Increasing Term Benefits With Annual Death Payments

Why the Benefit Schedule Matters

A term policy need not pay the same amount in every policy year. A hypothetical contract might pay one amount if death occurs in the first year of coverage, a larger amount if death occurs in the second year, and so on. When the benefit rises by the same fixed dollar amount each year, the main task is to combine that schedule with the probability and timing of each possible payment.

The benefit is assumed to be paid at the end of the policy year of death. If the insured survives the full $n$-year term, this coverage makes no payment. This is an $n$-year term insurance, not an endowment: there is no survival benefit at the end of the term.

Definition: Suppose the death benefit in the first policy year is $B_1$, and it increases by a fixed amount $c$ from one policy year to the next. The benefit payable at the end of policy year $k$, for $k=1,\ldots,n$, is $$ B_k=B_1+(k-1)c. $$ For a genuinely increasing schedule, $c>0$. The first benefit is $B_1$, not $B_1+c$; the increase has occurred zero times by the end of the first policy year.

The schedule is determined by two quantities: the first-year benefit and the fixed annual increase. For example, if $B_1=\$6{,}000$ and $c=\$3{,}000$, the first three benefits are $\$6{,}000$, $\$9{,}000$, and $\$12{,}000$. The benefit in year $k$ depends on the policy year, not the insured’s age alone.

Death-Year Probabilities and the Present Value

Let $K_x$ be the curtate future lifetime of a life aged $x$ at issue: $K_x=k-1$ means death occurs during policy year $k$. Write $q_{x+j}$ for the probability of death during the next year, conditional on being alive at age $x+j$, and ${}_jp_x$ for the probability of surviving $j$ years from age $x$. Thus ${}_0p_x=1$, and survival probabilities can be built successively from the annual mortality rates.

Key result: Under annual effective interest rate $i$, with $v=1/(1+i)$, the actuarial present value of an $n$-year term policy with end-of-year death benefit $B_k=B_1+(k-1)c$ is $$ \operatorname{APV}=\sum_{k=1}^{n}v^k\bigl({}_{k-1}p_xq_{x+k-1}\bigr)\bigl(B_1+(k-1)c\bigr). $$ The factor ${}_{k-1}p_xq_{x+k-1}$ is the probability that the insured survives to the start of policy year $k$ and dies during that year.

This expression follows by considering each possible year of death separately. Death during policy year $k$ has probability ${}_{k-1}p_xq_{x+k-1}$, and the benefit is paid $k$ years after issue, so its present value is $v^kB_k$. Multiplying the present value by the probability of that outcome gives its contribution to the expected present value. The death-year events are mutually exclusive, so adding their contributions gives the APV. Survival through year $n$ contributes zero because the contract has no maturity payment.

{}_0p_x=1,\qquad {}_kp_x={}_{k-1}p_x(1-q_{x+k-1}).

The recursion makes clear that the mortality rate for year $k$ is conditional on reaching that year. It is not, by itself, the unconditional probability of a payment in year $k$. The discount exponent is also $k$: a death in the first policy year produces a payment at the end of year 1, not at issue.

Worked Calculations

Worked Example: Three-Year Term With Constant Mortality

A hypothetical three-year term policy pays $\$10{,}000$ for death in year 1, with the benefit increasing by $\$2{,}000$ at the start of each later policy year. Find the APV. Assume annual effective interest $i=25\%$, and assume the conditional probability of death in each covered year is $q=0.20$, independent of attained age.

Method and assumptions. The curtate future lifetime has death-year probabilities determined by the constant annual mortality parameter $q=0.20$: the probability of death in year $k$ is $(1-q)^{k-1}q$. The discount factor is $v=1/(1.25)=0.8$. The benefit schedule has $B_1=\$10{,}000$ and $c=\$2{,}000$.

Step 1: List the benefits. Applying $B_k=B_1+(k-1)c$ gives $B_1=\$10{,}000$, $B_2=\$12{,}000$, and $B_3=\$14{,}000$.

Step 2: Find the death-year probabilities. In year 1, the probability is $0.20$. In year 2, it is $0.80(0.20)=0.16$. In year 3, it is $0.80^2(0.20)=0.128$.

Step 3: Discount and weight each benefit. The year 1 contribution is $0.20(0.8)(\$10{,}000)=\$1{,}600$. The year 2 contribution is $0.16(0.8^2)(\$12{,}000)=\$1{,}228.80$. The year 3 contribution is $0.128(0.8^3)(\$14{,}000)=\$917.504$.

Step 4: Add and check. The unrounded APV is $\$1{,}600+\$1{,}228.80+\$917.504=\$3{,}746.304$, or $\$3{,}746.30$ to the nearest cent. As a probability check, the probability of surviving all three years is $0.8^3=0.512$; the three death-year probabilities sum to $0.20+0.16+0.128=0.488$, and $0.488+0.512=1$.

Worked Example: Age-Varying Mortality and a Rising Benefit

A hypothetical three-year policy pays $\$6{,}000$ for death in year 1 and increases the benefit by $\$3{,}000$ each policy year. The annual conditional death probabilities in years 1, 2, and 3 are $0.10$, $0.20$, and $0.25$, respectively. If the annual effective interest rate is $25\%$, calculate the APV.

Method and assumptions. Here the mortality model is specified by the age-varying sequence $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Each rate applies only conditional on survival to that policy year. Use $v=0.8$.

Step 1: Construct the schedule. The fixed-increase formula gives benefits of $\$6{,}000$, $\$9{,}000$, and $\$12{,}000$ in years 1, 2, and 3.

