Why Condition on Survival?
A policy value at duration \(t\) is assessed given that the insured is alive at that duration. The future probabilities used in the valuation must therefore describe what can happen conditional on survival to time \(t\). Probabilities calculated from the original issue date do not automatically have that interpretation: they include the chance that the insured did not survive to the valuation date.
The adjustment is a conditional-probability calculation. Once survival to the valuation date is known, divide a later survival probability by the probability of reaching the valuation date. Death benefits are treated similarly, using the probability of death in a future interval divided by the probability of survival to the valuation date. Benefits triggered by survival to a future term date use the conditional probability of surviving from the valuation date to that date.
Conditional Probabilities for Future Policy Years
For annual benefits payable at the end of the policy year of death, it is useful to identify the probability of death in each future year, conditional on survival to the valuation date. Write \({}_{k-1|}q_{x+t}\) for the probability of death between durations \(t+k-1\) and \(t+k\), given survival to duration \(t\). This is not necessarily the same as the one-year death probability \(q_{x+t+k-1}\), which is conditional on survival to the start of that particular year.
The first expression combines two events: the insured must survive the first \(k-1\) years after the valuation date, then die during the next year. The ratio expression says the same thing directly: the probability of being alive at the start of future year \(k\) and dying in it, measured from issue, is divided by the probability of reaching the valuation date. In particular, a survival benefit payable at the end of \(n\) future years is payable with conditional probability \({}_np_{x+t}\), not \({}_{t+n}p_x\).
These future outcomes are mutually exclusive: death in year 1, death in year 2, and so on, or survival to the end of a stated term. If death benefits are paid at the end of the year of death and a survival benefit is paid at the end of year \(n\), their expected present values are calculated from the valuation date. The discount exponent is the number of years after the valuation date, not the duration since issue.
Worked Example: Death and Term-End Benefits After Survival
Worked Example: Death and Term-End Benefits After Survival
A hypothetical policyholder has survived to the valuation date. Over the next two years, the policy pays \( \$30{,}000\) at the end of the year of death. If the policyholder survives both years, it pays an additional \( \$10{,}000\) at the end of year 2. The effective annual interest rate is \(i=4\%\). Conditional on survival to the valuation date, the probability of death in year 1 is \(0.10\), death in year 2 is \(0.15\), and survival to the end of year 2 is \(0.75\). Find the conditional expected present value.
Method and event check. The three mutually exclusive outcomes have probabilities \(0.10+0.15+0.75=1.00\). The year-2 death probability \(0.15\) is already conditional on being alive at the valuation date; it is not the one-year death probability conditional on surviving year 1.
The discount factor is \(v=1/1.04\), so \(v^2=1/1.0816\). The year-1 death benefit has expected amount \( \$30{,}000(0.10)=\$3{,}000\), and its present value is \( \$3{,}000/1.04=\$2{,}884.615385\).
The year-2 death benefit has expected amount \( \$30{,}000(0.15)=\$4{,}500\). Its present value is \( \$4{,}500/1.0816=\$4{,}160.502959\).
The survival benefit has expected amount \( \$10{,}000(0.75)=\$7{,}500\). Its present value is \( \$7{,}500/1.0816=\$6{,}934.171598\).
Adding all three contributions gives \( \$2{,}884.615385+\$4{,}160.502959+\$6{,}934.171598=\$13{,}979.289942\). The conditional expected present value is \(\$13{,}979.29\), to the nearest cent.
Check. The two year-2 benefits are attached to different outcomes, so their expected amounts add: \( \$4{,}500+\$7{,}500=\$12{,}000\). Their combined present value is \( \$12{,}000/1.0816=\$11{,}094.674556\). Adding the year-1 contribution gives the same total, \( \$13{,}979.289941\) before rounding. The tiny difference in the last displayed decimal comes from rounding intermediate values.
Common mistake avoided: Omitting the year-2 death benefit, or discounting its expected amount using the survival-benefit probability, gives an incorrect total. Each payment must be paired with its own event probability.
Deriving the Conditioning Formula
The conditional survival formula follows directly from the definition of conditional probability. Let \(A\) be the event that the life survives from issue to duration \(t+s\), and let \(B\) be the event that the life survives to duration \(t\). Since survival to \(t+s\) implies survival to \(t\), \(A\) is a subset of \(B\). Thus \(P(A\mid B)=P(A)/P(B)\), which produces the ratio below. The same reasoning gives the conditional probability of death in a future policy year.
