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Whole-Life Values From Curtate-Lifetime Probabilities

Convert each curtate-lifetime probability into its discounted whole-life benefit contribution, then sum the contributions to obtain the actuarial present value.

Beginner 10 min read

What You'll Learn

  • Interpret probabilities for each value of curtate future lifetime.
  • Match each death year to its end-of-year benefit payment time.
  • Calculate a level-benefit whole-life actuarial present value from a probability table.
  • Check that supplied probabilities form a complete distribution.
  • Recognize and correct the off-by-one error in the discount exponent.

From Death-Year Probabilities to a Whole-Life Value

A discrete whole-life insurance pays a benefit at the end of the policy year in which the insured dies. If a probability distribution for curtate future lifetime is supplied directly, the actuarial present value can be found by discounting the benefit for each possible death year and weighting it by that year’s probability. The central task is to align the lifetime outcome with the correct payment time.

This approach is useful when the death-year probabilities are given in a table rather than through age-specific mortality rates or life-table counts. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let \(T_x\) be the future lifetime, in years, of a life aged \(x\), and let \(K_x=\lfloor T_x\rfloor\) be the curtate future lifetime. Thus, \(K_x=k\) means death occurs after completing \(k\) full years but before completing \(k+1\) full years. For a discrete whole-life insurance with benefit amount \(B\), payable at the end of the year of death, the payment occurs at time \(K_x+1\).

The event \(K_x=k\) is the \(k\)th policy-year death outcome when policy years are numbered starting with year 1: \(K_x=0\) corresponds to death in policy year 1, \(K_x=1\) to death in policy year 2, and so on. This indexing is important because the benefit is paid at the end of that policy year.

The Direct Valuation Formula

Assume a level benefit \(B\), an effective annual interest rate \(i\geq 0\), and payment at the end of the year of death. Write \(v=1/(1+i)\). If \(K_x=k\), the present value at issue is \(Bv^{k+1}\). Averaging over the possible values of \(K_x\) gives the actuarial present value.

$$ \operatorname{APV} =E\!\left[Bv^{K_x+1}\right] =B\sum_{k=0}^{\infty}v^{k+1}P(K_x=k). $$

For a unit benefit, the sum is the discrete whole-life insurance value \(A_x\), so that \(\operatorname{APV}=BA_x\). If the supplied distribution has only finitely many listed outcomes, use those terms and treat unlisted outcomes as having probability zero only when the problem states or implies that the table is complete.

Assumptions: The benefit is paid once, at the end of the policy year of death, and remains level at \(B\). The interest rate is an effective annual rate applying over each complete year. The probabilities describe the distribution of \(K_x\) for a life covered at age \(x\). If the insured can survive forever with positive probability, that outcome produces no death benefit; it contributes zero present value.

Why the Discount Exponent Is \(k+1\)

The formula follows by writing the present value as a random variable. On the event \(K_x=k\), the insured dies during the year from time \(k\) to time \(k+1\). Since this contract pays at the end of that year, the payment date is \(k+1\), not \(k\). Therefore the present value on that event is \(Bv^{k+1}\). The events \(K_x=0,1,2,\ldots\) are mutually exclusive, so the expected present value is the sum of each event’s present value multiplied by its probability:

$$ E[\text{present value}] =\sum_{k=0}^{\infty}P(K_x=k)\,Bv^{k+1} =B\sum_{k=0}^{\infty}v^{k+1}P(K_x=k). $$

This derivation also explains the payment-timing convention. A benefit payable immediately at death would generally require a different present-value model; it is not valued by simply using this end-of-year formula unchanged.

A Reliable Calculation Process

A probability table is easiest to use when each row records both the death year and the payment’s discount factor. Before calculating, confirm that the probabilities are nonnegative and sum to 1, subject to any stated rounding. Do not silently adjust probabilities if the total is not 1; first determine whether a probability is missing or whether the table is incomplete.

