Whole-Life Insurance Under Constant Mortality

Why Constant Mortality Gives a Closed-Form Value

A whole-life insurance benefit may be payable in any future year, so its present value accounts for both the chance of death in each year and the discount applied to a payment at that time. When the annual death probability and effective annual interest rate stay constant, the possible payment values form a geometric series. Summing that series gives a compact expression for the actuarial present value (APV).

This tutorial considers a level benefit paid at the end of the year of death. That timing matters: death during the first policy year produces a payment at time 1, not at time 0. The result applies to this discrete payment convention, not to a benefit paid immediately at death.

Definition: Suppose a life is insured at time 0. In each year, conditional on being alive at the start of that year, the probability of death during that year is the same constant $q$, where $0<q\leq1$. The annual survival probability is $p=1-q$. Let the effective annual interest rate be constant at $i\geq0$, with $v=1/(1+i)$. A unit whole-life insurance pays 1 at the end of the year of death. Its APV is denoted here by $A$.

The constant mortality assumption means that the probability of surviving each year, given survival to its start, is $p$. In particular, the probability of surviving $k$ full years and then dying in the following year is $p^kq$, for $k=0,1,2,\ldots$. No independence assumption about unrelated lives is needed; the calculation concerns the successive conditional survival probabilities for this one insured life.

Deriving the Whole-Life APV

If death occurs in year $k+1$, the unit benefit is paid at time $k+1$, so its present value is $v^{k+1}$. Multiplying that present value by the probability $p^kq$, then summing over all possible death years, gives the APV.

A=\sum_{k=0}^{\infty}p^kqv^{k+1} =qv\sum_{k=0}^{\infty}(pv)^k

Because $0<q\leq1$, $p<1$. Also $v\leq1$ when $i\geq0$, so $0\leq pv<1$. The geometric series therefore converges, and its sum is $1/(1-pv)$. Thus

A=\frac{qv}{1-pv} =\frac{q/(1+i)}{1-(1-q)/(1+i)} =\frac{q}{i+q}.

The last equality follows by multiplying the numerator and denominator of the middle fraction by $1+i$: its denominator becomes $(1+i)-(1-q)=i+q$. The same result can be understood by conditioning on the first year. Death in that year contributes $qv$; survival for one year, with probability $p$, leaves the same whole-life problem one year later, discounted by $v$.

A=qv+pvA \quad\Longrightarrow\quad A=\frac{qv}{1-pv} =\frac{q}{i+q}.

Theorem: Under constant annual death probability $q$, constant effective annual interest rate $i\geq0$, and a level unit benefit paid at the end of the year of death, the whole-life APV is $A=q/(i+q)$, provided $q>0$. For a level benefit $B$, the APV is $BA=Bq/(i+q)$, by multiplication of each possible payment by $B$.

If $q=0$, death never occurs under the model, so the insurance benefit is never paid and its APV is 0. The expression $q/(i+q)$ gives 0 when $q=0$ and $i>0$, but is undefined when both are zero; that case is handled directly by the zero probability of payment. When $i=0$ and $q>0$, the formula gives $A=1$. This is appropriate because the unit benefit is paid with probability 1 eventually and is not discounted.

Worked Applications

Worked Example: Checking the Closed Form by Conditioning

A hypothetical policy pays 1 at the end of the year of death. Assume the insured has a constant annual death probability $q=0.03$, and the effective annual interest rate is $i=0.02$. Calculate the whole-life APV using the closed form, then verify it by conditioning on the first year. Give the final APV to four decimal places.

Method. Set $p=1-q=0.97$ and $v=1/(1+i)=1/1.02$. The closed form gives

A=\frac{q}{i+q} =\frac{0.03}{0.02+0.03} =\frac{0.03}{0.05} =0.6.

For the first-year check, death in year 1 has probability 0.03 and a payment discounted by $v$. Survival through year 1 has probability 0.97; conditional on survival, the remaining APV is again $A=0.6$, discounted by $v$. Therefore

qv+pvA =\frac{0.03}{1.02}+\frac{0.97}{1.02}(0.60) =0.029411765+0.570588235 =0.600000000.

The two methods agree. Rounded only at the final step, the APV is 0.6000. The check also confirms the timing: both the first-year death payment and the continuation value are discounted for one year.

Common mistake avoided. The product $(0.97/1.02)(0.60)$ is approximately 0.5706, not 0.582. Incorrectly evaluating this product would make the first-year check fail even though the closed-form value is correct.

Worked Example: Valuing a Level Benefit

A hypothetical insurer offers a whole-life policy with a level death benefit of $80,000, payable at the end of the year of death. Assume a constant annual death probability of $q=0.05$ and a constant effective annual interest rate of $i=0.05$. Find the APV of the benefit, in dollars to the cent, and check the result using first-year conditioning.

Method. First calculate the APV for a unit benefit:

A=\frac{0.05}{0.05+0.05}=0.5.

A benefit of $80,000 is 80,000 times the unit benefit, so its APV is

80{,}000(0.5)=\$40{,}000.00.