The insured is known to be alive at duration \(t\), an event with probability \({}_tp_x\) from issue.
Survival to \(t+s\) has probability \({}_{t+s}p_x\) from issue, so its conditional probability given survival to \(t\) is \({}_{t+s}p_x/{}_tp_x={}_sp_{x+t}\).
From issue, survival to \(t+k-1\) followed by death before \(t+k\) has probability \({}_{t+k-1}p_x-{}_{t+k}p_x\). Divide by \({}_tp_x\) to condition on survival to \(t\).
Multiply each payment by its conditional event probability and discount by \(v^k\), where \(k\) is the number of years from the valuation date to payment.
Worked Example: Converting Life-Table Probabilities
Worked Example: Converting Life-Table Probabilities
A hypothetical insured was age \(x\) at issue and has survived to duration 3. The supplied survival probabilities from issue are \({}_3p_x=0.80\), \({}_4p_x=0.72\), and \({}_5p_x=0.60\). For the next two years, the policy pays \( \$9{,}000\) at the end of year 1 if death occurs in that year, or \( \$15{,}000\) at the end of year 2 if death occurs in that year. If the insured survives to the end of year 2, it pays \( \$4{,}000\). The effective annual interest rate is \(5\%\). Find the conditional expected present value at duration 3.
Step 1: Condition the survival probabilities. From duration 3 to duration 4, the conditional survival probability is \({}_1p_{x+3}={}_4p_x/{}_3p_x=0.72/0.80=0.90\). Therefore, the conditional probability of death in future year 1 is \(1-0.90=0.10\).
From duration 3 to duration 5, the conditional survival probability is \({}_2p_{x+3}={}_5p_x/{}_3p_x=0.60/0.80=0.75\). The conditional probability of death in future year 2 is \(0.90-0.75=0.15\). Equivalently, \(({}_4p_x-{}_5p_x)/{}_3p_x=(0.72-0.60)/0.80=0.15\). The probabilities \(0.10\), \(0.15\), and \(0.75\) sum to 1.
Step 2: Discount each benefit from duration 3. The year-1 death benefit contributes \( \$9{,}000(0.10)/1.05=\$900/1.05=\$857.142857\). The year-2 death benefit contributes \( \$15{,}000(0.15)/1.05^2=\$2{,}250/1.1025=\$2{,}040.816327\). The year-2 survival benefit contributes \( \$4{,}000(0.75)/1.1025=\$3{,}000/1.1025=\$2{,}721.088435\).
The total is \( \$857.142857+\$2{,}040.816327+\$2{,}721.088435=\$5{,}619.047619\). The conditional expected present value is \(\$5{,}619.05\), to the nearest cent.
Check and common mistake avoided. The year-2 outcomes have combined conditional probability \(0.15+0.75=0.90\), equal to the probability of reaching the start of future year 2. Using \(0.12\) as the year-2 death probability would incorrectly use the unconditional-from-issue difference \(0.72-0.60\) without dividing by the \(0.80\) probability of surviving to duration 3.
Benefits That Can Still Be Payable
Conditioning also clarifies which benefit events remain possible at the valuation date. A death benefit triggered by a death before duration \(t\) is not a future payment under the condition that the insured is alive at \(t\). A survival benefit payable at duration \(t\) has already reached its trigger date; it is not a benefit payable at a later future duration. By contrast, a death benefit for death after \(t\), and a survival benefit at a future term date, can remain payable. Their probabilities must be computed from the valuation date.
This is an event-and-timing issue, not merely an algebraic adjustment. First identify when a benefit is triggered and when it is paid; then identify the probability of that event given survival to the valuation date. The calculation must not include an event that is already ruled out by the conditioning information.
| Benefit trigger | Conditional probability from duration \(t\) | Typical payment time |
|---|---|---|
| Death during future year \(k\) | \({}_{k-1|}q_{x+t}\) | End of future year \(k\), under annual end-of-year timing |
| Survival to future duration \(t+n\) | \({}_np_{x+t}\) | End of future year \(n\) |
| Death before the valuation duration \(t\) | Zero given survival to \(t\) | Not a future payment under this condition |
Worked Example: Checking the Future Outcome Probabilities
Worked Example: Checking the Future Outcome Probabilities
At a valuation date, a hypothetical insured has survived to duration 2. From the original age, the supplied probabilities are \({}_2p_x=0.80\), \({}_3p_x=0.72\), \({}_4p_x=0.64\), and \({}_5p_x=0.56\). For each of the next three years, a benefit of \( \$10{,}000\) is payable at year-end if death occurs in that year. If the insured survives all three years, \( \$20{,}000\) is payable at the end of year 3. Assume \(i=0\%\). Find the conditional expected value.
