Why a Unit-Benefit APV Can Be Scaled
A discrete life insurance APV is often calculated for a benefit of one dollar. That unit value is useful because a policy may instead promise a benefit of thousands of dollars. When the same fixed amount is paid at the end of whichever policy year death occurs, the present value for that policy is the unit-benefit present value multiplied by the benefit amount. The mortality assumptions, payment timing, and interest assumptions stay the same; only the scale of the payment changes.
This result applies to level benefits for term or whole-life insurance. It does not mean that a benefit schedule that changes by policy year can always be valued by multiplying one unit APV by a single amount. Here, the benefit is constant whenever a covered death benefit is payable.
For an \(n\)-year term insurance on a life aged \(x\), let \(q_{x+k}\) be the probability of death during the year of age \(x+k\), conditional on being alive at that age. Write \({}_k p_x\) for the probability of surviving \(k\) years from age \(x\). Under a constant effective annual interest rate \(i\), let \(v=1/(1+i)\). The unit-benefit term APV is
The factor \({}_k p_x q_{x+k}\) is the unconditional probability that the insured survives to the start of policy year \(k+1\) and dies during that year. The factor \(v^{k+1}\) discounts a benefit payable at the end of that year. A fixed benefit \(B\) therefore has APV \(B\,{}_n A_x\). For whole-life insurance, the corresponding unit-benefit APV is \(A_x\), and the level benefit APV is \(B A_x\).
Why the Scaling Formula Holds
The justification is linearity of expectation. Each possible death year produces a payment that is \(B\) times the payment under a unit benefit. Discounting multiplies both payments by the same factor, so the entire present value random variable is multiplied by \(B\). Taking expectations preserves that constant multiplier.
The same reasoning can be seen by partitioning by death year. For an \(n\)-year term contract, the probability of a benefit being paid at the end of year \(k+1\) is \({}_k p_x q_{x+k}\). The expected present value is the sum, over covered years, of the benefit times the discount factor times that probability. Since \(B\) is the same in every covered year, it factors out of the sum. This does not require the mortality probabilities to be equal across years.
Use the term or whole-life death-year probabilities and the specified discount factors to find the APV for a one-dollar benefit.
Multiply the unit APV by the fixed benefit \(B\). The result is the APV in the same currency units as \(B\).
Calculate the discounted benefit in each possible death year and add the expected present values. The total should match the scaled APV.
Scaling a Term Insurance APV
The first example calculates the unit value from conditional mortality probabilities, then scales it to the stated face amount. The death benefit is payable at the end of the policy year of death, and there is no benefit if the insured survives the full term.
Worked Example: A Three-Year Level-Benefit Term Policy
A hypothetical three-year term policy on a life aged \(x\) pays $10,000 at the end of the year of death. The effective annual interest rate is \(i=25\%\). The conditional death probabilities in the three policy years are \(q_x=0.10\), \(q_{x+1}=0.20\), and \(q_{x+2}=0.25\). Find the APV and verify it by direct calculation.
Method. First find the death probability in each policy year and the unit-benefit APV. Then scale by $10,000 and check by summing the expected discounted benefit for each year. The common mistake to avoid is discounting again after using an APV that already includes discounting.
The discount factor is \(v=1/(1+0.25)=0.8\). The probability of death in year 1 is \(0.10\). The probability of surviving year 1 and dying in year 2 is \(0.90(0.20)=0.18\). The probability of surviving the first two years and dying in year 3 is \(0.90(0.80)(0.25)=0.18\).
Thus the unit-benefit APV is
Scaling by the $10,000 benefit gives
Direct check. The expected present values by year are \(10{,}000(0.8)(0.10)=\$800.00\), \(10{,}000(0.8^2)(0.18)=\$1{,}152.00\), and \(10{,}000(0.8^3)(0.18)=\$921.60\). Their sum is \(\$800.00+\$1{,}152.00+\$921.60=\$2{,}873.60\), matching the scaled APV.
Scaling a Whole-Life APV
The scaling rule is identical for whole-life insurance: the unit whole-life APV is multiplied by the fixed death benefit. The sum now includes every possible future death year. A constant mortality model provides a compact example in which both the unit value and direct verification use a geometric series.
Worked Example: Whole-Life Insurance Under Constant Mortality
A hypothetical whole-life policy pays $18,000 at the end of the year of death. The effective annual interest rate is \(i=25\%\), and the conditional probability of death in every year is \(q=0.20\), independently of attained age for this model. Find the APV and verify the result from the death-year payments.
Method. Use the constant mortality model to express the unit APV as an infinite geometric sum, scale by the benefit, and check the same series directly. The common mistake to avoid is treating the whole-life sum as a finite term sum or omitting the discount for the year in which the benefit is paid.
