Terminal Bonuses and Solvency at Maturity

Why a Terminal Bonus Needs Two Tests

A terminal bonus is an additional payment considered when a participating policy reaches maturity. It can allow some investment or other surplus to be shared with policyholders at that point. Unlike a guaranteed maturity benefit, however, a terminal bonus may be discretionary under the contract until it is declared or otherwise made binding. Its size can therefore depend on both a policy-level measure, such as asset share, and the insurer’s capacity to pay while maintaining its required solvency position.

Those are separate questions. An asset-share formula can indicate an amount that appears equitable for one policy, but that amount is not automatically affordable for the insurer as a whole. Conversely, a solvency surplus does not by itself establish how much of that surplus should be allocated to a particular policy. A sound calculation identifies the proposed payment, checks the contractual status of the benefit, and tests the resulting balance sheet.

Definition: A terminal bonus, denoted here by \(B_T\), is a maturity payment in addition to the policy’s guaranteed maturity benefit. Let \(G\) be that guaranteed benefit and \(A\) the policy’s asset share at maturity. Before a terminal bonus is declared or otherwise made binding under the contract, it is treated in this tutorial as discretionary, not as part of \(G\). The applicable contract determines when a bonus becomes guaranteed.

All amounts in the examples are stated at the maturity date, in thousands of dollars unless specified otherwise. Accumulation before maturity is reflected in the supplied amounts; there is no further time to discount or accumulate them at settlement, so the effective annual interest rate over that instant is 0%. Each example is deterministic: no probability distribution is used, and the policy is assumed to be in force and eligible for its stated maturity benefit. Mortality during the settlement instant is not modeled.

Asset-Share and Surplus Rules

One possible asset-share rule awards a specified proportion of the positive excess of asset share over the guaranteed benefit. With participation factor \(\alpha\), where \(0\leq\alpha\leq1\), the candidate bonus is

B_T^{(A)}=\alpha\max(A-G,0).

The positive-part function matters: an asset share below the guarantee does not create a negative bonus. This rule is a hypothetical allocation formula, not a statement that every participating contract uses the same method. A different hypothetical rule can allocate a specified fraction of distributable surplus, after allowance for guaranteed benefits and a solvency buffer.

Definition: Let \(R\) be assets available at maturity, \(L_G\) the total guaranteed liabilities payable or otherwise recognized at that valuation point, and \(K\) the required solvency buffer after those liabilities. If a proposed discretionary bonus is \(B_T\), the surplus remaining after the payment is \(R-L_G-B_T\). The payment preserves the buffer only if \(R-L_G-B_T\geq K\).

The distinction between the two methods is important. \(A\) is a policy-level measure used by the chosen asset-share rule. \(R\), \(L_G\), and \(K\) describe the solvency test. They need not be measured on the same basis, and the asset share is not automatically the insurer’s available assets. A bonus calculated from asset share must still pass the solvency test if the question requires it.

Why the Solvency Limit Has This Form

Suppose the insurer has assets \(R\) and must meet guaranteed liabilities \(L_G\). Before a discretionary payment, the residual surplus is \(R-L_G\). Paying \(B_T\) reduces that surplus dollar for dollar, leaving \(R-L_G-B_T\). Requiring at least \(K\) to remain gives

R-L_G-B_T\geq K \quad\Longleftrightarrow\quad B_T\leq R-L_G-K.

Since a bonus cannot be a negative payment, the maximum nonnegative amount allowed by this simplified solvency test is

B_T^{\max}=\max(R-L_G-K,0).

This is a cap, not necessarily a recommendation or an entitlement. If a proposed asset-share or surplus-rule bonus exceeds the cap, the calculation identifies a conflict between that proposal and the stated buffer requirement. It does not make a discretionary bonus guaranteed, nor does it tell the actuary how to resolve the conflict under a particular contract or governance process.

Establish the guaranteed amount.
Identify \(G\) and ensure that \(L_G\) includes the guaranteed liabilities relevant to the solvency test, without counting any item twice.

