Deriving Whole-Life APV From Curtate Future Lifetime

Why Curtate Future Lifetime Determines the Payment Time

For discrete whole-life insurance, the benefit is paid at the end of the policy year in which death occurs. The key calculation is therefore to pair each possible death year with its probability and discount the benefit from the correct payment time. Curtate future lifetime provides exactly this link.

Let $T_x$ be the complete future lifetime of a life aged $x$, measured in years, and let $K_x=\lfloor T_x\rfloor$ be the curtate future lifetime. Under the usual continuous-lifetime model, the probability of death at an exact integer duration is zero. Thus $K_x=k$ means the life survives for $k$ complete years and dies during the following year. The benefit is paid at time $k+1$, not at time $k$.

Definition: For a unit whole-life insurance benefit payable at the end of the year of death, the present-value random variable is $Z=v^{K_x+1}$, where $v=1/(1+i)$ and $i$ is the effective annual interest rate. Its expected present value is denoted by $A_x$. For a benefit amount $B$, the present-value random variable is $Bv^{K_x+1}$.

The model here assumes a level benefit payable once, at the end of the year of death, and a constant effective annual interest rate $i\geq 0$. It does not describe a continuous benefit payment at the instant of death. The exponent $K_x+1$ reflects the payment rule; it is not an extra assumption about when death occurs within the year.

Deriving the APV from the Probability Mass Function

Write $p_k=P(K_x=k)$ for $k=0,1,2,\ldots$. If $K_x=k$, the present value of a unit benefit is $v^{k+1}$. Taking the expectation over the possible values of $K_x$ gives the whole-life APV.

A_x=E[Z] =\sum_{k=0}^{\infty}v^{k+1}P(K_x=k) =\sum_{k=0}^{\infty}v^{k+1}p_k.

This is a probability-weighted sum: each discounted payment amount is multiplied by the probability that the payment occurs at that time. If the benefit is $B$, the same argument gives $E[BZ]=B A_x$. If the lifetime distribution has finite support, the sum stops at its last possible value of $K_x$. For an infinite support, it is an infinite series. With $i\geq0$, $0<v\leq1$, and total probability at most one, the series is bounded by $\sum_k P(K_x=k)$, so it converges.

A useful way to obtain the probability mass function is from survival probabilities at integer durations. Here ${}_kp_x$ is the probability that a life aged $x$ survives at least $k$ years. The event $K_x=k$ consists of surviving $k$ years but not surviving $k+1$ years. Consequently,

P(K_x=k)={}_kp_x-{}_{k+1}p_x.

The identity follows by subtracting the nested events: survival for $k+1$ years is contained in survival for $k$ years. Their difference is precisely survival through year $k$ followed by death before completing year $k+1$. This expression is a way to construct the curtate-lifetime probabilities; the APV formula itself only needs the probabilities $P(K_x=k)$.

Index the death year.
For each possible $k$, interpret $K_x=k$ as death during the $(k+1)$th policy year.

Find its probability.
Use the given probability mass function, or subtract consecutive integer-year survival probabilities.

Discount to the payment date.
Multiply the probability by $v^{k+1}$, because payment is at the end of that death year.

Sum and check.
Add the weighted terms and confirm that the probabilities account for the possible payment outcomes.

Worked Applications

Worked Example: Calculating the APV Directly from a Curtate-Lifetime Distribution

A hypothetical unit-benefit whole-life policy pays at the end of the year of death. Suppose $P(K_x=0)=0.20$, $P(K_x=1)=0.30$, and $P(K_x=2)=0.50$, with no other possible values. The effective annual interest rate is $i=5\%$. Find $A_x$, to four decimal places.

Method. Apply the probability-weighted present-value formula to the given mass function. Since $v=1/(1.05)=20/21$, death in the first, second, and third policy years corresponds respectively to payment times 1, 2, and 3.

\begin{aligned} A_x &=0.20v+0.30v^2+0.50v^3\\ &=0.20\left(\frac{20}{21}\right) +0.30\left(\frac{400}{441}\right) +0.50\left(\frac{8000}{9261}\right)\\ &=0.1904762+0.2721088+0.4319188\\ &=0.8945038. \end{aligned}

The terms are the contributions from the three possible payment times. The probabilities sum to $0.20+0.30+0.50=1.00$, so every possible death year is included. Therefore $A_x=0.8945$ to four decimal places.

Common mistake avoided: using $v^k$ instead of $v^{k+1}$. When $K_x=0$, death is during the first policy year and payment is at time 1, so its discount factor is $v$, not 1.

Worked Example: Converting Survival Probabilities into APV Terms

For a hypothetical insured life, suppose the integer-year survival probabilities are ${}_0p_x=1.00$, ${}_1p_x=0.80$, ${}_2p_x=0.55$, ${}_3p_x=0.30$, ${}_4p_x=0.10$, and ${}_5p_x=0$. A unit whole-life benefit is payable at the end of the year of death, and $i=5\%$. Find the APV to four decimal places.

