n-Year Term Insurance With Age-Varying Mortality

Why Age-Varying Mortality Changes the Calculation

In a term insurance calculation, a death in a later policy year can occur only if the insured survives every earlier year. When the annual mortality rate changes with attained age, each policy year therefore has its own conditional death probability, and the probability of a claim in that year must also reflect survival to the start of that year.

The key distinction is between a conditional probability, such as the probability of death during the next year given survival to that year, and an unconditional probability, such as the probability of death in policy year 3 measured from the start of the contract. The latter combines the conditional death probability at the attained age with the probability of reaching that age alive.

This tutorial considers a hypothetical $n$-year term insurance with a level benefit paid at the end of the policy year of death. It focuses on annual mortality rates that vary by attained age and a fixed effective annual interest rate. The treatment is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Age-Specific Mortality and Policy-Year Probabilities

Definition: Let $q_{x+k}$ be the probability that a life aged $x+k$ dies within one year, conditional on being alive at exact age $x+k$. Let $p_{x+k}=1-q_{x+k}$ be the corresponding one-year survival probability. The probability of surviving $k$ years from age $x$ is $$ {}_kp_x=\prod_{j=0}^{k-1}p_{x+j}, $$ where ${}_0p_x=1$. The probability of death in policy year $k+1$, measured from the start of the policy at age $x$, is $$ {}_{k|}q_x={}_kp_xq_{x+k},\qquad k=0,1,\ldots,n-1. $$

The notation ${}_{k|}q_x$ here denotes the probability of surviving $k$ complete years and then dying during the following year. It is not the one-year rate $q_{x+k}$ by itself. For example, death in policy year 3 requires survival through policy years 1 and 2, followed by death in the third year.

The formula follows directly from conditional probability. Let $A$ be the event of surviving the first $k$ years, and $D$ the event of dying in the next year. Then $P(A\cap D)=P(A)P(D\mid A)$. The first factor is ${}_kp_x$; conditional on reaching age $x+k$, the second factor is $q_{x+k}$. Thus $P(A\cap D)={}_kp_xq_{x+k}$. This argument also explains why the annual rates from earlier ages must be included.

P(\text{death in policy year }k+1) =p_xp_{x+1}\cdots p_{x+k-1}q_{x+k}.

The probabilities of death in the $n$ policy years add to the probability of death within the term:

\sum_{k=0}^{n-1}{}_kp_xq_{x+k} =1-{}_np_x.

This identity is a useful check: the only outcomes over the term are death during one of its policy years or survival to the end of the term. It does not say that the individual annual rates $q_x,q_{x+1},\ldots$ add to the probability of death during the term.

Expected Present Value of a Level-Benefit Term Insurance

Assume the benefit $B$ is paid at the end of the policy year of death, the contract pays nothing if the insured survives $n$ years, and the effective annual interest rate $i$ is fixed and nonnegative. The discount factor is $v=1/(1+i)$. A death in policy year $k+1$ produces a payment at time $k+1$, so its present value is $Bv^{k+1}$. The expected present value is therefore

\operatorname{EPV} =B\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} =B\sum_{k=0}^{n-1}v^{k+1} \left(\prod_{j=0}^{k-1}(1-q_{x+j})\right)q_{x+k}.

Each term is the benefit amount multiplied by the discount factor for its payment time and by the probability that the payment occurs. The expression assumes annual, end-of-policy-year payments and a constant interest rate. If the benefit is paid immediately at death, the discount time is not generally an integer number of years; if the interest rate changes by year, the fixed power $v^{k+1}$ must be replaced by the appropriate product of annual discount factors.

Worked Example: Three Age-Specific Rates and a Level Benefit

A hypothetical insurer issues a three-year term policy at age $x$ with a benefit of $B=\$100{,}000$, payable at the end of the policy year of death. The effective annual interest rate is $i=5\%$. The annual death probabilities are $q_x=0.01$, $q_{x+1}=0.02$, and $q_{x+2}=0.03$. Find the probability of death in each policy year and the expected present value of the benefit.

