The Whole-Life Identity with Annuity-Immediate Values

Why Payment Timing Changes the Identity

A discrete whole-life insurance pays a benefit at the end of the year of death. A life annuity-immediate, in contrast, pays at the end of each year in which the annuitant is alive. Because the insurance payment and annuity payments occur at different points in the policy timeline, the identity connecting their values includes a discount-factor adjustment. Using the annuity-due identity without adjusting for timing gives the wrong result.

The focus here is the identity for a unit whole-life insurance and a unit life annuity-immediate, valued at the same age, interest rate, and mortality basis. Once the identity is established, it can be applied to other benefit amounts by scaling the corresponding expected present values.

Definition: Suppose the effective annual interest rate is $i$, with $v=1/(1+i)$ and $d=i/(1+i)=1-v$. A discrete unit whole-life insurance pays 1 at the end of the year of death. Its expected present value is $A_x$. A unit life annuity-immediate pays 1 at the end of each year the person is alive; its expected present value is $a_x$. Both values use the same mortality basis and are evaluated at age $x$.

Let $K_x$ denote the curtate future lifetime: if $K_x=k$, death occurs in the year from age $x+k$ to age $x+k+1$. In that case, the insurance pays at time $k+1$, while the annuity-immediate pays at times $1,2,\ldots,k$. The annuity has no payment at time $k+1$, because the person does not survive to that payment date.

In terms of survival probabilities, the annuity value is

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

The insurance value is the expected discounted benefit at the end of the year of death. If death eventually occurs with probability 1, it can be written as

A_x=\sum_{k=0}^{\infty}v^{k+1}\,{}_kp_xq_{x+k},

where $q_{x+k}=1-p_{x+k}$ is the probability of death within one year for a life aged $x+k$. The path-by-path argument below also allows for a probability of surviving forever, provided the discounted cash flows are defined.

Deriving the Annuity-Immediate Identity

The key is to compare the cash flows for a fixed value of $K_x$ before taking expectations. If death occurs in year $k+1$, the present value of the insurance is $v^{k+1}$. The present value of the annuity payments is $v+v^2+\cdots+v^k$, interpreted as zero when $k=0$.

Fix the death year.
For $K_x=k$, list the insurance payment at time $k+1$ and the annuity-immediate payments at times $1$ through $k$.

Relate the annuity payments to the insurance payment.
Multiply the annuity present value by $d=1-v$ and simplify the finite geometric sum.

Take expectations.
The resulting pathwise equality holds for every death year, so its expected values give the identity for $A_x$ and $a_x$.

For a fixed $k$, the finite geometric sum gives

d\sum_{j=1}^{k}v^j =(1-v)\frac{v(1-v^k)}{1-v} =v(1-v^k) =v-v^{k+1}.

Adding the insurance present value $v^{k+1}$ therefore gives $v$, regardless of the year of death:

v^{k+1}+d\sum_{j=1}^{k}v^j=v.

Taking expectations on both sides yields the whole-life identity with an annuity-immediate value:

Theorem: For a unit discrete whole-life insurance paying at the end of the year of death and a unit life annuity-immediate paying at the end of each year survived, valued on the same mortality and interest basis,

A_x+d\,a_x=v, \qquad A_x=v-d\,a_x=v(1-i\,a_x).

The second form follows because $d=iv$. It makes the discount-factor adjustment explicit: the identity starts with $v$, not 1. The familiar whole-life relationship with an annuity-due value is $A_x+d\,\ddot{a}_x=1$. Since $\ddot{a}_x=1+a_x$, substituting gives $A_x+d(1+a_x)=1$, or equivalently $A_x+d a_x=1-d=v$. Thus both forms agree when the payment timing is handled consistently.

Worked Applications

Worked Example: Checking the Identity from a Survival Table

A hypothetical life has survival probabilities ${}_1p_x=0.60$, ${}_2p_x=0.30$, ${}_3p_x=0.30$, and ${}_4p_x=0$, with ${}_0p_x=1$. A unit whole-life insurance pays at the end of the year of death, and a unit annuity-immediate pays at each year-end while the life is alive. Using an effective annual interest rate of $i=25\%$, calculate $A_x$ and $a_x$ directly, then verify the identity.