1
Identify the payment timing.
For a discrete whole-life benefit paid at the end of the year of death, outcome \(K_x=k\) means payment at time \(k+1\).
2
Find the discount factor.
Use \(v=1/(1+i)\), then calculate \(v^{k+1}\) for each listed outcome.
3
Weight each discounted benefit.
Multiply \(Bv^{k+1}\) by \(P(K_x=k)\).
4
Sum and check.
Add the contributions, then check that the result is nonnegative and does not exceed \(B\) when \(i\geq0\).

Worked Valuations

Worked Example: Three Possible Death Years

A hypothetical discrete whole-life policy pays \(B=\$10{,}000\) at the end of the year of death. The effective annual interest rate is \(i=10\%\). For a life aged \(x\), suppose the complete distribution is \(P(K_x=0)=0.20\), \(P(K_x=1)=0.30\), and \(P(K_x=2)=0.50\). Find the actuarial present value. State the method and identify the indexing error to avoid.

Method. Use the direct expectation over curtate-lifetime outcomes. The probabilities sum to \(0.20+0.30+0.50=1.00\), so the distribution is complete. The discount factor is \(v=1/1.10=0.9090909\).

For \(K_x=0\), death is in policy year 1 and the payment is at time 1. Its contribution is \(0.20(\$10{,}000)v^1=\$2{,}000(0.9090909)=\$1{,}818.1818\).

For \(K_x=1\), payment is at time 2. Since \(v^2=0.8264463\), its contribution is \(0.30(\$10{,}000)v^2=\$3{,}000(0.8264463)=\$2{,}479.3388\).

For \(K_x=2\), payment is at time 3. Since \(v^3=0.7513148\), its contribution is \(0.50(\$10{,}000)v^3=\$5{,}000(0.7513148)=\$3{,}756.5740\).

Adding the contributions gives \(\$1{,}818.1818+\$2{,}479.3388+\$3{,}756.5740=\$8{,}054.0946\). The actuarial present value is \$8,054.09, rounded to the nearest cent. The result is below the face amount, as expected with positive interest and future payments.

Common mistake avoided. Discounting by \(v^k\) would value the \(K_x=0\) benefit as if it were paid immediately. That incorrect calculation is \(\$10{,}000[0.20+0.30v+0.50v^2]=\$8{,}859.50\), rather than \(\$8{,}054.09\). The correct exponent is \(k+1\) because every benefit is paid at a policy-year end.

Worked Example: A Four-Year Probability Table

A hypothetical insurer considers a discrete whole-life benefit of \(\$25{,}000\), payable at the end of the year of death. The effective annual interest rate is \(4\%\), and the complete distribution is \(P(K_x=0)=0.10\), \(P(K_x=1)=0.20\), \(P(K_x=2)=0.30\), and \(P(K_x=3)=0.40\). Calculate the actuarial present value.

Method. First check completeness: \(0.10+0.20+0.30+0.40=1.00\). The annual discount factor is \(v=1/1.04=0.9615385\). The four payment times are 1, 2, 3, and 4 years.

The needed powers are \(v=0.9615385\), \(v^2=0.9245562\), \(v^3=0.8889964\), and \(v^4=0.8548042\). The probability-weighted discount factors are:

\(0.10v=0.09615385\); \(0.20v^2=0.18491124\); \(0.30v^3=0.26669891\); and \(0.40v^4=0.34192168\).

Their sum is \(0.09615385+0.18491124+0.26669891+0.34192168=0.88968568\). Multiplying by the benefit gives \(\$25{,}000(0.88968568)=\$22{,}242.142\), so the actuarial present value is \$22,242.14, rounded to the nearest cent.

Common mistake avoided. The probability \(P(K_x=3)=0.40\) is the probability of death during policy year 4, not year 3. Its contribution therefore uses \(v^4\), not \(v^3\).

