Measuring uncertainty of solvency coverage ratio in ORSA for non-life insurance

Abstract

We apply a simple model to project the Solvency Capital Requirement (SCR) over several years, using an Own Risk Solvency Assessment (ORSA) perspective, in order to assess the probability of achieving a solvency coverage ratio. To do so, we rely on a simplified framework proposed in Guibert (Bulletin Français d’Actuariat 10(20), 2010) which provides a detailed explanation of the SCR. Then, we take into account temporal dynamics for liabilities, premiums and asset returns. Here, we consider guarantees in non-life insurance. This context, when simplified, allows us to use a lognormal distribution to approximate the distribution of the liabilities. It leads to a simple and tractable model for measuring the uncertainty of the solvency ratio in an ORSA perspective.

Appendix I

The calculation of the moments of the variable $\chi_{t + 1}$ is presented in this Section for one or more lines of business.

1.1 For one line of business

Upstream of the presentation of the multiple- lines of business, we start by introducing the calculation of the moments of the variable $\chi_{t + 1}$ in the presence of only one line of business.

Thanks to the following expression

$$ \begin{aligned} (h_{t} + \theta ) \times BEL_{t + 1} - (1 + \theta \times \beta_{t + 1} ) \times P_{t + 1} &= (h_{t} + \theta ) \times BEL_{t} \times e^{{\mu_{l} - \frac{{\sigma_{l}^{2} }}{2} + \sigma_{l} \times \varepsilon_{t + 1,l} }} \\ &\quad- (1 - h_{t} \times \beta_{t + 1} ) \times P_{t + 1} , \\ \end{aligned} $$

we find the value of the conditional expectancy

$$ \begin{aligned} E_{t} ((h_{t} + \theta ) \times BEL_{t + 1} - (1 + \theta \times \beta_{t + 1} ) \times P_{t + 1} ) &= (h_{t} + \theta ) \times BEL_{t} \times e^{{\mu_{l} }} \\ &\quad- (1 - h_{t} \times \beta ) \times P_{t} \times e^{{\mu_{p} }} . \\ \end{aligned} $$

Also, given that $\varepsilon_{t + 1,\beta }$, $\varepsilon_{t + 1,p}$, $\varepsilon_{t + 1,l}$ are independent, we have the conditional variance

$$ \begin{aligned} & V_{t} ((h_{t} + \theta ) \times BEL_{t + 1} - (1 + \theta \times \beta_{t + 1} ) \times P_{t + 1} ) \\ & \quad = V_{t} ((h_{t} + \theta ) \times BEL_{t} \times e^{{\mu_{l} - \frac{{\sigma_{l}^{2} }}{2} + \sigma_{l} \times \varepsilon_{t + 1,l} }} - (1 - h_{t} \times \beta_{t + 1} ) \times P_{t + 1} ) \\ & \quad = (h_{t} + \theta )^{2} \times BEL_{t}^{2} \times e^{{2\mu_{l} }} \times (e^{{\sigma_{l}^{2} }} - 1) + h_{t}^{2} \times \beta^{2} \times (e^{{\sigma_{\beta }^{2} }} - 1) \times P_{t}^{2} \times e^{{2\mu_{p} }} \times (e^{{\sigma_{p}^{2} }} - 1) \\ & \qquad + h_{t}^{2} \times \beta^{2} \times (e^{{\sigma_{\beta }^{2} }} - 1) \times P_{t}^{2} \times e^{{2\mu_{p} }} + P_{t}^{2} \times e^{{2\mu_{p} }} \times (e^{{\sigma_{p}^{2} }} - 1) \times (1 - h_{t} \times \beta )^{2} \\ & \quad = (h_{t} + \theta )^{2} \times BEL_{t}^{2} \times e^{{2\mu_{l} }} \times (e^{{\sigma_{l}^{2} }} - 1) + h_{t}^{2} \times \beta^{2} \times (e^{{\sigma_{\beta }^{2} }} - 1) \times P_{t}^{2} \times e^{{2\mu_{p} + \sigma_{p}^{2} }} \\ & \qquad + P_{t}^{2} \times e^{{2\mu_{p} }} \times (e^{{\sigma_{p}^{2} }} - 1) \times (1 - h_{t} \times \beta )^{2} , \\ \end{aligned} $$

and thus, we deduce of (5) the coefficient of variation of $(h_{t} + \theta ) \times BEL_{t + 1} - (1 + \theta \times \beta_{t + 1} ) \times P_{t + 1}$ is

$$ \omega_{t} = \frac{{\sqrt {\begin{array}{*{20}l} {(h_{t} + \theta )^{2} \times BEL_{t}^{2} \times e^{{2\mu_{t} }} \times (e^{{\sigma_{l}^{2} }} - 1) + h_{t}^{2} \times \beta^{2} \times (e^{{\sigma_{\beta }^{2} }} - 1) \times P_{t}^{2} \times e^{{2\mu_{p} + \sigma_{p}^{2} }} } \\ { + P_{t}^{2} \times e^{{2\mu_{p} }} \times (e^{{\sigma_{p}^{2} }} - 1) \times (1 - h_{t} \times \beta )^{2} } \\ \end{array} } }}{{(h_{t} + \theta ) \times BEL_{t} \times e^{{\mu_{l} }} - (1 - h_{t} \times \beta ) \times P_{t} \times e^{{\mu_{p} }} }} $$

