Bonus Declarations Across Asset Share and Solvency

Assessing a Bonus Declaration

A proposed bonus can look modest as a nominal addition to a policy benefit while creating a material obligation across a participating portfolio. The key is to evaluate the declaration from several perspectives at the same valuation date: the benefit promised to policyholders, the present value of the resulting liability, the available asset share, the shareholder margin, and the portfolio’s position relative to its capital requirement.

The calculations below use hypothetical policies and assumptions. They isolate the effect of a new declaration: unless stated otherwise, assets and existing liabilities are held fixed, and the bonus creates no immediate cash payment. This makes the incremental effect visible without treating an asset share as if it were a statutory reserve or a solvency measure.

Definition: A declared bonus is an allocation of participating-fund experience to policy benefits. A reversionary bonus, once declared under the assumed contract terms, increases a guaranteed benefit. Its incremental liability is the present value, at the valuation date, of the additional expected benefit payments caused by the declaration. An asset share is a retrospective accumulation of the premiums and other amounts allocated to a policy, less relevant outgo, with investment accumulation applied under the stated method. It is not automatically equal to the policy liability.

From Nominal Bonus to Incremental Liability

The amount declared and the liability created are different quantities. A nominal bonus is paid at a future date and may be conditional on the policy remaining in force. Its value at the declaration date therefore depends on the timing of payment, the probability of payment, and the valuation interest rate.

Suppose an additional amount $B$ is payable $n$ years from now only if the policy is then in force. Let ${}_np_x$ be the probability, on the valuation assumptions, that the policy remains in force for those $n$ years, and let $i$ be the effective annual valuation interest rate. If there are no other contingencies or payments associated with the addition, the incremental liability per policy is:

\Delta V = B\,{}_np_x\,v^n =\frac{B\,{}_np_x}{(1+i)^n}, \qquad v=\frac{1}{1+i}.

This formula assumes the benefit is paid at the end of year $n$, the survival probability and interest rate are appropriate for the liability valuation, and there are no expenses, taxes, lapses, or other cash flows attributable to the bonus. If the bonus is also payable on death before year $n$, or if several payments are added, the expected present value must include those payments as well. The simple formula would then be incomplete.

Why the formula holds: The added benefit is paid with probability ${}_np_x$, so its expected payment at time $n$ is $B\,{}_np_x$. A dollar payable at time $n$ has present value $v^n$ at the valuation date. Multiplying the expected payment by its discount factor gives $B\,{}_np_xv^n$. For a portfolio of $N$ identical policies, linearity of expectation gives an aggregate incremental liability of $N\Delta V$; independence between policies is not needed for this expected-value calculation.

A declaration does not, by itself, create investment assets. Under a fixed retrospective asset-share method, the current asset share therefore does not jump merely because a future benefit is declared. The declaration can, however, make the prospective liability larger than the asset share available to support it. If a declaration is paid immediately, or if a particular asset-share method explicitly deducts a bonus allocation, the asset-share calculation must reflect that cash flow or allocation.

Worked Valuation and Asset-Share Check

Worked Example: A One-Year Reversionary Addition

A hypothetical participating portfolio has 100 identical policies. At the declaration date, each policy has an asset share of $10,400 and an existing reserve of $10,000. A proposed reversionary bonus adds $500 to a benefit payable one year from now if the policy survives. Assume a one-year survival probability of 0.98 and an effective annual valuation interest rate of 5%. Assume the asset shares are unchanged by the declaration and that no other cash flows are affected. Find the incremental liability, the total liability after declaration, and the comparison with aggregate asset shares.

Method. Value the additional benefit as an expected payment discounted one year. Then compare the resulting liability with the existing asset shares; do not add the nominal $500 directly to the reserve.

The probability-weighted payment per policy is:

500(0.98)=\$490.00.

Discounting at 5% gives the incremental liability per policy:

\Delta V=\frac{490}{1.05}=\$466.6667\ldots

Rounded to the nearest cent, this is $466.67 per policy. For 100 policies, the aggregate increase is $100(466.6667\ldots)=\$46,666.67$, rounded to cents. The existing aggregate reserve is $100(\$10,000)=\$1,000,000$, so the new aggregate liability is $1,046,666.67.

