Whole-Life Annuity-Due Valuations

What a Whole-Life Annuity-Due Pays

A whole-life annuity-due makes level payments at the start of each year, for as long as the annuitant is alive. The first payment is made immediately, if the annuitant is alive at the valuation date. Later payments are made on anniversaries, but only if the annuitant is alive on those payment dates. The expected present value therefore depends on both the timing of each payment and the probability that the payment is made.

This tutorial assumes annual payments, an effective annual interest rate $i$, and a mortality model that gives annual survival probabilities. It does not include expenses, payment guarantees after death, or benefits paid continuously. Those features change the cash flows being valued.

Definition: For a life aged $x$, a unit whole-life annuity-due pays 1 at times $0,1,2,\ldots$, provided the annuitant is alive at the relevant payment time. Its expected present value is denoted by $\ddot{a}_x$. The discount factor for $k$ years is $v^k$, where $v=1/(1+i)$. If each payment is $B$, the expected present value is $B\ddot{a}_x$.

Write ${}_kp_x$ for the probability that a life aged $x$ survives for at least $k$ years. In particular, ${}_0p_x=1$, so the time-zero payment is certain under the assumption that the annuitant is alive at valuation. For $k\geq 1$, the payment at time $k$ is made only if the annuitant survives to that time.

Expected Present Value as a Survival-Weighted Sum

For each possible payment date, multiply the payment amount by its discount factor and by the probability it is made. Adding these expected present values gives the whole-life annuity-due value. If payments can occur only through time $n$, the formula is a finite sum. If payments can continue indefinitely, it is an infinite sum.

\ddot{a}_x = \sum_{k=0}^{\infty} v^k\,{}_kp_x = 1+\sum_{k=1}^{\infty} v^k\,{}_kp_x

If the mortality model specifies that no one can survive to time $n+1$, then ${}_{n+1}p_x=0$, and the sum reduces to $k=0,\ldots,n$. For a life table with survivor counts $\ell_x$, the survival probability at an integer duration $k$ is ${}_kp_x=\ell_{x+k}/\ell_x$, when $\ell_x>0$. Substitution gives the same valuation using life-table entries.

Key result: A whole-life annuity-due is valued by including the payment at time zero and then weighting each later payment by survival to its payment date. With $i>0$, the infinite sum converges: since ${}_kp_x\leq 1$, its terms are bounded above by the terms of the convergent series $\sum_{k=0}^{\infty}v^k$.

Why the Survival-Weighted Formula Holds

Let $I_k$ equal 1 if the payment at time $k$ is made and 0 otherwise. The present value of the payments is the random sum $\sum_{k=0}^{\infty}v^k I_k$. The payment indicator has expectation $E[I_k]={}_kp_x$, including $E[I_0]=1$. Taking expectations gives the survival-weighted sum. For an infinite stream, the terms are nonnegative, so the expected value of the increasing sequence of finite partial sums is the limit of their expected values.

E\left[\sum_{k=0}^{\infty}v^k I_k\right] = \sum_{k=0}^{\infty}v^k E[I_k] = \sum_{k=0}^{\infty}v^k\,{}_kp_x

There is also a useful age-indexed recursion. The first payment is certain. For the later payments, the life must first survive one year; conditional on that survival, the remaining payments have the value of a whole-life annuity-due at age $x+1$, discounted one year.

\ddot{a}_x=1+v\,{}_1p_x\,\ddot{a}_{x+1}

This recursion follows by separating the time-zero payment from all later payments. The probability of reaching the next anniversary is ${}_1p_x$, and discounting the value at that anniversary to time zero contributes $v$. The recursion applies to annual advance payments under the stated survival model; it does not describe payments made at the ends of years.

Worked Valuations

Worked Example: A Finite Whole-Life Mortality Model

A hypothetical contract pays $1,500 at the start of each year while a life aged 50 is alive. The effective annual interest rate is $i=25\%$, so $v=1/1.25=0.8$. The probabilities of survival from the valuation date to times 1, 2, and 3 are $0.80$, $0.60$, and $0.30$, respectively; survival to time 4 is impossible. Find the expected present value.

Method. Use the finite survival-weighted sum. Payments occur at times 0, 1, 2, and 3. The payment at time 0 is certain, and there can be no payment at time 4 or later.

For a unit payment, the four contributions are:

Time 0: $1\times 1=1$.
Time 1: $0.80\times 0.8=0.64$.
Time 2: $0.60\times 0.8^2=0.60\times0.64=0.384$.
Time 3: $0.30\times 0.8^3=0.30\times0.512=0.1536$.

Thus the unit annuity value is $1+0.64+0.384+0.1536=2.1776$. Multiplying by the annual benefit gives $1{,}500\times2.1776=$$3,266.40. The calculation uses the stated survival probabilities at each payment date and includes exactly four possible payments.

Check. The time-zero payment contributes $1,500 with certainty. The total value, $3,266.40, exceeds that first payment, as expected, while remaining below the present value of four certain payments, $1{,}500(1+0.8+0.64+0.512)=$$4,428.

