Combining Benefits by Present-Value Linearity

Why Decompose a Combined Contract?

A life contract may pay a benefit if the insured dies during a specified period, another amount if the insured survives to the end of that period, and a series of payments while the insured is alive. Valuing each cash-flow stream separately makes the contract easier to analyze and helps prevent one benefit from being omitted or counted twice.

The central tool is present-value linearity: when a contract’s present value is the sum of component present values, its expected present value (EPV) is the sum of the components’ EPVs. This remains true even when the benefits depend on the same future lifetime. The factors must, however, use the same interest and mortality assumptions and must match the contract’s actual payment timing. The decomposition approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: For an $n$-year discrete term insurance paying 1 at the end of the year of death, let $$ A^1_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}. $$ The $n$-year pure endowment pays 1 at time $n$ if the life aged $x$ then survives; its EPV is $v^n{}_np_x$. The $n$-year endowment insurance pays 1 at the end of the year of death if death occurs within $n$ years, or 1 at time $n$ if the life survives. Its EPV is $A^1_{x:\overline{n}|}+v^n{}_np_x$.

Here $v=1/(1+i)$, where $i$ is the effective annual interest rate. The probability ${}_kp_x$ is survival from age $x$ for $k$ years, and $q_{x+k}$ is the probability of death in the following year conditional on survival to age $x+k$. Thus, ${}_kp_xq_{x+k}$ is the unconditional probability of death in policy year $k+1$. A temporary life annuity-due paying 1 at times $0,1,\ldots,n-1$, provided the life is alive at each payment time, has EPV $ \ddot{a}_{x:\overline{n}|}=\sum_{j=0}^{n-1}v^j{}_jp_x $.

Linearity and a Useful Identity

Suppose a contract has component benefits with present values $Z_1,Z_2,\ldots,Z_m$, all evaluated for the same insured lifetime and interest rate. Its total present value is $Z=\sum_{r=1}^m Z_r$. By linearity of expectation,

E[Z]=E\!\left[\sum_{r=1}^m Z_r\right] =\sum_{r=1}^m E[Z_r].

The component present values can be dependent; for example, a death benefit and an annuity may both depend on when the same insured dies. Dependence does not change this expectation identity. It would matter for quantities involving variance or a nonlinear transformation of the total, but those are not needed to add EPVs.

A particularly useful check converts an endowment-insurance value into a temporary annuity-due value. Let $d=1-v=i/(1+i)$. Then $A^1_{x:\overline{n}|}+v^n{}_np_x=1-d\ddot{a}_{x:\overline{n}|}$. To see why, expand the annuity sum and collect successive survival probabilities:

Expand the discount term.
$d\ddot{a}_{x:\overline{n}|}=\sum_{j=0}^{n-1}(v^j-v^{j+1}){}_jp_x$.

Rearrange the finite sums.
Subtracting this expression from 1 gives $\sum_{k=1}^{n-1}v^k({}_{k-1}p_x-{}_kp_x)+v^n{}_{n-1}p_x$.

Identify the insurance and survival outcomes.
For $1\le k<n$, ${}_{k-1}p_x-{}_kp_x={}_{k-1}p_xq_{x+k-1}$, the probability of death in year $k$. The final term splits as $v^n{}_{n-1}p_x=v^n{}_{n-1}p_xq_{x+n-1}+v^n{}_np_x$. These are the year-$n$ death benefit and the pure endowment, respectively.

The resulting components are exactly the death benefits in the $n$-year term insurance plus the payment for survival to time $n$. This establishes the identity. It is a valuation check, not a license to ignore the payment conditions: the annuity-due payments occur at the beginnings of years, whereas the term insurance benefit occurs at the end of the year of death.

Worked Valuations

Worked Example: Combining Term, Survival, and Annuity Benefits

A hypothetical three-year contract pays $50,000 at the end of the year of death if the insured dies during the term, $20,000 at time 3 if the insured survives to then, and $1,000 at times 0, 1, and 2 if alive at each payment time. Assume an effective annual interest rate of $i=5\%$ and a constant annual mortality probability $q=0.10$, with $p=0.90$ each year. Find the EPV.