Method. First convert the survival probabilities into probabilities of death in each year. Then discount the insurance and annuity cash flows using $v=1/(1+i)$. The survival probabilities are nonincreasing, as required: $1$, $0.60$, $0.30$, $0.30$, $0$.

The probabilities of death in years 1 through 4 are the successive decreases in survival probability:

1-0.60=0.40,\qquad 0.60-0.30=0.30,\qquad 0.30-0.30=0,\qquad 0.30-0=0.30.

The interest quantities are

v=\frac{1}{1.25}=0.8, \qquad d=1-v=0.2.

For the insurance, multiply each year-of-death probability by the discount factor for that year:

\begin{aligned} A_x &=0.40(0.8)+0.30(0.8^2)+0(0.8^3)+0.30(0.8^4)\\ &=0.32+0.192+0+0.12288\\ &=0.63488. \end{aligned}

For the annuity-immediate, a payment at time $k$ is made only if the life survives to time $k$:

\begin{aligned} a_x &=0.60(0.8)+0.30(0.8^2)+0.30(0.8^3)+0(0.8^4)\\ &=0.48+0.192+0.1536+0\\ &=0.8256. \end{aligned}

Now check the identity:

v-d\,a_x=0.8-0.2(0.8256)=0.8-0.16512=0.63488=A_x.

The check agrees with the direct insurance calculation. In particular, the year-2 and year-3 survival probabilities sum to $0.60$; the year-3 survival probability is $0.30$, not $0$. The zero applies to survival through year 4.

Common mistake avoided: Treating the probability of death in year 3 as $0.30$ would be incorrect: that probability is ${}_2p_x-{}_3p_x=0$. The $0.30$ probability of death in this table is in year 4, because ${}_3p_x-{}_4p_x=0.30$.

Worked Example: Constant Annual Survival Probability

For a hypothetical life, suppose the probability of surviving each year, conditional on being alive at its start, is $q=0.80$, independently of the number of years already survived. Thus ${}_kp_x=0.80^k$. A unit whole-life insurance pays at the end of the year of death, and a unit annuity-immediate pays at year-end while the life is alive. At an effective annual interest rate of $i=25\%$, calculate both expected present values and verify the identity.

Method. Use the geometric series for the discounted survival probabilities and for the discounted probabilities of death in each year. The death probability in year $k+1$ is $0.20(0.80)^k$.

As above, $v=0.8$, so the annuity-immediate value is

\begin{aligned} a_x &=\sum_{k=1}^{\infty}v^k\,{}_kp_x =\sum_{k=1}^{\infty}(0.8\cdot0.8)^k\\ &=\sum_{k=1}^{\infty}0.64^k =\frac{0.64}{1-0.64} =\frac{0.64}{0.36} =\frac{16}{9}. \end{aligned}

For the insurance, sum the present values of the possible year-of-death benefits:

\begin{aligned} A_x &=\sum_{k=0}^{\infty}v^{k+1}(0.20)(0.80)^k\\ &=\frac{0.8(0.20)}{1-0.64} =\frac{0.16}{0.36} =\frac{4}{9}. \end{aligned}

Since $d=0.2$, the identity gives

v-d\,a_x=0.8-0.2\left(\frac{16}{9}\right) =\frac{36}{45}-\frac{16}{45} =\frac{20}{45} =\frac{4}{9}.

This equals the direct insurance value. The mortality model is valid for all years because it specifies a constant conditional survival probability; the infinite sums converge since $v q=0.64<1$.

Common mistake avoided: The probability of death in year $k+1$ is not simply $0.20$. It is $0.20(0.80)^k$, because the life must first survive the preceding $k$ years.

Worked Example: Recovering an Annuity Value and Scaling Payments

On a hypothetical valuation basis, the annual conditional probability of survival is $q=35/48$ at every year, and the effective annual interest rate is $i=25\%$. A unit whole-life insurance pays at the end of the year of death. First calculate its expected present value and use it to recover $a_x$. Then find the expected present value of a life annuity-immediate paying \$12,500 at each year-end while the person is alive.