Worked Example: Constant Annual Mortality

A hypothetical annuity pays $900 at the start of each year while a life is alive. The effective annual interest rate is $i=5\%$, and the annual probability of survival is $p=0.90$ at every age. Find the expected present value under this constant mortality model.

Method. Survival to time $k$ requires surviving each of the first $k$ years. Independence of the annual survival events under the model gives ${}_kp_x=p^k=0.90^k$. Here $v=1/1.05=20/21$, so $vp=18/21=6/7$.

The unit annuity value is the geometric series:

\ddot{a}_x = \sum_{k=0}^{\infty}(vp)^k = \frac{1}{1-vp} = \frac{1}{1-6/7} = 7

The expected present value of the $900 payments is $900\times7=$$6,300. This use of a geometric series requires the common survival probability to remain constant at each year and $vp<1$; the formula should not be applied to a mortality model with varying annual survival probabilities without first establishing a different sum.

Check. The value includes $900 at time zero. Each successive expected discounted payment is $6/7$ of the preceding term, so the infinite sum is larger than the first payment and equals $900/(1-6/7)=$$6,300.

Worked Example: Valuing from Survivor Counts

A hypothetical life table has $\ell_{60}=8{,}000$, $\ell_{61}=7{,}200$, $\ell_{62}=6{,}000$, and $\ell_{63}=0$. A whole-life annuity-due pays $1,000 annually to a life aged 60 while alive. The effective annual interest rate is $i=4\%$. Calculate the expected present value, rounding to the nearest cent.

Method. Convert each survivor count to a probability relative to the 8,000 lives aged 60. The survival probabilities to times 1, 2, and 3 are $7{,}200/8{,}000=0.90$, $6{,}000/8{,}000=0.75$, and $0/8{,}000=0$. Thus payments are possible at times 0, 1, and 2; no one survives to time 3. The discount factor is $v=1/1.04=25/26$.

The unit value is:

\ddot{a}_{60} = 1+0.90\left(\frac{25}{26}\right) +0.75\left(\frac{25}{26}\right)^2 = 1+0.865384615\ldots+0.694526627\ldots = 2.559911242\ldots

Multiplying by the annual payment gives $1{,}000\times2.559911242\ldots=$$2,559.91, rounded to the nearest cent. The zero survivor count at age 63 rules out a payment at time 3; it does not remove the payment at time 2, because 6,000 lives reach that date.

Check. The result exceeds the certain time-zero payment of $1,000 and is less than the value of three certain payments, $1{,}000(1+v+v^2)$, as it must be.

Common Exam Trap: Using Death Probabilities for Payment Dates

A payment at time $k$ is made if the annuitant survives to time $k$, not if the annuitant dies during the preceding year. In the first example, the one-year death probabilities are $0.20$, $0.25$, and $0.50$, while the survival probabilities to times 1, 2, and 3 are $0.80$, $0.60$, and $0.30$. The payment probabilities are the latter values.

A wrong calculation would substitute the annual death probabilities into the payment sum:

$$ 1+0.20(0.8)+0.25(0.8^2)+0.50(0.8^3)=1.576 $$

This does not value the annuity: the death probabilities describe deaths during particular intervals, not the chance of being alive on the payment dates. The correct unit value is $2.1776$, obtained from the survival probabilities to each date. Another frequent error is to omit the time-zero payment; an annuity-due includes it whenever the annuitant is alive at issue.

Key takeaway: For a whole-life annuity-due, list the payment times first. Include time zero, then multiply each later payment by the probability of survival to that date and by its discount factor. Sum the contributions and multiply by the payment amount.

Check Your Understanding

For each problem, identify the payment dates before applying the survival-weighted sum. Assume the annuitant is alive at the valuation date.

A hypothetical annuity pays $500 annually in advance. The effective annual interest rate is $25\%$; survival probabilities to times 1, 2, and 3 are $0.90$, $0.75$, and $0.40$, with no survival to time 4. Find the expected present value. (Answer: $1,202.40)
A hypothetical whole-life annuity-due pays $1,200 annually. The effective annual interest rate is $5\%$, and annual survival probability is constantly $0.85$. Find its expected present value. (Answer: $6,300)
A life table has $\ell_{60}=8{,}000$, $\ell_{61}=7{,}200$, $\ell_{62}=6{,}000$, and $\ell_{63}=0$. At $i=4\%$, find the value of an annuity-due paying $750 annually to a life aged 60, rounded to the nearest cent. (Answer: $1,919.93)
Under constant annual survival probability $p=0.80$, a unit whole-life annuity-due has value 4.2. What effective annual interest rate is consistent with this value? (Answer: $i=5\%$)
A hypothetical life has survival probability $0.90$ for the first year and $0.80$ in each subsequent year. At $i=4\%$, find the unit whole-life annuity-due value. (Answer: 4.75)

What You'll Learn