Worked Problems on Term and Whole-Life Insurance

Valuing Discrete Life Insurance Benefits

A discrete life insurance benefit is paid at the end of the policy year of death. The valuation therefore requires two steps for each possible year: find the probability that death occurs during that year, then discount that year-end benefit to the present. The same method applies to term and whole-life insurance; the difference is the number of years in which a benefit can be paid.

The problems below use annual mortality information. A mortality rate is conditional on survival to the start of its year, so it must be combined with the probability of reaching that year. Keeping those two parts separate helps avoid a common error: treating a one-year death probability as though it were the probability of death in a later policy year.

Definition: Let $K_x$ be the curtate future lifetime of a life aged $x$: if death occurs between exact ages $x+k$ and $x+k+1$, then $K_x=k$. Write $q_{x+k}={}_1q_{x+k}$ for the probability of death during the year of age $x+k$, conditional on survival to its start. The probability of surviving $k$ full years and then dying in the following year is $$ P(K_x=k)={}_kp_xq_{x+k}. $$ For an effective annual interest rate $i$, the discount factor is $v=1/(1+i)$. A benefit paid at the end of year $k+1$ has present value equal to that benefit multiplied by $v^{k+1}$.

Term Insurance: Set Up One Term for Each Possible Death Year

An $n$-year discrete term policy pays only if death occurs in one of the first $n$ policy years. If the benefit for death in policy year $k+1$ is $b_{k+1}$, its expected present value (EPV) is the sum of the discounted benefit over all covered death years:

\operatorname{EPV}=\sum_{k=0}^{n-1} b_{k+1}v^{k+1}{}_kp_xq_{x+k}.

For a level benefit $B$, set $b_{k+1}=B$ in every covered year. When mortality is supplied as annual rates, obtain ${}_kp_x$ by multiplying the conditional survival probabilities for the preceding years:

{}_kp_x=\prod_{j=0}^{k-1}(1-q_{x+j}), \qquad {}_0p_x=1.

This product matters because $q_{x+k}$ alone applies only to lives who reached age $x+k$. If the problem instead gives ${}_kp_x$ directly, use it directly rather than multiplying the earlier survival probabilities a second time.

Worked Example: Three-Year Term Cover With Annual Mortality Rates

A hypothetical three-year term policy covers a life aged $x$. The effective annual interest rate is $i=25\%$. The annual death probabilities are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. A benefit of $1,000 is paid at the end of the policy year of death. Calculate the EPV.

Method. Calculate each death-year probability as survival to that year multiplied by that year's conditional death probability. Then discount each $1,000 payment to its policy-year end. This avoids using a later year's conditional mortality rate as an unconditional death probability.

The discount factor is $v=1/(1.25)=0.8$. For policy year 1, ${}_0p_x=1$, so the probability of death is $1(0.10)=0.10$. Its discounted expected payment is $1{,}000(0.10)(0.8)=\$80.00$.

For policy year 2, survival through year 1 is ${}_1p_x=1-0.10=0.90$. The probability of death in year 2 is $0.90(0.20)=0.18$. Its discounted expected payment is $1{,}000(0.18)(0.8^2)=1{,}000(0.18)(0.64)=\$115.20$.

For policy year 3, survival through the first two years is ${}_2p_x=(0.90)(0.80)=0.72$. The probability of death in year 3 is $0.72(0.25)=0.18$. Its discounted expected payment is $1{,}000(0.18)(0.8^3)=1{,}000(0.18)(0.512)=\$92.16$.

Adding the three contributions gives $\$80.00+\$115.20+\$92.16=\boxed{\$287.36}$. As a check, the covered death probabilities sum to $0.10+0.18+0.18=0.46$, which is less than 1 because survival beyond the three-year term is possible.

Why the Term Formula Works

The death events for different policy years are mutually exclusive: a life cannot die in two different years. In year $k+1$, the life must first survive $k$ years, an event with probability ${}_kp_x$, and then die during the next year, conditional probability $q_{x+k}$. Thus the probability of that year's benefit is ${}_kp_xq_{x+k}$. The present value of the benefit in that event is $b_{k+1}v^{k+1}$. Taking the expectation over the mutually exclusive covered death years gives the sum:

E[\text{present value}] =\sum_{k=0}^{n-1} \bigl(b_{k+1}v^{k+1}\bigr)P(K_x=k) =\sum_{k=0}^{n-1}b_{k+1}v^{k+1}{}_kp_xq_{x+k}.

This argument relies on the stated timing: the benefit is paid at the end of the year of death. A benefit paid at a different time would have a different discount power.

Varying Benefits and Whole-Life Cover

The benefit need not be level. If it increases, decreases, or otherwise varies by policy year, use the benefit specified for that year as $b_{k+1}$. The mortality and discounting steps do not change; only the amount multiplied into each term changes.

Whole-life insurance uses the same calculation but includes every possible death year. For a level benefit of 1, the whole-life insurance APV is $A_x$. For a level benefit $B$, the EPV is $BA_x$. If death is certain by a finite ultimate age, the sum ends at the final possible year of death. Otherwise, it is an infinite sum, and its value is the limit of the partial sums when that limit exists.

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

The whole-life formula includes no payment for surviving a particular term. Each term represents a death benefit, and the probabilities of death over all possible years sum to 1 if death eventually occurs with probability 1.

Worked Example: Whole-Life Insurance From Life-Table Counts

A hypothetical life table has $l_{60}=1{,}000$, $l_{61}=900$, $l_{62}=720$, $l_{63}=504$, and $l_{64}=0$. A whole-life policy pays $1,000 at the end of the year of death. The effective annual interest rate is $i=25\%$. Find the EPV using the life-table counts.