For the first-year check, $p=0.95$ and $v=1/1.05$. The discounted contribution from death in year 1 is

80{,}000(0.05)v =\frac{4{,}000}{1.05} =\$3{,}809.5238\ldots.

If the insured survives year 1, the probability is 0.95 and the remaining benefit has the same APV, $40,000, at time 1. Its contribution at time 0 is

0.95v(40{,}000) =\frac{38{,}000}{1.05} =\$36{,}190.4762\ldots.

Adding the unrounded contributions gives $40,000. The APV, rounded only at the final step to the nearest cent, is $40,000.00.

Common mistake avoided. Multiplying the benefit by $q$ alone would give $4,000, which is only the undiscounted probability-weighted amount for death in the first year. It omits all later death years. The whole-life formula includes those years through the geometric series.

Worked Example: Solving for the Annual Death Probability

A hypothetical unit whole-life benefit is payable at the end of the year of death. Under constant mortality and a constant effective annual interest rate of $i=0.06$, its APV is 0.40. Find the implied annual death probability $q$, then verify the answer by first-year conditioning. Give the mortality rate to four decimal places.

Method. Start with $A=q/(i+q)$ and solve for $q$:

A(i+q)=q \quad\Longrightarrow\quad Ai=q(1-A) \quad\Longrightarrow\quad q=\frac{Ai}{1-A}.

Substitute $A=0.40$ and $i=0.06$:

q=\frac{(0.40)(0.06)}{1-0.40} =\frac{0.024}{0.60} =0.04.

Thus $q=0.0400$, or 4.00% per year, and $p=1-q=0.96$. Check the closed form:

\frac{q}{i+q} =\frac{0.04}{0.06+0.04} =0.40.

For a first-year check, $v=1/1.06$. The discounted death contribution is $qv=0.04/1.06=0.03773585\ldots$. The discounted continuation contribution is $pvA=(0.96/1.06)(0.40)=0.36226415\ldots$. Their sum is 0.40000000. The answer is consistent with both the closed form and the first-year decomposition.

Common mistake avoided. Rearranging $A=q/(i+q)$ as $q=A(i+q)$ and stopping would leave $q$ on both sides. Isolating the $q$ terms is necessary; dividing by $1-A$ gives the required solution.

Interpretation and Exam Checks

The closed form makes the role of the assumptions visible. For a fixed positive $q$, increasing $i$ increases the denominator $i+q$, and hence lowers the APV. The insurance payment is then discounted more heavily. For a fixed $i>0$, increasing $q$ raises $q/(i+q)$: payment is expected sooner, so its present value rises. These comparisons presume that the other assumption remains fixed.

The expression also satisfies basic checks. For $q>0$ and $i\geq0$, $0<A\leq1$. At $i=0$, $A=1$; with positive interest, $A<1$. Each possible unit payment is discounted by a factor no greater than 1, so the APV cannot exceed 1. When $q=1$, death occurs in the first year with certainty and $A=1/(1+i)$, which agrees with $q/(i+q)$.

On a problem, first verify the payment timing and assumptions before substituting. The formula requires a level benefit paid at the end of the year of death, a constant conditional annual death probability, and a constant effective annual interest rate. If mortality changes by year, the survival probabilities do not form powers of one constant $p$; if the benefit changes by year, the benefit cannot be factored out as one constant $B$. In either case, this closed form does not directly apply.

Identify the contract timing.
Confirm that the benefit is level and paid at the end of the year of death.

Check the model.
Confirm that the conditional annual death probability $q$ and effective annual interest rate $i$ are constant.

Calculate the unit value.
For $q>0$, use $A=q/(i+q)$; multiply by the benefit amount if the benefit is not 1.

Check the result.
Use $A=qv+pvA$ or confirm that the value is between 0 and 1, with $A=1$ when $i=0$ and $q>0$.

A frequent exam trap is to use $q$ as if it were the APV. The probability of death in the first year is $q$, but the whole-life APV sums discounted payments over every possible death year. Another trap is to discount the first-year benefit by 1: because the benefit is paid at the end of year 1, its discount factor is $v$. These distinctions are central to calculations aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, apply the constant-mortality, constant-interest model and respect the stated end-of-year payment timing.

A unit whole-life benefit is paid at the end of the year of death. If $q=0.02$ and $i=0.03$, find its APV. (Answer: 0.4000)
A hypothetical policy pays $25,000 at the end of the year of death. Assume $q=0.06$ and $i=0.04$. Find the APV in dollars to the cent. (Answer: $15,000.00)
A unit-benefit whole-life APV is 0.20 when $i=0.08$. Under constant mortality, find $q$ to four decimal places. (Answer: 0.0200)
A hypothetical whole-life policy pays $120,000 at the end of the year of death, with $q=0.04$ and $i=0.06$. Find the APV in dollars to the cent. (Answer: $48,000.00)
A unit-benefit whole-life APV is 0.50 under constant mortality with $q=0.03$. Find the effective annual interest rate $i$. (Answer: 0.0300, or 3.00%)

What You'll Learn