Aggregate asset shares are $100(\$10,400)=\$1,040,000$. Before the declaration, the asset-share amount exceeded the existing reserve by $40,000. After the declaration, the liability exceeds the aggregate asset share by $6,666.67, because the incremental liability of $46,666.67 is greater than the original $40,000 margin.

Check. The aggregate difference is $\$1,046,666.67-\$1,040,000=\$6,666.67$. The declared nominal benefit is $500 per policy, but its liability value is $466.67 under the stated survival and interest assumptions. The common mistake avoided here is treating the full nominal amount as a present liability at the declaration date.

Shareholder Margin and the Capital Constraint

An incremental liability reduces the excess of assets over liabilities if assets are unchanged. A capital requirement is a separate threshold: a portfolio may have positive assets in excess of liabilities and still fall short of the required capital. For a simplified valuation, define the pre-declaration solvency surplus as assets less liabilities. Define the margin above a stated capital requirement as that surplus less the required capital.

In a simplified shareholder-profit measure based on the change in this surplus, an increase in liability reduces current shareholder profit dollar-for-dollar unless another cash flow, asset movement, or valuation effect offsets it. This is a valuation statement, not a claim that the bonus causes an immediate cash payment. The treatment in an actual reporting or profit-testing framework depends on its specified definitions and timing.

Worked Example: Portfolio Declaration and Capital Requirement

A hypothetical fund has assets of $12,000,000 and existing liabilities of $10,800,000 for 1,000 policies. Their aggregate asset share is $11,500,000. The stated capital requirement is $500,000. A proposed bonus adds $1,000 to a benefit payable in one year if the policy survives; the one-year survival probability is 0.90 and the effective annual valuation interest rate is 5%. Assume assets and the capital requirement do not change at the declaration date. Calculate the post-declaration surplus and margin above required capital.

Method. Calculate the per-policy expected present value, multiply by the policy count, and deduct this increase from the pre-declaration surplus. Compare the resulting surplus with required capital as a separate step.

Per policy, the expected payment is $1,000(0.90)=\$900$. Its present value is:

\Delta V=\frac{900}{1.05}=\$857.142857\ldots

For 1,000 policies, the aggregate incremental liability is $857,142.86, rounded to cents. The initial surplus is:

\$12,000,000-\$10,800,000=\$1,200,000.

After the declaration, liabilities are $\$10,800,000+\$857,142.86=\$11,657,142.86$, and the surplus is:

\$12,000,000-\$11,657,142.86=\$342,857.14.

The margin above the $500,000 capital requirement is now $\$342,857.14-\$500,000=-\$157,142.86$. Thus, the portfolio has positive assets in excess of liabilities but falls short of the stated capital requirement by $157,142.86. Before the declaration, the corresponding margin was $\$1,200,000-\$500,000=\$700,000$. On this simplified measure, the shareholder margin falls by the same $857,142.86 as the incremental liability.

The aggregate asset share of $11,500,000 is not the fund’s solvency surplus: it is a separate retrospective measure. In this example the post-declaration liability is also greater than the aggregate asset share, but solvency is assessed here using the stated assets, liabilities, and capital requirement.

Check. The post-declaration surplus of $342,857.14 is below, not above, the $500,000 requirement. The common mistake avoided is to report a positive surplus as proof that the capital requirement is met.

Assumptions, Sensitivity, and the Decision to Declare

The effect of a proposed declaration depends on the basis used to value its future payments. A lower valuation interest rate increases the present value of a fixed future benefit. A higher probability that the policy remains eligible for the payment also increases its expected present value. These changes affect the liability and solvency margin even when the declared nominal amount is unchanged.

The comparison should keep its quantities distinct. The policy benefit is the contractual amount payable under the assumed terms. The liability is a prospective value of expected future payments. The asset share is a retrospective accumulation under its stated allocation method. Shareholder profit is a change in a defined profit measure, and solvency compares resources with liabilities and a capital requirement. None of these quantities can be substituted for another without a stated relationship.