The conditional probability of death in future year 1 is \((0.80-0.72)/0.80=0.10\). For future year 2 it is \((0.72-0.64)/0.80=0.10\). For future year 3 it is \((0.64-0.56)/0.80=0.10\). The probability of surviving all three future years is \(0.56/0.80=0.70\).
These outcomes sum to \(0.10+0.10+0.10+0.70=1.00\), so they account for all possibilities. With \(i=0\%\), \(v=1\), and discounting does not change any payment. The three death benefits contribute \( \$10{,}000(0.10+0.10+0.10)=\$3{,}000\). The survival benefit contributes \( \$20{,}000(0.70)=\$14{,}000\). The conditional expected value is therefore \(\$17{,}000.00\).
Common mistake avoided: The probability of surviving from issue to duration 5 is \(0.56\), but the relevant probability after survival to duration 2 is \(0.56/0.80=0.70\). Using \(0.56\) directly would fail to condition on the known survival.
Common Exam Traps
- Forgetting to divide by survival to the valuation date. If probabilities are supplied from issue, form the conditional ratio before valuing future payments.
- Confusing interval death probability with a one-year mortality rate. \({}_{k-1|}q_{x+t}\) is conditional on survival to the valuation date. \(q_{x+t+k-1}\) is conditional on survival to the start of future year \(k\). For \(k>1\), these are generally different.
- Discounting from issue instead of the valuation date. A payment at the end of the second future year is discounted by \(v^2\), even if the valuation date is several years after issue.
- Pairing a benefit with the wrong event. A death payment uses the probability of death in its specified interval; a term-end survival payment uses the probability of surviving to the term date.
- Counting an already excluded event. Given survival to duration \(t\), death before \(t\) is not a possible future claim in this conditional valuation.
Check Your Understanding
For each problem, begin by identifying the condition already known at the valuation date. Assume annual end-of-year payment timing unless the problem states otherwise.
- A life has survived to duration 3. Given \({}_3p_x=0.84\) and \({}_5p_x=0.63\), find the conditional probability of surviving two more years and the conditional probability of dying during those two years. (Answer: survival \(0.7500\); death \(0.2500\))
- At a valuation date, the conditional probability of death in future year 1 is \(0.05\), the probability of death in future year 2 is \(0.10\), and the probability of surviving to the end of year 2 is \(0.85\). A policy pays \( \$1{,}000\) at the end of year 1 if death occurs in year 1, \( \$2{,}000\) at the end of year 2 if death occurs in year 2, and \( \$500\) at the end of year 2 if the insured survives. If \(i=10\%\), find the conditional expected present value. (Answer: \(\$561.98\))
- A life has survived to duration 2. Given \({}_2p_x=0.80\) and \({}_4p_x=0.60\), a hypothetical policy pays \( \$6{,}000\) only if the life survives to duration 4. If \(i=5\%\), find the conditional expected present value at duration 2. (Answer: \(\$4{,}081.63\))
- At the valuation date, the conditional probabilities are \(0.08\) for death in year 1, \(0.12\) for death in year 2, and \(0.80\) for survival to year 2. A hypothetical contract pays \( \$25{,}000\) at the end of the year of death, or \( \$10{,}000\) at the end of year 2 if the insured survives. At \(i=4\%\), calculate the conditional expected present value. (Answer: \(\$12{,}093.20\))
- A life has survived to duration 3. The original-age survival probabilities are \({}_3p_x=0.75\), \({}_4p_x=0.675\), and \({}_5p_x=0.54\). A hypothetical policy pays \( \$12{,}000\) for death in future year 1, \( \$20{,}000\) for death in future year 2, or \( \$5{,}000\) for survival to the end of future year 2. At \(i=5\%\), find the conditional expected present value. (Answer: \(\$7{,}673.47\))