Why Premiums and Expenses Matter in a Participating Reserve
A prospective reserve is a value of future cash flows, conditional on the policy being in force at the valuation time. For a participating policy, the projected benefits may include guaranteed amounts and discretionary benefits on the stated projection basis. A gross prospective reserve also recognizes future premiums and expenses. Omitting either can materially change the result, even when the projected benefits are unchanged.
The essential sign convention is straightforward: benefits and expenses increase the amount needed to meet future outgo, while premiums provide future income and reduce that amount. Each cash flow must also be assigned its correct time and probability of occurrence. In particular, a premium and an expense payable only while the policy remains in force must not be treated as certain at later durations.
A benefit-only value is the expected present value of benefits, without premiums or expenses. A net prospective reserve is the benefit-only value less the expected present value of future premiums. Thus, “benefit-only value” and “net reserve” are not interchangeable: the former has no premium deduction, while the latter does.
Set the Valuation Time and Cash-Flow Basis
For the annual examples below, reserves are valued immediately before any premium due at the valuation time. Premiums and premium-related expenses are paid at the beginning of a year if the policy is then in force. Death benefits and claim expenses are paid at the end of the year of death. A maturity benefit and any maturity expense are paid at the end of the term. Interest is effective annually.
Let \(m\) be the number of years remaining at duration \(k\), and let \(v=1/(1+i)\). For a policy in force at duration \(k\), write \({}_jp_{x+k}\) for the probability of surviving the next \(j\) years and \({}_{j-1}p_{x+k}q_{x+k+j-1}\) for the probability of surviving \(j-1\) years and then dying in the \(j\)th year. Let \(D_j\) denote the projected death benefit paid at the end of that year, and let \(M\) denote the maturity benefit paid to a survivor at the end of year \(m\). These amounts may include projected participating benefits, provided the projection basis is consistent with the valuation.
If \(C_j^D\) is the expense incurred on a death claim in year \(j\), \(C^M\) is the maturity expense, \(e_j\) is an expense payable at the beginning of year \(j\) while the policy is in force, and \(P_j\) is the premium due at that time, the prospective value can be written as follows:
The benefit and expense terms are expected present values of outgo. The premium terms are expected present values of income. In the first sum, death benefits and claim expenses are weighted by the probability of death in the relevant year. The maturity amount is weighted by the probability of surviving to maturity. Beginning-of-year expenses and premiums are weighted by the probability of being in force at that time. This expression assumes no other cash-flow types; any additional policy cash flow must be included at its own timing and conditional probability.
Why the Annual Reserve Recursion Holds
The same valuation follows by splitting future cash flows into the current year and what remains after survival. For a policy with one year remaining, let \(P_k\) be the premium due now, \(e_k\) the expense due now, \(D\) the death benefit, and \(C^D\) the death-claim expense. Let \(M\) and \(C^M\) be the maturity benefit and expense if the policyholder survives the year. Then:
The premium is subtracted because it is income; the expense due now is added because it is outgo. At year-end, death and survival are mutually exclusive outcomes, so their expected payments are weighted by \(q_{x+k}\) and \({}_1p_{x+k}\), respectively. Discounting those year-end amounts by \(v\) brings them back to the valuation time. This argument requires the stated timing; if a premium or expense is paid at a different time, its discount factor and survival weight must be changed accordingly.
Separate death, maturity, and expense cash flows, and state when each is paid.
Use the probability the policy is in force or the relevant claim occurs, then discount from the payment time.
Weight premiums by the probability they are due and subtract their expected present value from outgo.
Confirm that expenses increase the reserve and premiums decrease it.
Worked Reserve Calculations
Worked Example: One Year Remaining With a Projected Bonus
A hypothetical participating endowment policy is valued immediately before its renewal premium. The effective annual interest rate is \(5\%\); the probability of death during the year is \(0.10\), and the probability of survival is \(0.90\). The projected death benefit is \$600, including a projected bonus, and the maturity benefit is \$520, also including a projected bonus. A claim or maturity expense of \$10 is paid with either outcome. The renewal premium is \$120, and a renewal expense of \$8 is payable now. Calculate the gross prospective reserve and compare it with the net benefit-only reserve.
Method. Value the two mutually exclusive end-of-year outcomes, add the expenses, and subtract the premium net of its associated expense. Here \(v=1/1.05\).
The expected end-of-year benefit and claim or maturity expense are:
Their present value is \(538/1.05=\$512.38095\). At the valuation time, the premium and renewal expense give net income of \(\$120-\$8=\$112\). Therefore:
For comparison, the benefit-only value excludes both expenses and premiums: \(\{0.10(600)+0.90(520)\}/1.05=528/1.05=\$502.86\). The net benefit-only reserve subtracts the premium, giving \(\$502.86-\$120=\$382.86\). The gross reserve is higher because it also includes the present value of the claim or maturity expense, \(\$10/1.05=\$9.52\), and the renewal expense of \(\$8\). The difference is \(\$17.52\), subject to rounding. The calculation avoids treating the maturity benefit as payable on death or the death benefit as payable on survival; it weights each by its own probability.
