Reserve Sensitivity to Guaranteed Benefit Changes

Why Guaranteed Benefit Changes Need Two Comparisons

A change to a linked product’s guarantee can affect both the expected amount payable and the uncertainty around that payment. Raising a maturity guarantee may increase the expected top-up, but the effect is not necessarily proportional to the increase in the guarantee: the top-up is zero in scenarios where the fund value exceeds the guarantee. Extending the guarantee period can also change expected fund values, the probability the contract remains eligible, and the time value of the payment.

A useful analysis therefore keeps two comparisons distinct. First, recalculate the expected benefit and its reserve on the stated valuation basis. Second, recalculate the reserve under a specified adverse stress and compare the stressed and best-estimate reserves. The second comparison is an illustrative stress measure in this tutorial, not a statement of a regulatory capital formula.

Definition: Let $G$ be a guaranteed maturity amount and $F_n$ the linked fund value at time $n$. If the guarantee is payable only when the contract remains eligible at maturity, the top-up is $(G-F_n)^+=\max(G-F_n,0)$. If $\pi_n$ is the probability that the benefit is payable, and eligibility is independent of $F_n$, the time-0 reserve on a deterministic effective annual interest rate $i$ is $$ V_0=v^n\pi_n E[(G-F_n)^+],\qquad v=\frac{1}{1+i}. $$ If eligibility and fund value are dependent, the expected payment must instead be evaluated jointly; multiplying separate probabilities and expectations is not generally valid.

The reserve formula assumes a single payment at time $n$, a known discount rate, and a clearly specified probability distribution for the fund value. For survival-contingent benefits, $\pi_n$ may be ${}_np_x$. For a product where survival and continued eligibility both matter, $\pi_n$ is the probability of satisfying all payment conditions. Any premium, fee, other benefit, or expense must be included separately if it is part of the liability being valued.

Changing the Guarantee Amount

For a fixed fund value $F_n$, raising $G$ cannot reduce the top-up. But the expected top-up is determined by applying the floor scenario by scenario, then averaging. In particular, $E[(G-F_n)^+]$ is generally not equal to $\max(G-E[F_n],0)$. This nonlinearity is central to valuing guarantees and to assessing how a change in guarantee amount affects a stressed reserve.

Worked Example: Increasing a Maturity Guarantee

A hypothetical equity-linked contract pays a maturity top-up after two years if the policyholder survives. The effective annual interest rate is 5%, annual survival probability is 0.98, and survival is independent of the fund value. Conditional on survival, the fund value in thousands of dollars is 80, 100, or 120, with probabilities 0.25, 0.50, and 0.25. Revalue the guarantee when it increases from 100 to 110 thousand dollars. Then calculate an illustrative stress measure using a specified adverse fund distribution.

Method. Calculate the top-up in each scenario, take its expectation, and multiply by the survival-and-discount factor. Keep full precision until reporting the final figures to four decimal places in thousands of dollars.

The two-year survival probability and discount factor are

{}_2p_x=0.98^2=0.9604,\qquad v^2=\frac{1}{1.05^2}=\frac{1}{1.1025}.

For a guarantee of 100, the top-ups at fund values 80, 100, and 120 are 20, 0, and 0. Thus

$$ E[(100-F_2)^+]=0.25(20)+0.50(0)+0.25(0)=5. $$

The reserve is

V_0(100)=\frac{0.9604(5)}{1.1025}=4.355555\ldots\text{ thousand dollars}\approx 4.3556\text{ thousand dollars}.

For a guarantee of 110, the scenario top-ups are 30, 10, and 0. Therefore

$$ E[(110-F_2)^+]=0.25(30)+0.50(10)+0.25(0)=7.5+5=12.5. $$

The revised reserve is

V_0(110)=\frac{0.9604(12.5)}{1.1025}=10.888888\ldots\text{ thousand dollars}\approx 10.8889\text{ thousand dollars}.

The expected top-up rises by $12.5-5=7.5$ thousand dollars, and the reserve rises by $10.888888\ldots-4.355555\ldots=6.533333\ldots$ thousand dollars, or approximately 6.5333 thousand dollars.

