Equity Return and Volatility Scenarios for Linked Business

Introduction

An equity-linked policyholder’s account moves with the specified investment returns, subject to charges and other contract cash flows. The insurer’s financial result is not the same as that account movement: a guarantee can require an additional payment when the account is insufficient, while charges and expenses affect the insurer’s margin. A sound scenario projection keeps these quantities separate.

This tutorial projects hypothetical linked business under stated return paths and volatility scenarios. It focuses on the arithmetic of the projection and on interpreting the results. A scenario result is conditional on its stated assumptions; it is not, by itself, a forecast or a probability-weighted measure of risk.

Model definition: Let $A_t$ be the policy account immediately after the end-of-year charge in year $t$, and let $r_t$ be the gross equity return during that year. With no new premiums or withdrawals and an end-of-year charge $c_t$, the account follows $A_t=A_{t-1}(1+r_t)-c_t$. If a maturity guarantee $G$ is payable at the end of year $n$, the insurer’s top-up is $C_n=\max(G-A_n,0)$. These equations assume the policy remains in force, charges are deducted as specified, and the guarantee is not reduced by other contract terms.

A return path means the ordered sequence of returns, not just their arithmetic average. The recurrence assumes the listed returns apply to the account before that year’s charge; applying a charge at the start instead would produce a different result. Unless a problem explicitly specifies otherwise, do not add mortality, lapse, taxes, or investment income on insurer cash flows to this simple projection.

Projecting an Account Through a Return Path

Calculate one year at a time. This makes the timing transparent and helps prevent the common error of applying the same return to the original account balance in every year. The following example uses hypothetical gross returns and fixed end-of-year charges.

Worked Example: Account Value and a Maturity Floor

A hypothetical policy begins with an account of $1,000. Its gross equity returns are 12% in year 1, -8% in year 2, and 5% in year 3. A $10 charge is deducted at the end of each year. The maturity guarantee is $1,075 at the end of year 3. Find the projected account and the guarantee top-up.

Method. Apply $A_t=A_{t-1}(1+r_t)-10$ in each year, then compare the maturity account with the guaranteed amount.

Year 1: the return is applied to $1,000, and then the $10 charge is deducted:

A_1=1{,}000(1.12)-10=1{,}120-10=1{,}110.

Year 2: the 8% loss applies to the year 1 account, not to the original $1,000:

A_2=1{,}110(0.92)-10=1{,}021.20-10=1{,}011.20.

Year 3:

A_3=1{,}011.20(1.05)-10=1{,}061.76-10=1{,}051.76.

The guarantee top-up is therefore $C_3=\max(1{,}075-1{,}051.76,0)=\$23.24$. The policy account at maturity is $1,051.76; the guaranteed maturity payment, including the top-up, is $1,075. The top-up is an insurer outgo, not an amount already included in the account. Check: the account is below the guarantee by exactly $23.24.

Common mistake avoided: applying each return to the initial account, or treating the guarantee as the account balance. The account is rolled forward from the prior year, and the top-up is calculated only after the maturity account has been projected.

Useful expansion: Repeated substitution in the account recurrence gives

A_n=A_0\prod_{j=1}^{n}(1+r_j)-\sum_{t=1}^{n}c_t\prod_{j=t+1}^{n}(1+r_j),

where a product with no factors equals 1. The initial account earns every subsequent return. A charge deducted at the end of year $t$ does not earn the return in year $t$, but it is reflected in the account exposed to later returns. For example, $c_1$ is multiplied by returns from years 2 through $n$, whereas $c_n$ is not multiplied by any return.

Why Return Order Can Matter

Without charges or other cash flows, multiplying the same annual growth factors in a different order gives the same final account. With fixed dollar charges, however, the amount left to earn later returns depends on when those charges are deducted. Consequently, two paths with the same returns, but in a different order, can produce different account values.

Worked Example: Reordering Two Equity Returns

A hypothetical policy starts with $100, earns gross returns of +20% and -20% over two years, and has a $2 end-of-year account charge in each year. The maturity guarantee is $95. Compare the paths +20%, then -20%, and -20%, then +20%. For this example only, define the undiscounted two-year insurer cash margin as collected charges less two $1 expenses and the maturity top-up.

Path A: +20%, then -20%. The account after year 1 is $100(1.20)-2=\$118$. After year 2 it is $118(0.80)-2=\$92.40$. The top-up is $\max(95-92.40,0)=\$2.60$. Charges total $4 and expenses total $2, so the specified cash margin is $4-2-2.60=-\$0.60$.

Path B: -20%, then +20%. The account after year 1 is $100(0.80)-2=\$78$. After year 2 it is $78(1.20)-2=\$91.60$. The top-up is $\max(95-91.60,0)=\$3.40$. The cash margin is $4-2-3.40=-\$1.40$.

Both paths have the same two returns and the same arithmetic average return, 0%. Yet the final accounts differ by $0.80, and the top-ups differ by $0.80. The charges are deducted at different account levels, which changes how much of the remaining balance receives the later return. Check: in each path, charges of $4 are deducted from the account in total; the difference arises from the returns earned after the first charge.

Common mistake avoided: concluding that equal average returns imply equal outcomes. They do not guarantee equal account values when cash flows occur during the projection, and they do not imply equal guarantee payments. The cash margin here is a deliberately simple, undiscounted measure, not a general profit-test present value.

Volatility Scenarios and Guarantee Costs

A volatility scenario specifies how returns vary across time or possible outcomes. To evaluate expected guarantee payments, the scenario must include probabilities as well as account projections. In contrast, a deterministic stress path has no probability weighting unless one is explicitly supplied.