Method and assumptions. Split the contract into a three-year term insurance, a three-year pure endowment, and a three-year temporary life annuity-due. Mortality is independent year to year with the stated constant conditional probabilities; all cash flows are discounted at the same annual effective rate.

First, $v=1/1.05=0.95238095$, $v^2=0.90702948$, and $v^3=0.86383760$. The death probabilities in years 1, 2, and 3 are $0.10$, $0.90(0.10)=0.09$, and $0.90^2(0.10)=0.081$. Therefore,

A^1_{x:\overline{3}|} =0.10v+0.09v^2+0.081v^3 =0.09523810+0.08163265+0.06997085 =0.24684159.

The probability of surviving three years is $0.90^3=0.729$. The pure endowment factor is $0.729v^3=0.62973761$. The annuity-due factor is $\ddot{a}_{x:\overline{3}|}=1+0.90v+0.81v^2=1+0.85714286+0.73469388=2.59183673$.

Multiplying each factor by its benefit amount gives a term-insurance EPV of $50{,}000(0.24684159)=\$12{,}342.08$, a pure-endowment EPV of $20{,}000(0.62973761)=\$12{,}594.75$, and an annuity EPV of $1{,}000(2.59183673)=\$2{,}591.84$. Using the unrounded factors, the three component EPVs sum to $\$27{,}528.67$, rounded to the nearest cent.

Check and mistake avoided. The three-year annuity factor includes a payment at time 0 and only includes later payments if the insured is then alive. Treating it as three guaranteed payments would incorrectly use an annuity-certain factor. The term benefit is also paid at the end of the death year, not at the instant of death.

Worked Example: Checking an Endowment Value with an Annuity

A hypothetical contract pays $25,000 at the end of the year of death if death occurs within three years, or $8,000 at time 3 if the insured survives. It also pays $300 at times 0, 1, and 2, conditional on being alive at each payment time. Assume $i=4\%$, with conditional one-year survival probabilities $p_x=0.90$, $p_{x+1}=0.80$, and $p_{x+2}=0.75$. Find the combined EPV.

Method and assumptions. Value the term insurance and pure endowment separately, then value the temporary annuity-due. Use the endowment-insurance/annuity identity as an independent check. The given conditional probabilities imply ${}_1p_x=0.90$, ${}_2p_x=0.90(0.80)=0.72$, and ${}_3p_x=0.72(0.75)=0.54$.

The discount factors are $v=1/1.04=0.96153846$, $v^2=0.92455621$, and $v^3=0.88899636$. The unconditional probabilities of death in years 1, 2, and 3 are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$. Thus,

A^1_{x:\overline{3}|} =0.10v+0.18v^2+0.18v^3 =0.09615385+0.16642012+0.16001934 =0.42259331.

The pure-endowment factor is $0.54v^3=0.48005803$. The annuity-due factor is $\ddot{a}_{x:\overline{3}|}=1+0.90v+0.72v^2=1+0.86538462+0.66568047=2.53106509$. The component EPVs are therefore $25{,}000(0.42259331)=\$10{,}564.83$, $8{,}000(0.48005803)=\$3{,}840.46$, and $300(2.53106509)=\$759.32$. Summing the unrounded component values gives $\$15{,}164.62$, to the nearest cent.

For the check, $d=1-v=0.03846154$. The endowment-insurance factor is $0.42259331+0.48005803=0.90265134$; the identity gives $1-d(2.53106509)=0.90265134$, agreeing to the displayed precision.

Check and mistake avoided. The survival probability at time 3 is a product of the three conditional survival probabilities, not just the last one. Using $0.75$ as ${}_3p_x$ would misstate both the pure endowment and the annuity factor.

Worked Example: Adding Whole-Life and Temporary Benefits

A hypothetical contract pays $18,000 at the end of the year of death, whenever death occurs, and pays $600 at the beginning of each year while the insured is alive, starting immediately. It also pays $4,000 at the end of the year of death if death occurs within three years. Assume $i=25\%$ and a constant annual conditional mortality probability $q=0.20$, so $p=0.80$. Find the EPV.