Now impose a hypothetical adverse fund distribution: fund values 70, 95, and 115, with probabilities 0.40, 0.40, and 0.20. Keep the interest and survival assumptions unchanged. For $G=100$, the expected stressed top-up is $0.40(30)+0.40(5)+0.20(0)=14$ thousand. Its stressed reserve is $0.9604(14)/1.1025=12.195555\ldots$ thousand, or 12.1956 thousand. The illustrative stress measure, stressed reserve less best-estimate reserve, is $12.195555\ldots-4.355555\ldots=7.8400$ thousand.

For $G=110$, the stressed expected top-up is $0.40(40)+0.40(15)+0.20(0)=22$ thousand. The stressed reserve is $0.9604(22)/1.1025=19.164444\ldots$ thousand, or 19.1644 thousand. The stress measure is $19.164444\ldots-10.888888\ldots=8.275555\ldots$ thousand, or 8.2756 thousand. Thus, the guarantee increase raises this illustrative stress measure by about $8.275555\ldots-7.8400=0.4356$ thousand dollars.

Check. In both distributions, each scenario’s top-up is nonnegative and the higher guarantee never reduces it. The higher guarantee also raises both the best-estimate reserve and the stressed reserve. The stress measure rises here because the stressed distribution’s expected top-up increases by 9.5 thousand for the higher guarantee, compared with a 7.5-thousand increase on the best-estimate distribution.

Why the Reserve Formula Holds

The formula follows by valuing the payment in each outcome. The contract pays $(G-F_n)^+$ at time $n$ if it is eligible, and pays zero otherwise. With eligibility independent of fund value, the expected payment is the probability of eligibility multiplied by the expected top-up conditional on eligibility. A payment at time $n$ is discounted by $v^n$. Combining these gives the stated reserve.

Determine the payment in each fund scenario.
For each possible $F_n$, calculate $\max(G-F_n,0)$. This applies the guarantee floor before averaging.

Average the contingent payment.
Weight each scenario top-up by its probability to obtain $E[(G-F_n)^+]$.

Allow for payment conditions and timing.
Multiply by $\pi_n$ when eligibility is independent of the fund value, then discount by $v^n$. If the variables are dependent, average the joint payment directly.

Changing the Guarantee Duration

A guarantee date change is not just a discounting adjustment. The fund-value distribution at the new date may differ, the probability of payment may change, and the payment is made at a different time. To isolate the sources of change, first compare the expected top-ups at the two dates. Then compare their reserves using each date’s own eligibility and discount factor. If useful, an algebraic decomposition can hold the earlier valuation factor fixed for one component and attribute the residual to timing and eligibility.

Worked Example: Extending a Guarantee by One Year

A hypothetical linked contract guarantees 100 thousand dollars at maturity, payable only if the policyholder survives. The effective annual interest rate is 5%, and annual survival probability is 0.98, independent of the fund value. At the one-year date, fund values are 80, 100, and 120 thousand dollars with probabilities 0.20, 0.50, and 0.30. If the guarantee is extended to year two, the corresponding values are 70, 100, and 130 thousand dollars with the same probabilities. Calculate the reserves and their difference.

At year one, the expected top-up is $0.20(20)+0.50(0)+0.30(0)=4$ thousand dollars. The one-year survival-and-discount factor is $0.98/1.05=14/15$. Therefore

V_0^{(1)}=4\left(\frac{14}{15}\right)=3.733333\ldots\text{ thousand dollars}\approx 3.7333\text{ thousand dollars}.

At year two, the expected top-up is $0.20(30)+0.50(0)+0.30(0)=6$ thousand dollars. The two-year factor is $0.98^2/1.05^2=0.871111\ldots$, so

V_0^{(2)}=6\left(\frac{0.98^2}{1.05^2}\right)=5.226666\ldots\text{ thousand dollars}\approx 5.2267\text{ thousand dollars}.

