Scenario Analysis for Capital Returns

Why Scenario Analysis Matters for Capital Returns

A business can appear to earn an acceptable risk-adjusted return under its central assumptions and still be vulnerable to adverse changes in claims, expenses, revenue, or investment income. Scenario analysis makes those assumptions explicit: it recalculates business profit and required capital under a specified set of conditions, then tests whether the resulting return clears a hurdle.

A scenario is a coherent set of assumed changes, not necessarily a forecast or a probability-weighted outcome. Its purpose is to ask what follows if those conditions occur. For a meaningful comparison, profit and capital must refer to the same business scope and measurement period. The analysis should also state how the scenario affects each quantity; a higher loss may reduce profit and increase required capital, but the capital effect must be supplied by the scenario’s capital method rather than inferred from profit alone.

Definition: For scenario $s$, let $\Pi_s$ be business profit before a capital charge, $K_s$ be the positive required capital assigned to the business, and $h$ be the hurdle rate. The scenario RAROC is $R_s=\Pi_s/K_s$. The business clears the hurdle when $R_s\ge h$. Profit, capital, and the hurdle must use consistent units and measurement bases.

In the examples below, amounts are hypothetical and stated in millions of dollars. Profit is calculated as revenue less claims or other losses and expenses, plus investment income where shown. Required capital is given for each scenario as an assumption from the applicable capital assessment. No probability is assigned to any scenario unless explicitly stated.

How to Recalculate and Test a Scenario

The calculation has two parts. First, recompute profit from the scenario’s operating assumptions. Second, use the scenario’s required capital to calculate RAROC and compare it with the hurdle. Keep a record of changes to profit separate from changes to capital: they affect the ratio through different channels.

Specify the scenario.
State which assumptions change and which remain fixed. For example, identify whether a claims shock applies to the whole claims amount or only to a specified component.

Recalculate profit.
Apply the scenario to each profit component, then sum the revised components. Do not apply a percentage change to profit if the scenario actually changes only one underlying component.

Set scenario capital.
Use the required capital specified or recalculated under the stated capital framework. Do not assume that capital is unchanged when the scenario says it rises or falls.

Calculate and test RAROC.
Divide scenario profit by scenario capital, then compare the result with the hurdle. Report the amount by which profit exceeds or falls short of the profit required at that capital level.

Hurdle test: If $K_s>0$, then $R_s\ge h$ exactly when $\Pi_s\ge hK_s$. Thus, the profit required to meet the hurdle is $hK_s$, and the profit shortfall (if positive) is $hK_s-\Pi_s$.

Worked Scenario Calculations

Worked Example: Claims Inflation and a Higher Capital Requirement

A hypothetical business has annual premium revenue of $12.00 million, claims of $7.20 million, expenses of $2.00 million, and investment income of $0.40 million. Its required capital is $25.00 million and its RAROC hurdle is 10%. In an adverse scenario, claims increase by 12%, expenses increase by $0.20 million, investment income falls by $0.15 million, and required capital rises by 20%. Premium revenue is unchanged. Calculate base and scenario RAROC, and quantify the scenario’s hurdle shortfall.

Method. Recalculate each profit component separately, then compare scenario profit with $hK_s$. This avoids the mistake of applying the claims shock to total profit or leaving capital at its base value.

Base profit. Revenue less claims is $12.00-7.20=4.80$. After expenses, $4.80-2.00=2.80$. Adding investment income gives $\Pi_0=2.80+0.40=\$3.20$ million.

Base RAROC. $R_0=3.20/25.00=0.128$, or 12.8%. The base business clears the 10% hurdle. At $25.00 million of capital, the required profit is $0.10(25.00)=\$2.50$ million, so the base profit cushion is $3.20-2.50=\$0.70$ million.

Scenario claims. The revised claims amount is $7.20(1.12)=\$8.064$ million. Scenario expenses are $2.00+0.20=\$2.20$ million, and investment income is $0.40-0.15=\$0.25$ million.

