Comparing Term and Whole-Life APVs at a Fixed Age

Why Compare Term and Whole-Life APVs?

A term policy pays a death benefit only if death occurs within a specified coverage period. A whole-life policy pays if death eventually occurs, subject to the mortality model. Comparing their expected present values helps isolate the effect of coverage duration: when the insured, level benefit, interest rate, and mortality assumptions are the same, the whole-life contract includes the term coverage and additional coverage after the term expires.

The comparison concerns the actuarial present value (APV) of benefits, not a premium, a policy value, or the amount paid for a policy. Assume throughout that death benefits are paid at the end of the policy year of death. A benefit is paid only if death occurs while coverage is in force. Premiums, expenses, and other contract features are not included.

Definition: For a life aged $x$, a level benefit of $B$, and an $n$-year term policy, let $A^{1}_{x:\overline{n}|}$ denote the APV per unit of benefit. Under annual death-benefit payments, $$ A^{1}_{x:\overline{n}|} = \sum_{k=0}^{n-1} v^{k+1}{}_kp_x q_{x+k}. $$ The APV of the benefit is $B A^{1}_{x:\overline{n}|}$. For whole-life insurance, $A_x$ denotes the APV per unit of benefit: $$ A_x = \sum_{k=0}^{\infty} v^{k+1}{}_kp_x q_{x+k}. $$ Here $v=1/(1+i)$, ${}_kp_x$ is the probability of surviving $k$ years from age $x$, and $q_{x+k}$ is the probability of death during the next year, conditional on survival to age $x+k$.

The first term of either sum is the expected discounted benefit if death occurs in the first policy year. In the term sum, the last covered death year is year $n$; a death after that year produces no term benefit. In the whole-life sum, later death years remain covered. The probabilities ${}_kp_x q_{x+k}$ are unconditional probabilities from age $x$: survive $k$ years, then die in the following year.

How the Two APVs Fit Together

The term APV is the first $n$ terms of the whole-life APV. The remaining terms can be expressed as the value, at age $x$, of surviving to the end of year $n$ and then having whole-life coverage from age $x+n$.

A_x = A^{1}_{x:\overline{n}|} + v^n{}_np_x A_{x+n}

This identity assumes the same interest and mortality basis is used in both parts, and that the whole-life continuation value $A_{x+n}$ is defined for a life alive at age $x+n$. Multiplying by the common benefit $B$ gives the corresponding identity for benefit APVs.

Separate the death years.
The term sum covers $k=0,\ldots,n-1$; the whole-life sum also includes $k=n,n+1,\ldots$.

Factor out survival to year $n$.
For each later death year $k\ge n$, ${}_kp_x={}_np_x\,{}_{k-n}p_{x+n}$.

Discount the continuation value.
The factor $v^n$ discounts from the end of year $n$ to age $x$; the remaining expected value is $A_{x+n}$.

More explicitly, the terms after year $n$ are $$ \sum_{k=n}^{\infty} v^{k+1}{}_kp_x q_{x+k} = v^n{}_np_x \sum_{j=0}^{\infty}v^{j+1}{}_jp_{x+n}q_{x+n+j} = v^n{}_np_x A_{x+n}. $$

This derivation also establishes an important check: for a nonnegative benefit and the same assumptions, whole-life APV cannot be less than the $n$-year term APV. The difference is precisely the nonnegative continuation component. This comparison is about expected values; it does not imply that either contract pays a benefit in every individual outcome.

Worked Example: Constant Mortality and a Two-Year Term

A hypothetical insurer considers a level benefit of $\$5{,}000$ for a life aged 45. Assume annual mortality is constant at $q_{45+k}=0.10$ for every $k$, the effective annual interest rate is $i=25\%$, and death benefits are paid at the end of the year of death. Compare the two-year term and whole-life benefit APVs.

Method. First calculate the two covered death-year probabilities. Then use the geometric series for whole life under constant mortality. This avoids treating the two-year term as though it covered deaths after year 2.

