Risk Margin calculation

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In the previous lessons, we explained why the Risk Margin exists and what it represents. In this lesson, we make the concept more concrete by showing how the Risk Margin is calculated in practice and by illustrating the calculation with a simple numerical example.

The Risk Margin reflects the cost of holding capital for non-hedgeable risks over the lifetime of insurance liabilities. As long as these liabilities exist, capital must be held to protect against adverse deviations from expected outcomes. The Risk Margin therefore takes a long-term view and looks beyond the current year.

The cost-of-capital formula

Under Solvency II, the Risk Margin is calculated using the cost-of-capital approach. The formula can be written as:

\( \mathrm{RM} = \mathrm{CoC} \cdot \sum_{t \ge 0} \frac{\mathrm{SCR}_{t}}{(1 + r_{t+1})^{t+1}} \)

where:

• \( \mathrm{CoC} \) is the cost-of-capital rate (set to 6% under Solvency II),
• \( \mathrm{SCR}_{t} \) is the Solvency Capital Requirement for non-hedgeable risks in year t,
• \( r_{t+1} \) is the risk-free interest rate for maturity t+1.

The formula takes the future capital required each year, applies a cost to that capital, and then discounts the resulting costs back to today.

Interpreting the formula intuitively

Each term in the formula represents the cost of holding capital in a single future year. These annual costs are then summed across all future years in which the liabilities are still present.

Three elements therefore determine the Risk Margin:

• how large the future capital requirements are,
• how long capital needs to be held,
• how strongly future costs are discounted.

This directly links the Risk Margin to the duration and riskiness of the liabilities.

A numerical example

Consider a simplified insurance portfolio with the following characteristics:

• Cost-of-capital rate: 6%
• Risk-free interest rate: 2% per year (flat)
• Non-hedgeable SCR profile:
• Year 0: 1000
• Year 1: 700
• Year 2: 300

Assume that after year 2, no liabilities remain.

Step 1: Calculate annual capital costs

For each year, multiply the SCR by the cost-of-capital rate:

• Year 0: \( 1000 \cdot 6\% = 60 \)
• Year 1: \( 700 \cdot 6\% = 42 \)
• Year 2: \( 300 \cdot 6\% = 18 \)

Step 2: Discount capital costs to today

Discount each annual cost using the 2% risk-free rate:

• Year 0: \( \frac{60}{(1 + 2\%)^1} = 58.8 \)
• Year 1: \( \frac{42}{(1 + 2\%)^2} = 40.4 \)
• Year 2: \( \frac{18}{(1 + 2\%)^3} = 17.0 \)

Step 3: Sum discounted values

\( \mathrm{RM} = 58.8 + 40.4 + 17.0 = 116.2 \)

The Risk Margin for this portfolio is approximately 116.

What this example shows

This simple example highlights several important points:

• Risk Margin depends on future capital, not just today's SCR.
• Capital requirements usually run off over time, reducing later contributions.
• Longer-lasting or riskier portfolios lead to higher Risk Margins.
• Lower interest rates would increase the Risk Margin by reducing discounting.

Now it's your turn to practise calculating the Risk Margin!