What a Prospective Reserve Measures
A participating policy may promise guaranteed benefits and may also pay declared bonuses, cash dividends, or other discretionary amounts. At a valuation date, the prospective reserve measures the present value of the future outgo included in the valuation, less the present value of future premiums. It is calculated conditional on the policy being in force at that date: the relevant probabilities describe what may happen from that in-force state onward.
The valuation basis must be explicit. It includes the interest rate, mortality assumptions, timing of cash flows, and the treatment of discretionary benefits. A declared bonus that has already become part of the policy benefit is generally included in the applicable guaranteed amount. A future discretionary payment is included only if the valuation basis calls for it; if included, it is valued using the projected amount and the probability and timing of payment. The resulting reserve is a valuation quantity, not a promise that a projected discretionary amount will be paid.
Use the effective annual interest rate \(i\) and \(v=1/(1+i)\) for annual cash flows. If a payment is due immediately at the valuation time, it has discount factor 1. If it is due one year later, its discount factor is \(v\). Always state whether the reserve is measured before or after any cash flow due at the valuation instant; that convention changes whether the immediate cash flow appears in the calculation.
Annual Term-Policy Formulation
Consider a policy valued at time \(t\), immediately before the cash flows due at that instant and before the next year of coverage. Suppose it can remain in force for at most \(n\) years. In year \(j\), let \(B_j^D\) be the benefit payable at the end of that year if death occurs during the year. Let \(B^M\) be the maturity benefit payable at time \(t+n\) if the policyholder survives to maturity. Let \(e_j\) and \(P_j\) be, respectively, an expense and a premium due at the start of year \(j\), provided the policy is then in force. All amounts may include only the guaranteed and discretionary components specified by the valuation basis.
Write \({}_{j-1}p_{x+t}\) for the probability, conditional on being in force at time \(t\), of surviving the first \(j-1\) years; write \(q_{x+t+j-1}\) for the conditional probability of death during year \(j\), given survival to its start. The annual death benefit is paid only if the policy survives to the beginning of that year and death then occurs during it. Maturity is paid only if all \(n\) years are survived. With the stated timing convention, the reserve is
Here the first two terms are the expected present values of death and maturity benefits, the third is the expected present value of expenses, and the last is the expected present value of premiums. An expense or premium due at the start of a year is weighted by the probability of being in force at that time, not by the probability of surviving to the end of the year. In particular, the probability weight for a time-zero cash flow is 1.
Why the Probability Weights Appear
The benefit part of the formula follows by dividing future outcomes into mutually exclusive events. Death in year \(j\) requires survival to the start of year \(j\) and death during that year. Its conditional probability is therefore \({}_{j-1}p_{x+t}q_{x+t+j-1}\), and its discounted contribution is that probability multiplied by \(v^jB_j^D\). Survival to maturity has probability \({}_np_{x+t}\), so its discounted contribution is \(v^n{}_np_{x+t}B^M\). Adding the contributions gives the expected present value of benefits because exactly one of the death-in-year-\(j\) events or the survival-to-maturity event occurs under this model.
The expense and premium terms use the same expected-value principle, but their timing differs. A start-of-year expense or premium is paid only if the policy is in force at the start of that year. Thus its probability weight is \({}_{j-1}p_{x+t}\), and its discount factor is \(v^{j-1}\). Subtracting premiums but adding expenses then gives the prospective reserve definition directly.
Specify whether cash flows due at that instant are included. Condition all future probabilities on the policy being in force then.
Separate death benefits, maturity benefits, expenses, and premiums. Identify which components are guaranteed, declared, or projected discretionary amounts.
Multiply by the probability the policy is in force when the cash flow occurs, then discount to the valuation date.
Add benefits and expenses; subtract premiums. Verify that each payment is weighted by the correct event probability and discounted for the correct time.
Worked Valuations
Worked Example: A One-Year Policy With a Declared Bonus
A hypothetical participating policy is valued immediately before a premium of $25 and an expense of $5 due at time 0. The effective annual interest rate is 5%. During the coming year, the probability of death is 0.20 and the probability of survival to maturity is 0.80. The death benefit is $350, comprising a $330 guaranteed benefit and a $20 declared bonus. The maturity benefit is $500, comprising a $470 guaranteed benefit and a $30 declared bonus. Calculate the prospective reserve.
Method. Since the declared bonuses are already specified as part of the benefits, include them in the relevant claim amounts. The claims are paid at time 1; the premium and expense are due immediately at time 0.
The expected benefit payable at time 1 is
Discounting one year gives an expected present value of benefits of
The time-zero premium reduces the reserve by $25, while the time-zero expense increases it by $5. Therefore,
Rounded to the nearest cent, the reserve is $427.62. As a check, the benefit value is positive, and the net time-zero cash outgo is $5-$25=-$20, so the immediate cash flows reduce the benefit-only value by $20, giving $427.62. The common mistake is to subtract the expense as though it were income; the expense must be added.
Worked Example: Including a Projected Terminal Bonus and Dividend
A hypothetical one-year policy is valued immediately before a $50 premium and a $10 expense due at time 0. The effective annual interest rate is 4%, and the probability of death is 0.15. The guaranteed death benefit is $600 and the guaranteed maturity benefit is $800. The valuation projection includes a $60 terminal bonus payable on death and a $25 dividend payable only on survival to maturity. Calculate the reserve including both projected amounts, then calculate it excluding them.