1.2 For several lines of business

We continue with model containing several lines of business supposing the lognormal approximation is validated. First, we calculate the first two moments of the numerator of $\chi_{t + 1}$, and then we deduce those of $\chi_{t + 1}$. We have calculated the conditional expectation of the variable $\sum\nolimits_{j = 1}^{n} {((h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} )}$

$$ \begin{aligned} & E_{t} \left( {\sum\limits_{j = 1}^{n} {((h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} )} } \right) \\ & \quad = \sum\limits_{j = 1}^{n} {(E_{t} ((h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} ))} , \\ & \quad = \sum\limits_{j = 1}^{n} {((h_{t} + \theta^{j} ) \times BEL_{t}^{j} \times e^{{\mu_{l}^{j} }} - (1 - h_{t} \times \beta^{j} ) \times P_{t}^{j} \times e^{{\mu_{p}^{j} }} )} . \\ \end{aligned} $$

and its conditional variance

$$ \begin{aligned} & V_{t} \left( {\sum\limits_{j = 1}^{n} {((h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} )} } \right) \\ & \quad = \sum\limits_{j = 1}^{n} {(V_{t} ((h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} ))} \\ & \quad \; + 2\sum\limits_{1 \le i < j \le n} {Cov_{t} ((h_{t} + \theta^{i} ) \times BEL_{t + 1}^{i} - (1 + \beta_{t + 1}^{i} \times \theta^{i} ) \times P_{t + 1}^{i} ,(h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} )} . \\ \end{aligned} $$

The first component of the conditional variance is obtained simply

$$ \begin{aligned} & \sum\limits_{j = 1}^{n} {(V_{t} ((h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} ))} \\ & \quad = \sum\limits_{j = 1}^{n} {\left( \begin{gathered} (h_{t} + \theta^{j} )^{2} \times BEL_{t}^{j2} \times e^{{2\mu_{l}^{j} }} \times (e^{{\sigma_{l}^{j2} }} - 1) \hfill \\ + P_{t}^{j2} \times e^{{2\mu_{p}^{j} }} \times (h_{t}^{2} \times \beta^{j2} \times (e^{{\left( {\sigma_{p}^{j2} + \sigma_{\beta }^{j2} } \right)}} - 1) + (e^{{\sigma_{p}^{j2} }} - 1)(1 - 2h_{t} \times \beta^{j} )) \hfill \\ \end{gathered} \right)} . \\ \end{aligned} $$

The second term is obtained by noting that the covariance of the two random lognormal variables $(Y_{1} ,Y_{2} )$ with parameters $(\mu_{1} ,\sigma_{1} )$ and $(\mu_{2} ,\sigma_{2} )$, respectively, are obtained from the covariance of the underlying normal variables $(\varepsilon_{1} ,\varepsilon_{2} )$

$Cov(Y_{1} ,Y_{2} ) = E(Y_{1} ) \times E(Y_{2} )(e^{{Cov\left( {\varepsilon_{1} ,\varepsilon_{2} } \right)}} - 1)$.

Denoting that the correlation coefficients $\rho_{l}^{ij}$, $\rho_{p}^{ij}$ and $\rho_{\beta }^{ij}$ are associated with the variables $(BEL_{t}^{i} ,BEL_{t}^{j} )$, $(C_{t}^{i} ,C_{t}^{j} )$ and $(\beta_{t}^{i} ,\beta_{t}^{j} )$, respectively, we get

$$ \begin{aligned} & Cov_{t} ((h_{t} + \theta^{i} ) \times BEL_{t + 1}^{i} - (1 + \beta_{t + 1}^{i} \times \theta^{i} ) \times P_{t + 1}^{i} ,(h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} ) \\ & \quad = (h_{t} + \theta^{i} )(h_{t} + \theta^{j} ) \times BEL_{t}^{i} \times BEL_{t}^{j} \times e^{{\mu_{l}^{i} + \mu_{l}^{j} }} \times (e^{{\rho_{l}^{ij} \sigma_{l}^{i} \sigma_{l}^{j} }} - 1) \\ & \quad \; + h_{t}^{2} \times Cov_{t} (\beta_{t + 1}^{i} \times P_{t + 1}^{i} ,\beta_{t + 1}^{j} \times P_{t + 1}^{j} ) - h_{t} \times Cov_{t} (\beta_{t + 1}^{i} \times P_{t + 1}^{i} ,P_{t + 1}^{j} ) \\ & \quad \; - h_{t} \times Cov_{t} (\beta_{t + 1}^{j} \times P_{t + 1}^{j} ,P_{t + 1}^{i} ) + Cov_{t} (P_{t + 1}^{i} ,P_{t + 1}^{j} ). \\ \end{aligned} $$