The guarantee payment is nonlinear in the account value: shortfalls create payments, while account values above the floor do not create negative payments. Therefore, averaging account values first and then applying the floor can miss guarantee costs. The following example compares a low-variation path with an explicitly defined higher-variation distribution.

Worked Example: Expected Top-Up Under a Two-Year Return Distribution

A hypothetical account starts at $100. In each of two years, the gross return is independently +10% or -10%, each with probability $0.5$. A $1 charge is deducted at each year-end. The maturity guarantee is $95. Find the expected account value, expected top-up, and the simplified expected insurer cash margin if total expenses are $1 and charges are collected regardless of returns.

Method. Enumerate the four equally likely return paths, project each account, and calculate the top-up path by path. The probability of each path is $0.5\times0.5=0.25$.

For path ++, $A_1=100(1.10)-1=109$, then $A_2=109(1.10)-1=118.90$, so the top-up is $0. For path +-, $A_1=109$, then $A_2=109(0.90)-1=97.10$, so the top-up is $0. For path -+, $A_1=100(0.90)-1=89$, then $A_2=89(1.10)-1=96.90$, so the top-up is $0. For path --, $A_1=89$, then $A_2=89(0.90)-1=79.10$, so the top-up is $95-79.10=\$15.90$.

The expected maturity account is:

E[A_2]=0.25(118.90+97.10+96.90+79.10)=0.25(392.00)=\$98.00.

The expected top-up is:

E[C_2]=0.25(0+0+0+15.90)=\$3.975.

The insurer collects $2 in charges and has $1 of expenses, so the simplified expected cash margin is $2-1-3.975=-\$2.975$. This is an undiscounted expectation under the stated four-path model. It is not the same as an account-value expectation or a discounted profit measure.

Check and interpretation: applying the floor to the expected account would give $\max(95-98,0)=\$0$, which is not the expected top-up of $3.975. The top-up must be calculated for each outcome before averaging. The calculation also shows why a positive expected account surplus over the guarantee does not rule out positive expected guarantee cost.

Common mistake avoided: replacing $E[\max(G-A_2,0)]$ with $\max(G-E[A_2],0)$. These quantities are generally unequal because the maximum operation is nonlinear.

Takeaway: A return path determines an account projection; a probability distribution over paths determines expected scenario outcomes. Guarantee outgo must be calculated from each path’s account value before it is averaged. Neither an arithmetic average return nor an average account alone describes the distribution of guarantee payments.

Reading the Results Correctly

For each scenario, report the account value and insurer cash flows as separate quantities. A useful presentation distinguishes at least the following:

Policy account: the value remaining after returns and account deductions.
Guarantee top-up: any additional payment needed to meet the specified floor.
Charges and expenses: amounts received by the insurer and amounts it pays, kept distinct from the account unless the contract explicitly says otherwise.
Insurer result: a stated margin or valuation measure, with its timing and discounting convention identified.

A change in the account is not automatically an insurer loss. If an equity return is negative, the policyholder account may fall; the insurer’s direct exposure depends on the product terms, including whether a guarantee is triggered and how charges respond. Conversely, an account above its floor does not mean every insurer cash flow is positive: expenses may exceed charges.

When comparing scenarios, preserve the contract mechanics and change only the return inputs unless the scenario explicitly changes another assumption. State whether returns are annual, whether charges are fixed dollars or a percentage of the account, when deductions occur, and whether probabilities are assigned. These details determine what is being compared.

Common Exam Traps

Using the arithmetic average as a compounded return. Compound the stated annual factors in sequence and apply intervening charges at their specified times.
Applying a return to the wrong balance. Each year’s return applies to the balance entering that year, before any end-of-year deduction in this model.
Subtracting a guarantee from the account during every year. In the examples, the guarantee is a maturity payment; calculate its top-up only at maturity.
Averaging before applying the guarantee. For a distribution of outcomes, calculate $\max(G-A_n,0)$ separately in each outcome, then probability-weight the payments.
Calling a scenario margin a present value. A cash margin that adds nominal amounts is not discounted profit. State the measurement basis before interpreting the result.

Check Your Understanding

For each problem, follow the stated timing convention and calculate the requested quantities before checking the answer.

A hypothetical account begins at $200. Gross returns are -10% in year 1 and +10% in year 2. A $3 charge is deducted at each year-end, and the maturity guarantee is $190. Find the maturity account and top-up. (Answer: account $191.70; top-up $0.00)
A hypothetical account begins at $500, earns +15% and then -10%, and has a $5 end-of-year charge in each year. The maturity floor is $520. Find the account after each year and the maturity top-up. (Answer: year 1 $570.00; year 2 $508.00; top-up $12.00)
A one-year hypothetical account is $100 with no charges. Its return is +20% or -20%, each with probability $0.5$, and the maturity guarantee is $90. Find the expected account and expected top-up. (Answer: expected account $100.00; expected top-up $5.00)
A hypothetical account starts at $100, earns 0% in each of two years, and has a $2 end-of-year charge in each year. The maturity guarantee is $95. Charges are received by the insurer, and expenses total $2. Using the undiscounted cash-margin definition of charges less expenses less top-up, find the account, top-up, and margin. (Answer: account $96.00; top-up $0.00; margin $2.00)
A hypothetical account starts at $50. In each of two years, its gross return is independently +10% or -10%, each with probability $0.5$; a $0.50 charge is deducted at each year-end. The guarantee is $45. Find the expected top-up. (Answer: $1.3625)

What You'll Learn