October 2011

The Simple Algebra of the Square Root Formula Behind RBC and Solvency II

• Introduction

A core element of all systems of prudential financial regulation is a risk-sensitive method of assessing the capital adequacy and solvency of regulated financial institutions.  The more advanced methods currently in use or under development involve an aggregation of the different risks facing financial institutions to determine their solvency capital requirements. Two of the main risk aggregation methodological approaches are used by the U.S. RBC (Risk-Based Capital) and the European Solvency II insurance regulatory frameworks. While the risk measures used may be different, the generic formula at the center of both approaches works similarly by adding up the main risks commonly faced by insurance companies, taking into account potential dependencies among these risks and allowing for diversification benefits. In this article, we describe how the standard risk aggregation formula works and we explore the similarities and the differences of the formulas used in both solvency frameworks.

The Generic Square Root Formula

The generic formula is used to aggregate all risks and establish the relationship between separate risk categories. To illustrate how this formula, commonly known as the “square root formula,” works under a variety of formulations, we present a series of four simple examples of its use. Let’s assume a formula with three risk components—x, y and z—whose hypothetical measures (values) are:

x=1, y=2, and z=3.

Then, if we just simply add the three risk components in order to calculate the risk-based regulatory capital requirement we get


where every risk component is given its full effect. In this risk aggregation formula there is no diversification benefit.

Next, let’s assume that the risk factors y and z are uncorrelated but risk x is still present at full effect. The formula then becomes


By accounting for the absence of correlation between the y and z risks, we avoid overestimating the cumulative effect of these risk components. While there is still no allowance for diversification benefit for risk x, some diversification benefits are assessed by the lack of correlation between risk components y and z. Thus, the overall risk charge drops to 4.61, significantly less than the earlier formulation.

In the third formulation, we assume that risk components x and y are correlated with each other but they are uncorrelated with risk factor z under the square sign.


In this formulation, more diversification benefits are assessed, further reducing the total risk charge to 4.24.

Lastly, we assume that all the risk components under the square sign are uncorrelated with each other, assessing full diversification benefits.


These examples of aggregating measured risks demonstrate how different correlation assumptions can dramatically change (overstate or understate) the outcome of the calculations of the total risk capital charge. The significance of the gradual introduction of diversification benefits in the four formulations (from no benefits whatsoever in the first to full benefits in the last) is reflected by the drop from the highest (6) to the lowest (3.74), or about a 38% decline in the total risk charges.    

The U.S. RBC Formula

The formula developed in the U.S. by the NAIC to calculate RBC follows the same logic as the generic “square root” formula described above. Adding up all the risk measures in a formula designed to account for the lack of covariance between risks, as in the numerical illustration above, results in a lower aggregate risk.  A separate RBC formula exists for the three primary types of insurance (life, property/casualty and health), but while the risk components may differ, the formulation is exactly the same. The formula tries to capture all the material risks that are common to the particular type of insurance. Each risk is measured by applying a factor to an amount pulled from insurers’ statutory financial statements. The factors themselves are based upon relevant statistics. Once the risk measures are calculated for all risk components, they are included in the RBC formula, which closely resembles the square root formula. Measured risks in the three RBC formulas are aggregated as:

In the Life RBC formula, there are three asset risk components ( is affiliated asset risk,  is the asset risk for other investments and  is the common stock risk). The remaining risk components are insurance risk (), interest rate risk (), health provider credit risk (), market risk () and two components of business risk—guaranty fund assessment and separate account related () and health administrative expense risk (). The P/C RBC formula also includes three asset risk components ( is affiliated investments, is fixed income, and  is common stock). It also includes credit risk assets  and two underwriting risk components ( for reserving risk and  for net written premium risk). The Health RBC formula includes two asset risk components ( is affiliated assets and  is other assets). The other risk components included are underwriting risk, credit risk , and business risk .

The RBC formula recognizes the existence of both systematic risk and idiosyncratic risk assessing both inter-risk and intra-risk diversification benefits. Systematic risk is determined by common underlying drivers of risk across different risk components.