Worked Example: Completing a Distribution Before Valuation

A hypothetical discrete whole-life contract pays \(\$20{,}000\) at the end of the year of death. The effective annual interest rate is \(5\%\). The listed probabilities are \(P(K_x=0)=0.12\), \(P(K_x=1)=0.18\), \(P(K_x=2)=?\), \(P(K_x=3)=0.25\), and \(P(K_x=4)=0.15\). Assume these are the only possible outcomes. Find the missing probability and the actuarial present value.

Method. Use total probability to complete the distribution, then apply the direct whole-life formula. The known probabilities sum to \(0.12+0.18+0.25+0.15=0.70\). Thus \(P(K_x=2)=1-0.70=0.30\). The completed probabilities are nonnegative and sum to 1.

Here \(v=1/1.05=0.9523810\). The relevant powers are \(v^1=0.9523810\), \(v^2=0.9070295\), \(v^3=0.8638376\), \(v^4=0.8227025\), and \(v^5=0.7835262\). The weighted factors, in order, are \(0.12v=0.11428571\), \(0.18v^2=0.16326531\), \(0.30v^3=0.25915128\), \(0.25v^4=0.20567562\), and \(0.15v^5=0.11752893\).

Their sum is \(0.11428571+0.16326531+0.25915128+0.20567562+0.11752893=0.85990685\), with the small final-digit difference due to displayed rounding. Multiplying the unrounded sum by \(\$20{,}000\) gives approximately \(\$17{,}198.14\). The actuarial present value is \$17,198.14, rounded to the nearest cent.

Common mistake avoided. The missing probability must be found before valuation. Treating it as zero would leave only \(0.70\) total probability and would omit a valid death-year outcome.

Checks and Interpretation

For a nonnegative effective annual interest rate, \(0<v\leq1\). If the insured eventually dies with probability 1 and the benefit is level, each possible present value is between 0 and \(B\); its expected value must therefore also lie between 0 and \(B\). This is a useful arithmetic check, though it does not establish that each row was discounted at the correct time.

At \(i=0\), \(v=1\), so the APV equals \(B\) if the probabilities sum to 1 and death eventually occurs. At a positive interest rate, later payment years receive smaller discount factors. Consequently, the probability distribution matters not only through its total probability, but also through how much probability it assigns to earlier versus later death years.

Keep the three quantities distinct: \(K_x\) counts completed years lived after issue, the policy-year number of death is \(K_x+1\), and the payment time in years is also \(K_x+1\) under this benefit convention. The most frequent error is using \(K_x\) as the discount exponent and thereby shifting every payment one year too early.

Check Your Understanding

For each problem, assume a discrete whole-life benefit paid at the end of the year of death, with the stated effective annual interest rate and a complete curtate-lifetime distribution.

  1. A policy pays \(\$10{,}000\) and \(i=5\%\). The probabilities are \(P(K_x=0)=0.25\) and \(P(K_x=1)=0.75\). Find the APV. (Answer: \(\$9,183.67\))
  2. A policy pays \(\$30{,}000\) and \(i=0\%\). The probabilities for \(K_x=0,1,2,3\) are \(0.10,0.20,0.30,0.40\), respectively. Find the APV. (Answer: \(\$30,000.00\))
  3. A policy pays \(\$5{,}000\) and \(i=8\%\). The probabilities for \(K_x=0,1,2\) are \(0.40,0.35,0.25\), respectively. Find the APV. (Answer: \(\$4,344.49\))
  4. A policy pays \(\$12{,}000\) and \(i=4\%\). The probabilities for \(K_x=0,1,2,3\) are \(0.20,0.10,0.30,0.40\), respectively. Find the APV. (Answer: \(\$10,720.61\))
  5. A policy pays \(\$40{,}000\) and \(i=6\%\). The probabilities for \(K_x=0,1,2,3,4\) are \(0.05,0.15,0.25,0.25,0.30\), respectively. Find the APV. (Answer: \(\$32,511.00\))

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