Why Compare Product Designs Under the Same Scenarios?
A market loss does not create the same obligation for every long-term product. In a universal life contract, the credited rate and any minimum benefit affect what the insurer owes. In an equity-linked contract, the policy value follows an investment account, but a maturity guarantee can leave the insurer responsible for a shortfall. In a participating contract, a vested bonus may remain payable even when assets fall. Comparing these designs is useful only if the asset amounts, scenario returns, timing, and valuation basis are consistent.
This tutorial uses a deliberately small, deterministic one-year model to isolate those differences. It is a scenario comparison, not a complete regulatory capital calculation. In particular, it does not include mortality, lapse, expenses after issue, diversification, or management actions. These exclusions allow the effect of each benefit design to be seen directly.
A Common Basis for the Comparison
Assume each hypothetical contract receives a single premium of $1,000 at time 0. An issue expense of $50 is paid immediately, leaving $950 of supporting assets. All contracts remain in force and survive to the end of the year, when the relevant benefit is paid. There are no other cash flows. The supporting assets earn one of three annual returns: \(+10\%\), \(0\%\), or \(-20\%\). Therefore, for every design,
where \(r\) is the scenario return. The asset amounts are $1,045, $950, and $760, respectively.
The product rules are hypothetical and intentionally explicit:
- Universal life (UL): the account starts at $950 and is credited the greater of the scenario return and a 2% minimum rate. The maturity benefit is the greater of the resulting account value and a $1,000 minimum maturity benefit.
- Equity-linked: the linked account receives the scenario return without a credited-rate floor. The maturity benefit is the greater of the linked account value and a $900 maturity guarantee.
- Participating: the maturity benefit is a $900 guaranteed amount plus a $40 bonus already declared and vested. The benefit is therefore $940 in every scenario; it is not reduced when the investment return is negative.
The scenario return determines investment performance, but it does not by itself determine the liability. The contract rules turn that performance into an account value or a benefit. That distinction is central to interpreting the comparison.
Worked Example: Calculate All Three Designs Under a Common Stress Set
Using the assumptions above, calculate each design’s account value, end-of-year benefit reserve, and stressed funding shortfall in all three scenarios. The method is to project assets and benefits separately, then subtract the available assets from the benefit only when measuring the shortfall.
Step 1: Project supporting assets. For returns of \(+10\%\), \(0\%\), and \(-20\%\),
Step 2: Calculate UL account values and benefits. The credited rates are \(10\%\), \(2\%\), and \(2\%\), because the 2% minimum applies when the scenario return is below 2%. The account values are
The maturity benefit is the greater of account value and $1,000, giving benefits of $1,045, $1,000, and $1,000. The corresponding shortfalls are
Step 3: Calculate equity-linked benefits. The linked account values are the asset amounts: $1,045, $950, and $760. Applying the $900 guarantee gives benefits of $1,045, $950, and $900. The shortfalls are \(0\), \(0\), and \(900-760=140\).
Step 4: Calculate participating benefits. The vested benefit is $900+$40=$940 in each scenario. The shortfalls are \(0\), \(\max(0,940-950)=0\), and \(940-760=180\).
The resulting comparison is:
| Design and measure | Return +10% | Return 0% | Return -20% |
|---|---|---|---|
| UL account value | $1,045 | $969 | $969 |
| UL benefit reserve | $1,045 | $1,000 | $1,000 |
| UL shortfall | $0 | $50 | $240 |
| Equity-linked benefit reserve | $1,045 | $950 | $900 |
| Equity-linked shortfall | $0 | $0 | $140 |
| Participating benefit reserve | $940 | $940 | $940 |
| Participating shortfall | $0 | $0 | $180 |
Check. In every cell, the shortfall is nonnegative and equals the benefit in excess of assets, if any. The UL has the largest shortfall in the downturn in this particular example because its minimum credited rate and maturity benefit support a higher obligation. This is a result of the stated assumptions, not a general ranking of product types.
Why the Shortfall Formula Works
Immediately before the end-of-year benefit is paid, the insurer has assets \(A_1\) and owes benefit \(B_1\). The difference \(A_1-B_1\) is the amount remaining after payment. If that difference is negative, additional funds of \(B_1-A_1\) are needed to meet the obligation. If it is zero or positive, no additional funds are needed for this single payment. Combining these two cases gives the shortfall formula.
This reasoning also explains why a guarantee amount alone is not a capital measure. The relevant comparison depends on the benefit actually payable under the contract and the assets available in the same scenario and at the same time.
Worked Example: Measure the Effect of Raising an Equity-Linked Guarantee
Suppose the equity-linked account still begins at $950 and the assets earn \(-20\%\), but the maturity guarantee is increased from $900 to $1,000. Find the change in the end-of-year reserve and stressed shortfall. The method is to hold the scenario and linked account fixed and change only the contractual guarantee.
Step 1: Find the account and assets. The linked account and supporting assets are each
Step 2: Apply the original guarantee. With a $900 guarantee, the benefit is \(\max(760,900)=900\). The reserve is $900 and the shortfall is \(900-760=140\).
Step 3: Apply the higher guarantee. With a $1,000 guarantee, the benefit is \(\max(760,1{,}000)=1{,}000\). The reserve is $1,000 and the shortfall is \(1{,}000-760=240\).
