Scaling Identities for Nonunit Benefits

Why Benefit Scaling Matters

Insurance-annuity identities are commonly stated for a benefit of one. A contract, however, may promise a benefit of tens of thousands of dollars. The key is to scale the present value by the benefit amount while preserving the contract’s payment timing. The amount of an insurance benefit is not the same thing as the number or frequency of annuity payments.

This tutorial applies that distinction to discrete and continuous insurance values. The identities themselves are familiar from the earlier material on insurance-annuity relationships; the emphasis here is how to use them correctly when the benefit is not one.

Definition: If a contract pays a benefit of $B$ dollars at the same time and under the same contingency as a unit benefit, its present-value random variable is $B$ times the unit-benefit present-value random variable. Consequently, its expected present value is $B$ times the unit-benefit expected present value. This scaling does not change the benefit’s timing or the probability of payment.

Scaling Discrete Insurance Values

For a discrete whole-life insurance, suppose the benefit is paid at the end of the year of death and the effective annual interest rate is $i$. The unit-benefit actuarial present value is $A_x$. A benefit of $B$ dollars has actuarial present value $B A_x$. The standard whole-life insurance-annuity identity is $A_x=1-d\ddot{a}_x$, where $d=i/(1+i)$ and $\ddot{a}_x$ is the value of a unit annuity-due payable annually while the life is alive.

\text{APV of the }B\text{-dollar whole-life benefit} = B A_x = B\left(1-d\ddot{a}_x\right).

For a discrete $n$-year endowment insurance with a unit benefit payable at death during the term or at time $n$ if the insured survives, let $A_{x:\overline{n}|}$ denote its unit-benefit value and $\ddot{a}_{x:\overline{n}|}$ the $n$-year temporary annuity-due value. Then $A_{x:\overline{n}|}=1-d\ddot{a}_{x:\overline{n}|}$. Scaling the benefit gives $B A_{x:\overline{n}|}=B(1-d\ddot{a}_{x:\overline{n}|})$.

For term insurance alone, the complementary value is the pure endowment ${}_nE_x=v^n{}_np_x$, where $v=1/(1+i)$ and ${}_np_x$ is the probability of surviving $n$ years. Thus, for a unit benefit payable at the end of the year of death, $A^1_{x:\overline{n}|}+{}_nE_x=1-d\ddot{a}_{x:\overline{n}|}$. For a $B$-dollar term benefit, multiply the term value by $B$; do not treat the pure endowment as part of the term benefit unless the contract also pays on survival.

Why the Scaling Works

The reason is linearity of present value and expectation. Let $Z_1$ be the discounted payment from a unit-benefit contract. If the contract pays $B$ times as much under exactly the same circumstances, its discounted payment is $Z_B=BZ_1$. Therefore:

$$ E[Z_B]=E[BZ_1]=B E[Z_1]. $$

The identity for a unit benefit can therefore be scaled as a whole. For example, if $A_x=1-d\ddot{a}_x$, multiplying both sides by $B$ gives $B A_x=B-Bd\ddot{a}_x$. This does not say that an annuity paying a different amount has value $Bd\ddot{a}_x$; that term appears only as part of the scaled algebraic identity.

More generally, if a contract has several benefit amounts or different payment times, a single multiplier may not describe it. Each payment must be valued according to its own amount, timing, and condition. Scaling by $B$ is valid when the entire unit-benefit payment pattern is multiplied by $B$, with no other change.

Worked Example: Scaling a Two-Year Term Benefit

A hypothetical two-year term policy pays $40{,}000$ dollars at the end of the year of death. The effective annual interest rate is $5\%$. Mortality is specified by $q_x=0.10$ and $q_{x+1}=0.20$, with the second probability conditional on surviving the first year. Find the actuarial present value. Then verify the result using the temporary-annuity complement identity.

Method. First calculate the unit-benefit term value from the unconditional probabilities of death in each year, then multiply by $40{,}000$. For the check, calculate the temporary annuity-due and pure endowment values using the same assumptions.

