Testing Equity-Linked Values Under Investment Shocks

Why Test Equity-Linked Values Under Investment Shocks?

An equity-linked policy connects at least part of its benefit to an investment fund. A change in investment returns can therefore affect the policyholder’s fund, the insurer’s guarantee cost, and the reserve held for future payments. But the effect depends on the contract: a benefit based only on the terminal fund may respond differently from one that records a high-water mark along the way.

An investment shock test recalculates the relevant values after changing one or more assumed returns. Comparing scenarios helps distinguish investment sensitivity—how much a value changes when returns change—from path dependence—whether the order of returns matters, even when the returns have the same cumulative effect. These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: For a fund with value $F_0$ at time 0 and annual fund returns $r_1,\ldots,r_n$, with no intervening cash flows or deductions, the value after $n$ years is $F_n=F_0\prod_{j=1}^{n}(1+r_j)$. A maturity guarantee with amount $G_n$ pays total benefit $\max(F_n,G_n)$; the insurer’s guarantee top-up is $\max(G_n-F_n,0)$. The actuarial discount rate $i$ is used to value future insurer cash flows and need not equal any fund return.

This basic projection assumes the stated annual returns apply over one-year periods and that the fund remains invested throughout. It does not account for mortality, premiums, withdrawals, fees, or other cash flows unless these are included explicitly. In the examples below, mortality is ignored: each policy survives to the stated maturity with certainty. The fund return scenarios are deterministic unless a scenario probability is specified.

Return Order: What Changes and What Does Not?

When a fund has no intervening cash flows or deductions, its terminal value depends on the product of the annual growth factors. Multiplication is commutative, so reordering the same returns does not change that product. However, a benefit that records the highest fund value at any anniversary can change when the returns are reordered.

Result: For a benefit depending only on $F_n$, reordering the same set of annual returns leaves the terminal fund and benefit unchanged. For a benefit depending on interim fund values, such as a ratchet, the order can change the guarantee and the benefit even when the terminal fund is identical.

The reason can be seen directly from the fund projection. For two returns $r_1$ and $r_2$, the fund at time 2 is $F_0(1+r_1)(1+r_2)$. Reversing them gives $F_0(1+r_2)(1+r_1)$, which is equal. But the intermediate values are $F_0(1+r_1)$ and $F_0(1+r_2)$, which need not be equal. A contract that uses the larger anniversary value can therefore produce different guarantees under the two orderings.

Worked Example: Same Terminal Fund, Different Ratchet

A hypothetical policy has an initial fund of $100 and a maturity guarantee that ratchets to the highest fund value at an anniversary, including time 0. Its maturity benefit is the greater of the terminal fund and the ratcheted guarantee. Compare returns of +20% and −16⅔% in both orders. Use a 5% effective annual actuarial discount rate. There are no charges, contributions, or deaths before maturity, and no probability distribution is assigned because each return path is tested deterministically.

Positive return first. After year 1, the fund is $100(1.20)=\$120$. After year 2, the fund is $120(5/6)=\$100$. The highest anniversary fund, including the initial $100, is $120. Thus the ratcheted guarantee is $120, the maturity benefit is $\max(100,120)=\$120$, and the insurer top-up is $20.

Negative return first. After year 1, the fund is $100(5/6)=\$83.33$ to the nearest cent. After year 2, it is $83.333\ldots(1.20)=\$100$. The highest anniversary fund is $100, so the guarantee and maturity benefit are each $100. The insurer top-up is zero.

Both paths have the same terminal value because $1.20(5/6)=1$. The present value at time 0 of the top-up under the first path is $20/(1.05)^2=\$18.14$; under the reversed path it is zero. The calculation is checked by noting that the terminal fund is $100 under each path, while only the positive-first path records an anniversary fund above $100. The method is to project each anniversary before applying the ratchet; the common mistake is to infer the guarantee from the terminal fund alone.

