Why Benefit Timing Matters
A survival benefit is payable only if the insured is alive on a specified date. A maturity benefit is a survival benefit tied to the end of the contract term. In a profit vector, the amount and the date both matter: the payment date determines which policy year receives the cash flow, while the survival condition determines the probability that the cash flow occurs.
The key distinction is between a benefit paid at the end of a policy year and one paid earlier in that year. A benefit due at the end of year \(k\) is conditional on survival to that year-end. If the payment is instead due at an earlier time, it is conditional on survival only to that earlier time. The amount is an insurer outgo, so it enters a signed profit vector as a negative item.
Putting a Benefit into the Correct Year
Let a policy be issued at age \(x\), and suppose the policy is in force at the start of year \(k\). If a benefit of amount \(B_k\) is payable at the end of that year provided the insured survives, its expected outgo per policy in force at the start of the year is \(-B_k\,{}_1p_{x+k-1}\). This is a year-\(k\) cash-flow amount; it is not yet discounted to issue.
For a benefit paid at time \(k-1+s\), where \(0<s<1\), the corresponding probability is \({}_sp_{x+k-1}\), not the full-year survival probability. The expected outgo per policy in force at the start of year \(k\) is \(-B\,{}_sp_{x+k-1}\), with payment date \(k-1+s\). If the benefit is conditional on survival from issue, its expected outgo per policy issued is weighted by survival to the start of year \(k\) as well.
The second expression applies when the benefit is paid at the end of year \(k\) only if the policyholder survives from issue to that date, and when the payment amount is \(B_k\). The two expressions use different exposure bases: one is conditional on being in force at the start of the year, while the other is per policy originally issued. Do not multiply by survival to issue twice.
Worked Example: Annual Survival Benefits and a Separate Maturity Benefit
A hypothetical three-year policy pays a survival benefit of $1,000 at the end of each year the insured is alive. In addition, it pays a maturity benefit of $4,000 at the end of year 3 if the insured is alive then. Assume independent annual mortality with \(q=0.10\) in each policy year, so \(p=0.90\), and an effective annual interest rate \(i=5\%\). Find the expected benefit outgo entries per policy in force at issue, and the present value at issue of the total expected outgo.
Method. First calculate survival probabilities to each payment date. Then determine the expected outgo at each date, keeping the annual survival benefit and the additional maturity benefit separate.
The probabilities of survival from issue to the ends of years 1, 2, and 3 are \(0.90\), \(0.90^2=0.81\), and \(0.90^3=0.729\). The expected year-end survival-benefit outgoes per policy issued are therefore \(-\$900\), \(-\$810\), and \(-\$729\). The maturity benefit is also payable at the end of year 3, so its expected outgo is \(-\$4{,}000(0.729)=-\$2{,}916\). The total expected outgo at the end of year 3 is \(-\$729-\$2{,}916=-\$3{,}645\).
The year-by-year expected cash-flow entries are thus \(-\$900\) at time 1, \(-\$810\) at time 2, and \(-\$3{,}645\) at time 3. They are not discounted amounts. Since \(v=1/1.05\), the present value at issue is
Rounded to cents, the expected present value of benefit outgo is \(-\$4{,}743.54\). The calculation checks by treating the year-3 survival and maturity payments as one total outgo of $5,000 conditional on survival to year 3: its expected amount is \(0.729(5{,}000)=\$3{,}645\), as used above. Common mistake avoided: the maturity benefit is additional to the stated annual survival benefit here because the contract explicitly promises both; it should not replace the year-3 survival benefit.
Deriving the Expected Outgo and Present Value
The survival weighting follows directly from the definition of expectation. Consider a benefit \(B\) paid at time \(t\) only if the insured is alive then. Let \(I_t\) equal 1 if the insured survives to time \(t\), and 0 otherwise. The signed payment is \(-BI_t\). Since \(E[I_t]=P(\text{alive at }t)={}_tp_x\), its expected amount per policy issued is \(-B\,{}_tp_x\).
If the payment is made at a fixed time \(t\) and the effective annual interest rate is \(i\), its value at issue is the expected payment discounted from that date. For fractional times, use the effective annual accumulation factor \((1+i)^t\). This valuation assumes a fixed effective annual rate and that the payment date is known; if payment timing is uncertain, the valuation must account for the possible payment dates separately.
This expression values an expected cash flow at issue. It is not the same thing as the realized payment on an individual policy: on a particular policy the benefit is either paid or not paid. Nor is a present value automatically the profit-vector entry for that year. Keep the dated expected cash-flow amount in its policy-year entry, and discount it only when calculating a present value or another quantity that requires discounting.
Worked Example: Auditing a Three-Date Benefit Present Value
A hypothetical policy has annual mortality probability \(q=0.10\) in each of three years, independently, and an effective annual interest rate \(i=5\%\). It pays survival benefits of $400 at the end of year 1 and \(430/0.81=\$530.8642\) at the end of year 2, plus a maturity benefit of \(2{,}700/0.729=\$3{,}703.7037\) at the end of year 3, each only if the insured survives to the payment date. Calculate the expected present value of benefit outgo per policy issued.
Method. Apply the appropriate cumulative survival probability to each benefit, then discount each expected payment from its actual year-end date. Here \({}_1p_x=0.9\), \({}_2p_x=0.81\), and \({}_3p_x=0.729\).
