Term Insurance APVs From Survival Probabilities

From Survival Probabilities to a Term Insurance APV

A term insurance benefit depends on whether death occurs during a specified coverage period. When survival probabilities are given at integer durations, the essential step is to find the probability of death in each policy year. The difference between survival to the start and survival to the end of that year gives this probability directly.

Once the policy-year probabilities are known, calculate the expected present value (APV) by discounting each possible year-end benefit and weighting it by the probability that it is paid. This approach uses the information in the survival probabilities without requiring a separate mortality table or an assumed continuous lifetime distribution.

Definition: Consider an $n$-year term insurance issued at age $x$, with a level benefit $B$ paid at the end of the year of death if death occurs within the $n$-year term. Write ${}_kp_x$ for the probability that a life aged $x$ survives at least $k$ years, with ${}_0p_x=1$. The unconditional probability of death during policy year $k+1$ is ${}_kp_x-{}_{k+1}p_x$, for $k=0,\ldots,n-1$.

The benefit is paid at times 1 through $n$, measured in years from issue. With an effective annual interest rate $i$, the discount factor for a payment at time $k+1$ is $v^{k+1}$, where $v=1/(1+i)$. If the life survives the full term, this policy pays no death benefit.

Theorem: Under an annual effective interest rate $i$, and with a level benefit $B$ payable at the end of the policy year of death, the term insurance APV is $$ \operatorname{APV} =B\sum_{k=0}^{n-1}v^{k+1}\left({}_kp_x-{}_{k+1}p_x\right). $$ For unit benefit, the sum is the discrete $n$-year term insurance APV.

Why Consecutive Survival Probabilities Give the Death Probability

The event “death during policy year $k+1$” consists of surviving the first $k$ years but not surviving the first $k+1$ years. Thus it is the difference between the event of survival to time $k$ and the smaller event of survival to time $k+1$. Because the latter event is contained in the former, the probability of their difference is

P(\text{death in year }k+1) =P(\text{survive }k\text{ years})-P(\text{survive }k+1\text{ years}) ={}_kp_x-{}_{k+1}p_x.

For each possible year of death, the present value is $Bv^{k+1}$. The APV is the expected present value, so it is the sum of each possible present value multiplied by the probability of that outcome. This gives the formula above. The policy-year death probabilities must add to the probability of death within the term:

\sum_{k=0}^{n-1}\left({}_kp_x-{}_{k+1}p_x\right)=1-{}_np_x.

This telescoping check is useful: the term probabilities should account for exactly the lives that do not survive to the end of the term. It does not replace discounting, since benefits paid in different years have different present values.

Calculation Procedure

Record the survival values.
List ${}_0p_x,{}_1p_x,\ldots,{}_np_x$, including ${}_0p_x=1$. Confirm that they are probabilities of survival from issue to each stated duration.

Difference adjacent values.
For year $k+1$, calculate ${}_kp_x-{}_{k+1}p_x$. This is the unconditional probability, from issue, of a payment in that year.

Discount each benefit.
Multiply the year-$k+1$ death probability by $Bv^{k+1}$. The exponent is the year-end payment time, not the start of the policy year.

Sum and check.
Add the discounted terms. Separately verify that the death probabilities sum to $1-{}_np_x$, then round the APV as requested.

Worked Examples

Worked Example: Three-Year Term From Survival Values

A hypothetical insurer issues a three-year term policy with a benefit of $100,000 payable at the end of the year of death. The effective annual interest rate is $i=4\%$. Under the hypothetical mortality model, the survival probabilities from issue are ${}_0p_x=1.00$, ${}_1p_x=0.98$, ${}_2p_x=0.95$, and ${}_3p_x=0.91$. Find the APV to the nearest cent.

Method and assumptions. Use the discrete year-end benefit formula. The given values specify survival at integer durations; no assumption about death timing within a year is needed. The discount factor is $v=1/1.04$.

Year 1. The unconditional probability of death is $1.00-0.98=0.02$. Its discounted expected benefit is $100{,}000(0.02)v=100{,}000(0.02/1.04)=\$1{,}923.0769$.

Year 2. The unconditional probability is $0.98-0.95=0.03$. Its discounted expected benefit is $100{,}000(0.03)v^2=100{,}000(0.03/1.0816)=\$2{,}773.6686$.

Year 3. The unconditional probability is $0.95-0.91=0.04$. Its discounted expected benefit is $100{,}000(0.04)v^3=100{,}000(0.04/1.124864)=\$3{,}555.9854$.

Adding the unrounded contributions gives an APV of $100{,}000(0.0825273099)=\$8{,}252.73099$, or $8,252.73 to the nearest cent. As a check, the death probabilities sum to $0.02+0.03+0.04=0.09=1-0.91$. The common mistake avoided here is discounting every benefit for the same duration; the year-three benefit is discounted for three years, not one.

Worked Example: Do Not Treat a Conditional Rate as an Unconditional Probability

A hypothetical two-year term policy pays $10,000 at the end of the year of death. The effective annual interest rate is $i=5\%$, and the survival probabilities are ${}_0p_x=1.00$, ${}_1p_x=0.96$, and ${}_2p_x=0.88$. Calculate the APV to the nearest cent.

Method and assumptions. The survival values are unconditional probabilities from issue under the specified discrete mortality model. Difference adjacent values to obtain the probabilities of payment in each policy year.

