One-Year Term Insurance From the Mortality Rate

Why Mortality and Interest Both Matter

A one-year term policy pays a benefit only if the insured dies during the policy year. When the benefit is paid at the end of that year, its expected present value depends on two quantities: the probability that the benefit is paid and the value today of a dollar paid one year from now. Keeping those quantities separate makes the calculation straightforward and helps prevent timing errors.

The mortality input is the insured’s probability of death during the year, conditional on being alive at the start. The interest input is the effective annual rate. This tutorial assumes the benefit is level and is paid at the end of the policy year if death occurs during that year. It does not value a benefit paid immediately at death.

Definition: For an insured aged $x$, the one-year death probability is $q_x={}_1q_x$, the probability that the insured dies within one year given survival to age $x$. For a level benefit $B$ payable at the end of the year of death, the present value random variable $Z$ is $Bv$ if death occurs during the year and $0$ if the insured survives the year, where $v=1/(1+i)$ and $i$ is the effective annual interest rate.

The One-Year Expected Present Value

There are just two possible outcomes for the benefit present value: $Bv$, with probability $q_x$, or zero, with probability $1-q_x$. The expected present value (EPV) is therefore the probability-weighted average of these outcomes.

\operatorname{EPV}(Z)=E[Z]=q_x(Bv)+(1-q_x)(0)=Bq_xv=\frac{Bq_x}{1+i}.

This is the expected present value of the benefit, not necessarily the premium charged by an insurer. It excludes expenses, profit margins, and any other contract provisions. If a problem gives the one-year survival probability ${}_1p_x$ instead of the death probability, use ${}_1p_x+{}_1q_x=1$, so $q_x=1-{}_1p_x$.

Identify the benefit and payment time.
For this one-year contract, the benefit $B$ is paid at the end of the policy year if death occurs during the year.

Find the death probability.
Use $q_x={}_1q_x$, or calculate $q_x=1-{}_1p_x$ if the survival probability is given.

Discount the benefit.
Use $v=1/(1+i)$ for one year at effective annual interest rate $i$.

Weight by the probability of payment.
Calculate $Bq_xv$. Keep full precision through comparisons, then round the final dollar amount to cents.

Worked Calculations

Worked Example: Starting With a Survival Probability

A hypothetical one-year policy covers a person aged $x$ for a level death benefit of $150,000, payable at the end of the policy year if death occurs in that year. The probability of surviving the year is ${}_1p_x=0.986$. The effective annual interest rate is $i=0.04$. Find the EPV of the benefit.

Method. Convert the given survival probability to the conditional death probability, discount the year-end benefit, and multiply by the probability that it is paid. This avoids treating survival probability as the probability of a claim.

First calculate the death probability:

q_x=1-{}_1p_x=1-0.986=0.014.

The one-year discount factor is:

v=\frac{1}{1+i}=\frac{1}{1.04}=0.9615384615\ldots

The undiscounted expected benefit is $150{,}000(0.014)=\$2{,}100$. Discounting this one-year expectation gives:

E[Z]=Bq_xv =150{,}000(0.014)\left(\frac{1}{1.04}\right) =\frac{2{,}100}{1.04} =2{,}019.230769\ldots

Rounded to the nearest cent, the EPV is $2,019.23. As a check, the benefit present value is $150,000/1.04 = $144,230.7692… if death occurs, and it is zero otherwise; weighting $144,230.7692… by 0.014 gives the same result.

Worked Example: Solving for a Benefit Under an EPV Limit

A hypothetical insurer is evaluating a one-year policy with a death probability of $q_x=0.025$ and an effective annual interest rate of $i=0.06$. The policy pays a level benefit at the end of the year of death. First find the EPV for a $64,000 benefit. Then find the benefit that would produce an EPV of $1,250 under the same assumptions.

Method. Apply $Bq_x/(1+i)$ to the stated benefit. For the second part, rearrange that same relationship to solve for $B$; do not omit the one-year discount factor.

For a $64,000 benefit, the undiscounted expected benefit is:

$$ 64{,}000(0.025)=1{,}600. $$

The discount factor is $v=1/1.06=0.9433962264\ldots$, so:

E[Z]=\frac{1{,}600}{1.06} =1{,}509.433962\ldots =\$1{,}509.43.

For a target EPV of $1,250, solve $1{,}250=B(0.025)/1.06$:

B=\frac{1{,}250(1.06)}{0.025} =\frac{1{,}325}{0.025} =\$53{,}000.

Check. A $53,000 benefit has an undiscounted expected benefit of $53{,}000(0.025)=\$1{,}325$, and $1{,}325/1.06=\$1{,}250$. The result is exact.