Method. First calculate each year's unconditional death probability by multiplying the rate at that attained age by the probability of surviving to the start of that year. Then discount each possible payment at its payment time.

The one-year survival probabilities are $p_x=0.99$, $p_{x+1}=0.98$, and $p_{x+2}=0.97$. The policy-year death probabilities are:

Policy year 1: ${}_0p_xq_x=1(0.01)=0.01$.
Policy year 2: ${}_1p_xq_{x+1}=0.99(0.02)=0.0198$.
Policy year 3: ${}_2p_xq_{x+2}=0.99(0.98)(0.03)=0.9702(0.03)=0.029106$.

The probability of surviving all three years is $0.99(0.98)(0.97)=0.941094$. The death probabilities sum to $0.01+0.0198+0.029106=0.058906$, which equals $1-0.941094$, as required.

The discount factors for payments at the ends of years 1, 2, and 3 are $v=1/1.05$, $v^2=1/1.1025$, and $v^3=1/1.157625$. The expected present value per dollar of benefit is

\frac{0.01}{1.05} +\frac{0.0198}{1.1025} +\frac{0.029106}{1.157625} =0.0095238095+0.0179591837+0.0251428571 =0.0526258503.

Multiplying by the benefit gives $100{,}000(0.0526258503)=\$5{,}262.58503$, or \$5,262.59 to the nearest cent. The probability check confirms the mortality calculation; the discounting check uses the payment times of one, two, and three years.

Worked Example: Probability of Death in Later Policy Years

Worked Example: Later-Year Claims in a Four-Year Term

A hypothetical four-year term contract begins at age $x$. Its annual death probabilities are $q_x=0.04$, $q_{x+1}=0.05$, $q_{x+2}=0.06$, and $q_{x+3}=0.08$. The effective annual interest rate is $4\%$. Calculate the probability of death in each policy year and the probability of death during policy years 3 or 4.

Method. Build the probability of reaching each policy year alive, then multiply by the death probability for that year. Interest does not affect these probabilities; it would matter if the question asked for a present value.

The survival probabilities for one year at each attained age are $0.96$, $0.95$, $0.94$, and $0.92$. Hence the probability of death in policy year 1 is $0.04$. For year 2, the probability of reaching its start is $0.96$, so its death probability is $0.96(0.05)=0.048$.

For policy year 3, survival to its start is $0.96(0.95)=0.912$. Thus the probability of death in that year is $0.912(0.06)=0.05472$. For policy year 4, survival to its start is $0.96(0.95)(0.94)=0.85728$. Its death probability is $0.85728(0.08)=0.0685824$.

Therefore, the probability of death in policy year 3 or 4 is $0.05472+0.0685824=\mathbf{0.1233024}$. As a check, the probability of surviving all four years is $0.85728(0.92)=0.7886976$. The probability of death during the term is $1-0.7886976=0.2113024$, which equals the sum $0.04+0.048+0.05472+0.0685824$.

The conditional probability of death during policy year 4, given survival to its start, is $0.08$. The unconditional probability of death in policy year 4 from the start of the contract is $0.0685824$. These answer different questions.

Worked Example: Solving for an Unknown Annual Rate

Worked Example: An Unknown Final-Year Mortality Rate

A hypothetical three-year term policy has $q_x=0.02$ and $q_{x+1}=0.04$. The probability of death during the three-year term is $0.10$. The benefit is $\$75{,}000$, paid at the end of the policy year of death, and the effective annual interest rate is $5\%$. Find $q_{x+2}$ and the expected present value of the benefit.

Method. First use the cumulative term-death probability to find survival through the third year. Then calculate the probability of death in each year and discount its payment.

Survival through the first two years is $0.98(0.96)=0.9408$. The probability of survival through all three years is $1-0.10=0.90$. Therefore, the one-year survival probability in the third year is $0.90/0.9408=0.9566326531$, and $q_{x+2}=1-0.9566326531=0.0433673469$.

