Whole-Life Annuity-Immediate Valuations

Why Payment Timing Matters

A whole-life annuity-immediate pays at the end of each year in which the annuitant is alive at that payment date. Its value therefore depends on two factors for every possible payment: the probability the payment is made and the interest discount back to the valuation date. The first payment is one year from now, not immediately.

This tutorial develops the expected present value of that payment stream from annual survival probabilities. The method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. In the examples, the annuity pays one dollar per year unless a different payment amount is stated; a level annual payment of $b$ dollars has value $b a_x$.

Definition: A whole-life annuity-immediate issued to a life aged $x$ pays one unit at the end of each year $k=1,2,\ldots$, provided the life is alive at that time. With effective annual interest rate $i$, discount factor $v=1/(1+i)$, and ${}_kp_x$ denoting the probability that a life aged $x$ survives $k$ years, its expected present value is

a_x=\sum_{k=1}^{\infty} v^k\,{}_kp_x.

The factor ${}_kp_x$ weights the payment at time $k$ by the probability it is received. The factor $v^k$ discounts that payment for $k$ years. By convention, ${}_0p_x=1$, but there is no payment at time zero for an annuity-immediate.

For an annual payment of $b$, the present value on a particular lifetime is random: it is the sum of discounted payments actually received. Taking its expectation gives $b\sum_{k=1}^{\infty}v^k{}_kp_x=b a_x$, since the expected value of the indicator that the life survives to time $k$ is ${}_kp_x$. When $i>0$, the series converges because $0\leq {}_kp_x\leq1$ and $\sum_{k=1}^{\infty}v^k$ converges. If $i=0$, a finite value additionally requires a finite expected number of payments.

Deriving the Recursive Valuation

A whole-life value can be calculated either by summing survival-weighted discount factors or by conditioning on survival through the first year. The recursive approach is useful when one-year survival probabilities or values at successive ages are available.

Result: For a life aged $x$, the whole-life annuity-immediate value satisfies

a_x=v\,{}_1p_x\left(1+a_{x+1}\right).

Here $a_{x+1}$ is the value at age $x+1$, under the mortality and interest assumptions applicable from that age onward.

To justify the relation, condition on whether the life survives one year. If the life does not survive, no annuity payment is made at time 1 and there are no later payments. If the life survives, the payment of one unit at time 1 is made, and the remaining payments have the same form as a whole-life annuity-immediate valued at age $x+1$, one year later. At time 1, that combined value is $1+a_{x+1}$. Discount it one year and multiply by the probability of reaching that date:

a_x ={}_1p_x\,v(1+a_{x+1})+ (1-{}_1p_x)\,v(0) =v\,{}_1p_x(1+a_{x+1}).

The annuity-due value $\ddot{a}_x$ includes a payment at time zero, when the life is alive at the valuation date, as well as the later payments of the annuity-immediate. Thus, under the same payment and survival assumptions, $\ddot{a}_x=1+a_x$. This relation is a timing check, not a replacement for identifying which payment dates the contract specifies.

Worked Valuations

The first example uses a constant survival probability, so the annuity value is a geometric series. The next examples use the recursive relation when mortality varies by age.

Worked Example: Constant Annual Survival Probability

A hypothetical annuity pays one dollar at each year-end while the annuitant is alive. The effective annual interest rate is $25\%$. Assume the probability of surviving each year is $0.75$, independently of the previous years, so ${}_kp_x=(0.75)^k$. Find the expected present value.

Method. Substitute the survival and discount factors in the survival-weighted series. The discount factor is $v=1/(1.25)=0.8$. Therefore,

a_x=\sum_{k=1}^{\infty}(0.8)^k(0.75)^k =\sum_{k=1}^{\infty}(0.6)^k.

This is a geometric series with first term $0.6$ and common ratio $0.6$. Since $0.6<1$, its sum is $0.6/(1-0.6)=0.6/0.4=1.5$. The expected present value is \$1.50 per dollar of annual payment.

As a check, the first three expected discounted payments are $0.6$, $0.36$, and $0.216$, each smaller than the last. Their infinite sum is $1.5$, as given by the geometric-series formula. The calculation avoids the common mistake of discounting payments but omitting their survival probabilities.

Worked Example: Mortality Changes After the First Year

A hypothetical whole-life annuity-immediate pays one dollar annually. The effective annual interest rate is $25\%$. A life aged $x$ has one-year survival probability ${}_1p_x=0.8$; at every age from $x+1$ onward, the one-year survival probability is $0.5$. Find $a_x$.

Method. First find the value at age $x+1$. From that age, the one-year survival probability remains $0.5$, so the value is a geometric sum with ratio $v(0.5)=(0.8)(0.5)=0.4$:

a_{x+1}=\sum_{k=1}^{\infty}(0.4)^k =\frac{0.4}{1-0.4} =\frac{2}{3}.