The extension increases the reserve by $5.226666\ldots-3.733333\ldots=1.493333\ldots$ thousand dollars, or 1.4933 thousand dollars. The expected top-up itself increases by 2 thousand dollars, from 4 to 6. To separate that cash-flow effect from the timing and survival factor, use the year-one factor as a reference:

V_0^{(2)}-V_0^{(1)} =(6-4)\left(\frac{14}{15}\right) +6\left(\frac{0.98^2}{1.05^2}-\frac{14}{15}\right) =1.866666\ldots-0.373333\ldots =1.493333\ldots.

The first component is the increase in expected top-up valued at the earlier factor. The second is the adjustment from using the later factor on the new expected top-up. It is negative in this example because the year-two payment is more heavily discounted and less likely to be paid. The net reserve still increases because the expected top-up also increases.

For a separate stress comparison, suppose the year-one adverse fund values are 60, 90, and 120 with probabilities 0.20, 0.50, and 0.30. The stressed expected top-up is $0.20(40)+0.50(10)=13$ thousand. The stressed reserve is $13(14/15)=12.133333\ldots$ thousand, giving an illustrative stress measure of $12.133333\ldots-3.733333\ldots=8.4000$ thousand.

At year two, suppose the adverse values are 50, 80, and 120 with those same probabilities. The stressed expected top-up is $0.20(50)+0.50(20)=20$ thousand. The stressed reserve is $20(0.871111\ldots)=17.422222\ldots$ thousand, giving a stress measure of $17.422222\ldots-5.226666\ldots=12.195555\ldots$ thousand, or 12.1956 thousand. The extension therefore raises this stress measure by approximately 3.7956 thousand dollars.

Check. The year-two best-estimate reserve exceeds the year-one reserve, and the year-two stressed reserve also exceeds its best-estimate reserve. The extension’s effect on the stress measure is not inferred from the reserve increase alone; both bases were recalculated using their specified fund distributions.

Separating Expected Cash Flows from Risk Exposure

A reserve is a present value on a specified best-estimate basis. A stress measure compares that reserve with a reserve calculated under a defined adverse basis. This distinction matters: a larger reserve does not, by itself, establish that risk exposure has increased by the same amount. The cash-flow distribution may change differently under stress, or the payment probability and timing may change.

The following example includes a combined eligibility condition. The annual probability of remaining eligible is specified directly, so it need not be decomposed into survival and persistency assumptions. It is assumed independent of fund value.

Worked Example: Raising a Guarantee with an Eligibility Condition

A hypothetical linked contract pays a maturity top-up after two years only if it remains eligible. The annual probability of remaining eligible is 0.95, independently in each year and independently of the fund value. The effective annual interest rate is 5%. On the best-estimate basis, the two-year fund values are 80, 100, and 120 thousand dollars with probabilities 0.25, 0.50, and 0.25. Recalculate the reserve and illustrative stress measure when the guarantee rises from 100 to 110 thousand dollars. For stress, use fund values 70, 95, and 120 with probabilities 0.40, 0.40, and 0.20.

The two-year eligibility probability is $0.95^2=0.9025$, and the discount factor is $1/1.05^2=1/1.1025$. Their product is

\frac{0.9025}{1.1025}=\frac{361}{441}=0.818594104\ldots.

For $G=100$, the best-estimate expected top-up is $0.25(20)=5$ thousand dollars. The reserve is $5(361/441)=1805/441=4.092970\ldots$ thousand, or 4.0930 thousand dollars.

For $G=110$, the scenario top-ups are 30, 10, and 0, giving expected top-up $0.25(30)+0.50(10)=12.5$ thousand dollars. The reserve is $12.5(361/441)=4512.5/441=10.232426\ldots$ thousand, or 10.2324 thousand dollars. The guarantee change raises the best-estimate reserve by $7.5(361/441)=6.139455\ldots$ thousand, or 6.1395 thousand dollars.