The policy-year death probabilities are $0.02$ in year 1, $0.98(0.04)=0.0392$ in year 2, and $0.9408(0.0433673469)=0.0408$ in year 3. They sum to $0.02+0.0392+0.0408=0.10$, matching the stated term-death probability.

At $i=5\%$, the expected present value per dollar of benefit is

\frac{0.02}{1.05} +\frac{0.0392}{1.1025} +\frac{0.0408}{1.157625} =0.0190476190+0.0355555556+0.0352445734 =0.0898477480.

Thus the expected present value is $75{,}000(0.0898477480)=\$6{,}738.5811$, or \$6,738.58 to the nearest cent. The inferred $q_{x+2}$ lies between 0 and 1, and the reconstructed death probabilities total $0.10$; both checks support the result.

Common Exam Trap: Using the Later-Year Rate Alone

A frequent error is to treat $q_{x+k}$ as the probability of a claim in policy year $k+1$ from the contract's start. It is instead conditional on reaching age $x+k$ alive. In the first example, the incorrect calculation for the third policy year would be $q_{x+2}=0.03$. That omits the requirement to survive the first two years.

The correction is to multiply by the probability of reaching the third year: $0.99(0.98)(0.03)=0.029106$. The error also overstates the third-year expected present-value contribution: it would use $100{,}000(0.03)/1.05^3$, instead of $100{,}000(0.029106)/1.05^3$. Another related trap is discounting the year-3 benefit by $v^2$; an end-of-year-3 payment must be discounted by $v^3$.

List the annual rates.
Assign the correct $q_{x+k}$ to each policy year using attained age.

Find survival to each year.
Multiply the earlier one-year survival probabilities to obtain ${}_kp_x$.

Find each year's death probability.
Use ${}_kp_xq_{x+k}$; do not use $q_{x+k}$ alone.

Discount the benefit at its payment time.
For an end-of-year payment in policy year $k+1$, use $v^{k+1}$.

Check the result.
The death-year probabilities must sum to $1-{}_np_x$.

Age-varying rates matter because mortality at the attained age for a later policy year does not by itself describe the chance of a claim from the issue date. Keeping conditional rates separate from unconditional policy-year probabilities makes it possible to calculate both term-death probabilities and expected present values consistently. The same survival products also give a direct check on the total probability of a claim during the coverage period.

Check Your Understanding

For each problem, calculate the requested quantity using the stated annual mortality rates and payment timing.

A two-year term begins at age $x$, with $q_x=0.03$ and $q_{x+1}=0.04$. Find the probability of death in each policy year and the probability of death during the term. (Answer: Year 1: $0.03$; year 2: $0.0388$; during the term: $0.0688$.)
A three-year term has $q_x=0.02$, $q_{x+1}=0.03$, and $q_{x+2}=0.05$. Find the unconditional probability of death in policy year 3. (Answer: $0.04753$.)
A two-year term has $q_x=0.02$, $q_{x+1}=0.03$, benefit $\$1{,}000$, and effective annual interest $i=5\%$. Find the expected present value, with benefits paid at the end of the year of death. (Answer: $\$45.71$ to the nearest cent.)
A two-year term has $q_x=0.04$, unknown $q_{x+1}$, and probability of death during the term $0.10$. If the benefit is $\$20{,}000$, paid at the end of the year of death, and $i=5\%$, find $q_{x+1}$ and the expected present value. (Answer: $q_{x+1}=0.0625$; expected present value $=\$1{,}850.34$ to the nearest cent.)
A four-year term has annual death probabilities $0.02,0.03,0.04,0.05$ at successive attained ages. Find the unconditional probability of death in policy year 4 and the probability of death during the four-year term. (Answer: Year 4: $0.0456288$; during the term: $0.1330528$.)

What You'll Learn