Step 4: Compare. The reserve increases by \(1{,}000-900=100\), and the shortfall increases by \(240-140=100\). The entire increase reaches the shortfall because the account remains below both guarantee levels. If the account were above both guarantees, increasing the guarantee within that range would not change the benefit in that scenario.
Check. Assets are unchanged at $760, so the $100 increase in the payable benefit produces exactly a $100 increase in the deficit. The mistake to avoid is treating the guarantee increase as a $100 increase in every scenario: when the linked account exceeds both guarantee amounts, the guarantee does not determine the benefit.
Discounting a One-Year Stress Result
A time-1 reserve and a time-0 present value are different quantities. To compare a one-year obligation with values at time 0, use the same valuation date and discount basis for the benefit and the stressed asset amount. For the next example, assume a risk-free effective annual interest rate \(i=4\%\), so \(v=1/1.04\). This deterministic calculation is a present-value translation of the scenario shortfall; it is not a claim that a full solvency capital requirement is obtained by discounting one stress.
Worked Example: Discount Downturn Reserves and Shortfalls Consistently
Under the \(-20\%\) scenario, calculate the time-0 present value of the benefit and the present value of the funding shortfall for each design. Use the one-year benefits and assets from the common comparison, and discount at \(i=4\%\). The method is to discount both time-1 amounts by \(1.04\).
Step 1: Identify time-1 amounts. The supporting assets are $760. The benefits are $1,000 for UL, $900 for equity-linked, and $940 for participating.
Step 2: Discount each benefit. The present-value reserves are
Step 3: Discount each shortfall. At time 1, the shortfalls are \(1{,}000-760=240\), \(900-760=140\), and \(940-760=180\). Their present values are
Rounded to the nearest cent, the present-value reserves are $961.54, $865.38, and $903.85; the present values of the shortfalls are $230.77, $134.62, and $173.08. The arithmetic is consistent because discounting the benefit-asset difference gives the same result as subtracting the discounted amounts:
Check. In each case the present value of the benefit exceeds the present value of the stressed assets by the present value of the shortfall. Do not compare an undiscounted time-1 reserve with time-0 assets: that mixes valuation dates.
Interpreting Differences Between Designs
A useful comparison separates three questions. First, what benefit is contractually payable? The account value, guarantee, and any vested bonus determine that amount. Second, what assets are available in the scenario? Here those assets are calculated from the same initial amount and the same scenario return for every design. Third, what is the valuation measure? A benefit at time 1, its present value at time 0, and a stressed funding shortfall are related but not interchangeable.
The example also shows why product labels do not determine a universal solvency ranking. With the selected terms, the equity-linked guarantee produces the smallest downturn shortfall, while UL produces the largest. Changing the guarantee, the credited-rate floor, the bonus commitment, or the asset backing could change the ranking. A sound comparison states those terms rather than relying on a general claim such as “linked products have less guarantee risk.”
The approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus: translate product features into projected benefits and reserves, then interpret the result under a common stress basis. For a fuller solvency assessment, additional scenarios, decrements, expenses, capital definitions, and interactions must be specified.
Common Exam Traps
- Comparing only the guarantee amounts. The benefit may be greater than the guarantee if the account value is higher. Apply the contract benefit formula before calculating a reserve.
- Using one design’s asset amount for another scenario. Project supporting assets from the same starting amount and the return assigned to that scenario.
- Subtracting in the wrong direction. A shortfall is \(\max(0,\text{benefit}-\text{assets})\), not \(\max(0,\text{assets}-\text{benefit})\).
- Ignoring vested benefits. In this example, the declared participating bonus is payable even in the downturn. It cannot be removed merely because assets performed poorly.
- Mixing dates or measures. State whether a number is an end-of-year benefit reserve, a present value, or a funding shortfall. Discount consistently when converting between dates.
Check Your Understanding
For each problem, use the stated assumptions and distinguish the benefit reserve from the stressed funding shortfall.
- A hypothetical equity-linked contract has a $1,140 account at the start of the year, a \(-15\%\) investment return, and a $1,000 maturity guarantee. Supporting assets equal the starting account and earn the same return. Find the maturity benefit and shortfall. (Answer: benefit $1,000; shortfall $31.)
- A hypothetical UL account is $800 at the start of the year. The credited rate is the greater of the scenario return and 3%, while the maturity benefit is at least $850. The scenario return is \(-10\%\), and supporting assets earn \(-10\%\). Find the account value, benefit, and shortfall. (Answer: account value $824; benefit $850; shortfall $130.)
- A participating contract has supporting assets of $1,500 before a \(-12\%\) return. Its guaranteed maturity amount is $1,250, and a $70 bonus is already vested. Find the benefit and shortfall. (Answer: benefit $1,320; assets $1,320; shortfall $0.)
- At time 1, a scenario reserve is $945 and stressed assets are $810. If \(i=5\%\), find the present value of the reserve and of the stressed shortfall. (Answer: reserve present value $900.00; shortfall present value $128.57.)
- An equity-linked account is $1,000 at maturity and supporting assets are $820 under a stress. Compare the shortfall with a $950 guarantee and a $1,100 guarantee. (Answer: shortfalls are $180 and $280, respectively; the higher guarantee increases the shortfall by $100.)