Method and assumptions. Apply whole-life factors to the first death benefit and the life-annuity-due factor to the recurring payments. Add the three-year term factor for the additional death benefit in the first three years. Mortality is constant throughout life; the whole-life calculations use the resulting infinite geometric sums.

Here $v=1/1.25=0.80$, so $vp=0.64$. The whole-life insurance factor is $A_x=\sum_{k=0}^{\infty}qv(vp)^k=0.16/(1-0.64)=4/9$. The whole-life annuity-due factor is $\ddot{a}_x=\sum_{j=0}^{\infty}(vp)^j=1/(1-0.64)=25/9$. The three-year term factor is

A^1_{x:\overline{3}|} =qv\{1+vp+(vp)^2\} =0.16(1+0.64+0.4096) =0.327936.

The respective EPVs are $18{,}000(4/9)=\$8{,}000.00$, $600(25/9)=\$1{,}666.67$, and $4{,}000(0.327936)=\$1{,}311.74$. Using unrounded values, the total is $\$10{,}978.41$, rounded to the nearest cent.

As a factor check, the whole-life insurance and annuity factors satisfy $1-d\ddot{a}_x=1-0.20(25/9)=4/9=A_x$. The temporary term benefit remains a separate component because it is paid only for deaths in the first three years.

Check and mistake avoided. The $4,000 benefit is additional to the whole-life benefit for deaths in the first three years. It should be added as a separate term-insurance component, not treated as a replacement for the $18,000 benefit during those years.

When Component Addition Works—and When to Pause

Component addition is reliable when the contract cash flow really is the sum of the stated components and every factor is evaluated on the same basis. Before calculating, identify for each component the amount, the event that triggers payment, the payment time, and whether the benefit continues beyond the stated term. Then choose the matching factor. A term-insurance factor, a pure-endowment factor, and an annuity factor do not represent interchangeable payment patterns.

A frequent exam trap is to add factors that look familiar without matching their conditions. For example, a pure endowment is paid only on survival to the specified time; it is not an unconditional payment at that time. Likewise, an annuity-due payment at time $j$ requires survival to time $j$, while an annuity-immediate payment at the end of a year has a different survival condition and discount factor. These distinctions are part of defining the component correctly before using linearity.

The direct sum also requires care if a feature changes the benefit based on the combined outcome, such as a cap on total payments or a rule that pays only the larger of two benefits. Such a contract is not automatically the sum of its separate benefits. First express its actual cash flow correctly; only then apply expectation to a valid sum of components.

Key takeaway: Write the contract as a sum of clearly defined contingent cash flows. Value each component using its own matching insurance or annuity factor, then add the unrounded EPVs. Linearity handles dependence between components, but it does not repair a mismatch in survival conditions, benefit amount, or payment timing.

Check Your Understanding

For each problem, decompose the benefits and use a consistent annual effective interest and mortality basis. Round dollar answers to the nearest cent.

A two-year contract pays $10,000 at the end of the year of death if death occurs within two years, and $2,000 at time 2 if the insured survives. Assume $i=5\%$ and constant annual $q=0.20$. Find its EPV. (Answer: $4,517.01)
A hypothetical contract pays $500 at times 0 and 1 if the insured is alive at each time, and $5,000 at time 2 if alive then. Assume $i=10\%$ and constant annual $q=0.10$. Find the combined EPV. (Answer: $4,256.20)
A three-year endowment insurance pays $7,500 at the end of the year of death or at time 3 on survival. It also pays $250 at times 0, 1, and 2 while the insured is alive. Assume $i=25\%$ and constant annual $q=0.20$. Find the combined EPV. (Answer: $4,938.00)
A contract pays $9,000 at the end of the year of death for the insured’s whole life and $200 at the beginning of each year while alive, starting immediately. Assume $i=5\%$ and constant annual $q=0.10$. Find the EPV. (Answer: $7,400.00)
A two-year endowment insurance pays $20,000 at the end of the year of death within two years or at time 2 on survival. A separate annuity pays $500 at times 0 and 1 if alive. Assume $i=4\%$, $p_x=0.90$, and $p_{x+1}=0.80$. Find the combined EPV. (Answer: $19,497.78)

What You'll Learn