Calculate a candidate bonus.
Apply the stated asset-share or surplus rule, keeping its inputs distinct from solvency resources.

Test the payment.
Calculate \(R-L_G-B_T\) and compare it with \(K\). Report any shortfall separately from the bonus amount.

Worked Example: An Asset-Share Bonus That Exceeds the Solvency Capacity

A hypothetical participating policy has asset share \(A=1{,}300\) and guaranteed maturity benefit \(G=1{,}100\), both in thousands of dollars. The proposed asset-share rule pays 80% of positive excess asset share. For the insurer’s maturity-date solvency test, assets are \(R=1{,}230\), guaranteed liabilities are \(L_G=1{,}100\), and the required buffer is \(K=45\). Find the candidate bonus, test it, and determine the maximum bonus consistent with the buffer. All values are maturity-date amounts, with no further interest accumulation or discounting; the case is deterministic and assumes an eligible in-force policy.

Method. First apply the asset-share rule; then perform the separate solvency test. Avoid treating the policy’s asset share as if it were the insurer’s solvency assets.

The asset-share excess is \(1{,}300-1{,}100=200\). Applying the 80% factor gives the candidate bonus:

B_T^{(A)}=0.80(200)=160.

The post-payment surplus is \(R-L_G-B_T=1{,}230-1{,}100-160=-30\). Thus the proposed payment leaves a deficit of 30 thousand dollars relative to the guaranteed liabilities. It also falls short of the required buffer by \(45-(-30)=75\) thousand dollars.

The maximum bonus consistent with the buffer is \(1{,}230-1{,}100-45=85\). The asset-share rule proposes 160, which is \(160-85=75\) above that limit. The solvency test therefore fails; the calculation does not convert the 160 candidate into a guaranteed liability.

Check. With a bonus of 85, the remaining surplus is \(1{,}230-1{,}100-85=45\), exactly the required buffer.

Worked Example: Allocating a Fraction of Distributable Surplus

For a hypothetical maturity portfolio, assets are \(R=1{,}500\) thousand dollars. Other guaranteed liabilities are 1,100 thousand dollars, and the guaranteed maturity benefit for the policy under consideration is \(G=250\) thousand dollars. The required buffer after meeting those guarantees is \(K=60\) thousand dollars. The proposed rule allocates 70% of positive distributable surplus to the policy. Find the bonus and the surplus remaining after payment. All figures are maturity-date values; no further interest applies, and the calculation is deterministic with the policy in force at maturity.

Method. Calculate distributable surplus only after deducting both the other guaranteed liabilities and the policy’s guarantee, then apply the allocation fraction.

The guaranteed liabilities included in the solvency test total \(1{,}100+250=1{,}350\). After reserving the required buffer, distributable surplus is

D=\max(1{,}500-1{,}350-60,0)=90.

The terminal bonus under the stated rule is \(0.70(90)=63\) thousand dollars. After paying the guarantee and bonus, the remaining surplus is

$$ 1{,}500-1{,}350-63=87. $$

This is \(87-60=27\) thousand dollars above the required buffer. Equivalently, the payment uses 63 of the 90 thousand dollars distributable above the buffer, leaving \(90-63=27\) thousand dollars unallocated. The result checks: \(60+27=87\) thousand dollars remains after guaranteed liabilities.

Common mistake avoided. Applying 70% to assets, or to assets less only the other liabilities, would ignore either the guaranteed maturity benefit or the required buffer. Both must be deducted before applying this particular surplus rule.

Worked Example: A Shortfall Is Not a Negative Terminal Bonus

Under a simplified surplus rule, available assets at maturity are 100 dollars, the guaranteed maturity benefit is 100 dollars, and the required buffer is 10 dollars. The candidate discretionary bonus is the positive part of the surplus after both the guarantee and buffer. Find the raw residual, the bonus, and the solvency position. The figures are deterministic maturity-date amounts, with no further interest or discounting; mortality is not modeled because the policy is assumed to have reached maturity in force.