Scenario profit. $12.00-8.064=\$3.936$ million after claims; $3.936-2.20=\$1.736$ million after expenses; and $1.736+0.25=\Pi_s=\$1.986$ million after investment income.

Scenario capital and RAROC. $K_s=25.00(1.20)=\$30.00$ million, so $R_s=1.986/30.00=0.0662$, or 6.62%. The required profit is $0.10(30.00)=\$3.00$ million. The shortfall is $3.00-1.986=\$1.014$ million.

Check. Scenario profit falls by $3.20-1.986=\$1.214$ million, and capital rises by $5.00 million. The computed RAROC is below 10%, and the profit is $1.014 million below the hurdle requirement, as expected.

Worked Example: Testing a Hypothetical Reinsurance Response

A hypothetical business has annual revenue of $20.00 million, claims of $12.00 million, expenses of $3.00 million, and investment income of $0.60 million. Under an adverse scenario, claims rise to $15.60 million, expenses rise to $3.30 million, investment income falls to $0.30 million, and required capital increases from $40.00 million to $52.00 million. Revenue remains $20.00 million. The hurdle is 6%. A proposed reinsurance arrangement would reduce the adverse-scenario claims by $2.40 million, cost $1.10 million, and reduce required capital to $43.00 million. Calculate the adverse RAROC before and after the proposed response.

Method. First calculate the adverse result without reinsurance. Then include both the claims recovery and the reinsurance cost in profit, and use the separately stated post-reinsurance capital. The claims reduction is not free, and the capital reduction does not itself count as profit.

Base profit for comparison. $20.00-12.00=\$8.00$ million after claims; $8.00-3.00=\$5.00$ million after expenses; and $5.00+0.60=\$5.60$ million after investment income.

Adverse profit before reinsurance. $20.00-15.60=\$4.40$ million after claims; $4.40-3.30=\$1.10$ million after expenses; and $1.10+0.30=\Pi_s=\$1.40$ million after investment income.

Adverse RAROC before reinsurance. $1.40/52.00=0.026923\ldots$, or 2.69% rounded to two decimal places. The hurdle profit is $0.06(52.00)=\$3.12$ million, so the profit shortfall is $3.12-1.40=\$1.72$ million.

Profit after reinsurance. The claims reduction increases profit by $2.40 million, while the reinsurance cost reduces it by $1.10 million. Therefore, revised profit is $1.40+2.40-1.10=\$2.70$ million.

Adverse RAROC after reinsurance. Using post-reinsurance capital of $43.00 million, $R_s=2.70/43.00=0.062791\ldots$, or 6.28% rounded to two decimal places. The hurdle profit is $0.06(43.00)=\$2.58$ million, so profit exceeds the requirement by $2.70-2.58=\$0.12$ million.

Check. The response raises profit by a net $1.30 million and reduces capital by $9.00 million. Both changes improve the ratio in this scenario; the resulting RAROC is just above the 6% hurdle. This conclusion depends on the stated claims reduction, cost, and capital effect.

Separating Profit and Capital Effects

A RAROC change cannot be explained by looking only at the profit change. If both profit and capital change, each affects the ratio. Let the base values be $\Pi_0$ and $K_0$, and write the scenario values as $\Pi_0+\Delta\Pi$ and $K_0+\Delta K$. Provided $K_0>0$ and $K_0+\Delta K>0$, the exact change in RAROC is:

\frac{\Pi_0+\Delta\Pi}{K_0+\Delta K}-\frac{\Pi_0}{K_0} = \frac{K_0\Delta\Pi-\Pi_0\Delta K}{K_0(K_0+\Delta K)}.

