Decomposing Composite Benefits to Price Premiums

Why Decompose a Benefit?

A contract may provide one benefit if death occurs during a specified period and another if the insured survives that period. It may also combine temporary coverage with coverage that begins later. Rather than valuing every possible payment from scratch, we can express the contract as a sum of simpler insurance benefits whose present values are familiar.

The central condition is that the components must reproduce the contract’s payment in every possible outcome. Once that is true, the net single premium is the sum of the component present values. This approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: A composite benefit is a benefit that can be represented as the sum of two or more simpler benefit amounts, each tied to a specified event or time. Under the net single premium principle, its expected present value equals the sum of the expected present values of those component benefits, provided the components describe the actual payments and use the same interest and mortality assumptions.

Notation and the Additivity Principle

Assume a discrete annual model: premiums, if any, are paid at time 0, and a death benefit is paid at the end of the year of death. Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Write ${}_kp_x$ for the probability a life aged $x$ survives $k$ years. In a given year, death and survival probabilities must be interpreted consistently: a conditional death probability for that year is multiplied by the probability of reaching the year to obtain the unconditional probability of death then.

The net single premium is the expected present value of the promised benefits. In particular, if two benefit components have present values $Z_1$ and $Z_2$, the total present value is $Z_1+Z_2$ in every outcome. Taking expectations gives $E[Z_1+Z_2]=E[Z_1]+E[Z_2]$. This is why component premiums add; the components need not be independent.

Key notation: $A_x$ denotes the net single premium for a unit whole-life insurance benefit paid at the end of the year of death. $A^1_{x:\overline{n}|}$ denotes the net single premium for a unit $n$-year term insurance benefit. The pure endowment factor is ${}_nE_x=v^n{}_np_x$, the present value factor for a unit payment at time $n$ if the life survives to then. An $n$-year endowment insurance, denoted $A_{x:\overline{n}|}$, pays on death during the term or on survival to its end.

The superscript $1$ distinguishes term insurance from endowment insurance: term insurance pays only on death during the term, while endowment insurance also pays at the end of the term if the insured survives.

Endowment Insurance as Term Insurance Plus a Pure Endowment

An $n$-year endowment with benefit amount $B$ has two mutually exclusive payment events: death in one of the first $n$ years, or survival to time $n$. The first set of events is exactly the payment pattern of $n$-year term insurance. The survival payment is a pure endowment. Therefore,

B A_{x:\overline{n}|} = B A^1_{x:\overline{n}|} + B\,{}_nE_x = B A^1_{x:\overline{n}|} + Bv^n{}_np_x.

This equality also follows by writing the present value as a sum over outcomes. For death in year $k$, the payment is discounted by $v^k$; for survival to year $n$, it is discounted by $v^n$. The death outcomes and survival outcome are disjoint, so each payment event is counted once.

Worked Example: Splitting a Two-Year Endowment

A hypothetical policy pays $10,000 at the end of the year of death if death occurs within two years, or $10,000 at the end of year 2 if the insured survives. The effective annual interest rate is 5%. For a life aged $x$, the unconditional probabilities of death in years 1 and 2 are 0.10 and 0.15, respectively, and the probability of survival to year 2 is 0.75. Find the net single premium.

Method. Split the endowment into two-year term insurance and a pure endowment. The stated event probabilities sum to $0.10+0.15+0.75=1.00$, so they account for all outcomes. The second-year death probability is unconditional; it is not a conditional $q$ for someone who has already survived year 1.

The year-1 death benefit has expected present value $10{,}000(0.10)v=10{,}000(0.10/1.05)=\$952.38095$. The year-2 death benefit has expected present value $10{,}000(0.15)v^2=10{,}000(0.15/1.1025)=\$1{,}360.54422$.

The pure endowment payment has expected present value $10{,}000(0.75)v^2=10{,}000(0.75/1.1025)=\$6{,}802.72109$.

Adding the unrounded amounts gives $\$952.38095+\$1{,}360.54422+\$6{,}802.72109=\$9{,}115.64626$. The net single premium is $9,115.65, rounded to the nearest cent.

