Deriving the Term Insurance APV by Death-Year Partition

Why Partition by Death Year?

A discrete term insurance benefit is paid only if death occurs during a specified number of policy years. Its present value depends on which covered year contains the death: a benefit paid later is discounted for longer. The expected present value (APV) therefore combines two ingredients for each possible death year: the probability that death occurs in that year and the present value of the benefit paid then.

This tutorial derives that calculation by separating the covered death years into mutually exclusive events. The method makes the timing explicit and works when mortality changes from one age to the next. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Consider a life aged $x$ at issue and an $n$-year term policy that pays a level benefit $B$ at the end of the policy year of death, provided death occurs within the $n$-year term. If the insured survives to the end of the term, the policy pays no death benefit. The discrete term insurance APV, denoted here by $A^1_{x:\overline{n}|}$ for a unit benefit, is the expected present value of that benefit. For benefit $B$, the APV is $B A^1_{x:\overline{n}|}$.

Let $K_x$ be the curtate future lifetime: the number of complete years lived after age $x$. Thus, $K_x=k$ means death occurs during the $(k+1)$th policy year. For an $n$-year term, the covered death-year events are $K_x=0,1,\ldots,n-1$. Define $q_{x+k}$ as the probability that a life aged $x+k$ dies during the following year, given survival to that age, and let $p_{x+k}=1-q_{x+k}$. In particular, ${}_kp_x$ is the probability of surviving the first $k$ years, with ${}_0p_x=1$.

Deriving the Death-Year Sum

Assume a constant effective annual interest rate $i$, with $v=1/(1+i)$. A death in the first policy year produces a payment at time 1, a death in the second year a payment at time 2, and a death in the $(k+1)$th year a payment at time $k+1$. The event $K_x=k$ requires survival through the first $k$ years and death in the next year. Its probability is therefore ${}_kp_x q_{x+k}$.

P(K_x=k)={}_kp_xq_{x+k},\qquad k=0,1,\ldots,n-1

Let $Z$ be the present value of the unit death benefit. On event $K_x=k$, the benefit is paid at time $k+1$, so $Z=v^{k+1}$. If the life survives the term, $Z=0$. The expected value of a random variable with finitely many mutually exclusive outcomes is the sum of each outcome value multiplied by its probability. Applying that definition gives the APV.

A^1_{x:\overline{n}|} =E[Z] =\sum_{k=0}^{n-1}v^{k+1}P(K_x=k) =\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}

For a benefit $B$, the present value in each covered death year is $Bv^{k+1}$, so every term in the sum is multiplied by $B$. The sum stops at $n-1$: a death after the term does not trigger a term death benefit. The probability of surviving to the end of the term is not an additional benefit outcome; it contributes zero to the expected present value.

Why the formula holds: The events $K_x=0,\ldots,K_x=n-1$ cannot happen together, and their union is exactly the event of death during the term. On each event, the present value is known: $Bv^{k+1}$. On survival through the term, it is zero. Summing present value times event probability over these mutually exclusive outcomes is precisely the definition of expected present value.

The one-year mortality rates can vary by age. The survival probability in a term for year $k+1$ is the product of the preceding conditional one-year survival probabilities:

{}_kp_x=\prod_{j=0}^{k-1}p_{x+j},\qquad {}_0p_x=1

This product is not an assumption of constant mortality; it accounts for the succession of age-specific rates. The APV sum applies when the benefit is paid at the end of the year of death and the interest rate used for discounting is constant and effective annually. A benefit paid immediately at death, at the beginning of the death year, or at a term-end maturity date has a different present-value timing and requires a different calculation.

Worked Calculations

Worked Example: Three-Year Coverage With Changing Mortality

A hypothetical insurer issues a three-year term policy to a life aged $x$. The annual effective interest rate is $i=25\%$. The one-year death probabilities at ages $x$, $x+1$, and $x+2$ are $0.10$, $0.20$, and $0.25$, respectively. The benefit is $10,000, payable at the end of the year of death. Calculate the APV.

Method. Partition the possible claims into deaths in years 1, 2, and 3. First calculate the discount factor: $v=1/(1+0.25)=0.8$. The relevant powers are $v=0.8$, $v^2=0.64$, and $v^3=0.512$.

Death in year 1. This event has probability $q_x=0.10$. Its discounted expected benefit is $0.10\times 0.8\times \$10{,}000=\$800.00$.

Death in year 2. The life must survive year 1 and then die in year 2. The event probability is $p_xq_{x+1}=(1-0.10)(0.20)=0.90\times0.20=0.18$. Its discounted expected benefit is $0.18\times0.64\times\$10{,}000=\$1{,}152.00$.

Death in year 3. The life must survive the first two years and die in year 3. The event probability is $p_xp_{x+1}q_{x+2}=0.90\times0.80\times0.25=0.18$. Its discounted expected benefit is $0.18\times0.512\times\$10{,}000=\$921.60$.

Adding the three contributions gives an APV of $\$800.00+\$1{,}152.00+\$921.60=\$2{,}873.60$. The covered-death probability is $0.10+0.18+0.18=0.46$; the survival probability is $0.90\times0.80\times0.75=0.54$. The probabilities total $1.00$, as required. The method avoids treating the age-specific death probabilities as unconditional probabilities.

