Capital Effects of Universal Life Guarantees

Introduction

A universal life contract links its account value to premiums, charges, and credited interest, while also promising benefits that may not be limited to that account value. When investment performance is weak, a minimum crediting rate can require the insurer to credit more than its assets earned. A minimum account value or death-benefit floor can create an additional payment when the account is insufficient. These guarantees can increase liabilities precisely in adverse conditions.

The capital effect is not automatically equal to the amount of a guarantee or to its expected payment. It depends on the valuation basis, the stress being tested, the other contract cash flows, and the change in asset values. The calculations below isolate specific guarantee effects under explicitly stated hypothetical assumptions. They are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $A_t$ denote a universal life policy’s account value at time $t$. A minimum crediting guarantee requires the credited rate to be at least a stated rate, even if the insurer’s investment return is lower. A minimum account-value guarantee pays an additional amount if an account value at a specified date is below a stated floor. A death-benefit floor makes the death benefit at least a stated amount, even when $A_t$ is smaller.

From Contract Guarantees to Capital Effects

For a period with no premiums or charges, a guaranteed annual crediting rate $g$ gives the account value $A_{t+1}=A_t(1+g)$, if the guarantee determines the credited rate. With premiums or charges, their timing must be specified before applying the crediting rate; for example, a contract may credit interest to the account after a premium and charge have been applied. A stress to the insurer’s earned investment return does not, by itself, change the contract’s guaranteed rate.

If a maturity guarantee sets a minimum value $G$, its incremental payment at maturity is the shortfall, not the entire floor:

(G-A_T)^+ = \max(G-A_T,0).

If a death-benefit floor is $F$, the death benefit is $\max(A_t,F)$. When the ordinary account-value benefit is already included in the liability, the incremental amount due to the floor is $(F-A_t)^+$. This separation prevents counting the account value twice.

Capital analysis compares the insurer’s position under a specified valuation basis or stress. In a simplified comparison, the decrease in own funds is the increase in liabilities plus any decrease in asset value:

\text{own-funds loss} = \Delta L-\Delta A,

where $\Delta L$ is the change in liabilities and $\Delta A$ is the change in the value of assets backing the business. Thus an asset-value decrease has $\Delta A<0$ and adds to the own-funds loss. This relationship is an accounting comparison, not a universal formula for a regulatory capital requirement. A prescribed capital measure can apply specified stresses, valuation rules, and other adjustments; a guarantee’s liability increase alone does not determine the full requirement.

Key distinction: The account value is an ordinary contract obligation when it is payable under the policy. A guarantee’s incremental cost is the additional amount above that obligation, valued under the stated assumptions. A stressed investment return may affect both the account value and the insurer’s assets, so both sides of the balance sheet may need attention.

Why the Incremental Guarantee Formula Works

Consider a policy with account value $A$ and a death-benefit floor $F$. The contract pays $\max(A,F)$ on death, whereas the account-value-only benefit would pay $A$. The incremental payment is therefore

\max(A,F)-A = \begin{cases} F-A, & F>A,\\ 0, & F\leq A, \end{cases} = (F-A)^+.

This piecewise result explains two important features. First, the floor has no incremental cost when the account already exceeds it. Second, the guarantee’s value is sensitive to assumptions that affect the account or the likelihood and timing of the guaranteed payment. For a random account value and a random time of death, the expected incremental payment depends on the joint distribution of those quantities, not just on the stated floor.

Worked Applications

Worked Example: One-Year Crediting and Death-Benefit Guarantees

A hypothetical insurer has 100 identical in-force universal life policies. Each has a current account value of $10,000. There are no premiums, charges, lapses, or expenses during the next year. The contract guarantees 2% annual crediting, although the stressed investment return is 0%. At year-end, the death benefit is the greater of the account value and $11,000. Assume the one-year mortality probability is 0.0100, death and survivor account payments occur at year-end, and the valuation discount rate is 4%. Calculate the present-value increase relative to a contract with neither guarantee, under which the account remains $10,000 and the death benefit equals the account value.

Method. Project the guaranteed account value, calculate the expected year-end payment under each contract, and discount their difference. This comparison includes both the crediting guarantee and the death-benefit floor.

The guaranteed account value is

A_1=10{,}000(1.02)=\$10{,}200.

Under the guaranteed contract, a survivor receives the $10,200 account value. A death claim is $11,000 because the floor exceeds $10,200. The one-policy expected year-end payment is

(1-0.0100)(10{,}200)+0.0100(11{,}000) =0.99(10{,}200)+0.01(11{,}000) =10{,}098+110 =\$10{,}208.

For 100 policies, the expected payment is $100(10{,}208)=\$1{,}020{,}800$. Without either guarantee, the payment is $10,000 per policy regardless of whether the policyholder survives or dies, so the expected portfolio payment is $100(10{,}000)=\$1{,}000{,}000$. The increase at year-end is $20,800. At a 4% effective annual discount rate, its present value is

\frac{1{,}020{,}800-1{,}000{,}000}{1.04} =\frac{20{,}800}{1.04} =\$20{,}000.00.

Check. The guaranteed-contract expected payment per policy is $10,208, which exceeds the $10,000 counterfactual by $208. Across 100 policies, $208 multiplied by 100 is $20,800, consistent with the portfolio calculation. The present value is lower than the year-end increase because the discount rate is positive.

Common mistake avoided. Applying the 2% credit only to survivors would miss the account increase in the death benefit where the floor does not bind. Here the guaranteed account is $10,200 for all policies; the death benefit is then raised to $11,000 only on death.