Valuing a Guarantee Under Scenario Weights

A deterministic stress test shows what happens under a specified path. If a valuation basis assigns weights to several possible paths, the expected guarantee top-up is the probability-weighted average of the top-ups under those paths. The reserve at time 0 is the expected future top-up discounted at the actuarial rate, provided the payment is at the stated future time and the valuation uses those scenario weights.

These weights are assumptions of the valuation basis. They are not automatically market prices or a claim that actual returns follow a particular statistical distribution. A reserve based only on the insurer’s guarantee top-up also does not include the fund portion of the policyholder benefit.

Worked Example: Scenario-Weighted Reserve After Upward and Downward Paths

A hypothetical two-year policy starts with a fund of $100 and has a maturity floor of $100. The fund earns either +10% in each year, with valuation weight 0.75, or −10% in each year, with valuation weight 0.25. These are the only two complete paths in the valuation distribution. Use an effective annual actuarial discount rate of 5%. Assume no charges, cash flows, or deaths before maturity.

Upward path. The terminal fund is $100(1.10)(1.10)=\$121$. The top-up is $\max(100-121,0)=\$0$.

Downward path. The terminal fund is $100(0.90)(0.90)=\$81$. The top-up is $\max(100-81,0)=\$19$.

The probability-weighted expected top-up at maturity is $0.75(0)+0.25(19)=\$4.75$. The time-0 reserve for this top-up is $4.75/(1.05)^2=4.75/1.1025=\$4.31$, rounded to the nearest cent. The result is checked by confirming that the expected payment is below the $19 payment in the downward path and above zero because that path has positive weight. The method is to apply the guarantee separately to each terminal fund, then weight the resulting payments; the common mistake is to apply the floor to an average fund value instead of averaging the path-specific top-ups.

The order of those operations matters because the top-up function $\max(G_n-F_n,0)$ is nonlinear. In general, the expected top-up is not the top-up calculated from the expected fund:

E[\max(G_n-F_n,0)] \ne \max(G_n-E[F_n],0).

In the example, the expected fund is $0.75(121)+0.25(81)=\$111$, which would suggest a zero top-up if the floor were applied only to that average. The correct expected top-up is $4.75. Applying the guarantee path by path preserves the effect of the downward scenario.

From a Shocked Claim to a Profit Result

A profit comparison requires a stated convention for reserves and cash flows. In a simple one-year guarantee test, suppose the insurer holds a reserve $V_0$ at the start of the year, the reserve earns the effective annual rate $i$, and the only maturity cash flow is the actual guarantee top-up $C_1$. With no premiums, expenses, or other cash flows, the insurer’s profit at the end of the year is the accumulated reserve less the actual top-up. A negative result is a loss on this simplified basis.

\text{Profit at time 1}=V_0(1+i)-C_1.

The formula follows from the cash-flow definition: the opening reserve grows to $V_0(1+i)$ by the end of the year, and the insurer then pays $C_1$. Subtracting the payment from the accumulated amount gives the remaining profit or loss. This is a scenario profit calculation, not a full profit test: premiums, expenses, taxes, and other contract cash flows would need to be added if present.

Worked Example: Actual Downward Shock Produces a Guarantee Loss

A hypothetical one-year policy starts with a fund of $100 and has a maturity floor of $105. For the valuation basis, the fund return is +10% with probability 0.75 or −10% with probability 0.25. The actuarial reserve earns an effective annual rate of 5%. There are no charges, premiums, expenses, or deaths during the year. After calculating the reserve, test an actual return of −20%.

Calculate the valuation-basis top-ups. Under +10%, the terminal fund is $100(1.10)=\$110$, so the top-up is zero. Under −10%, the fund is $100(0.90)=\$90$, so the top-up is $105-90=\$15$.

Calculate the opening reserve. The expected maturity top-up is $0.75(0)+0.25(15)=\$3.75$. Discounting one year at 5% gives $V_0=3.75/1.05=\$3.5714$, or $3.57 to the nearest cent. The reserve accumulated at the end of the year is $3.5714(1.05)=\$3.75$, before rounding.