Method. Calculate the residual before applying the zero floor, then interpret the result. The raw amount above the guarantee and buffer is

$$ 100-100-10=-10. $$

The raw residual is negative 10 dollars. Therefore the terminal bonus under this rule is \(\max(-10,0)=0\), not negative 10 dollars. After paying the guaranteed benefit, assets remaining are \(100-100=0\), which is 10 dollars below the required buffer.

Check. The 10-dollar buffer shortfall agrees with the magnitude of the raw residual, while the discretionary payment remains zero. A solvency shortfall and a terminal bonus are different quantities.

Guarantees, Declarations, and Common Traps

For the solvency test, include benefits that are guaranteed under the stated contract or valuation basis in \(L_G\). Before a discretionary terminal bonus is declared or otherwise binding, keep it separate from those guarantees and test its proposed payment as \(B_T\). If the bonus becomes guaranteed under the contract’s terms, its status changes: it must then be treated consistently with the guaranteed obligations in the relevant calculation. This distinction is about the benefit’s status, not just its label.

A useful way to check a maturity calculation is to write down three amounts separately: the guaranteed benefit \(G\), the candidate discretionary bonus \(B_T\), and the post-payment surplus \(R-L_G-B_T\). This makes it less likely that the bonus is counted both inside and outside the guaranteed liabilities, or that an asset-share figure is mistaken for a solvency resource.

Common exam trap: Do not report a negative residual as a negative bonus. First calculate the raw amount available after guarantees and the buffer; then apply the rule’s zero floor to the discretionary payment. Report any remaining shortfall separately. Also do not conclude that a bonus is affordable merely because an asset-share rule produces it: verify that the payment leaves at least \(K\).

Terminal-bonus questions often combine a policy-level allocation rule with an insurer-level constraint. The reliable approach is to calculate each on its own terms, state whether the result is only a proposed discretionary payment or a guaranteed liability, and then reconcile the amounts through the solvency equation. This distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated rule and distinguish the calculated bonus from any solvency shortfall. All amounts are deterministic values at maturity; assume no further interest or discounting and that the policy is in force and eligible for its stated benefit.

A policy has asset share \(A=720\), guarantee \(G=600\), and an asset-share bonus factor of 75%. The insurer has \(R=800\), \(L_G=650\), and required buffer \(K=50\), all in thousands of dollars. Find the candidate bonus, the post-payment surplus, and whether the buffer is met. (Answer: Candidate bonus \(90\); post-payment surplus \(60\); buffer met by \(10\).)
A surplus rule allocates 60% of distributable surplus. Assets are \(R=2{,}000\), other guaranteed liabilities are 1,400, the policy guarantee is \(G=300\), and \(K=100\), all in thousands of dollars. Find the bonus and remaining surplus above the buffer. (Answer: Distributable surplus \(200\); bonus \(120\); remaining surplus \(180\), or \(80\) above the buffer.)
A proposed asset-share bonus is 140 thousand dollars. The insurer has \(R=1{,}050\), \(L_G=900\), and \(K=40\), all in thousands of dollars. Find the maximum affordable bonus and the amount by which the proposal exceeds it. (Answer: Maximum bonus \(110\); proposal exceeds it by \(30\).)
Available assets are 500 thousand dollars, guaranteed liabilities are 470 thousand dollars, and the required buffer is 40 thousand dollars. A rule sets the bonus equal to the positive part of the residual after liabilities and buffer. Find the raw residual, bonus, and buffer shortfall if no bonus is paid. (Answer: Raw residual \(-10\); bonus \(0\); buffer shortfall \(10\).)
A portfolio has assets of 1,800 thousand dollars, guaranteed liabilities of 1,500 thousand dollars, and a required buffer of 100 thousand dollars. An asset-share rule proposes a bonus of 160 thousand dollars. How much surplus remains after the proposal, and by how much does it miss the buffer? (Answer: Surplus after payment \(140\); buffer exceeded by \(40\).)

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