The modeling of the RBC formula involves a covariance calculation that reduces the aggregate risk of various risk components by assuming less-than-perfect correlation. As in the generic square root formula, the result of this covariance calculation is a total RBC charge that is less than the sum of the individual RBC risk charges.

Furthermore, while it is incorrect to assume perfect correlation for all risk components, it is equally inappropriate to assume zero correlation for all of them. Some risk components, like assets and interest rate and common stock and market risk, are correlated and the life RBC formula accounts for that.

Idiosyncratic risk, which is the uncorrelated nonsystematic risk unique to specific elements within a risk component, is dealt with by intra-risk diversification. The RBC formula involves risk factor calculations, such as asset and issuer concentrations, and asset credit quality, which provide intra-risk diversification benefits.

The EU Solvency II Formula

The formula for the solvency capital requirement (SCR) developed under the Solvency II directive in Europe is another version of the generic square root formula. Just as in the generic and the RBC formulas, Solvency II aggregates all risks commonly faced by insurance companies.  Instead of factors, each component’s capital charge is calculated to reflect the value at risk (VaR), which is the maximum loss that could be expected in one-year period with a given probability. The total SCR is the sum of all capital charges calculated for each risk component. Solvency II uses a confidence level of 99.5% to estimate VaR.  Stated plainly, the VaR measurement for the overall SCR denotes that there is a 1-in-200 or 0.5% probability over one year that an insurer will go insolvent. The use of VaR in this context is somewhat controversial, as a number of researchers and practitioners believe that such use has no underlying (theoretical) basis.  Moreover, even the EU makes no claim as to the calibration of the overall value of SCR when all of the component risks are combined into the final formula.

Just as in the RBC formula, Solvency II’s SCR accounts for correlations among the various risk components. The SCR employs the variance-covariance approach to aggregate all risks through a correlation matrix to ensure that the overall SCR is less than the sum of the individual risk components. The correlations between risks are based both on statistical studies and on expert judgment.

The Solvency II SCR formula is written as


where i and j denote the gamut of the component risks (market risk, counterparty default risk, life underwriting risk, non-life underwriting risk and health underwriting risk) and  represents the parameters in the correlation matrix.

Inter-risk diversification is recognized by the variance-covariance aggregation method. Correlation between credit and non-life risks, for example, is 0.5 while most other correlations are 0.25.

The recognition of intra-risk diversification occurs at the level of the sub-risk aggregation, where each risk component’s SCR is calculated. The aggregation of the sub-risks uses the same variance-covariance approach through a correlation matrix to estimate each individual SCR. To estimate the market risk SCR, for example, six sub-risks (interest rate, equity, property, spread, currency and concentration risk) are aggregated using the market risk correlation matrix where the correlations between equity and the property and spread risks are 0.75 and all other correlations are 0.5.

The SCR does not allow for diversification between the operational risk and the other risks.  To calculate the total SCR for an insurer, the overall SCR is added to the operational risk and the adjustment for the loss-absorbing capacity of technical provisions and deferred taxes.

One other difference between the U.S. RBC regulatory framework and Solvency II is that the latter explicitly allows insurers to develop internal models themselves. These internal models can be used by the companies for risk management as well as for the determination of their regulatory capital requirements.


The square root formula underpins the risk modeling approaches of most banking and insurance regulatory frameworks around the world. As illustrated in this article, the formula provides the foundation upon which the key tools (i.e., RBC company action level, SCR) in the risk-based capital systems of both Solvency II and NAIC’s RBC are built. We have shown how the aggregation of risks in the square root formula is not simply additive, but allows for the recognition of diversification effects between risk categories. At the same time, while the calculation of the required capital is fundamentally based on the square root formula, Solvency II and RBC systems differ in regard to the use of different risk measures (factor-based vs. probabilistic measures), the  application of internal models, the treatment of diversification effect, and the limits imposed to investments. In a future article, we will explore these and other differences of the two regulatory frameworks as insurance regulators work towards global harmonization and supervisory convergence.

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