The one-year survival probability is \(p=1-q=0.80\), and \(v=1/(1.25)=0.80\). The probability of death in policy year \(k+1\) is \(p^kq=(0.80)^k(0.20)\). The unit present value in that event is \(v^{k+1}\), so
The APV for an $18,000 benefit is therefore
Direct check. The present value contribution in death year \(k+1\), before multiplying by the benefit, is \(0.16(0.64)^k\). With the $18,000 benefit, the first three expected contributions are $2,880.00, $1,843.20, and $1,179.648. The full infinite sum is
The direct series agrees with the scaled unit value. The displayed third contribution is $1,179.648 before rounding; the full APV is exact to the cent.
Using a Target APV to Find a Benefit Amount
Scaling also works in reverse. If the unit-benefit APV is known and a policy has a specified APV target, divide that target by the unit APV to find the level benefit consistent with it. This is an algebraic valuation step, not a premium calculation: it does not add expenses, profit margins, or other loadings.
Worked Example: Solving for a Benefit From an APV Target
A hypothetical three-year term contract has an APV target of $7,534.08 for its death benefit. The effective annual interest rate is \(i=25\%\), and the conditional death probabilities are \(0.05\), \(0.10\), and \(0.20\) in policy years 1, 2, and 3. The benefit is level and payable at the end of the year of death. Find the benefit amount and verify it directly.
Method. Calculate the unit term APV, divide the target by that unit value, and substitute the resulting benefit into the death-year calculation. The common mistake to avoid is dividing by the probability of death alone; the unit APV also reflects survival to each year and discounting.
Here, \(v=0.80\). The unconditional death probabilities by policy year are \(0.05\), \((0.95)(0.10)=0.095\), and \((0.95)(0.90)(0.20)=0.171\). Therefore
Let \(B\) be the level benefit. The target equation is \(B(0.188352)=\$7{,}534.08\), so
Direct check. With a $40,000 benefit, the expected discounted payments by year are \(40{,}000(0.80)(0.05)=\$1{,}600.00\), \(40{,}000(0.80^2)(0.095)=\$2{,}432.00\), and \(40{,}000(0.80^3)(0.171)=\$3{,}502.08\). Their total is \(\$1{,}600.00+\$2{,}432.00+\$3{,}502.08=\$7{,}534.08\), as required.
Common Scaling Errors and a Practical Check
The benefit amount changes the size of each payment, not its timing or probability. The discount factors and death-year probabilities belong in the unit APV. Once that value has been calculated, multiplying by \(B\) already gives the APV for a benefit of \(B\) dollars.
- Discounting twice. In the three-year example, the scaled APV is $2,873.60. Multiplying by \(v=0.8\) again would give $2,298.88, which incorrectly discounts the entire contract value by an additional year.
- Using conditional death probabilities as unconditional probabilities. For example, year 2 death probability is not \(q_{x+1}\) alone; it is \({}_1 p_x q_{x+1}\), because the insured must first survive year 1.
- Scaling a non-level schedule as if it were constant. The rule \(B A\) applies when every covered death payment has the same amount \(B\). If the benefit differs by policy year, each year’s amount must remain with its own discounted death probability.
- Mixing units. A unit APV is measured per dollar of benefit. Multiplying it by a benefit stated in dollars produces a dollar APV; do not interpret the unit APV itself as a dollar amount.
A useful final check is to compare the result with the direct death-year sum for a short term, or with the corresponding infinite sum when it has a simple form. If the two methods disagree, check the payment time, the survival factor before each death probability, and whether a discount factor has been applied more than once.
Check Your Understanding
For each hypothetical policy, use the stated discrete payment timing and show how the unit-benefit value scales. Amounts in the answers are rounded to the nearest cent.
- A one-year term policy pays $25,000 at the end of the year of death. The effective annual interest rate is 25%, and the one-year death probability is 0.08. Find the APV. (Answer: $1,600.00)
- A two-year term policy pays $10,000 at the end of the year of death. The effective annual interest rate is 25%, with conditional death probabilities 0.10 in year 1 and 0.20 in year 2. Find the APV. (Answer: $1,952.00)
- A three-year term policy pays $20,000 at the end of the year of death. The effective annual interest rate is 25%, with conditional death probabilities 0.05, 0.10, and 0.20 in successive years. Find the APV. (Answer: $3,767.04)
- A whole-life policy pays $12,000 at the end of the year of death. The effective annual interest rate is 25%, and the conditional death probability is 0.25 in every year. Find the APV. (Answer: $6,000.00)
- A deferred two-year term policy covers deaths in policy years 2 and 3, paying $25,000 at the end of the year of death. The effective annual interest rate is 25%, and the conditional death probabilities in years 1, 2, and 3 are 0.10, 0.20, and 0.25. Find the APV, accounting for survival through year 1. (Answer: $5,184.00)