Method. First take consecutive differences to find the curtate-lifetime probabilities:

\begin{aligned} P(K_x=0)&=1.00-0.80=0.20,\\ P(K_x=1)&=0.80-0.55=0.25,\\ P(K_x=2)&=0.55-0.30=0.25,\\ P(K_x=3)&=0.30-0.10=0.20,\\ P(K_x=4)&=0.10-0=0.10. \end{aligned}

These probabilities sum to $0.20+0.25+0.25+0.20+0.10=1.00$. With $v=1/1.05=20/21$, discount each probability according to its payment time:

\begin{aligned} A_x &=0.20v+0.25v^2+0.25v^3+0.20v^4+0.10v^5\\ &=0.1904762+0.2267574+0.2159594+0.1645405+0.0783526\\ &=0.8760861. \end{aligned}

Thus the APV is $0.8761$ to four decimal places. As a check on the timing, the $0.10$ probability at $K_x=4$ corresponds to death in the fifth year, so its contribution uses $v^5$.

Common mistake avoided: treating the listed survival probabilities themselves as death-year probabilities. For example, $0.80$ is the probability of surviving one year, not the probability of dying during the first year. That first-year death probability is $1.00-0.80=0.20$.

Worked Example: Applying a Face Amount and Checking the Payment Shift

A hypothetical policy pays \$25,000 at the end of the year of death. Suppose $P(K_x=0)=0.15$, $P(K_x=1)=0.35$, and $P(K_x=2)=0.50$, and the effective annual interest rate is $i=10\%$. Find the APV in dollars to the nearest cent.

Method. First calculate the unit-benefit APV, then multiply by the face amount. Here $v=1/1.10=10/11$, and the payment times are 1, 2, and 3 years.

\begin{aligned} A_x &=0.15v+0.35v^2+0.50v^3\\ &=0.15\left(\frac{10}{11}\right) +0.35\left(\frac{100}{121}\right) +0.50\left(\frac{1000}{1331}\right)\\ &=0.1363636+0.2892562+0.3756574\\ &=0.8012772. \end{aligned}

The policy APV is therefore

\$25{,}000(0.8012772)=\$20{,}031.93.

The probabilities sum to 1, and scaling the unit-benefit APV by \$25,000 gives an amount in dollars, as required. The answer is \$20,031.93 to the nearest cent.

Common mistake avoided: using $0.15+0.35v+0.50v^2$ discounts each benefit one year too little. That incorrect expression gives approximately $0.8814$ per unit of benefit. The correct expression starts with $v$, because $K_x=0$ means payment at time 1.

Interpreting and Checking the Result

The APV is not the probability of a claim and is not the expected number of years until death. It is the expected value of the discounted benefit. Each term in the sum combines two distinct pieces of information: how likely that death year is and how much a payment at its year-end is worth at the valuation date.

Several checks help catch errors before a result is reported:

Check the payment date. The event $K_x=k$ leads to a payment at time $k+1$, hence the factor $v^{k+1}$.
Check the probability mass. If death eventually occurs with probability 1, the probabilities $P(K_x=k)$ across all possible $k$ must sum to 1. In a finite table, include the final possible death interval.
Check the units. The sum gives an APV per unit of benefit. Multiply by the face amount only after calculating the unit-benefit APV.
Check the interest convention. The annual discount factor is $v=1/(1+i)$ for effective annual rate $i$; do not substitute a different rate convention without converting it.

A common source of confusion is the difference between the curtate lifetime and the payment time. $K_x$ counts complete years lived, while the insurer pays one year-end after those complete years. Keeping those quantities separate makes the exponent in the APV formula immediate.

Key result: For a unit whole-life benefit paid at the end of the year of death, $A_x=\sum_{k=0}^{\infty}v^{k+1}P(K_x=k)$. The exponent is one more than the curtate future lifetime because the payment occurs at the end of the death year.

Check Your Understanding

For each problem, use the end-of-year payment rule and keep the payment time distinct from curtate future lifetime.

A unit benefit is payable at the end of the year of death. If $P(K_x=0)=0.40$, $P(K_x=1)=0.60$, and $i=5\%$, find the APV to four decimal places. (Answer: $0.9252$)
A hypothetical unit-benefit policy has $P(K_x=0)=0.15$, $P(K_x=1)=0.35$, and $P(K_x=2)=0.50$. At $i=10\%$, state the correct APV expression and its value to four decimal places. (Answer: $0.15v+0.35v^2+0.50v^3=0.8013$)
Suppose ${}_0p_x=1.00$, ${}_1p_x=0.70$, ${}_2p_x=0.40$, and ${}_3p_x=0$. A unit benefit is paid at the end of the year of death, with $i=5\%$. Find the APV to four decimal places. (Answer: $0.9034$)
A hypothetical policy pays \$50,000 at the end of the year of death. Its curtate-lifetime probabilities are $P(K_x=0)=0.25$, $P(K_x=1)=0.25$, and $P(K_x=2)=0.50$, and $i=4\%$. Find the APV to the nearest cent. (Answer: \$45,801.09)

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