The discount factor is $$ v=\frac{1}{1.25}=0.8. $$ The probability of death in year 1 is $0.10$. The probability of death in year 2 is survival through year 1 followed by death: $$ {}_1p_{45}q_{46}=0.90(0.10)=0.09. $$ Thus the two-year term APV is $$ 5000\left[0.10(0.8)+0.09(0.8)^2\right] = 5000(0.0800+0.0576) = 5000(0.1376) = \$688.00. $$

For whole life, the probability of death in year $k+1$ is $0.9^k(0.1)$. Therefore, the APV per unit of benefit is $$ A_{45} = \sum_{k=0}^{\infty}0.8^{k+1}(0.9)^k(0.1) = \frac{0.08}{1-0.72} = \frac{2}{7}. $$ The whole-life benefit APV is $5000(2/7)=\$1{,}428.57$, rounded to the nearest cent.

The difference is $\$1{,}428.57-\$688.00=\$740.57$. As a check, the continuation expression gives $$ 5000(0.8)^2(0.9)^2\left(\frac{2}{7}\right) = \$740.57 $$ to the nearest cent. The sum of the term and continuation components is $\$1{,}428.57$, as required.

Using Age-Specific Mortality

Constant mortality is not necessary for a comparison. When mortality changes by age, calculate each covered death-year probability using the mortality rate for that year. For a term policy, include only the years it covers. For whole life, either continue the death-year sum or calculate a continuation APV using the model specified for ages beyond the term.

Worked Example: Changing Mortality and Zero Interest

A hypothetical two-product comparison is made for a life aged $x$, with a level benefit of $\$100{,}000$. Assume $i=0\%$, $q_x=0.04$, $q_{x+1}=0.06$, and $q_{x+2}=0.08$. From age $x+3$ onward, assume the annual mortality rate is $0.10$, so death eventually occurs with probability 1. Compare a three-year term with whole life.

Method. Use the age-specific mortality rates to find the unconditional probability of death in each covered year. At zero interest there is no discounting, so each covered death probability is multiplied directly by the benefit.

The death probabilities in the three covered years are $$ \begin{aligned} P(\text{death in year 1})&=0.04,\\ P(\text{death in year 2})&=0.96(0.06)=0.0576,\\ P(\text{death in year 3})&=0.96(0.94)(0.08)=0.072192. \end{aligned} $$ The term APV is therefore $$ 100{,}000(0.04+0.0576+0.072192) = 100{,}000(0.169792) = \$16{,}979.20. $$

The probability of surviving all three years is $$ {}_3p_x=0.96(0.94)(0.92)=0.830208. $$ For a life alive at age $x+3$, eventual death is certain under the stated continuation assumption. Since $i=0\%$, the whole-life APV from age $x+3$ is $\$100{,}000$. The continuation component at age $x$ is thus $$ {}_3p_x(\$100{,}000)=0.830208(\$100{,}000)=\$83{,}020.80. $$ The whole-life APV is $\$16{,}979.20+\$83{,}020.80=\$100{,}000.00$. This also checks directly: the whole-life benefit is paid with probability 1 and is not discounted.

The zero interest assumption makes this example especially transparent, but it matters: with positive interest, the timing of death affects the APV even when eventual death is certain.

Life-Table Counts and a Continuation Tail

A life table provides another way to obtain death probabilities. If $\ell_x$ lives are alive at age $x$, then $\ell_x-\ell_{x+1}$ deaths occur in the following year in the table. Dividing by $\ell_x$ gives the conditional mortality rate $q_x$, while dividing the deaths by the original $\ell_x$ gives the unconditional probability of death in that year for a life selected from age $x$.

Worked Example: Life-Table Counts With a Whole-Life Tail

For a hypothetical life aged $x$, a life table has $\ell_x=10{,}000$, $\ell_{x+1}=9{,}700$, and $\ell_{x+2}=9{,}409$. A level benefit of $\$20{,}000$ is payable at the end of the year of death. Assume $i=10\%$. For ages $x+2$ and later, assume constant annual mortality $q=0.03$. Find the two-year term and whole-life APVs.

Method. Convert the life-table decrements to unconditional death probabilities from age $x$, calculate the term APV, and then value the continuation for a life surviving to age $x+2$. The continuation model applies from age $x+2$, not from the original age.