Apply the actual shock. At −20%, the actual terminal fund is $100(0.80)=\$80$, so the actual top-up is $C_1=105-80=\$25$. The profit is $3.75-25=-\$21.25$. Thus this shock produces a $21.25 loss on the stated simplified profit basis. The calculation is checked against the reserve basis: the reserve accumulated to the expected claim of $3.75, but the actual claim is $25. The method is to revalue the actual shortfall under the shocked return and compare it with accumulated reserve; the common mistake is to use the expected claim of $3.75 as if it were the actual claim.

A Practical Shock-Test Sequence

A consistent process reduces errors when several values must be recalculated under alternative investment paths. Keep the contract definition, the investment projection, and the actuarial discounting separate. In particular, do not discount fund returns at the actuarial rate unless the contract or valuation basis explicitly says to do so.

Specify the contract and basis.
State the initial fund, benefit formula, valuation rate, any scenario weights, and which cash flows, charges, and decrements are included.

Project the fund by period.
Apply each return in its stated order and record anniversary values. Include any specified contributions, deductions, or withdrawals at their contractual times.

Apply the benefit formula to each path.
Calculate the guarantee, total benefit, or top-up separately for each scenario. Do not average fund values before applying a nonlinear guarantee.

Value and compare.
Weight payments if a scenario distribution is given, discount them using the stated actuarial basis, and compare any actual shocked payment with the reserve accumulated to the payment date.

Common Traps in Investment Sensitivity Calculations

Assuming every benefit depends only on the terminal fund. A ratchet or other anniversary-based feature can respond to the order of returns. Project all relevant intermediate values.
Applying the guarantee to an average fund. For a nonlinear top-up, calculate the payment in each scenario first and then take the weighted average.
Confusing the fund return with the valuation rate. Use returns to project the policy fund and $i$ to discount insurer cash flows, unless the problem explicitly defines a different relationship.
Comparing a total benefit with a top-up reserve. If the reserve is for the insurer’s guarantee cost, compare it with the insurer’s top-up cash flow, not with the fund amount paid from the policyholder’s account.
Changing several assumptions without tracking them. A sensitivity result is interpretable only when the shock and the assumptions held fixed are clear.

Key takeaway: Reproject the fund under each stated path, apply the contract’s benefit formula path by path, and only then calculate weighted values or profits. Return order is irrelevant to a terminal-only benefit when there are no intervening cash flows, but it can matter whenever the contract uses interim fund values.

Check Your Understanding

For each problem, assume the policy survives to the stated maturity and that no charges or other cash flows apply unless specified. Round money to the nearest cent.

A fund starts at $250 and earns +10% followed by −10%. The maturity floor is $250. What are the terminal fund and insurer top-up? (Answer: terminal fund $247.50; top-up $2.50.)
A $100 fund has a guarantee that ratchets to the greatest anniversary fund, including the initial fund. Compare returns of −10% then +11⅑% with the reversed order. What are the two ratcheted guarantees and terminal funds? (Answer: both terminal funds are $100; the first-order guarantee is $100 and the reversed-order guarantee is $111.11.)
A $100 fund has a two-year maturity floor of $100. With probability 0.40 it earns +10% in each year; with probability 0.60 it earns −10% in each year. If the effective annual actuarial rate is 5%, calculate the time-0 reserve for the top-up. (Answer: $10.34.)
A one-year policy starts with a $500 fund and a $520 maturity floor. The valuation returns are +8% with probability 0.60 and −4% with probability 0.40. The reserve earns 4%. If the actual return is −10%, calculate the opening reserve and the one-year profit or loss. (Answer: opening reserve $15.38; loss $54.00.)
A $100 fund receives a $20 contribution at the end of year 1, after the first year’s return. Compare +10% then −10% with −10% then +10%. A maturity floor is $120. Calculate both terminal funds and top-ups. (Answer: terminal funds $117 and $121, respectively; top-ups $3 and $0, respectively.)

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