Under stress, the expected top-up for $G=100$ is $0.40(30)+0.40(5)=14$ thousand dollars. The stressed reserve is $14(361/441)=5054/441=11.460317\ldots$ thousand. Subtracting the best-estimate reserve gives a stress measure of $9(361/441)=7.367346\ldots$ thousand, or 7.3673 thousand dollars.

For $G=110$, the stressed expected top-up is $0.40(40)+0.40(15)=22$ thousand dollars. The stressed reserve is $22(361/441)=7942/441=18.009070\ldots$ thousand. The stress measure is $(22-12.5)(361/441)=9.5(361/441)=7.776643\ldots$ thousand, or 7.7766 thousand dollars. Thus the guarantee increase raises the stress measure by about $7.776643\ldots-7.367346\ldots=0.4093$ thousand dollars.

Check. Both reserves use the same two-year eligibility-and-discount factor, so the difference in best-estimate reserves comes solely from the changed expected top-up. The stress measure changes by a different amount because the stressed and best-estimate distributions respond differently to the higher guarantee.

Common Mistakes and Practical Checks

A frequent error is to average the fund value before applying the guarantee floor. In the first worked example, the best-estimate mean fund value is $0.25(80)+0.50(100)+0.25(120)=100$. The incorrect shortcut would give $\max(100-100,0)=0$, and hence a zero expected top-up. The correct scenario calculation gives $0.25(20)+0.50(0)+0.25(0)=5$ thousand dollars. The shortcut loses the value created by the guarantee in the low-fund scenario.

Other checks help prevent errors in an integrated valuation:

Apply the floor scenario by scenario. A guarantee protects against low fund outcomes, so replacing a distribution with its mean generally changes the answer.
Use the correct payment date. A year-$n$ payment is discounted by $v^n$, not $v$ unless $n=1$.
Include every payment condition once. If eligibility probability already includes survival and continued coverage, do not multiply by those probabilities again.
State what the stress measure represents. A stressed reserve less a best-estimate reserve is useful for comparison, but is not automatically a prescribed capital requirement.
Check units and precision. If benefits are in thousands of dollars, reserves and stress measures are also in thousands until converted. Keep unrounded values through subtraction.

These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In a product-level analysis, the key discipline is to recalculate each basis consistently: identify the benefit, apply its guarantee floor, account for eligibility, discount to the valuation date, and then compare the best-estimate and stressed results.

Check Your Understanding

For each problem, calculate the requested amount before checking the answer in parentheses. Monetary answers are in thousands of dollars unless stated otherwise.

A one-year maturity top-up is payable only on survival. The effective annual interest rate is 4%, survival probability is 0.97, and the guarantee is 90. Conditional fund values are 70, 90, and 110 with probabilities 0.20, 0.50, and 0.30. Calculate the reserve. (Answer: 3.73 thousand dollars.)
A two-year guarantee of 100 is payable if the contract remains eligible. The annual probability of eligibility is 0.96, independently in each year, and the effective annual interest rate is 5%. Fund values at maturity are 80, 100, and 120 with probabilities 0.25, 0.50, and 0.25. Calculate the reserve. (Answer: 4.18 thousand dollars.)
A one-year guarantee rises from 100 to 108. The effective annual interest rate is 4%, payment probability is 0.98, and fund values 90, 100, and 120 have probabilities 0.25, 0.50, and 0.25. By how much does the reserve increase? (Answer: 5.65 thousand dollars.)
For a one-year guarantee of 100, the effective annual interest rate is 5% and payment probability is 0.90. The best-estimate fund values 80, 100, and 120 have probabilities 0.25, 0.50, and 0.25. Under stress, fund values are 80, 90, and 120 with probabilities 0.50, 0.20, and 0.30. Calculate stressed reserve less best-estimate reserve. (Answer: 6.00 thousand dollars.)
A guarantee of 100 is extended from year two to year three. The effective annual interest rate is 0%, and the annual eligibility probability is 0.95, independently across years and of fund value. The expected top-up is 4 thousand dollars at year two and 6 thousand dollars at year three. Calculate the reserve increase. (Answer: 1.53 thousand dollars.)

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