Now use ${}_1p_x=0.8$, $v=0.8$, and the recursive relation:

a_x=(0.8)(0.8)\left(1+\frac{2}{3}\right) =0.64\left(\frac{5}{3}\right) =\frac{16}{15} =1.066666\ldots.

Rounded only at the final step to four decimal places, $a_x=1.0667$, or \$1.07 per dollar of annual payment when expressed in cents. The first expected discounted payment is $0.8(0.8)=0.64$; the value of all subsequent payments is $0.64(2/3)=0.426666\ldots$. Their sum is $1.066666\ldots$, checking the recursive result. The method avoids applying the age-$x$ survival probability repeatedly after mortality changes.

Worked Example: Using a Short Mortality Table and a Tail Assumption

A hypothetical annuity-immediate pays one dollar at each year-end. The effective annual interest rate is $25\%$. The one-year survival probabilities are $0.9$ at age $x$, $0.8$ at age $x+1$, and $0.5$ at every age from $x+2$ onward. Find the value at age $x$, showing how the later-age value is used.

Method. At age $x+2$, the one-year survival probability remains $0.5$ at all future ages. As in a geometric series with ratio $0.8(0.5)=0.4$, $a_{x+2}=0.4/(1-0.4)=2/3$.

Move back one age using ${}_1p_{x+1}=0.8$:

a_{x+1}=(0.8)(0.8)\left(1+\frac{2}{3}\right) =0.64\left(\frac{5}{3}\right) =\frac{16}{15}.

Then use ${}_1p_x=0.9$:

a_x=(0.8)(0.9)\left(1+\frac{16}{15}\right) =0.72\left(\frac{31}{15}\right) =1.488.

Thus the value is \$1.49 per dollar of annual payment, rounded to the nearest cent. To check directly, the first two expected discounted payments are $0.8(0.9)=0.72$ and $0.8^2(0.9)(0.8)=0.4608$. The third is $0.8^3(0.9)(0.8)(0.5)=0.18432$; from then on, each additional expected discounted payment is $0.4$ times the previous one. The total is $0.72+0.4608+0.18432/(1-0.4)=1.488$, agreeing with the recursion. The calculation avoids treating the first-year mortality probability as if it applied at every future age.

Timing Checks and Common Traps

A whole-life annuity-immediate is not valued by counting every year as a certain payment. Each year-end payment requires survival to that date. In particular, the payment at time $k$ is weighted by ${}_kp_x$, not by ${}_{k-1}p_x$. Using ${}_{k-1}p_x$ would count a payment at time $k$ even for a life that dies during year $k$.

A second frequent error is to include a payment at time zero. That changes the contract to an annuity-due. For example, in the first worked example the immediate value is $1.5$; the corresponding due value would be $1+1.5=2.5$, because it adds one certain payment at issue. A third error is to use $1+i$ as the discount factor: present values use $v=1/(1+i)$, so a payment one year from now is discounted by $v$, not multiplied by $1+i$.

Key takeaway: For a whole-life annuity-immediate, payment $k$ occurs at the end of year $k$, is made only if the annuitant survives $k$ years, and is discounted by $v^k$. Hence $a_x=\sum_{k=1}^{\infty}v^k{}_kp_x$, and conditioning on first-year survival gives $a_x=v\,{}_1p_x(1+a_{x+1})$.

Check Your Understanding

For each problem, identify the payment timing and apply the relevant survival-weighted valuation. Answers are given for checking.

A hypothetical annuity-immediate pays one dollar per year. The effective annual interest rate is $25\%$, and the annual survival probability is $0.8$ at every future age. Find its value. (Answer: $16/9$, or $1.7778$ to four decimal places.)
The effective annual interest rate is $25\%$. For a hypothetical annuitant, ${}_1p_x=0.6$, and the annual survival probability at every age after $x$ is $0.5$. Find $a_x$. (Answer: $0.80$.)
A hypothetical annuity-immediate has an effective annual interest rate of $10\%$, ${}_1p_x=0.8$, and $a_{x+1}=3$. Find $a_x$ to four decimal places. (Answer: $32/11=2.9091$.)
The effective annual interest rate is $25\%$. At age $x$, ${}_1p_x=0.75$, at age $x+1$, ${}_1p_{x+1}=0.5$, and at age $x+2$, ${}_1p_{x+2}=0$. Find the whole-life annuity-immediate value at age $x$. (Answer: $0.84$.)
A hypothetical annuity-immediate has an effective annual interest rate of $25\%$ and constant annual survival probability $p$. Its value is $1$. Find $p$. (Answer: $0.625$.)

What You'll Learn