**Paul Huber and Nick Kinrade explain how risk margins used in Solvency II and IFRS 17 should be understood and calibrated**

One topic that is hotly debated is the cost of a pint of beer. However, actuaries are typically more involved in the consumption of this product than its pricing. In manufacturing and services, the cost of producing a product (the cost of the beer from the brewery) is generally known and the challenge lies in allocating fixed costs (the cost of running the pub). In insurance, working out these costs of production is far more challenging, and actuaries are involved in both their pricing and consumption.

The costs of production of an insurance company can be considered by thinking about the stakeholders involved:

- Expected claims and the cost of servicing policies, benefiting the policyholder
- Taxes paid to the government, and
- Capital, usually provided by shareholders or debtholders, which comes with a return expectation.

Actuarial methodology and models are well established for the first item, which is termed the best estimate liability and is equal to the present value of best estimate cashflows discounted at risk-free rates. The second item is represented on the economic balance sheet as the deferred tax liability and is typically defined as tax on temporary differences between the value of the insurance liabilities for tax purposes and their economic value. Estimating the cost associated with the provision of capital is more challenging.

One way of looking at this is to consider how a shareholder or debtholder would value expected payments from an insurer.

**What are risk margins?**

The standard finance approach to valuing cashflows is to discount them at the risk-free rate plus a spread calibrated to replicate observed market prices. This spread is interpreted as compensation for systemic risk, uncertainty, asset illiquidity, and various frictional costs shareholders incur when investing in insurers. The risk margin reflects the cost of financing this spread. This can be computed in one of two ways (see *Figure 1*):

- Present value of outstanding capital amounts multiplied by a spread, which is the standard approach specified in Solvency II, or
- Difference between the market value of the cashflows to service this capital (discounted at the cost-of-capital rate) and their risk-free discounted value (which is the amount needed to replicate the capital cashflows).

Assume capital requirements of 10 at inception amortising over five years, a 0% risk-free rate, and a 6% cost-of-capital spread. Under the standard approach, the risk margin is the discounted value of the capital amounts multiplied by the 6% spread, which is equal to 1.8.

Under the alternative method, the capital cashflows are the capital amounts at the end of the interval, minus the capital amounts at the start of the interval accumulated at the cost-of-capital. For example, for the first year the capital cashflow is 8 minus 10 multiplied by 1.06 or -2.60. The present value of the capital cashflows at the cost-of-capital rate is equal to the initial capital investment of 10 (the market value) and at the risk-free rate is 11.8 (the cost of replicating the capital servicing costs). The difference between these two values is the risk margin of 1.8.

Both approaches are equivalent; however, the second approach is more aligned with standard economic theory and is thus more accessible to analysts and investors. In addition, it can be used to avoid circular calculations, if the capital amounts are indirectly derived from a non-economic capital requirement (such as rating agency capital), and when estimating return on capital.

Typically, the return on capital is approximated as the profit divided by the present value of allocated capital. For instance, if in the above example the profit is 1.5 then the return on capital would be reported as 5% (1.5/30). The alternative approach facilitates a more standard method for determining return on capital, namely by adjusting the capital cashflows for the profit and then determining the internal rate of return. If the initial profit of 1.5 counts towards available capital at inception (which is the case under Solvency II), then the initial capital cashflow becomes 8.5 and the internal rate of return on the capital cashflows is 13%. As a result, the profit margin from a shareholder perspective is actually 7%, reflecting the additional spread over the cost-of-capital.

**How should the cost-of-capital rate be calibrated?**

The next key question is what is an appropriate cost-of-capital spread for determining the risk margin? In our recent article, 'A Generic Framework for the Economic Valuation of Insurance Liabilities', we firstly derive the risk margin generically and show that it can be derived from the insurer's weighted average cost-of-capital with adjustments for franchise value, investment risk, and tax. These adjustments are made for the following reasons:

- Franchise value is the difference between the market capitalisation of an insurer and its economic equity. This adjustment is needed as the insurer's equity cost-of-capital is based on market capitalisation as opposed to economic equity, which does not allow for the future new business.
- Investment risk. This adjustment is needed as investment risk does not have the same risk profile as insurance risk.
- Taxation. Equity cost-of-capital reflects the after-tax return to shareholders, while risk margins are typically treated as temporary differences in the deferred tax calculations. As a result, the resulting cost-of-capital rate needs to be grossed-up for tax. In addition to adjusting the cost-of-capital in the risk margin, a frictional tax margin is needed to cover tax on projected risk-free investment income on investments in excess of the tax value of liabilities (not taken into account in
*Figure 2*).

FIGURE 2. Cost-of-capital calibration:

Assume an insurer has a market capitalisation of 6, investments with a market value of 25, economic equity of 5, and capital requirements of 3.6 backed with 0.6 of debt and 3.0 of equity.

In addition, assume...

- Insurer's equity cost-of-capital spread of 6%
- Debt spread of 3%
- Excess investment return of 0.15%
- New business margin of 1%
- Tax rate of 25%

...where the new business margin is the economic profit on new business divided by franchise value minus insurer's equity cost-of-capital. The excess investment return is the after-tax expected return on total investments, less the expected return allocated to support insurance liabilities (including any matching adjustment or illiquidity spread), less the cost of any excess economic equity (economic equity not backing insurance liabilities multiplied by the insurer's equity cost-of-capital).

As a result, the franchise value adjustment is: the price-to-equity ratio (market capitalisation to economic equity) of 1.2 minus one, multiplied by the capital leverage ratio (economic equity to equity requirements backing insurance liabilities) of 1.67 multiplied by the new business margin of 1%, which is 0.33%.

The investment adjustment is the investment leverage (market value of total investments to economic equity) of 5 multiplied by the capital leverage of 1.67 multiplied by the excess investment return of 0.15%, which equals 1.25%.

The post-tax equity cost-of-capital spread is then 4.42% (6% minus 0.33% minus 1.25%). Dividing this by one minus the tax rate results in a pre-tax spread of 5.89%.

The debt spread is 3% (no adjustment is needed for tax as interest expenses are tax deductible) and thus the weighted average cost-of-capital spread used in the risk margin calculation is 5.41%.

**Application to IFRS 17 and Solvency II**

Although insurance liabilities are not traded in an active market, by calibrating the cost-of-capital within an appropriately calibrated risk margin framework, it is possible to estimate the market value of an insurance contract. We believe that this approach can be used by insurers to specify and justify their cost-of-capital rate applied in IFRS 17 and to assess the suitability of the Solvency II rate of 6%.

**Nick Kinrade i**s a director at KPMG in Switzerland. He leads the firm's multi-valuation accounting

**Paul Huber** works for Swiss Re

*The opinions expressed in this article are the authors' own views*