But we have

$$ Cov_{t} (P_{t + 1}^{i} ,P_{t + 1}^{j} ) = P_{t}^{i} \times P_{t}^{j} \times e^{{\mu_{p}^{i} + \mu_{p}^{j} }} \times (e^{{\rho_{p}^{ij} \sigma_{p}^{i} \sigma_{p}^{j} }} - 1), $$

$$ \begin{aligned} Cov_{t} (P_{t + 1}^{i} ,\beta_{t + 1}^{j} \times P_{t + 1}^{j} ) &= E_{t} (\beta_{t + 1}^{j} ) \times Cov_{t} (P_{t + 1}^{i} ,P_{t + 1}^{j} ) \\ &= \beta^{j} \times P_{t}^{i} \times P_{t}^{j} \times e^{{\mu_{p}^{i} + \mu_{p}^{j} }} \times (e^{{\rho_{p}^{ij} \sigma_{p}^{i} \sigma_{p}^{j} }} - 1), \\ \end{aligned} $$

And

$$ \begin{aligned} & Cov_{t} (\beta_{t + 1}^{i} \times P_{t + 1}^{i} ,\beta_{t + 1}^{j} \times P_{t + 1}^{j} ) \\ & \quad = E_{t} (\beta_{t + 1}^{i} ) \times E_{t} (\beta_{t + 1}^{j} ) \times Cov_{t} (P_{t + 1}^{i} ,P_{t + 1}^{j} ) + E_{t} (P_{t + 1}^{i} ) \times E_{t} (P_{t + 1}^{j} ) \times Cov_{t} (\beta_{t + 1}^{i} ,\beta_{t + 1}^{j} ) \\ & \quad = \beta^{i} \times \beta^{j} \times P_{t}^{i} \times P_{t}^{j} \times e^{{\mu_{p}^{i} + \mu_{p}^{j} }} (e^{{\rho_{p}^{ij} \sigma_{p}^{i} \sigma_{p}^{j} + \rho_{\beta }^{ij} \sigma_{\beta }^{i} \sigma_{\beta }^{j} }} - 1). \\ \end{aligned} $$

We then deduce

$$ \begin{aligned} & Cov_{t} ((h_{t} + \theta^{i} ) \times BEL_{t + 1}^{i} - (1 + \beta_{t + 1}^{i} \times \theta^{i} ) \times P_{t + 1}^{i} ,(h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} ) \\ & \quad = (h_{t} + \theta^{i} )(h_{t} + \theta^{j} ) \times BEL_{t}^{i} \times BEL_{t}^{j} \times e^{{\mu_{l}^{i} + \mu_{l}^{j} }} \times (e^{{\rho_{l}^{ij} \sigma_{l}^{i} \sigma_{l}^{j} }} - 1) \\ & \qquad + P_{t}^{i} \times P_{t}^{j} \times e^{{\mu_{p}^{i} + \mu_{p}^{j} }} \left[ \begin{gathered} h_{t}^{2} \times \beta^{i} \times \beta^{j} \times (e^{{\rho_{p}^{ij} \sigma_{p}^{i} \sigma_{p}^{j} + \rho_{\beta }^{ij} \sigma_{\beta }^{i} \sigma_{\beta }^{j} }} - 1) \hfill \\ + (e^{{\rho_{p}^{ij} \sigma_{p}^{i} \sigma_{p}^{j} }} - 1)(1 - h_{t} \times (\beta^{i} + \beta^{j} )) \hfill \\ \end{gathered} \right]. \\ \\ \end{aligned} $$

We finally infer the coefficient of variation of $\sum\nolimits_{j = 1}^{n} {((h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} )}$

$$ \omega_{t} = \frac{{\sqrt {V_{t} \left( {\sum\nolimits_{j = 1}^{n} {((h_{t} + \theta^{j} ) \times BEL_{t + 1}^{j} - (1 + \beta_{t + 1}^{j} \times \theta^{j} ) \times P_{t + 1}^{j} )} } \right)} }}{{\left( {\sum\nolimits_{j = 1}^{n} {((h_{t} + \theta^{j} ) \times BEL_{t}^{j} \times e^{{\mu_{l}^{j} }} - (1 - h_{t} \times \beta^{j} ) \times P_{t}^{j} \times e^{{\mu_{p}^{j} }} )} } \right)}}, $$

and then the parameters of the lognormal approximation

$$ \sigma_{t}^{2} = \ln (1 + \omega_{t}^{2} ),\mu_{t} = \ln \left( {\frac{{\sum\nolimits_{j = 1}^{n} {((h_{t} + \theta^{j} ) \times BEL_{t}^{j} \times e^{{\mu_{l}^{j} }} - (1 - h_{t} \times \beta^{j} ) \times P_{t}^{j} \times e^{{\mu_{p}^{j} }} )} }}{{\sqrt {1 + \omega_{t}^{2} } }}} \right). $$

As in the case of a single line of business (see Appendix I), the distribution of $\chi_{t + 1}$ conditional on the information available at time t is approximated by a lognormal distribution with parameters

$$ \mu_{t} (\chi ) = \mu_{t} - \mu_{a} + \frac{{\sigma_{a}^{2} }}{2},\,\sigma_{t}^{2} (\chi ) = \sigma_{t}^{2} + \sigma_{a}^{2} . $$