Gross Reserves With Declared Bonuses

Why Gross Reserves Need a Consistent Basis

A participating policy may have a guaranteed benefit increased by bonuses already declared. A gross prospective reserve must value that benefit alongside the future gross premiums and expenses associated with the policy. Calling the reserve “gross” means that the valuation includes expenses and uses gross premiums; it does not mean that premiums are added to the liability. Future premiums reduce the prospective reserve, while future expenses increase it.

The main practical challenge is consistency. The valuation must use the same policy status, payment dates, survival conditions, and interest basis for each future cash flow. A premium due at the reserve date is included if the reserve is measured just before that premium. A future renewal premium is included only if the policy is still in force when it falls due.

Definition: A gross prospective reserve at a specified policy duration is the expected present value, conditional on the policy being in force at that duration, of future guaranteed benefits and expenses, less the expected present value of future gross premiums. A declared bonus is included in guaranteed benefits when it has vested or otherwise become payable under the contract.

For clarity, the examples use annual time steps and reserves measured immediately before the premium due at the start of each policy year. At time $k$, the current premium and any expense due at that time are therefore included. Death benefits are paid at the end of the year of death. If the policy survives to maturity, the maturity benefit is paid at the end of the final year. Expenses at claim are paid with the claim.

The examples assume a fixed effective annual interest rate $i$, with $v=1/(1+i)$. Mortality probabilities refer to the stated policy year. Future discretionary bonuses are not projected: the benefit basis includes only declared bonuses specified in the example. Actual contract terms determine when bonuses vest and which benefits they increase.

Prospective Valuation and the Annual Recursion

Let $V_k$ be the gross prospective reserve immediately before the premium at time $k$. Let $P_k$ be the gross premium due then and $e_k$ the expense due then. Let $q_k$ be the probability of death during the following year, conditional on being in force at time $k$. Write $B_k$ for the death benefit paid at the end of that year and $C_k$ for the associated claim expense. If the policy survives, its reserve at time $k+1$ is $V_{k+1}$, measured before the next premium. For a nonterminal policy year, the annual reserve recursion is:

V_k=-P_k+e_k+v\left[q_k(B_k+C_k)+(1-q_k)V_{k+1}\right].

At the final year, replace the survival-branch reserve by the maturity benefit and any maturity expense, both payable at the end of that year. Premiums and expenses due at time $k$ are not discounted in this formula because they occur at the valuation instant. The amounts in the bracket occur one year later and are discounted once.

Why the recursion holds: Conditional on the policy being in force at time $k$, the immediate net outgo is $e_k-P_k$. At the end of the year, death occurs with probability $q_k$, producing the death benefit and claim expense. With probability $1-q_k$, the policy survives and has reserve $V_{k+1}$. Taking the probability-weighted value of these two outcomes and discounting one year gives the displayed recursion.

The recursion and the prospective definition are two views of the same valuation. The recursion is useful for checking a calculation year by year; direct prospective valuation makes the full set of future premiums, benefits, and expenses visible. A declared reversionary bonus must be reflected in every benefit to which it applies. It must not be counted as a separate future cash payment as well as being included in the increased sum assured.

Worked Valuations

Worked Example: A Two-Year Endowment With a Vested Bonus

A hypothetical two-year endowment is valued at issue, immediately before the premium at time 0. The base sum assured is $10,000, and a declared reversionary bonus of $1,000 has already vested and is payable on death or maturity. Thus the claim benefit is $11,000 whenever the policy ends. The effective annual interest rate is 5%. The annual mortality probabilities are $q_1=q_2=0.10$. A claim expense of $200 is paid with any death or maturity benefit. The gross premium is $6,000 at times 0 and 1 while the policy is in force. Acquisition expense is $800 at time 0, and renewal expense is $300 at time 1 if the policy survives the first year. Calculate the gross reserve at time 0.

Method. Calculate the expected present value of the claim outgo, premiums, and expenses on the stated timing basis. Since the declared bonus applies to both death and maturity, each claim outgo is $11,000+$200=$11,200.