Step 2: Calculate death-year probabilities. Year 1 has probability $0.10$. Year 2 has probability $(1-0.10)(0.20)=0.90(0.20)=0.18$. Year 3 has probability $0.90(1-0.20)(0.25)=0.90(0.80)(0.25)=0.18$.

Step 3: Calculate the present-value contributions. For year 1, $0.10(0.8)(\$6{,}000)=\$480$. For year 2, $0.18(0.8^2)(\$9{,}000)=0.18(0.64)(\$9{,}000)=\$1{,}036.80$. For year 3, $0.18(0.8^3)(\$12{,}000)=0.18(0.512)(\$12{,}000)=\$1{,}105.92$.

Step 4: Add and check. The APV is $\$480+\$1{,}036.80+\$1{,}105.92=$ $\$2{,}622.72$. The probability of surviving all three years is $0.90(0.80)(0.75)=0.54$. The death probabilities sum to $0.10+0.18+0.18=0.46$, so the total is $1.00$.

Worked Example: Four-Year Schedule With a Final-Year Benefit

A hypothetical four-year term contract pays $\$5{,}000$ for death in the first policy year, and its benefit increases by $\$2{,}000$ each year. The conditional probabilities of death in years 1 through 4 are $0.10$, $0.15$, $0.20$, and $0.25$. At annual effective interest $i=25\%$, find the APV, including the possibility of death in the final covered year.

Method and assumptions. Use the specified age-varying mortality sequence, with each $q$ conditional on survival to the relevant year, and $v=1/1.25=0.8$. The contract pays for death in any of its four policy years, including year 4.

Step 1: Find each benefit. The schedule is $\$5{,}000$, $\$7{,}000$, $\$9{,}000$, and $\$11{,}000$. The fourth-year amount has three annual increases added to the first-year benefit: $\$5{,}000+3(\$2{,}000)=\$11{,}000$.

Step 2: Find each death-year probability. The four probabilities are $0.10$, $0.90(0.15)=0.135$, $0.90(0.85)(0.20)=0.153$, and $0.90(0.85)(0.80)(0.25)=0.153$.

Step 3: Discount each possible payment. The contributions are:

\begin{aligned} \text{Year 1: }&0.10(0.8)(\$5{,}000)=\$400,\\ \text{Year 2: }&0.135(0.64)(\$7{,}000)=\$604.80,\\ \text{Year 3: }&0.153(0.512)(\$9{,}000)=\$705.024,\\ \text{Year 4: }&0.153(0.4096)(\$11{,}000)=\$689.3568. \end{aligned}

Step 4: Add and check. The unrounded total is $\$400+\$604.80+\$705.024+\$689.3568=\$2{,}399.1808$, so the APV is $\$2{,}399.18$ to the nearest cent. Survival through all four years is $0.90(0.85)(0.80)(0.75)=0.459$. The four death probabilities sum to $0.541$, and $0.541+0.459=1$. The final-year benefit is included because death during that year is still within the term.

Common Exam Trap: Counting the First Increase Too Early

A frequent error is to treat $c$ as though it has already been added to the benefit in year 1. For the year-1 benefit of $\$6{,}000$ and annual increase of $\$3{,}000$ in the second example, the incorrect schedule $B_k=B_1+kc$ gives $\$9{,}000$, $\$12{,}000$, and $\$15{,}000$. Using the correct death probabilities and discount factors, that incorrect schedule produces contributions of $\$720$, $\$1{,}382.40$, and $\$1{,}382.40$, for an incorrect APV of $\$3{,}484.80$.

The correction is to count the number of increases before the policy year in question: there are zero before year 1, one before year 2, and two before year 3. Thus $B_k=B_1+(k-1)c$, giving the correct schedule $\$6{,}000$, $\$9{,}000$, and $\$12{,}000$, and the correct APV of $\$2{,}622.72$. Also check the term boundary: death in year $n$ is covered, but survival to the end of year $n$ does not create a payment under this term contract.

Key takeaway: Build the benefit schedule first, then calculate the unconditional probability of death in each policy year, discount that year’s benefit by $v^k$, and sum only over the $n$ covered years. This method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each hypothetical policy, identify the benefit schedule or calculate the requested APV using the stated annual mortality and interest assumptions.

A three-year policy pays $\$4{,}800$, $\$6{,}300$, and $\$7{,}800$ for death in years 1, 2, and 3. Is this a fixed-increase schedule, and what is the annual increase? (Answer: Yes; the increases are $\$1{,}500$ and $\$1{,}500$.)
A two-year policy pays $\$4{,}000$ for death in year 1 and increases by $\$2{,}000$ in year 2. Assume annual effective interest $i=25\%$ and constant conditional annual death probability $q=0.20$. Calculate the APV. (Answer: $\$1{,}254.40$.)
A three-year policy pays $\$3{,}000$ in year 1 and increases by $\$1{,}000$ each year. The conditional death probabilities are $0.05$, $0.10$, and $0.20$, and the annual effective interest rate is $0\%$. Calculate the APV. (Answer: $\$1{,}385.00$.)
A three-year policy pays $\$10{,}000$ in year 1 and increases by $\$2{,}500$ each year. The conditional death probabilities in years 1, 2, and 3 are $0.10$, $0.20$, and $0.30$, and $i=25\%$. Calculate the APV, including the year-3 death benefit. (Answer: $\$3{,}898.88$.)

What You'll Learn