Method. Convert the successive counts into conditional death probabilities. For each year, the number dying is the difference between the count at the start and the count at the next age. Divide by the starting count to get the conditional rate; multiply by survival from age 60 to get the unconditional death probability for that year.

From age 60 to 61, $q_{60}=(1{,}000-900)/1{,}000=0.10$, so the death probability in year 1 is $0.10$. From age 61 to 62, $q_{61}=(900-720)/900=0.20$; the probability of death in year 2 from age 60 is $(900/1{,}000)(0.20)=0.18$.

From age 62 to 63, $q_{62}=(720-504)/720=0.30$; the probability of death in year 3 from age 60 is $(720/1{,}000)(0.30)=0.216$. Finally, $q_{63}=(504-0)/504=1$, so the probability of death in year 4 from age 60 is $(504/1{,}000)(1)=0.504$.

The discount factors for years 1 through 4 are $0.8$, $0.64$, $0.512$, and $0.4096$. The expected present values of the four possible payments are, respectively, $1{,}000(0.10)(0.8)=\$80.00$, $1{,}000(0.18)(0.64)=\$115.20$, $1{,}000(0.216)(0.512)=\$110.592$, and $1{,}000(0.504)(0.4096)=\$206.4384$.

The unrounded total is $\$512.2304$, so the EPV is $\boxed{\$512.23}$, rounded to the nearest cent. The death probabilities sum to $0.10+0.18+0.216+0.504=1$, confirming that the table accounts for the entire cohort by age 64.

Worked Example: Increasing Benefits on a Term Policy

A hypothetical three-year term policy pays $1,000 for death in year 1, $2,000 for death in year 2, and $3,000 for death in year 3. The insured is age $x$, with $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. The effective annual interest rate is $i=25\%$. Calculate the EPV.

Method. Use each year’s stated benefit with the same death probabilities and discount factors as in the level-benefit calculation. The common mistake is to multiply an increasing benefit by the probability of death but discount it using the wrong policy-year power.

Here $v=0.8$. The year 1 contribution is $1{,}000(0.10)(0.8)=\$80.00$. Survival to year 2 is $0.90$, so the year 2 death probability is $0.90(0.20)=0.18$; its contribution is $2{,}000(0.18)(0.64)=\$230.40$.

Survival to year 3 is $(0.90)(0.80)=0.72$, so the year 3 death probability is $0.72(0.25)=0.18$. The year 3 contribution is $3{,}000(0.18)(0.512)=\$276.48$. Therefore, the EPV is $\$80.00+\$230.40+\$276.48=\boxed{\$586.88}$.

A useful check is to compare this with the level-$1,000 EPV of $287.36 from the same mortality and interest assumptions. The increasing-benefit EPV is higher, as expected, because the benefit is at least $1,000 in every covered year and is larger in years 2 and 3.

Checks That Catch Setup Errors

Before accepting a result, check the probability and timing structure as well as the arithmetic. In particular, distinguish conditional mortality from unconditional death-year probabilities, and match the discount exponent to the payment date.

Check the probabilities. Each death-year probability is nonnegative. The sum for an $n$-year term is $1-{}_np_x$, not necessarily 1. For a finite whole-life table ending with zero survivors, the death-year probabilities should sum to 1.
Check the benefit timing. End-of-year-of-death benefits in year $k+1$ are multiplied by $v^{k+1}$, not $v^k$. In particular, the first-year benefit is discounted one year.
Check the mortality input. If $q_{x+k}$ is conditional, multiply it by ${}_kp_x$. If an unconditional death-year probability is already supplied, do not multiply by survival again.
Check the stated term. An $n$-year term includes years 1 through $n$, but not year $n+1$. Whole-life cover has no fixed final policy year unless the model specifies a finite ultimate age.

When life-table counts are used, the conditional rate at age $x+k$ is $q_{x+k}=(l_{x+k}-l_{x+k+1})/l_{x+k}$, provided $l_{x+k}>0$. Counts give a direct check: the unconditional probability of death in a policy year is the number dying in that year divided by the original count $l_x$. The resulting probabilities should agree with the survival-times-conditional-rate calculation.

Key takeaway: For each covered death year, multiply that year’s benefit by the probability of reaching the year and dying during it, then discount to the year-end payment date. Sum those contributions over the term, or over all possible death years for whole-life cover.

Check Your Understanding

For each problem, use end-of-year-of-death benefits and show the death-year probabilities before discounting.

A two-year term policy pays $5,000 at the end of the year of death. Let $q_x=0.05$, $q_{x+1}=0.10$, and $i=4\%$ effective annually. Find the EPV, rounded to the nearest cent. (Answer: $679.55)
A whole-life policy pays $2,000 at the end of the year of death. The probabilities of death in years 1, 2, and 3 from issue are $0.20$, $0.24$, and $0.56$, respectively; death by the end of year 3 is certain. If $i=10\%$, find the EPV to the nearest cent. (Answer: $1,601.80)
A two-year term policy pays $1,000 for death in year 1 and $3,000 for death in year 2. Given $q_x=q_{x+1}=0.10$ and $i=25\%$, find the EPV. (Answer: $252.80)
A life table has $l_x=800$, $l_{x+1}=640$, $l_{x+2}=320$, and $l_{x+3}=0$. A whole-life policy pays $1,000 at the end of the year of death, and $i=25\%$. Find the EPV. (Answer: $620.80)
A two-year term policy pays $1,000 at the end of the year of death. The first-year death probability is $0.10$, the second-year conditional death probability is $q_{x+1}$, $i=25\%$, and the EPV is $200. Find $q_{x+1}$, to four decimal places. (Answer: 0.2083)

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