The probability of a claim in year 1 is 0.10. The probability the policy remains in force to time 1 is 0.90. In year 2, death and maturity together exhaust the surviving policies, so the probability of an end-of-year-2 payment is 0.90. With $v=1/1.05=0.95238095$, the expected present value of claim outgo is:

11{,}200\left(0.10v+0.90v^2\right) =11{,}200\left(0.09523810+0.81632653\right) =\$10{,}209.52.

The time-0 premium is certain. The time-1 premium is payable only if the policy survives the first year. Therefore:

\operatorname{EPV}(\text{premiums}) =6{,}000+0.90(6{,}000)v =6{,}000+5{,}142.86 =\$11{,}142.86.

The acquisition expense is due immediately; the renewal expense is due at time 1 if the policy survives:

\operatorname{EPV}(\text{expenses}) =800+0.90(300)v =800+257.14 =\$1{,}057.14.

Thus the reserve is claim outgo plus expenses less premiums:

V_0=10{,}209.52+1{,}057.14-11{,}142.86 =\boxed{\$123.81}.

Check and common mistake. The reserve is positive because the expected present value of claim outgo and expenses exceeds the expected present value of premiums by $123.81. Omitting the vested $1,000 bonus would understate claim outgo; counting that bonus separately after already using the $11,000 benefit would instead double-count it.

Worked Example: A One-Year Reserve After Bonus Declaration

A hypothetical one-year endowment is valued at time 1, immediately before its final annual premium. A declared bonus has increased the amount payable on death or maturity from $12,000 to $13,500. The effective annual interest rate is 4%, and the conditional probability of death during the year is 0.08. A claim expense of $250 is paid with either death or maturity. The gross premium due at time 1 is $5,000, and a renewal expense of $400 is due at the same time. Find the gross reserve.

Method. Both possible policy endings produce a payment at the end of the year, and the declared-bonus benefit is the same in either case. Therefore the total end-of-year outgo is $13,500+$250=$13,750 with certainty. Mortality does not change the expected amount in this particular setup, although it determines whether the payment is a death or maturity claim.

The current premium and expense are due at the valuation instant. Only the benefit and claim expense are discounted:

V_1=13{,}750\left(\frac{1}{1.04}\right)-5{,}000+400 =13{,}221.15-4{,}600 =\boxed{\$8{,}621.15}.

Check and common mistake. The reserve is the discounted future claim outgo less the premium, plus expense. Discounting the premium or expense would be wrong because both are due at time 1, the reserve date. The bonus is part of the guaranteed amount payable, not a premium or a separate expense.

Worked Example: A Three-Year Policy With Mortality Varying by Year

A hypothetical three-year endowment is valued at issue, immediately before the time-0 premium. Its base benefit is $10,000, and a declared bonus of $500 applies to death and maturity benefits, making each benefit $10,500. The effective annual interest rate is 5%. The annual death probabilities are $q_1=0.10$, $q_2=0.20$, and $q_3=0.25$, conditional on reaching the respective year. A claim expense of $150 is payable with each death or maturity claim. Gross premiums of $4,000 are due at times 0, 1, and 2 while the policy is in force. Acquisition expense is $700 at time 0; renewal expense is $100 at times 1 and 2 if the policy is in force. Calculate the time-0 reserve.

Method. Add the claim expense to the benefit to obtain $10,650 per claim. The probabilities of death in years 1 and 2 are 0.10 and $0.90(0.20)=0.18$. The probability of reaching year 3 is $0.90(0.80)=0.72$. In year 3, death has probability $0.72(0.25)=0.18$, and maturity has probability $0.72(0.75)=0.54$. Both year-3 outcomes pay at time 3, so their probabilities combine to 0.72.

With $v=1/1.05=0.95238095$, the expected present value of claim outgo is:

10{,}650\left(0.10v+0.18v^2+0.72v^3\right) =10{,}650(0.88046647) =\$9{,}376.97.

The probability of being in force at time 1 is 0.90 and at time 2 is 0.72. Hence:

\begin{aligned} \operatorname{EPV}(\text{premiums}) &=4{,}000+0.90(4{,}000)v+0.72(4{,}000)v^2\\ &=4{,}000+3{,}428.57+2{,}612.24 =\$10{,}040.82. \end{aligned}

Renewal expenses are subject to the same in-force probabilities as the corresponding future policy years, whereas the acquisition expense is immediate:

\begin{aligned} \operatorname{EPV}(\text{expenses}) &=700+0.90(100)v+0.72(100)v^2\\ &=700+85.71+65.31 =\$851.02. \end{aligned}

Therefore:

V_0=9{,}376.97+851.02-10{,}040.82 =\boxed{\$187.17}.