Check. The combined year-2 probability is $0.15+0.75=0.90$. Direct valuation gives $10{,}000[0.10/1.05+0.90/(1.05)^2]=\$9{,}115.65$, the same result. A common mistake is to treat the year-2 death probability as conditional and multiply it by survival to year 1; the problem gives an unconditional probability, so that extra multiplication would be incorrect.

Whole-Life Insurance Split at a Fixed Duration

A whole-life benefit can be separated at any chosen duration $n$. The benefit is paid either on death during the first $n$ years, or after year $n$, provided the insured survives to that point. The first component is $n$-year term insurance; the second is a deferred whole-life insurance.

Theorem: Under the discrete annual model, the unit whole-life value decomposes as $$ A_x=A^1_{x:\overline{n}|}+{}_{n|}A_x, \qquad {}_{n|}A_x=v^n{}_np_xA_{x+n}. $$ Here ${}_{n|}A_x$ is the value at age $x$ of a unit whole-life benefit payable only if death occurs after the first $n$ years.

To justify the deferred value, the benefit can be paid only if the life survives $n$ years. That survival has probability ${}_np_x$, and the value at time $n$, conditional on survival, is $A_{x+n}$. Discounting that conditional value back $n$ years multiplies it by $v^n$. Thus the deferred component has value $v^n{}_np_xA_{x+n}$. The term and deferred components cover disjoint death periods, together accounting for every possible death time.

Worked Example: Combining Term and Deferred Whole-Life Coverage

A hypothetical contract pays $20,000 at the end of the year of death if death occurs within the first three years. If the insured survives three years, it instead pays $50,000 at the end of the year of any later death. The effective annual interest rate is 5%. Assume the conditional probability of death in each year, at every age in this model, is 0.10. Find the net single premium.

Method. Value the three-year term component and the three-year deferred whole-life component separately. The annual survival probability is $p=0.90$, and $v=1/1.05=20/21$, so $vp=6/7$.

For the term component, the probability of death in year $k$ is $0.10(0.90)^{k-1}$. Its value per dollar of benefit is $(0.10)v[1+(vp)+(vp)^2]$. Since $0.10v=2/21$ and $1+6/7+36/49=127/49$, this factor is $(2/21)(127/49)=254/1029$. The term component is therefore $\$20{,}000(254/1029)=\$4{,}936.83188$.

For the deferred component, the probability-weighted discount factor for surviving three years is $(vp)^3=(6/7)^3=216/343$. At age $x+3$, the unit whole-life value is \[ A_{x+3}=\frac{0.10v}{1-0.90v} =\frac{2/21}{1-6/7} =\frac{2}{3}. \] This geometric series is valid because the conditional death probability and interest rate are constant from year to year. The deferred value per dollar is therefore $(6/7)^3(2/3)=144/343$, and the $50,000 component is $\$50{,}000(144/343)=\$20{,}991.25364$.

Adding the unrounded component values gives $\$4{,}936.83188+\$20{,}991.25364=\$25{,}928.08552$. The net single premium is $25,928.09, rounded to the nearest cent.

Check. The first component covers death in years 1–3; the second covers death after year 3, conditional on survival to that point. These periods do not overlap, and together they cover every possible death time. A common mistake is to value the $50,000 deferred benefit as if it were payable at year 3; it is payable on a later death, so the post-year-3 whole-life value $A_{x+3}$ must be included.

Adding Components with Different Benefit Amounts

Decomposition is useful even when a contract does not look like one standard policy. A benefit can be described as a base amount payable on a range of events plus an additional amount payable on a narrower range. The total benefit for each event is the sum of the amounts triggered by that event. To avoid double-counting, write down the event and payment amount for each component before calculating its present value.

List payment events.
Separate death years, survival dates, and any deferred period that change whether a payment is due.

Match each event to a simpler benefit.
For example, death during a fixed term is term insurance; survival to a fixed date is a pure endowment; death after a deferred period is deferred insurance.

Value each component under the stated assumptions.
Use the same interest and mortality model throughout, and account for survival to any deferred starting date.