Worked Example: Minimum Account Value at Maturity

A hypothetical universal life contract has an account value of $10,000 today and a 1% minimum annual crediting rate for five years. At the end of year five, the contract guarantees a minimum account value of $11,000. Assume the stressed investment return is 0% each year, there are no premiums, charges, or lapses, and the probability that the policy is in force at maturity is 0.8000. Value only the incremental maturity guarantee, using an effective annual discount rate of 3%.

Method. Accumulate the account at the contractual guaranteed rate, find the maturity shortfall, weight by the probability of reaching maturity in force, and discount five years.

The account value under the minimum crediting rate is

A_5=10{,}000(1.01)^5 =10{,}000(1.0510100501\ldots) =\$10{,}510.100501\ldots.

The guarantee pays the amount by which the $11,000 floor exceeds that value:

(11{,}000-A_5)^+ =11{,}000-10{,}510.100501\ldots =\$489.899499\ldots.

The expected incremental payment at maturity is

0.8000(489.899499\ldots)=\$391.919599\ldots.

Since $(1.03)^5=1.1592740743\ldots$, the present value is

\frac{391.919599\ldots}{(1.03)^5} =\$338.07.

The answer is rounded to the nearest cent only at the final step. It is the value of the incremental floor payment, not the present value of the entire account value or the full $11,000 floor.

Common mistake avoided. Crediting 0% because the stressed investment return is 0% contradicts the contract guarantee. The account must be projected at 1% under these assumptions; the stress can also affect asset values, but that is not part of this isolated liability calculation.

Worked Example: Mortality Stress on a Death-Benefit Floor

A hypothetical portfolio has 10,000 identical universal life policies, each with an account value of $50,500 at the end of the coming year. The death benefit is the greater of the account value and $60,000. Compare a base one-year mortality probability of 0.0050 with a stressed probability of 0.0100. Assume the probabilities apply independently to each policy, the account-value portion is already included in the liability under both bases, there are no other changed cash flows, and the discount rate is 4%. Find the present-value increase attributable to the death-benefit floor under the mortality stress.

Method. Isolate the floor excess per death, multiply by the change in expected death count, and discount one year. Because the account value is already included, only the excess over it is counted.

The incremental payment on a death claim is

60{,}000-50{,}500=\$9{,}500.

The increase in mortality probability is $0.0100-0.0050=0.0050$. Across 10,000 policies, the increase in expected death claims is $10{,}000(0.0050)=50$. Therefore, the increase in expected floor payments at year-end is

50(9{,}500)=\$475{,}000.

Discounting one year gives

\frac{475{,}000}{1.04}=\$456{,}730.77.

The result is the isolated present-value increase due to the floor under this mortality comparison. It does not include the account-value death benefit, which was assumed to be present in both liability calculations.

Common mistake avoided. Multiplying the change in expected deaths by $60,000 would count the account-value amount again. The incremental floor exposure per death is $9,500, not the full death benefit.

Interpreting the Results for Capital

The examples show three different channels. A minimum crediting rate can raise account values relative to a low-return stress. A maturity floor pays only when the projected account falls below the specified minimum. A death-benefit floor creates an excess payment only on deaths for which the account value is below the floor. Each guarantee is contingent on the contract remaining in force and on the relevant shortfall or claim occurring.

For a capital assessment, the actuary must align the comparison with the valuation question. The assumed in-force population, mortality, lapses, expenses, premiums, charges, interest rates, and timing of cash flows all matter. If a guarantee changes policyholder behavior or the duration of the liability, those effects may also matter when the model includes them. The simple examples deliberately hold other items fixed so that a particular guarantee effect can be identified.

A liability increase is not necessarily the total capital impact. For example, if the same adverse scenario also reduces the value of assets, the own-funds loss includes both effects. Conversely, asset changes may partly offset a liability increase. The capital requirement itself depends on the applicable framework and its prescribed measurement; it should not be inferred solely from an isolated expected guarantee payment.

Common Errors to Avoid

Counting the whole floor as an extra payment. If the account value is already included, the incremental amount is only the positive shortfall below the floor.
Ignoring the contract minimum rate. A stressed earned return below the minimum does not allow the projected credited rate to fall below the guarantee.
Applying a floor when it does not bind. If the account value exceeds the death-benefit floor, the floor creates no additional death payment.
Mixing up expected payment and present value. Apply the stated probability and timing, then discount cash flows to the valuation date.
Calling a liability change the full capital requirement. Capital analysis also depends on the asset side and the governing valuation and stress rules.

Key takeaway: Project the guaranteed account under the contract terms, identify only the payment above the ordinary account-value obligation, and value that contingent excess under the stated assumptions. Then assess how the liability effect and any associated asset change alter the insurer’s position.

Check Your Understanding

For each problem, isolate the quantity requested and apply the stated timing and assumptions.

A policy has an $8,000 account value, a 1.5% minimum annual credit, a $9,000 death-benefit floor, and one-year mortality probability 0.0200. There are no other cash flows. Find the present value of the expected year-end payment using a 5% effective annual discount rate. (Answer: $7,750.10)
A $20,000 account earns the guaranteed 2% annually for five years. At maturity, a $23,000 minimum account value is payable if the policy is in force; the probability of being in force is 0.9000. Find the present value of the incremental floor using a 4% effective annual discount rate. (Answer: $679.36)
A portfolio has 5,000 policies with a $30,000 account value and a $35,000 death-benefit floor. One-year mortality rises from 0.0080 to 0.0120 under stress. Find the present-value increase in the floor liability at a 3% discount rate, assuming the account-value liability is unchanged. (Answer: $97,087.38)
Under a specified stress, universal life liabilities increase by $1.2 million and backing assets decrease in value by $0.3 million. What is the resulting decrease in own funds, before any other adjustments? (Answer: $1.5 million)

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