The first-year deaths are $10{,}000-9{,}700=300$, giving probability $300/10{,}000=0.03$. The second-year deaths are $9{,}700-9{,}409=291$, giving probability $291/10{,}000=0.0291$ from the original age. The discount factor is $v=1/1.10=10/11$. Therefore, $$ \begin{aligned} \text{two-year term APV} &=20{,}000\left[0.03\left(\frac{10}{11}\right) +0.0291\left(\frac{10}{11}\right)^2\right]\\ &=20{,}000(0.0272727273+0.0240495868)\\ &=\$1{,}026.45, \end{aligned} $$ rounded to the nearest cent.

Under the constant continuation mortality assumption, the whole-life APV per unit of benefit at age $x+2$ is $$ A_{x+2} = \frac{qv}{1-(1-q)v} = \frac{0.03(10/11)}{1-0.97(10/11)} = \frac{3}{13}. $$ The probability of surviving to age $x+2$ is $9{,}409/10{,}000=0.9409$. Thus the continuation component, valued at age $x$, is $$ 20{,}000\left(\frac{10}{11}\right)^2(0.9409)\left(\frac{3}{13}\right) = \$3{,}588.94 $$ to the nearest cent. The whole-life APV is consequently $\$1{,}026.45+\$3{,}588.94=\$4{,}615.39$. Using unrounded components, the total is $\$4{,}615.3846\ldots$, which rounds to $\$4{,}615.38$; the one-cent difference in adding the separately rounded components is solely due to rounding. The correctly rounded whole-life APV is $\$4{,}615.38$.

Common Mistakes in the Comparison

The comparison is simple only when the contract terms and timing are kept consistent. Several errors can change the result substantially.

Using the wrong years in the term sum. An $n$-year term covers death in years 1 through $n$, so its sum runs from $k=0$ to $n-1$. The discount factor for death in year $k+1$ is $v^{k+1}$, not $v^k$.
Using conditional mortality as an unconditional death probability. In year 2, the probability from age $x$ is ${}_1p_xq_{x+1}$, not just $q_{x+1}$. The latter is conditional on reaching year 2.
Forgetting the survival factor in the continuation value. $A_{x+n}$ is conditional on being alive at age $x+n$. From age $x$, the tail must include ${}_np_x$, as well as $v^n$.
Comparing different bases. A term APV and a whole-life APV are not a clean coverage-duration comparison if they use different benefits, interest rates, ages, or mortality assumptions.

For example, in the first worked example, a common incorrect term calculation is $5000[0.10(0.8)+0.10(0.8)^2]=\$720.00$. It incorrectly uses the conditional second-year mortality rate $0.10$ without multiplying by the $0.90$ probability of surviving the first year. The correct second-year contribution is $5000[0.90(0.10)(0.8)^2]=\$288.00$, giving the correct total $\$688.00$.

Key takeaway: With the same age, level benefit, interest rate, and mortality basis, the whole-life APV equals the term APV plus the discounted continuation value for those who survive the term. Use this decomposition both to interpret the difference and to check the arithmetic.

Check Your Understanding

For each problem, use end-of-year death benefits and compare only the benefit APVs requested.

A hypothetical one-year term policy provides $\$8{,}000$ to a life with $q_x=0.05$. If $i=0\%$, find the term APV. (Answer: $\$400.00$)
For a life with constant annual mortality $q=0.20$, a hypothetical two-year term and whole-life policy each provide $\$10{,}000$. If $i=25\%$, find both APVs. (Answer: term $\$2{,}624.00$; whole life $\$4{,}444.44$, rounded to the nearest cent)
A hypothetical two-year term policy provides $\$50{,}000$. Assume $q_x=0.03$, $q_{x+1}=0.04$, and $i=5\%$. Find its APV. (Answer: $\$3{,}188.21$, rounded to the nearest cent)
A life table has $\ell_x=5{,}000$, $\ell_{x+1}=4{,}900$, and $\ell_{x+2}=4{,}802$. A two-year term benefit is $\$10{,}000$, $i=0\%$, and mortality from age $x+2$ onward has constant $q=0.02$. Assuming eventual death is certain, find the term and whole-life APVs. (Answer: term $\$396.00$; whole life $\$10{,}000.00$)
A hypothetical four-year term and whole-life policy each provide $\$12{,}000$. Mortality is constant at $q=0.10$, and $i=0\%$. Find the term APV, whole-life APV, and difference. (Answer: term $\$4{,}126.80$; whole life $\$12{,}000.00$; difference $\$7{,}873.20$)

What You'll Learn