The discount factor is $v=1/1.05=20/21$, so $v^2=400/441$. The probability of death in year 1 is $0.10$. The probability of death in year 2 is the probability of surviving year 1 and then dying in year 2: $0.90(0.20)=0.18$.

A^1_{x:\overline{2}|} =0.10v+0.18v^2 =0.10\left(\frac{20}{21}\right) +0.18\left(\frac{400}{441}\right) =\frac{2}{21}+\frac{8}{49} =\frac{38}{147}.

The policy’s present value is therefore $40{,}000(38/147)=10{,}340.136\ldots$, or $10,340.14 to the nearest cent.

For the identity check, the two-year temporary annuity-due value is $\ddot{a}_{x:\overline{2}|}=1+v\,{}_1p_x=1+(20/21)(0.90)=13/7$. Also, $d=i/(1+i)=1/21$, and the pure endowment value is ${}_2E_x=v^2{}_2p_x=(400/441)(0.90)(0.80)=32/49$. Thus:

1-d\ddot{a}_{x:\overline{2}|}-{}_2E_x =1-\frac{13}{147}-\frac{96}{147} =\frac{38}{147}.

This agrees with the direct unit-benefit calculation, and scaling by $40{,}000$ gives the same dollar value.

Benefit Amounts and Annuity Payments

A benefit amount describes what the insurer pays when the benefit condition occurs. An annuity payment amount describes each payment in a stream. These amounts may differ. Under the equivalence principle, if an annual premium $P$ is paid at the beginning of each year while the insured is alive, the expected present value of that premium stream is $P\ddot{a}_{x:\overline{n}|}$. For a $B$-dollar endowment benefit, the benefit present value is $B A_{x:\overline{n}|}$. Equivalence sets these two present values equal; it does not make $P$ equal to $B$.

Payment frequency is a separate feature. Replacing one annual payment with several installments changes the payment times and usually the present-value factor. It does not mean that the insurance benefit is multiplied by the number of installments. If an annuity pays $C$ dollars per installment, its value is based on $C$ and the actual schedule of installments. The frequency is reflected in the schedule and its value, not by changing the insurance benefit amount.

Worked Example: Endowment Benefit with Annual Premiums

A hypothetical two-year endowment policy pays $60{,}000$ dollars at the end of the year of death if death occurs in the term, or at the end of year 2 if the insured survives. The effective annual interest rate is $5\%$, $q_x=0.10$, and $q_{x+1}=0.20$, conditional on survival to year 2. Annual premiums are paid in advance at time 0 and, if the insured survives, at time 1. Find the benefit present value and the level annual net premium under the equivalence principle.

Method. Scale the unit endowment value to the benefit amount, then divide by the present-value factor for the two annual premiums. The premium amount is not the insurance benefit amount.

From the previous example’s assumptions, the endowment value per unit is the term value plus the pure endowment value:

A_{x:\overline{2}|} =\frac{38}{147}+\frac{32}{49} =\frac{38}{147}+\frac{96}{147} =\frac{134}{147}.

The benefit present value is $60{,}000(134/147)=54{,}693.8775\ldots$, or $54,693.88 to the nearest cent. The annual premium present-value factor is $\ddot{a}_{x:\overline{2}|}=13/7$, since the time-0 premium is certain and the time-1 premium is payable only if the insured survives the first year.

P\left(\frac{13}{7}\right) =60{,}000\left(\frac{134}{147}\right), \qquad P=60{,}000\left(\frac{134}{147}\right)\left(\frac{7}{13}\right) =\frac{8{,}040{,}000}{273} =29{,}450.549\ldots.

The level annual net premium is $29,450.55 to the nearest cent. As a check, multiplying this unrounded premium by $13/7$ reproduces the unrounded benefit present value. The premium is paid in two possible annual installments; the benefit is a single amount payable under the death-or-survival condition.

Continuous Benefits Also Scale

For a continuous whole-life insurance paying one dollar at the moment of death, the actuarial present value is $\bar{A}_x$. If the benefit is $B$ dollars at that same moment, its value is $B\bar{A}_x$. The continuous insurance-annuity identity is $\bar{A}_x=1-\delta\bar{a}_x$, where $\delta$ is the force of interest and $\bar{a}_x$ is the continuous whole-life annuity value with payment rate one while the insured is alive.