The expected outgo amounts are \(-\$400(0.9)=-\$360\), \(-\$530.8642(0.81)=-\$430\), and \(-\$3{,}703.7037(0.729)=-\$2{,}700\). Using the exact benefit expressions supplied in the question gives those expected amounts exactly. Discount the three amounts:
The present value is \(-\$3{,}065.24\) rounded to cents. In particular, \((1.05)^3=1.157625\), so \(2{,}700/(1.05)^3=\$2{,}332.3615160\) to the displayed precision. The check is that the undiscounted expected outgo is \(-\$3{,}490\), and discounting each payment at a positive interest rate reduces the absolute present value below that amount. Common mistake avoided: do not place all three items at one common date, and do not discount an amount before applying its survival probability unless the algebra retains both factors.
Payments Made Before the End of a Policy Year
A benefit payable earlier in a year belongs to the policy year containing its payment date, but its survival condition and discount period end at that earlier date. Suppose a benefit is paid \(s\) years after the start of a policy year, where \(0<s<1\). If mortality follows a constant force \(\mu\) during that period, then \({}_sp=e^{-\mu s}\). With an effective annual interest rate \(i\), the payment is discounted by \((1+i)^{-s}\), not by \(v\) or \(v^2\) merely because it occurs within a numbered policy year.
Worked Example: Comparing Midyear and Year-End Survival Payments
A hypothetical policy pays $1,000 at the half-year point if the insured is then alive. For comparison, consider a separate version paying $1,000 at the end of the year if the insured is alive then. Assume constant force of mortality \(\mu=\ln 2\) per year and effective annual interest \(i=21\%\). Calculate the expected present value of each benefit per policy in force at the start of the year.
Method. For each version, use survival to its own payment date and discount over the same period to that date. Under the constant-force assumption, survival to time \(t\) is \(e^{-\mu t}\).
For the midyear benefit, survival to time \(0.5\) is \(e^{-(\ln 2)(0.5)}=1/\sqrt{2}\). The half-year discount factor is \((1.21)^{-0.5}=1/1.1\). Therefore,
to cents. For the year-end version, survival is \(e^{-\ln 2}=0.5\), and the discount factor is \(1/1.21\). Thus,
These are different benefit promises, each conditional on being alive at its own payment date. The earlier payment has a higher expected present-value outgo here because survival to half a year is more likely than survival to one year, and the payment is discounted for less time. The calculation checks that the half-year survival probability \(1/\sqrt{2}\) is greater than the year-end probability \(0.5\). Common mistake avoided: using the full-year death probability for the midyear payment would apply a later survival test than the contract requires.
Practical Checks and Common Traps
When adding a survival or maturity benefit to an annual profit vector, identify the payment date first, then the event that makes the payment due, and finally the exposure basis of the vector. The payment amount is an outgo and therefore negative. Its probability weight is conditional survival to the payment date from the relevant starting point. Its discount factor, if a present value is needed, must use that same payment date.
- Wrong year: a benefit paid at the end of year \(k\) belongs to year \(k\), even if the insured entered that year alive.
- Wrong survival period: a benefit paid halfway through a year is weighted by survival to halfway through that year, not automatically by full-year survival.
- Wrong exposure basis: distinguish an amount per policy in force at the start of a year from an amount per policy issued.
- Double counting or omission: a maturity benefit is not automatically the same as a scheduled survival benefit at maturity. Follow the contract wording to determine whether one or both payments are due.
- Wrong timing in present value: an expected cash flow at time \(t\) is discounted by \((1+i)^{-t}\). Do not treat its policy-year index as its payment time if it is paid earlier.
Once the benefit entries have been placed and signed correctly, they can be combined with the other components of the annual profit vector. Keeping the dated cash-flow entries distinct from discounted values makes the later profit calculation auditable and helps prevent both timing errors and accidental duplicate payments.
Check Your Understanding
For each problem, identify the payment timing and survival condition before calculating.
- A two-year hypothetical contract pays $600 at each year-end if the insured is alive. Annual mortality probabilities are \(q_1=0.05\) and \(q_2=0.10\). Find the expected benefit outgo at the end of each year per policy issued. (Answer: year 1, \(-\$570\); year 2, \(-\$513\))
- A three-year policy pays $8,000 at maturity if the insured survives. Annual mortality probability is \(q=0.20\), independent by year. What is the expected maturity outgo per policy issued? (Answer: \(-\$4{,}096\))
- A benefit of $2,000 is payable one quarter-year after the start of a policy year if the insured is alive. Assume constant force of mortality \(\mu=\ln 2\) per year. What is the expected benefit outgo per policy in force at the start of that year, before discounting? (Answer: \(-\$2{,}000e^{-(\ln 2)/4}\approx-\$1{,}681.79\))
- A hypothetical two-year policy pays $1,500 at the end of year 1 if alive and a separate $6,000 maturity benefit at the end of year 2 if alive. Let \(q=0.10\) each year and \(i=4\%\). Find the present value at issue of both expected benefit outgoes. (Answer: \(-\$1{,}500(0.9)/1.04-\$6{,}000(0.81)/(1.04)^2=-\$5{,}026.26\))
- A $900 survival benefit is paid at time \(0.5\) if the insured is alive then. Assume constant force \(\mu=\ln 2\) per year and effective annual interest \(i=10\%\). Find its expected present value per policy in force at the start of the year. (Answer: \(-900(1/\sqrt{2})(1.10)^{-0.5}\approx-\$608.03\))