The probability of death in year 1 is $1.00-0.96=0.04$. The probability of death in year 2 is $0.96-0.88=0.08$. The year-two value $0.08$ is unconditional from issue.

The discount factor is $v=1/1.05$. Therefore,

\begin{aligned} \operatorname{APV} &=10{,}000\left(\frac{0.04}{1.05}+\frac{0.08}{1.05^2}\right)\\ &=10{,}000(0.0380952381+0.0725623583)\\ &=\$1{,}106.57596. \end{aligned}

The APV is $1,106.58 to the nearest cent. The probability check is $0.04+0.08=0.12=1-0.88$.

Common mistake and correction. The conditional probability of dying in year 2 given survival through year 1 is $1-0.88/0.96=0.0833333$. Using that conditional rate directly as the year-two payment probability gives the wrong calculation $10{,}000(0.04/1.05+0.0833333/1.05^2)\approx \$1{,}136.81$. To use a conditional rate, it must first be multiplied by the probability of reaching the start of year 2: $0.96(0.0833333)=0.08$. Differencing the survival values already gives the correct unconditional probability.

Worked Example: A Three-Year Term With a Different Interest Rate

A hypothetical three-year policy pays $20,000 at the end of the year of death. Its effective annual interest rate is $i=3\%$. The survival probabilities under the hypothetical mortality model are ${}_0p_x=1.00$, ${}_1p_x=0.97$, ${}_2p_x=0.92$, and ${}_3p_x=0.86$. Find the APV to the nearest cent.

Method and assumptions. Apply the level-benefit term formula using the supplied integer-duration survival probabilities. The discount factors for years 1, 2, and 3 are $v=1/1.03$, $v^2=1/1.0609$, and $v^3=1/1.092727$, respectively.

The year-specific unconditional death probabilities are:

Year 1: $1.00-0.97=0.03$.
Year 2: $0.97-0.92=0.05$.
Year 3: $0.92-0.86=0.06$.

The discounted expected benefit contributions are:

Year 1: $20{,}000(0.03/1.03)=\$582.5243$.
Year 2: $20{,}000(0.05/1.0609)=\$942.5959$.
Year 3: $20{,}000(0.06/1.092727)=\$1{,}098.1700$.

Using unrounded terms, the APV is $20{,}000(0.0291262136+0.0471297955+0.0549084995)=\$2{,}623.29017$, or $2,623.29 to the nearest cent. The death probabilities sum to $0.14=1-0.86$, as required. This example highlights a common timing trap: survival through the first year is used to form the year-two death probability, but the year-two payment is discounted for two years.

Checks and Exam Traps

A sound calculation keeps the probability and timing of each payment paired. For policy year $k+1$, the probability is ${}_kp_x-{}_{k+1}p_x$, and the discount factor is $v^{k+1}$. Using the right probability with the wrong discount exponent still produces an incorrect APV.

Do not confuse conditional and unconditional probabilities. The conditional mortality rate for a year is conditional on surviving to that year. The difference of consecutive survival probabilities is instead an unconditional probability from issue.
Do not omit the first duration. Set ${}_0p_x=1$; the first-year death probability is $1-{}_1p_x$.
Do not include a benefit at the end of the term for survival. An $n$-year term policy pays only for death within the term. The probability ${}_np_x$ contributes no death benefit under this contract.
Do not round intermediate terms prematurely. Keep sufficient precision through the sum and round the final money amount to cents.

Key takeaway: For a level $n$-year term benefit paid at the end of the year of death, difference adjacent survival probabilities to get the unconditional payment probabilities, discount each by its year-end payment time, and check that the probabilities sum to $1-{}_np_x$.

Check Your Understanding

For each hypothetical policy, use the stated annual effective interest rate and survival probabilities. Give money answers to the nearest cent.

A two-year term policy pays $5,000 at the end of the year of death. The interest rate is $i=4\%$, and the survival probabilities are $1.00, 0.95, 0.90$. Find both policy-year death probabilities and the APV. (Answer: probabilities $0.05, 0.05$; APV $471.52)
A three-year term policy pays $12,000 at the end of the year of death. The interest rate is $i=0\%$, and the survival probabilities are $1.00, 0.99, 0.96, 0.90$. Find the APV and verify the total probability of payment. (Answer: APV $1,200.00; total probability of payment $0.10$)
A two-year term policy pays $25,000 at the end of the year of death. The interest rate is $i=5\%$, and the survival probabilities are $1.00, 0.98, 0.94$. Find the year-two unconditional death probability and the APV. (Answer: year-two probability $0.04$; APV $1,383.22)
A two-year policy pays $10,000 at the end of the year of death. The interest rate is $i=10\%$, and the survival probabilities are $1.00, 0.90, 0.81$. Find the conditional probability of death in year 2 given survival through year 1, the unconditional probability of death in year 2, and the APV. (Answer: conditional probability $0.10$; unconditional probability $0.09$; APV $1,652.89)
A four-year term policy pays $10,000 at the end of the year of death. The interest rate is $i=2\%$, and the survival probabilities are $1.00, 0.96, 0.93, 0.89, 0.84$. Find the four policy-year death probabilities and the APV. (Answer: probabilities $0.04, 0.03, 0.04, 0.05$; APV $1,519.36)

What You'll Learn