Worked Example: Valuation-Basis Sensitivity for a Two-Year Addition

A hypothetical portfolio has 400 policies. A proposed guaranteed addition of $600 is payable two years from now if the policy is then in force. The cumulative two-year survival probability is 0.95. Assets are $6,500,000, existing liabilities are $5,900,000, and the capital requirement is $400,000. Assume these assets, existing liabilities, and the requirement are held fixed for this sensitivity comparison. Calculate the incremental liability and capital margin using first a 4% effective annual valuation interest rate and then a 2% rate.

Method. Apply the same benefit and survival probability on both bases, changing only the discount factor. Then deduct each incremental liability from the initial surplus before comparing with required capital.

At 4%, the probability-weighted payment per policy is $600(0.95)=\$570$. Since $1.04^2=1.0816$, the incremental liability per policy is:

\Delta V_{4\%}=\frac{570}{1.0816}=\$526.9970\ldots

For 400 policies, this is $210,798.82, rounded to cents. Initial surplus is $\$6,500,000-\$5,900,000=\$600,000$. Post-declaration surplus is $\$600,000-\$210,798.82=\$389,201.18$. The margin above the $400,000 capital requirement is therefore -$10,798.82.

At 2%, $1.02^2=1.0404$, so:

\Delta V_{2\%}=\frac{570}{1.0404}=\$547.8662\ldots

For 400 policies, the liability increase is $219,146.48. Post-declaration surplus is $\$600,000-\$219,146.48=\$380,853.52$, giving a margin above required capital of -$19,146.48. The lower rate increases the incremental liability by $\$219,146.48-\$210,798.82=\$8,347.66$, to cents.

Check. The 2% basis produces the larger liability because the same expected payment is discounted less. The capital shortfall therefore increases by $8,347.66 when moving from the 4% basis to the 2% basis, with all other stated quantities held fixed.

Common Exam Trap

A frequent error is to value the nominal bonus as though it were certain to be paid, or to add its nominal amount directly to the reserve. In the portfolio example above, using $1,000/1.05=\$952.38$ per policy ignores the 0.90 survival probability. The correct value is:

\frac{1,000(0.90)}{1.05} =\frac{900}{1.05} =\$857.14 \quad \text{per policy, to cents.}

The incorrect calculation overstates the liability by $\$952.38-\$857.14=\$95.24$ per policy, or approximately $95,238.10 for 1,000 policies. The correction is not merely a rounding adjustment: it follows from the stated condition that payment is made only if the policy survives. Conversely, if the benefit is guaranteed regardless of survival, applying a survival factor would understate its value.

A sound declaration assessment therefore states the payment condition, timing, valuation assumptions, and portfolio count before calculating. It then reports both the post-declaration surplus and the comparison with required capital; those are different conclusions.

Key takeaway: A bonus declaration increases future policy obligations by the present value of its additional expected payments. Under unchanged assets, that increase reduces the surplus and a simplified current shareholder margin, while the asset share changes only if the stated asset-share method or an actual cash flow changes it. Test capital adequacy separately from whether surplus remains positive. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

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

A proposed $400 benefit is payable one year from now only if the policy survives. The one-year survival probability is 0.95 and the effective annual valuation interest rate is 4%. What is the incremental liability per policy? (Answer: $365.38)
A portfolio has 250 identical policies. Each receives a proposed $800 addition payable in one year if in force; the one-year survival probability is 0.95 and the effective annual valuation interest rate is 5%. What is the aggregate incremental liability? (Answer: $180,952.38)
A fund has assets of $4,200,000, existing liabilities of $3,700,000, and a capital requirement of $400,000. A declaration increases liabilities by $300,000 with assets and the requirement unchanged. What is the post-declaration surplus and margin above required capital? (Answer: surplus $200,000; margin -$200,000, a $200,000 shortfall)
A portfolio has 600 policies. A proposed $500 addition is payable in two years if the policy is then in force; the cumulative two-year survival probability is 0.96 and the effective annual valuation interest rate is 4%. What is the aggregate incremental liability? (Answer: $266,272.19)

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