To see why, combine the two fractions over the common denominator $K_0(K_0+\Delta K)$. The numerator becomes $K_0(\Pi_0+\Delta\Pi)-\Pi_0(K_0+\Delta K)$. Expanding gives $K_0\Pi_0+K_0\Delta\Pi-\Pi_0K_0-\Pi_0\Delta K$; the $K_0\Pi_0$ terms cancel, leaving $K_0\Delta\Pi-\Pi_0\Delta K$. The formula shows that a profit increase does not guarantee a RAROC increase: a sufficiently large capital increase can outweigh it. Conversely, reducing capital can improve RAROC even if profit is unchanged, though the business must still be tested against its hurdle.

Worked Example: Finding the Drivers of a Hurdle Breach

A hypothetical business earns annual revenue of $24.00 million, has claims of $13.00 million, expenses of $6.00 million, and investment income of $0.40 million. Required capital is $45.00 million and the hurdle is 10%. An adverse scenario reduces revenue by $1.20 million, increases claims by $2.60 million, increases expenses by $0.50 million, reduces investment income by $0.20 million, and raises required capital to $54.00 million. Assess the breach and then test a proposed package: increase revenue by $1.50 million, reduce expenses by $0.70 million, obtain a $1.80 million claims reduction through reinsurance costing $0.60 million, and reduce capital to $48.00 million.

Method. Build profit from its components, attribute the difference from base, and then recalculate the package as a new set of assumptions. This avoids assuming that an improvement in one driver automatically restores the hurdle.

Base profit and RAROC. Revenue less claims is $24.00-13.00=\$11.00$ million. After expenses, $11.00-6.00=\$5.00$ million; after investment income, $\Pi_0=\$5.40$ million. Base RAROC is $5.40/45.00=12.0\%$. The base hurdle profit is $0.10(45.00)=\$4.50$ million, a $0.90 million cushion.

Scenario profit. Revenue is $24.00-1.20=\$22.80$ million. Claims are $13.00+2.60=\$15.60$ million, expenses are $6.00+0.50=\$6.50$ million, and investment income is $0.40-0.20=\$0.20$ million. Therefore, profit is $22.80-15.60=\$7.20$ million after claims, $7.20-6.50=\$0.70$ million after expenses, and $0.70+0.20=\$0.90$ million after investment income.

Scenario RAROC and breach. $R_s=0.90/54.00=0.016666\ldots$, or 1.67% rounded to two decimal places. The hurdle profit is $0.10(54.00)=\$5.40$ million, so the scenario is short by $5.40-0.90=\$4.50$ million.

Driver attribution. Relative to base, revenue reduces profit by $1.20 million, claims reduce it by $2.60 million, expenses reduce it by $0.50 million, and investment income reduces it by $0.20 million. Together these changes reduce profit by $1.20+2.60+0.50+0.20=\$4.50$ million, matching $5.40-0.90=\$4.50$ million. The $9.00 million increase in capital raises the hurdle profit from $4.50 million to $5.40 million, an additional $0.90 million hurdle requirement.

Test the package. Starting from scenario profit of $0.90 million, the revenue increase adds $1.50 million, the expense reduction adds $0.70 million, and reinsurance changes profit by $1.80-0.60=\$1.20$ million. Package profit is $0.90+1.50+0.70+1.20=\$4.30$ million. At capital of $48.00 million, the hurdle profit is $0.10(48.00)=\$4.80$ million. Package RAROC is $4.30/48.00=8.96\%$ rounded to two decimal places, so it still misses the hurdle by $4.80-4.30=\$0.50$ million.

Check. The four adverse profit effects sum to $4.50 million, and the proposed package improves profit by $3.40 million while reducing capital by $6.00 million. The package substantially improves RAROC, but the final hurdle test shows it is not sufficient on the stated assumptions.

Using Scenario Results for Decisions

A scenario breach is a diagnostic, not an automatic instruction to accept or reject a business. It identifies how far the result is from a stated hurdle under a specified set of assumptions. Management can then examine which assumptions drive the result and consider responses such as revising underwriting terms, changing exposure, reducing expenses, using risk transfer, or holding additional capital. Each response should be assessed for its effect on both profit and capital, including any direct cost.