Check and common mistake. The renewal premiums and expenses use the same survival probabilities because each is conditional on the policy being in force at its due date. Applying the year-2 mortality probability directly to the time-2 premium, rather than the probability $0.72$ of reaching time 2, would use the wrong condition.

Interpreting and Checking the Result

A declared bonus affects the reserve through the benefit cash flows to which it applies. If the bonus has already vested, it belongs in the guaranteed benefit basis used for the valuation. Its effect may differ by claim type or timing if the contract specifies different death and maturity benefits. In that case, each possible payment must be valued using its own amount and payment date.

A useful general check is to classify each future cash flow before calculating its present value:

Identify the reserve instant.
Decide whether the reserve is just before or just after a premium and expense due at that time.

Set the guaranteed benefit basis.
Add declared bonuses to the benefits they increase, and distinguish death payments from maturity payments where their amounts differ.

Attach probabilities to due dates.
Use the probability the policy is in force when a premium or renewal expense is due, and the probability of each claim outcome at its payment date.

Discount and combine.
Discount each cash flow from its payment date to the reserve date, then calculate benefits plus expenses less premiums.

A reserve can be negative under a specified gross basis if the expected present value of future premiums exceeds expected benefits and expenses. That result is not, by itself, an arithmetic error; it should prompt a check of the contract, timing, and valuation assumptions. Conversely, a positive reserve does not mean that the policy has a cash surrender value of the same amount. Surrender values and other policyholder options are distinct valuation questions.

The annual recursion provides a second check when a calculation spans multiple years. Starting from the terminal-year benefit and working backward should reproduce the direct prospective value, provided each reserve is measured at the same point relative to premiums and expenses. Mixing a reserve immediately after premium with a recursion defined immediately before premium creates a one-year cash-flow mismatch.

Key takeaway: A gross reserve with declared bonuses is the prospective value of guaranteed bonus-inclusive benefits and future expenses, less future gross premiums. The valuation is reliable only when benefit amounts, premium and expense timing, survival conditions, and discounting all use the same basis.

Check Your Understanding

For each problem, use the stated reserve timing and round money to the nearest cent.

A one-year endowment is valued just before a premium of $3,000. Its declared-bonus benefit is $8,400 at death or maturity, and claim expense is $100. The interest rate is 5%; the death probability is 0.10. A maintenance expense of $150 is due immediately. Find the gross reserve. (Answer: $5,245.24)
A two-year endowment is valued at issue just before its time-0 premium. It pays $9,000 at death or maturity, has no claim expense, and has annual death probabilities of 0.20 in both years. Interest is 0%. Premiums of $3,000 are due at times 0 and 1 while in force. Acquisition expense is $500 at time 0 and renewal expense is $200 at time 1 if in force. Find the gross reserve. (Answer: $4,260.00)
A two-year policy is valued at issue just before the time-0 premium. Its declared-bonus benefit is $10,000 at either year-end claim. Interest is 5%; $q_1=0.10$ and $q_2=0.20$. Premiums of $4,000 are due at times 0 and 1 while in force. Expenses are $300 at time 0 and $100 at time 1 if in force. There is no claim expense. Find the gross reserve. (Answer: $2,072.79)
A two-year endowment is valued just before its time-0 premium, with 0% interest. It pays $7,000 for a year-1 death and $9,000 for a year-2 death or maturity. The death probabilities are $q_1=0.10$ and $q_2=0.20$. Premiums of $5,000 are due at times 0 and 1 while in force. Expenses are $200 at time 0 and $100 at time 1 if in force. Find the gross reserve. (Answer: −$410.00)
A two-year endowment pays $10,000 at death or maturity and is valued at issue with 5% interest. Its death probabilities are 0.10 in each year. A declared bonus increases both benefits by $500. Ignoring premiums and expenses, by how much does the expected present value of claim benefits increase because of the bonus? (Answer: $455.78)

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