Add and check.
Confirm that the component benefits sum to the original payment in every outcome, then add their expected present values.

Worked Example: A Two-Stage Benefit with an Additional Deferred Component

A hypothetical contract pays $30,000 on death during the first two years or on survival to the end of year 2. If the insured survives year 2, it also pays $40,000 at the end of the year of any later death. The effective annual interest rate is 4%. The conditional death probabilities are 0.05 in year 1, 0.10 in year 2, and 0.10 in every year after year 2. Find the net single premium.

Method. The first promise is a two-year endowment; the second is a two-year deferred whole-life benefit. The probability of surviving year 2 is $0.95(0.90)=0.855$. The unconditional death probabilities are 0.05 in year 1 and $0.95(0.10)=0.095$ in year 2.

The $30,000 endowment pays on death in year 1, death in year 2, or survival to year 2. Its value is \[ \$30{,}000\left(\frac{0.05}{1.04} +\frac{0.095+0.855}{(1.04)^2}\right). \] Here $0.095+0.855=0.950$, and $(1.04)^2=1.0816$. The factor is $0.05/1.04+0.95/1.0816=0.0480769231+0.8783284024=0.9264053255$. Thus this component is $\$27{,}792.15976$.

For the later-death component, the unit whole-life value after year 2 is \[ A_{x+2}=\frac{0.10v}{1-0.90v} =\frac{5/52}{1-45/52} =\frac{5}{7}, \] where $v=25/26$. The deferred factor is $v^2{}_2p_x=(0.855)/(1.0816)=4275/5408$. Consequently, the $40,000 deferred component is $\$40{,}000(4275/5408)(5/7)=\$22{,}585.58741$.

The total is $\$27{,}792.15976+\$22{,}585.58741=\$50{,}377.74717$, so the net single premium is $50,377.75, rounded to the nearest cent.

Check. If death occurs during the first two years, only the $30,000 component pays. If the insured survives year 2, the $30,000 endowment pays then, and the additional $40,000 is payable only upon later death. The deferred component therefore does not add a second payment at year 2. Confusing the survival payment with a later death payment would change the contract being valued.

Common Exam Traps

Using conditional and unconditional probabilities interchangeably. If a year-2 death probability is conditional on survival through year 1, multiply it by the probability of surviving year 1 to get the probability of death in year 2 for the original cohort. Do not multiply if the problem already gives the unconditional probability.
Counting the endowment survival benefit twice. The term component covers death during the term only. The pure endowment component covers survival to the term’s end.
Starting a deferred benefit too early. A deferred whole-life benefit requires both survival to the deferral date and a later death. Its value includes the survival probability, discounting, and the whole-life value at the deferral age.
Adding values without checking the contract. Component present values add only when the component payments sum to the original benefit in each outcome. Overlapping components are acceptable only when the contract truly pays both amounts on that outcome.

Key takeaway: Decompose by payment event, not merely by the product’s name. Identify which component pays, when it pays, and the probability of reaching that event; then value each component and add the results.

Check Your Understanding

For each problem, identify the relevant components before calculating. Round monetary answers to the nearest cent.

A hypothetical two-year endowment pays $5,000 on death in either year or on survival to year 2. The effective annual interest rate is 0%, and the unconditional probabilities of death in years 1 and 2 are 0.10 and 0.20; survival to year 2 is 0.70. Find its net single premium. (Answer: $5,000.00)
A hypothetical two-year term policy pays $12,000 at the end of the year of death. The effective annual interest rate is 10%, with conditional death probabilities $q_x=0.05$ and $q_{x+1}=0.10$. Find its net single premium. (Answer: $1,487.60)
A hypothetical two-year endowment pays $8,000 on death during the term or survival to year 2. The effective annual interest rate is 5%, and the conditional death probabilities are 0.10 in year 1 and 0.20 in year 2. Find its net single premium. (Answer: $7,292.52)
A hypothetical contract pays $10,000 on death during the first three years and, if the insured survives three years, $15,000 on any later death. The effective annual interest rate is 0%, and the conditional death probability is 0.20 each year. Find its net single premium. (Answer: $12,560.00)

What You'll Learn