of three guaranteed payments, since survival is not certain. Using $0.90$ for every payment would incorrectly ignore the additional mortality exposure before years 2 and 3.

Worked Example: Using Life-Table Values

A hypothetical two-year temporary life annuity-immediate pays $1,000 at each year-end if the annuitant is alive. At age $x$, a mortality table gives $\ell_x=1{,}000$, $\ell_{x+1}=800$, and $\ell_{x+2}=500$. The effective annual interest rate is 10%. Calculate the expected present value.

Method. In a life table, the probability of surviving from age $x$ to age $x+k$ is ${}_kp_x=\ell_{x+k}/\ell_x$. Therefore, ${}_1p_x=800/1{,}000=0.80$ and ${}_2p_x=500/1{,}000=0.50$. The two payments are due at times 1 and 2, so they are discounted by $1.10$ and $1.10^2=1.21$, respectively.

The first expected present value is $1000(0.80)/1.10=\$727.272727\ldots$. The second is $1000(0.50)/1.21=\$413.223140\ldots$. Thus the total is $\$727.272727\ldots+\$413.223140\ldots=\$1,140.495867\ldots$, or $1,140.50 to the nearest cent.

Check and trap avoided. The second survival probability is $\ell_{x+2}/\ell_x$, not $\ell_{x+2}/\ell_{x+1}$. The latter is a conditional one-year survival probability from age $x+1$, not the probability of being alive at the second payment date from the contract’s issue age.

Worked Example: Solving for the Level Payment

A hypothetical three-year annuity-immediate has an expected present value of $1,700. The effective annual interest rate is 10%. Under a discrete mortality model, the conditional probabilities of dying during years 1, 2, and 3, given survival to the start of each respective year, are $0.10$, $1/6$, and $0.20$. Find the level annual payment, rounded to the nearest cent.

Method. Convert the conditional mortality probabilities into cumulative survival probabilities at the payment dates. Survival to time 1 is $1-0.10=0.90$. Survival to time 2 is $0.90(1-1/6)=0.90(5/6)=0.75$. Survival to time 3 is $0.75(1-0.20)=0.60$.

The annuity factor is therefore $0.90/1.10+0.75/1.10^2+0.60/1.10^3$. The three terms are $0.818181818\ldots$, $0.619834710\ldots$, and $0.450788881\ldots$, giving a factor of $1.888805409\ldots$. If the level payment is $B$, then $1700=B(1.888805409\ldots)$. Hence $B=1700/(1.888805409\ldots)=\$900.0398\ldots$, or $900.04 to the nearest cent.

Check and trap avoided. Substituting the unrounded payment back into the valuation gives an expected present value of $1,700. Using the conditional survival probabilities as though they were cumulative probabilities would give incorrect weights for years 2 and 3.

Timing, Interpretation, and Common Errors

A temporary life annuity-immediate differs from a temporary annuity-due in payment timing. The immediate annuity’s first payment is at time 1, so it is discounted for one year and weighted by the probability of surviving one year. A due annuity has a payment at time 0; that payment is neither discounted nor conditional on survival through the first year. The distinction changes both the discount exponents and the survival probabilities.

The finite term also matters. Even if the annuitant survives beyond year $n$, the contract described here makes no payment after year $n$. The formula therefore has exactly $n$ terms. It is not a whole-life valuation, and survival probabilities after time $n$ do not affect its value.

Starting the sum at time 0. This inserts a payment at issue, which is not part of an annuity-immediate.
Using the wrong survival date. A payment at time $k$ is weighted by ${}_kp_x$, since the life must be alive at that date.
Discounting by $k-1$ years. A payment at the end of year $k$ is discounted for $k$ full years.
Extending beyond the term. Payments after year $n$ are excluded even if the annuitant survives.
Mixing conditional and cumulative survival probabilities. Multiply successive conditional survival probabilities to obtain survival from age $x$ to a later payment date.

For a level benefit, the factor $a_{x:\overline{n}|}$ isolates the effect of interest and mortality from the dollar amount of the payment. This makes it useful both for valuing a stated benefit and for solving for a benefit consistent with a specified expected present value. The method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Key takeaway: For a temporary life annuity-immediate, value each possible year-end payment at its payment date, discount it by $v^k$, weight it by ${}_kp_x$, and sum from $k=1$ through the fixed term $n$.

Check Your Understanding

For each problem, use the stated annual interest and mortality assumptions. Round dollar amounts to the nearest cent.

A hypothetical two-year annuity-immediate pays $600 annually. The effective annual interest rate is 4%, and cumulative survival probabilities at years 1 and 2 are 0.95 and 0.85. Find its expected present value. (Answer: $1,019.60)
A life table has $\ell_x=800$, $\ell_{x+1}=760$, and $\ell_{x+2}=680$. A hypothetical two-year annuity-immediate pays $1,000 annually, and the effective annual interest rate is 5%. Find its expected present value. (Answer: $1,675.74)
A hypothetical two-year annuity-immediate pays $2,000 annually. The effective annual interest rate is 6%, and the conditional probability of surviving each year is 0.90. Find its expected present value. (Answer: $3,139.91)
A hypothetical two-year annuity-immediate has an expected present value of $1,232. The effective annual interest rate is 25%, and the cumulative survival probabilities at years 1 and 2 are 0.90 and 0.80. Find its level annual payment. (Answer: $1,000.00)

What You'll Learn