Common Errors in Discrete Life Insurance Valuation

Why Discrete Life Insurance Calculations Go Wrong

A discrete life insurance present value is a sum of possible benefits, each weighted by its probability and discounted to the valuation date. The arithmetic is often straightforward; the difficulty is aligning three details: which policy year the death occurs in, the probability of reaching that year and dying in it, and when the benefit is paid. A mismatch in any one of these can produce a plausible-looking but incorrect answer.

Throughout this tutorial, assume a hypothetical policy issued at age $x$, with a deterministic effective annual interest rate $i$. Let $v=1/(1+i)$. Write $q_{x+k}$ for the probability that a person aged $x+k$ dies within the following year, conditional on being alive at that age. Unless a problem states otherwise, a discrete death benefit is paid at the end of the policy year of death. These conventions are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: The probability of surviving $k$ years from age $x$ and then dying during the following year is ${}_{k|}q_x={}_kp_xq_{x+k}$, where ${}_kp_x$ is the probability of surviving $k$ years. In particular, ${}_{0|}q_x=q_x$. This is an unconditional probability measured from the policy issue date; $q_{x+k}$ alone is conditional on being alive at age $x+k$.

The Correct Year-by-Year Structure

For a level benefit $B$ on an $n$-year term policy, the benefit is payable only if death occurs during the term. Since death in policy year $k+1$ means surviving $k$ years and then dying in the next year, its probability is ${}_kp_xq_{x+k}$. The payment is at time $k+1$, so its present value is multiplied by $v^{k+1}$.

\operatorname{EPV} = B\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} = B\sum_{k=0}^{n-1}v^{k+1}{}_{k|}q_x.

For a benefit $B_{k+1}$ that varies by policy year, replace $B$ in each term with the benefit payable in that year. The formula assumes the stated annual mortality probabilities apply to the insured and that the benefit is paid at the end of the year of death. It does not apply unchanged to a benefit paid at a different time or to a contract that also pays on survival to the end of the term.

Key takeaway: For death in policy year $k+1$, use the probability of surviving the first $k$ years and dying in the next year, then discount the year-end payment by $v^{k+1}$. The index starts at $k=0$, not $k=1$.

Why the Survival Factor Is Necessary

The year-$(k+1)$ death event requires two events in sequence: the insured must be alive at the start of that policy year, and must die during it. Conditional probability gives the product ${}_kp_xq_{x+k}$. The events “survive $k$ years” and “die in the next year” therefore cannot be replaced by the one-year conditional probability $q_{x+k}$ alone when calculating from the issue date.

The present value of a benefit paid at time $k+1$ is its amount times $v^{k+1}$. Partitioning possible outcomes by policy year gives mutually exclusive death events. Multiplying the payment’s present value by each event probability and adding across the term yields the expected present value:

\operatorname{EPV} = \sum_{k=0}^{n-1} \left(B_{k+1}v^{k+1}\right) P(\text{death in policy year }k+1) = \sum_{k=0}^{n-1}B_{k+1}v^{k+1}{}_kp_xq_{x+k}.

This derivation also identifies a practical check: every term must pair the benefit for a particular policy year with both that year’s death probability and its payment-time discount factor.

Worked Valuations and Error Checks

Worked Example: Conditional Versus Unconditional Death Probabilities

A hypothetical three-year term policy pays $10,000 at the end of the policy year of death. The effective annual interest rate is $5\%$. The conditional one-year death probabilities are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Find the expected present value. Keep full precision until the final answer, which should be rounded to cents.

Method. Calculate each policy-year death probability from the issue date, then discount the corresponding year-end benefit. Here $v=1/1.05$.

In year 1, the death probability is $0.10$. The contribution is $10{,}000(0.10)v=1{,}000/1.05=\$952.380952$.

In year 2, the insured must first survive year 1. Thus the probability is $(1-0.10)(0.20)=0.90(0.20)=0.18$. The contribution is $10{,}000(0.18)v^2=1{,}800/1.1025=\$1{,}632.653061$.

In year 3, the insured must survive both earlier years. The probability is $0.90(1-0.20)(0.25)=0.90(0.80)(0.25)=0.18$. The contribution is $10{,}000(0.18)v^3=1{,}800/1.157625=\$1{,}554.907459$.

Adding the contributions gives $\$952.380952+\$1{,}632.653061+\$1{,}554.907459=\$4{,}139.941472$, or $4,139.94 to the nearest cent.

Check. The year-by-year death probabilities sum to $0.10+0.18+0.18=0.46$, leaving probability $0.54$ of surviving the full term. They sum to one when that term-survival probability is included.

Common mistake avoided. Treating each conditional probability as an unconditional probability would give $10{,}000(0.10v+0.20v^2+0.25v^3)=\$4{,}926.03$. This overstates the value because it incorrectly allows an insured who died in an earlier year to generate a later-year benefit.

Worked Example: Matching Benefit Year and Discount Exponent

A hypothetical two-year term policy pays $20,000 if death occurs in year 1 and $30,000 if death occurs in year 2, with each benefit paid at the end of its death year. The effective annual interest rate is $4\%$, and the conditional death probabilities are $q_x=0.08$ and $q_{x+1}=0.12$. Find the expected present value, rounded to cents.