Method. Calculate the insurance value from its geometric series, apply $A_x=v-d a_x$ to recover the unit annuity value, and multiply by the annual payment amount.

Here $v=0.8$, and the product of the annual survival probability and discount factor is

vq=0.8\left(\frac{35}{48}\right)=\frac{7}{12}.

The annual death probability conditional on being alive is $1-q=13/48$. Therefore,

\begin{aligned} A_x &=\frac{v(1-q)}{1-vq} =\frac{0.8(13/48)}{1-7/12}\\ &=\frac{13/60}{5/12} =\frac{13}{25} =0.52. \end{aligned}

With $d=0.2$, solve the identity for the unit annuity-immediate value:

\begin{aligned} a_x &=\frac{v-A_x}{d} =\frac{0.8-0.52}{0.2}\\ &=\frac{0.28}{0.2} =1.4. \end{aligned}

A payment of \$12,500 each year has expected present value

\$12{,}500(1.4)=\$17{,}500.

As a check, the direct geometric annuity calculation uses $vq=7/12$:

a_x=\sum_{k=1}^{\infty}\left(\frac{7}{12}\right)^k =\frac{7/12}{1-7/12} =\frac{7}{5} =1.4.

The recovered value agrees with the direct calculation, and the dollar scaling is linear because each annuity payment is a fixed multiple of the unit payment.

Common mistake avoided: The \$12,500 is an annual payment, not a single payment. Multiplying the insurance value by \$12,500 would value an insurance benefit, not the annuity; the annuity value must be obtained from $a_x$.

Common Timing Error and Practical Checks

A frequent error is to use $A_x=1-d a_x$. That formula substitutes an immediate annuity value into an identity whose corresponding annuity is due. For example, in the previous example, this incorrect calculation would give

$$ 1-0.2(1.4)=0.72, $$

which does not equal the insurance value $0.52$. The correction follows from the timing relation $\ddot{a}_x=1+a_x$:

A_x=1-d\ddot{a}_x =1-d(1+a_x) =(1-d)-d a_x =v-d a_x =0.52.

Before applying the identity, check that the insurance and annuity use the same interest and mortality assumptions, that the insurance benefit is paid at the end of the year of death, and that the annuity is paid at the end of each year survived. The formula does not apply unchanged if either payment timing or benefit structure differs. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Key takeaway: For the specified discrete whole-life insurance and annuity-immediate, $A_x=v-d a_x$, not $1-d a_x$. The starting value is $v$ because the annuity-immediate omits the time-zero payment included in an annuity-due.

Check Your Understanding

For each problem, use the stated payment timing and interest basis. Answers are provided for checking.

A hypothetical life has ${}_1p_x=0.75$, ${}_2p_x=0.50$, and ${}_3p_x=0$. A unit insurance pays at the end of the year of death and a unit annuity-immediate pays at year-end while alive. At $i=25\%$, calculate $A_x$ and verify it from $a_x$. (Answer: $a_x=0.9200$, $A_x=0.6160$.)
Under a constant conditional annual survival probability of $0.60$, calculate $A_x$ and $a_x$ at $i=25\%$. (Answer: $A_x=8/13\approx0.6154$, $a_x=12/13\approx0.9231$.)
At $i=4\%$, a unit whole-life insurance has $A_x=0.70$. Find the value of the unit annuity-immediate on the same basis. (Answer: $a_x=6.8000$.)
A hypothetical mortality table has ${}_1p_x=0.80$, ${}_2p_x=0.50$, ${}_3p_x=0.20$, and ${}_4p_x=0$. At $i=10\%$, calculate $A_x$ directly from the year-of-death probabilities and check it using $a_x$. (Answer: $a_x\approx1.2908$, $A_x\approx0.7917$.)
At $i=25\%$, the expected present value of a unit whole-life insurance is $A_x=0.64$. On the same basis, a life annuity-immediate pays \$12,500 at each year-end while the person is alive. Find the unit annuity value, the annuity's expected present value, and the corresponding annuity-due value. (Answer: $a_x=0.8000$, expected present value $=\$10{,}000.00$, and $\ddot{a}_x=1.8000$.)

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