A useful reverse calculation is the capital level consistent with the hurdle for a given positive profit: $K_s\le\Pi_s/h$, when $h>0$. Alternatively, for a given capital amount, the required profit is $hK_s$. These calculations quantify what would need to change; they do not establish that a proposed change is feasible or that it will have the assumed effect.

Scenario analysis also has limits. A set of adverse assumptions is not automatically a joint probability model, and a scenario RAROC is not an expected RAROC. Do not average scenario ratios without a stated and appropriate aggregation rationale. Report the scenario assumptions, the profit and capital basis, the hurdle, and the unrounded calculation before presenting a rounded result.

Common Mistakes and Exam Traps

Applying a shock to the wrong quantity. A 10% claims increase applies to claims if claims are the named driver, not automatically to total profit.
Holding capital fixed by habit. If the scenario specifies a change in required capital, use that amount in the RAROC denominator and hurdle-profit calculation.
Counting a risk-transfer benefit but omitting its price. Include both the reduction in losses and the reinsurance cost, and use the scenario capital after risk transfer only when that effect is stated or supported by the capital method.
Reporting only the ratio. Show the dollar profit required at the hurdle and the resulting cushion or shortfall; this makes the decision implication clear.
Treating a scenario as a forecast. Unless probabilities and a suitable aggregation method are specified, a scenario result is conditional, not an expected outcome.

Key takeaway: A sound capital-return scenario analysis changes profit and required capital only according to stated assumptions, computes RAROC using the scenario values, and tests the result against the hurdle in both percentage and dollar terms.

Check Your Understanding

For each hypothetical case, calculate the requested scenario result and compare it with the stated hurdle.

A business has revenue of $14.00 million, claims of $8.00 million, expenses of $3.00 million, and investment income of $0.20 million. Capital is $25.00 million and the hurdle is 10%. In a scenario, claims rise by $1.00 million and capital rises to $28.00 million; all other items remain fixed. Find scenario profit, RAROC, and the hurdle shortfall or cushion. (Answer: Profit $2.20 million; RAROC 7.86%; hurdle profit $2.80 million; shortfall $0.60 million.)
A hypothetical business has revenue of $16.00 million, claims of $9.00 million, expenses of $4.00 million, and investment income of $0.50 million. In an adverse scenario, claims increase by $1.80 million, expenses by $0.30 million, and investment income falls by $0.20 million; capital rises from $30.00 million to $36.00 million. The hurdle is 8%. Calculate scenario profit, RAROC, and profit shortfall. (Answer: Profit $1.20 million; RAROC 3.33%; hurdle profit $2.88 million; shortfall $1.68 million.)
A business has revenue of $10.00 million, claims of $5.00 million, and expenses of $2.00 million, with no investment income. Under a scenario, revenue falls by $1.00 million, claims rise by $1.00 million, expenses rise by $0.20 million, and capital increases from $20.00 million to $24.00 million. The hurdle is 5%. Find scenario profit and the additional profit needed to meet the hurdle. (Answer: Profit $0.80 million; hurdle profit $1.20 million; additional profit needed $0.40 million.)
Under an adverse scenario, a business has profit of $2.40 million and capital of $30.00 million. A hypothetical reinsurance arrangement reduces claims by $1.20 million, costs $0.50 million, and reduces capital to $24.00 million. Calculate post-reinsurance profit and RAROC, and determine whether it clears an 8% hurdle. (Answer: Profit $3.10 million; RAROC 12.92%; it clears the hurdle, with a $1.18 million profit cushion.)
A scenario produces profit of $4.20 million with required capital of $38.00 million. If the hurdle is 12%, what is the maximum capital consistent with meeting the hurdle, and by how much must capital fall from its scenario amount? (Answer: Maximum capital $35.00 million; required reduction $3.00 million.)

What You'll Learn