Method. Index the first policy year by $k=0$ and the second by $k=1$. The end-of-year payment times are 1 and 2, so the discount factors are $v$ and $v^2$, where $v=1/1.04$.

The year 1 death probability is $0.08$, giving a contribution of $20{,}000(0.08)v=1{,}600/1.04=\$1{,}538.461538$.

For year 2, survival through year 1 has probability $1-0.08=0.92$. Thus the probability of death in year 2 is $0.92(0.12)=0.1104$. Its contribution is $30{,}000(0.1104)v^2=3{,}312/1.0816=\$3{,}062.130178$.

The expected present value is $\$1{,}538.461538+\$3{,}062.130178=\$4{,}600.591716$, or $4,600.59 to the nearest cent.

Check. The probability of a death benefit is $0.08+0.1104=0.1904$, and the probability of surviving the two-year term is $0.92(0.88)=0.8096$. These probabilities sum to one. The second-year benefit is discounted two years because it is paid at time 2, not time 1.

Common mistake avoided. Using $v^0$ for the first-year benefit or $v^1$ for the second-year benefit treats an end-of-year payment as though it were paid one year earlier. The first policy year begins at time 0, but its benefit is paid at time 1.

Worked Example: Checking a Finite Mortality Model

For a hypothetical insured aged 70, the conditional annual death probabilities are $q_{70}=0.10$, $q_{71}=0.25$, $q_{72}=0.50$, and $q_{73}=1.00$. A whole-life policy pays $1,000 at the end of the year of death. The effective annual interest rate is $25\%$. Find the expected present value, rounded to cents.

Method. Convert each conditional probability into the probability of death in that policy year, discount by the year-end payment time, and verify that the finite mortality model accounts for all outcomes. Here $v=1/1.25=0.8$.

The probability of death in year 1 is $0.10$. The probability of death in year 2 is $(1-0.10)(0.25)=0.90(0.25)=0.225$. The probability of death in year 3 is $0.90(1-0.25)(0.50)=0.90(0.75)(0.50)=0.3375$.

The probability of reaching age 73 is $0.90(0.75)(0.50)=0.3375$. Since $q_{73}=1.00$, everyone reaching that age dies in policy year 4. The year 4 death probability is therefore $0.3375(1.00)=0.3375$.

The four death probabilities sum to $0.10+0.225+0.3375+0.3375=1.00$. Thus the model assigns probability one to death by the end of year 4.

The discounted contributions are $1{,}000(0.10)(0.8)=\$80.00$, $1{,}000(0.225)(0.8^2)=\$144.00$, $1{,}000(0.3375)(0.8^3)=\$172.80$, and $1{,}000(0.3375)(0.8^4)=\$138.24$. Their sum is $535.04.

Check. Each discount exponent matches the end of the relevant policy year, and the probabilities of death in years 1 through 4 exhaust all possible outcomes. In particular, the year 4 contribution is not omitted simply because the model has a finite ultimate age.

Common mistake avoided. Ending the calculation after year 3 would leave the $0.3375$ probability of surviving to age 73 without a benefit, even though $q_{73}=1$ requires death during the following year.

Common Errors and Practical Checks

When reviewing a discrete insurance valuation, identify the event, the payment time, and the starting point for every probability. These checks are especially useful when probabilities are given in a table or when benefits vary by policy year.

Using conditional $q_{x+k}$ as an issue-date probability. For $k>0$, multiply by the probability of surviving the first $k$ years.
Starting the index at one. If $k=0$ denotes the first policy year, its payment is at time 1 and has discount factor $v^1$.
Discounting by the year number before checking the payment time. A year-2 death benefit paid at the end of that year is discounted by $v^2$.
Mixing conditional and unconditional mortality data. If a problem directly supplies probabilities of death in each policy year measured from issue, those are already unconditional; do not multiply by survival a second time.
Omitting a final policy year. For a finite mortality table, check whether the probability of reaching its ultimate age leads to a later death benefit.

A useful probability check is to add the probabilities of death in every modeled policy year and the probability of surviving beyond the modeled period. The total should equal one. A useful valuation check is to verify that each discounted term uses the benefit, probability, and payment time for the same policy year.

Check Your Understanding

For each problem, apply the stated end-of-year benefit convention and round dollar answers to cents.

A two-year term policy pays $12,000 at the end of the year of death. The effective annual interest rate is $5\%$, with $q_x=0.10$ and $q_{x+1}=0.20$. Find its expected present value. (Answer: $3,102.04)
A three-year term policy pays $8,000 at the end of the year of death. The effective annual interest rate is $4\%$, and $q_x=0.05$, $q_{x+1}=0.10$, and $q_{x+2}=0.20$. Find the expected present value. (Answer: $2,303.43)
A two-year term policy pays $5,000 at the end of the year of death. The effective annual interest rate is $25\%$, with $q_x=0.20$ and $q_{x+1}=0.25$. Calculate the year 1 and year 2 death probabilities and the expected present value. (Answer: Probabilities $0.20$ and $0.20$; expected present value $1,440.00)
A three-year term policy pays $15,000 at the end of the year of death, and the effective annual interest rate is $5\%$. The probabilities of death in policy years 1, 2, and 3, measured from issue, are $0.12$, $0.22$, and $0.18$. Find the expected present value. (Answer: $7,039.84)

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