Worked Example: Two-Year Term Reserve With Conditional Premiums
A hypothetical participating term policy is valued at the beginning of a policy year, just before the premium is due, with two years remaining. The effective annual interest rate is \(5\%\), and the death probability in each year is \(0.10\), conditional on being in force at the start of that year. The death benefit is \$400 in the first year and \$440 in the second year; these projected amounts include the assumed participating benefits. A death claim expense of \$4 is paid with each benefit. A premium of \$40 and an expense of \$6 are due at the beginning of each year while the policy is in force. There is no maturity benefit. Calculate the gross reserve.
Method. Weight first-year death outgo by \(0.10\). Second-year death outgo requires survival through the first year and death in the second, with probability \(0.90(0.10)=0.09\). The second premium and expense are also conditional on survival to that payment time.
The present value of death benefits alone is:
The present value of claim expenses is:
The premium-date expenses have present value \(6+0.90(6)v=6+5.14286=\$11.14286\). The premiums have present value \(40+0.90(40)v=40+34.28571=\$74.28571\). Thus:
As a check, the net benefit-only reserve is \(\$74.01361-\$74.28571=-\$0.27\). Adding claim and premium-date expenses, totaling \(\$11.85034\) in present value, gives \(\$11.58\) after rounding. A negative benefit-only net value is possible; it is not a reason to change the sign convention. The key avoided error is treating the second premium as certain: it is payable only if the policy survives the first year.
Worked Example: Solving for a Gross Premium and a Later Reserve
A hypothetical two-year participating endowment has an effective annual interest rate of \(0\%\). The probability of death in each year, conditional on being in force at the start of that year, is \(0.20\). The first-year death benefit is \$500, including a projected bonus. At the end of the second year, the death benefit and maturity benefit are both \$600, including projected bonuses. An acquisition expense of \$20 is paid at issue, and a renewal expense of \$5 is paid at the start of year two if the policy is in force. Level premiums \(P\) are paid at issue and at the start of year two if in force. Find the premium under the equivalence principle, then find the reserve at the start of year two immediately before its premium.
Method. At issue, equate the expected present value of premiums to that of benefits and expenses. Since \(i=0\%\), no discounting is needed. The probability of surviving the first year is \(0.80\).
Expected benefits are \(0.20(500)+0.80(600)=100+480=\$580\). Expected expenses are \(20+0.80(5)=\$24\). Expected premiums are \(P+0.80P=1.80P\). Therefore:
At the start of year two, the policy is in force and the second-year benefit of \$600 is payable at year-end whether death or survival to maturity occurs. The renewal expense of \$5 and premium \(P\) are due immediately. Hence:
The issue-date equivalence check is \(1.80(335.555\ldots)=\$604\), equal to expected benefits plus expenses. The reserve calculation uses the unrounded premium internally and rounds only the final monetary answers to the nearest cent. This avoids both omitting the renewal expense from the gross premium and subtracting a premium that is not due at the stated reserve time.
Common Mistakes and Practical Checks
A frequent error is to calculate a benefit-only net reserve and label it gross. In the first example, that incorrect calculation is \(\$502.86-\$120=\$382.86\). It omits the expected claim or maturity expense and the renewal expense. The correction is to include both expense cash flows: \(\$382.86+\$9.52+\$8=\$400.38\).
- Check the valuation instant. A premium due at the valuation time is included if the reserve is immediately before it; it is not included if the reserve is immediately after it.
- Check conditionality. A later premium or renewal expense is payable only if the policy remains in force to that payment time.
- Check the signs. Benefits and expenses add to a gross reserve; premiums are subtracted.
- Check the participating basis. Projected discretionary benefits must match the stated valuation projection; do not silently change them while adding premiums or expenses.
- Check the scope of the calculation. Acquisition costs already paid before the valuation time are not future expenses in a prospective reserve. Include them only when they are still future cash flows at the stated valuation time.
Check Your Understanding
For each problem, use the stated timing and round the final monetary answer to the nearest cent.
- A one-year endowment reserve is valued immediately before a renewal premium. The effective annual interest rate is \(4\%\); death probability is \(0.25\), and survival probability is \(0.75\). The death and maturity benefits are \$300 and \$260, respectively. An expense of \$5 is paid with either outcome. The premium due now is \$80, and the renewal expense due now is \$3. Find the gross reserve. (Answer: \$187.42)
- A two-year term policy is valued at issue with \(i=0\%\) and annual death probability \(0.20\). Death benefits are \$200 in year one and \$250 in year two; the premium is \$30 at the start of each year if in force. A premium-date expense of \$4 is payable with each premium, and a claim expense of \$2 is paid with each death claim. Find the gross reserve at issue. (Answer: \$33.92)
- At a one-year remaining valuation time, \(i=5\%\), death probability is \(0.10\), and survival probability is \(0.90\). The death benefit is \$500 and the maturity benefit is \$450. An expense of \$10 is paid with either outcome. A premium of \$100 and a renewal expense of \$5 are due immediately. Find the gross reserve. (Answer: \$347.86)
- A two-year endowment has \(i=0\%\), annual conditional death probability \(0.10\), a first-year death benefit of \$400, and a second-year death or maturity benefit of \$500. An acquisition expense of \$12 is paid at issue and a renewal expense of \$3 at the start of year two if in force. Level premiums \(P\) are due at issue and at the start of year two if in force. Find the gross premium under the equivalence principle. (Answer: \$265.63)