Worked Example: A Two-Year Term With a Different Mortality Pattern

A hypothetical two-year policy pays $5,000 at the end of the year of death. The effective annual interest rate is $25\%$. The death probabilities are $q_x=0.20$ and $q_{x+1}=0.10$. Find the APV and the probability of a claim during the term.

Method. There are two covered events. The discount factors are $v=0.8$ and $v^2=0.64$.

Death in year 1 has probability $0.20$, so its contribution is $0.20\times0.8\times\$5{,}000=\$800.00$.

Death in year 2 requires survival in year 1. Its probability is $(1-0.20)(0.10)=0.80\times0.10=0.08$, and its contribution is $0.08\times0.64\times\$5{,}000=\$256.00$.

The APV is $\$800.00+\$256.00=\$1{,}056.00$. The probability of a claim is $0.20+0.08=0.28$. As a check, the probability of surviving both years is $0.80\times0.90=0.72$, and $0.28+0.72=1.00$. The second-year death probability is $0.08$, not $0.10$, because $0.10$ is conditional on reaching age $x+1$.

Worked Example: Calculating From Life-Table Counts

A hypothetical life table has $\ell_x=2{,}000$, $\ell_{x+1}=1{,}600$, $\ell_{x+2}=1{,}200$, $\ell_{x+3}=900$, and $\ell_{x+4}=600$. A four-year term policy pays $12,000 at the end of the year of death. Assume a constant effective annual interest rate of $25\%$. Calculate the APV.

Method. The number dying in each year is found by subtracting consecutive survivor counts. The counts dying in years 1 through 4 are $2{,}000-1{,}600=400$, $1{,}600-1{,}200=400$, $1{,}200-900=300$, and $900-600=300$. Dividing each count by the initial $2{,}000$ gives the unconditional probabilities of each death-year event: $0.20$, $0.20$, $0.15$, and $0.15$.

With $v=0.8$, the discount factors are $0.8$, $0.64$, $0.512$, and $0.4096$. The four APV contributions are:

Year 1: $0.20\times0.8\times\$12{,}000=\$1{,}920.00$.
Year 2: $0.20\times0.64\times\$12{,}000=\$1{,}536.00$.
Year 3: $0.15\times0.512\times\$12{,}000=\$921.60$.
Year 4: $0.15\times0.4096\times\$12{,}000=\$737.28$.

The total APV is $\$1{,}920.00+\$1{,}536.00+\$921.60+\$737.28=\$5{,}114.88$. The covered-death probability is $(400+400+300+300)/2{,}000=0.70$. The $600$ survivors at the end of year 4 represent probability $600/2{,}000=0.30$, so the covered-death and survival probabilities sum to $1.00$. The life-table counts already provide unconditional death-year probabilities when each year's deaths are divided by the original $\ell_x$.

Timing Trap and Practical Checks

A frequent error is to discount a first-year death benefit by $v^0=1$, as if it were paid at issue. Under the stated discrete benefit timing, death during year 1 produces payment at the end of that year, at time 1. In the first worked example, the incorrect calculation using $1$, $v$, and $v^2$ for years 1, 2, and 3 would give:

\$10{,}000\left(0.10+0.18(0.8)+0.18(0.64)\right) =\$10{,}000(0.3592) =\$3{,}592.00

That is not the stated contract's APV. The correction uses $v^1$, $v^2$, and $v^3$, giving $\$2{,}873.60$. The exponent is the time of payment, not the number of complete years lived before death.

A second common error is to use $q_{x+k}$ alone as the probability of death in year $k+1$. That rate is conditional on survival to age $x+k$. For $k>0$, multiply it by the probability of reaching that age: ${}_kp_xq_{x+k}$. When using a life table, an equivalent check is that covered death-year probabilities sum to $1-{}_np_x$; they should not sum to 1 unless survival through the term has probability zero.

Key takeaway: For a level benefit paid at the end of the year of death, calculate one term for each covered year: unconditional probability of death in that year, multiplied by the benefit and by the discount factor for the payment time. Include no benefit for survival to the end of the term.

Check Your Understanding

For each problem, partition covered deaths by policy year and account for the payment time. Round money to the nearest cent.

A two-year term has benefit $1,000, effective annual interest rate $25\%$, and $q_x=0.20,\ q_{x+1}=0.10$. Find its APV. (Answer: $211.20)
A three-year term has benefit $5,000, effective annual interest rate $25\%$, and death probabilities $0.10,\ 0.20,\ 0.25$ at ages $x,\ x+1,\ x+2$. Find the APV. (Answer: $1,436.80)
A two-year policy pays $8,000 at the end of the year of death. Assume $i=25\%$, $q_x=0.15$, and $q_{x+1}=0.20$. Find the APV. (Answer: $1,830.40)
A life table has $\ell_x=5{,}000,\ \ell_{x+1}=4{,}500,\ \ell_{x+2}=3{,}600,\ \ell_{x+3}=2{,}700$. A three-year term pays $4,000 at the end of the year of death, with $i=25\%$. Find the APV. (Answer: $1,149.44)
A four-year term pays $10,000 at the end of the year of death. Assume $i=25\%$ and age-specific death probabilities $0.05,\ 0.10,\ 0.15,\ 0.20$ for the four successive years. Find the APV. (Answer: $2,259.99)

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