Worked Example: Comparing Two One-Year Policies

A hypothetical applicant compares two one-year policies, each paying its level benefit at the end of the policy year if death occurs during that year. Policy A offers a $100,000 benefit, has annual death probability $q_x=0.010$, and uses effective annual interest $i=0.03$. Policy B offers an $80,000 benefit, has annual death probability $q_x=0.015$, and uses effective annual interest $i=0.04$. Which policy has the greater benefit EPV, and by how much?

Method. Calculate $Bq_x/(1+i)$ separately for each policy, retain unrounded values, and then take the difference. This avoids a cent-rounding error when subtracting the two results.

For Policy A, the undiscounted expected benefit is $100{,}000(0.010)=\$1{,}000$. Therefore:

E[Z_A]=\frac{1{,}000}{1.03} =970.873786\ldots

For Policy B, the undiscounted expected benefit is $80{,}000(0.015)=\$1{,}200$. Therefore:

E[Z_B]=\frac{1{,}200}{1.04} =1{,}153.846154\ldots

Subtract before rounding:

E[Z_B]-E[Z_A] =1{,}153.846154\ldots-970.873786\ldots =182.972368\ldots

Policy B has the greater EPV, by $182.97 to the nearest cent. The rounded individual EPVs are $1,153.85 and $970.87, but subtracting those rounded amounts gives $182.98. That one-cent discrepancy is why the comparison should use unrounded values until the final step.

Why the Formula Holds—and When It Applies

The formula follows directly from the two-outcome structure of the contract. In the event of death during the year, the payment occurs one year from the valuation date, so its present value is $Bv$. If the insured survives the year, no death benefit is paid and the benefit present value is zero. Taking the expectation over those mutually exclusive outcomes gives the result:

Specify the outcomes.
Death during the year has probability $q_x$ and benefit present value $Bv$. Survival has probability $1-q_x$ and benefit present value zero.

Take the expectation.
For a discrete random variable, multiply each possible value by its probability and add: $E[Z]=q_x(Bv)+(1-q_x)(0)$.

Simplify.
The zero-benefit term vanishes, leaving $E[Z]=Bq_xv=Bq_x/(1+i)$.

The calculation assumes the policy is in force for the full year, the benefit is payable only on death during that year, the benefit amount is fixed, and the payment is made exactly at the end of the policy year. Because the payment date is fixed at year-end, the precise date of death within the year does not change its discount factor. The mortality probability must refer to this same one-year period and be conditional on survival to its beginning.

If a problem instead specifies payment at the moment of death, discounting depends on the timing of death within the year; the end-of-year formula above does not apply as written. Similarly, a multi-year contract requires accounting for death probabilities and payment times across its coverage period. For the one-year end-of-year benefit considered here, the annual death probability and the one-year discount factor are sufficient.

Common Exam Traps

Using the survival probability as the claim probability. If ${}_1p_x=0.986$, the death probability is $1-0.986=0.014$, not $0.986$.
Forgetting to discount. In the first example, $2,100 is the undiscounted expected benefit. The year-end payment has EPV $2{,}100/1.04=\$2{,}019.23$, not $2,100.
Discounting for the wrong time. A benefit payable at the end of the policy year is discounted one year, using $v=1/(1+i)$. It is not discounted for a fraction of a year merely because death may occur before year-end.
Rounding before comparing. Compute the difference from unrounded EPVs, then round that difference. In the policy comparison, subtracting the separately rounded values gives $182.98, whereas the correctly rounded difference from full precision is $182.97.
Confusing a benefit EPV with a premium quote. The calculation values the expected present value of the stated benefit only. It does not add expenses or other pricing components.

Key takeaway: For a level benefit $B$ payable at the end of the one-year policy year if death occurs, calculate $Bq_x/(1+i)$. Confirm that $q_x$ is the death probability for that year, use the effective annual rate, and round only after completing the calculation.

Check Your Understanding

For each question, assume a level death benefit is paid at the end of the policy year if death occurs during that year, and use the stated effective annual interest rate.

A hypothetical policy pays $90,000. The insured’s one-year death probability is $0.008$, and $i=0.02$. Find the benefit EPV to the nearest cent. (Answer: $705.88)
A person has a one-year survival probability of $0.97$. A policy pays $50,000 at year-end if death occurs, and $i=0.04$. Find the benefit EPV to the nearest cent. (Answer: $1,442.31)
A one-year policy has $q_x=0.014$ and $i=0.05$. What level benefit gives an EPV of $2,100? (Answer: $157,500)
Compare a $125,000 benefit with $q_x=0.012$, $i=0.03$, against a $100,000 benefit with $q_x=0.016$, $i=0.04$. Which has the greater EPV, and by how much? Calculate the difference before rounding. (Answer: The second policy, by $82.15)
A policy pays $80,000, with $q_x=0.025$ and $i=0.05$. A candidate reports $80{,}000(0.025)=\$2{,}000$ as the EPV. Give the correct EPV and identify the missing calculation. (Answer: $1,904.76; discount the expected year-end benefit by $1.05$)

What You'll Learn