Worked Example: Continuous Whole-Life Benefit Under Constant Forces

A hypothetical continuous whole-life policy pays $250{,}000$ dollars immediately at death. Assume a constant force of mortality $\mu=0.04$ and a constant force of interest $\delta=0.06$. Find the actuarial present value and verify it using the continuous insurance-annuity identity.

Method. Under a constant force of mortality, the future lifetime $T$ has density $f_T(t)=\mu e^{-\mu t}$ for $t\geq0$. The present value of a unit benefit is $e^{-\delta T}$. Therefore:

\bar{A}_x =E[e^{-\delta T}] =\int_0^\infty e^{-\delta t}\mu e^{-\mu t}\,dt =\mu\int_0^\infty e^{-(\mu+\delta)t}\,dt =\frac{\mu}{\mu+\delta}.

Substituting the stated forces gives $\bar{A}_x=0.04/(0.04+0.06)=0.40$. The $250{,}000$-dollar benefit therefore has present value $250{,}000(0.40)=$ $100,000.00.

For an independent identity check, the continuous annuity value under these constant forces is $\bar{a}_x=\int_0^\infty e^{-(\mu+\delta)t}\,dt=1/0.10=10$. Hence $1-\delta\bar{a}_x=1-(0.06)(10)=0.40$, matching the unit insurance value. Multiplying by the benefit amount again gives $100,000.00.

This calculation depends on the stated constant-force mortality and interest assumptions. The integral expression for a different mortality model must use that model’s future-lifetime distribution; the value $\mu/(\mu+\delta)$ does not apply automatically.

Common Scaling Errors

A frequent error is to scale the benefit correctly but use the wrong probability or payment factor. In the two-year term example, a mistaken calculation might use $40{,}000(0.10v+0.20v^2)$. It treats $q_{x+1}=0.20$ as an unconditional probability of death in year 2. But that probability is conditional on reaching year 2. The unconditional probability is $0.90(0.20)=0.18$, so the correct calculation is $40{,}000(0.10v+0.18v^2)$, giving $10,340.14.

Another mistake is to multiply by the number of premium installments instead of using the present value of the actual payment stream. A two-year policy with two possible annual premiums does not have a premium present value of $2P$ unless both payments are certain and undiscounted. Here the factor is $1+v\,{}_1p_x=13/7$, which accounts for both survival-contingent payment and discounting.

Scale the entire unit-benefit value. If benefit timing and contingencies are unchanged, multiply the unit value by $B$.
Keep the payment schedule separate. A benefit amount does not determine the premium amount or annuity payment frequency.
Use the right survival or death probability. Conditional probabilities must be combined with survival to the relevant year when calculating unconditional payment probabilities.

Key takeaway: A nonunit benefit scales the expected present value linearly only when its payment pattern is the unit-benefit pattern multiplied by that amount. Benefit size, benefit timing, and the frequency and amount of annuity payments are distinct contract features.

Check Your Understanding

For each problem, identify the benefit scaling and preserve the stated timing and probability assumptions.

A hypothetical three-year term policy pays $25,000 at the end of the year of death. The effective annual interest rate is $4\%$, with conditional death probabilities $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Find the actuarial present value. (Answer: $10,564.83)
A hypothetical continuous whole-life policy pays $400,000 at the moment of death. Assume constant forces of mortality $\mu=0.03$ and interest $\delta=0.05$. Find the actuarial present value. (Answer: $150,000.00)
A hypothetical two-year endowment pays $30,000 at the end of the year of death or at time 2 if the insured survives. Let $i=5\%$, $q_x=0.10$, and $q_{x+1}=0.20$, with annual premiums in advance while alive. Find the level annual net premium. (Answer: $14,725.27)
A hypothetical two-year term policy pays $100,000 at the end of the year of death. Let $i=4\%$, $q_x=0.04$, and $q_{x+1}=0.06$, with the latter conditional on survival through year 1. Find the actuarial present value. (Answer: $9,171.60)

What You'll Learn