Method. Build the total payment in each outcome first, then probability-weight and discount. The death payment including the projected terminal bonus is \(600+60=660\); the maturity payment including the projected dividend is \(800+25=825\). The survival probability is \(1-0.15=0.85\).
Including the projected amounts, the expected benefit payable at time 1 is
Its present value is \(800.25/1.04=769.4711538\). Adding the $10 expense and subtracting the $50 premium, both due at time 0, gives
Excluding the projected amounts, the expected benefit payable at time 1 is
The benefit present value is \(770/1.04=740.3846154\), so
The increase of $29.09 is the present value of the included projected amounts: \([0.15(60)+0.85(25)]/1.04=30.25/1.04=\$29.09\). This provides a useful check. A common error is to subtract the $10 expense; doing so would understate each reserve by $20 relative to the correct result, because the proper term is \(+10\), not \(-10\).
Worked Example: A Two-Year Participating Endowment
A hypothetical two-year policy is valued at issue immediately before its first premium and expense. The effective annual interest rate is 5%. The probability of death during year 1 is 0.10. Conditional on surviving year 1, the probability of death during year 2 is 0.20. A death at the end of year 1 results in a $420 payment, comprising a $400 guaranteed benefit and a $20 declared bonus. A death at the end of year 2 results in a $530 payment, comprising a $500 guaranteed benefit and a $30 declared bonus. Survival to maturity results in a $740 payment, comprising a $700 guaranteed benefit and a projected $40 terminal bonus. A $30 premium and a $4 expense are due at time 0. If the policyholder survives year 1, a further $30 premium and $4 expense are due at time 1, before year 2 coverage starts. Calculate the prospective reserve, including the projected terminal bonus.
Method. Partition the outcomes into death in year 1, death in year 2, and survival to maturity. Their probabilities are, respectively,
These add to \(0.10+0.18+0.72=1.00\), as required. The expected present value of benefits is
The first term is \(42/1.05=40\). The second is \(628.2/1.1025=569.7959184\). Thus the benefit present value is \(609.7959184\).
The time-zero expense is $4. The time-1 expense is paid only if the policyholder survives year 1, so its present value is \(0.90(4)/1.05=3.4285714\). Total expense present value is \(4+3.4285714=7.4285714\). The premium present value is \(30+0.90(30)/1.05=30+25.7142857=55.7142857\). Therefore,
The check on signs is that expenses add $7.43 to the benefit value and premiums subtract $55.71. The time-1 premium and expense are weighted by 0.90 because they are due only if the policy is still in force at that time. Weighting them by the year-2 survival probability of 0.72 would incorrectly exclude policies that survive to time 1 but die during year 2.
Interpreting the Result and Avoiding Traps
A prospective reserve depends on both the contract and the valuation basis. Changing the interest rate changes discount factors; changing mortality changes the probability weights; changing the treatment or projection of future discretionary benefits changes the amounts included. If a future dividend is excluded from the basis, it does not belong in the benefit cash flow. If it is included, it should be included once, at the time and in the event in which it is projected to be paid. A declared bonus already included in a stated benefit should not be added a second time.
Expenses must be assigned their actual payment dates and conditions. A maintenance expense payable only while the policy is in force is weighted by the probability the policy remains in force when that expense falls due. A claim expense payable only on death would instead be weighted by the relevant death probability. In either case, an expense is outgo and enters with a plus sign. Premiums enter with a minus sign because they are future income to the policy.
The reserve can be positive, zero, or negative under a particular basis. Its sign alone does not establish whether the valuation is correct. The appropriate checks are that the outcome probabilities are consistent, the payment amounts match the stated benefit components, every cash flow has the right date and condition, and the signs reflect whether it is an inflow or outflow.
Check Your Understanding
For each problem, use the stated valuation time and cash-flow timing. Round dollar answers to the nearest cent.
- A one-year policy is valued immediately before a $35 premium and a $6 expense due at time 0. At 4% effective annual interest, the death probability is 0.10. The death benefit is $500 and the maturity benefit is $700, both paid at time 1. Find the reserve. (Answer: $624.85)
- A one-year policy is valued immediately before a $20 premium and a $3 expense at time 0. The effective annual interest rate is 5%; the death probability is 0.25. The death benefit is $440 and the maturity benefit is $600, payable at time 1. Find the reserve. (Answer: $516.33)
- A two-year policy has no premiums or expenses. The effective annual interest rate is 5%. The death probabilities are 0.20 in year 1 and 0.10 in year 2 conditional on surviving year 1. Benefits are $300 for death in year 1, $500 for death in year 2, and $800 for survival to maturity. Find the reserve at issue. (Answer: $615.87)
- A one-year policy is valued immediately before a $10 premium and a $2 expense. The interest rate is 0%, and the death probability is 0.20. Guaranteed benefits are $400 on death and $500 on survival; the valuation basis includes a $40 projected death bonus and a $20 projected maturity dividend. Find the reserve including the projections and the reserve excluding them. (Answer: $496.00 including; $472.00 excluding)
- A two-year policy has 0% interest. Death probabilities are 0.10 in year 1 and 0.20 in year 2 conditional on surviving year 1. Benefits are $300 for death in year 1, $500 for death in year 2, and $700 for maturity. A $20 premium and $5 expense are due at time 0; another $20 premium and $5 expense are due at time 1 only if the policyholder survived year 1. Find the reserve at issue. (Answer: $595.50)