#### October 2011

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

*By Tom Herzog, Ph.D.,
CIPR Distinguished Scholar, and CIPR Staff*

**• 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.

**• Conclusion**

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.