Portfolio VaR and portfolio volatility

Hello anyone knows where can I find out more about the formula for the above?

I saw this in last year's Q&A Bank Part 5, Q5.28iii. The solution talks about undiversified VaR and diversified VaR. Not sure where can I find out more about these?

 

Hi Alex

Solvency II SCR is a Value at Risk measure, there's quite a lot of guidance on the SCR diversification available in the level 2 advice for Standard Formula firms, you may find this document useful to provide some context: https://eiopa.europa.eu/CEIOPS-Archive/Documents/Advices/CEIOPS-L2-Advice-Correlation-Parameters.pdf

Broadly speaking, the VaR is calculated for each component risk, then the diversified SCR is determined with reference to the correlation between each component. This is done by taking the sum over all risks of the square root of the product of their SCRs multiplied by the correlation coefficient (shown on page 4 of the referenced guidance).

Diversification occurs wherever the risks are not perfectly correlated. (Hedging would occur where they were negatively correlated.) In EIOPA's case they have defined a correlation matrix for each risk type. In the context of this document, Basic SCR is simply the SCR before considering Operational Risk, this is because no diversification benefit is assumed on Operational Risk, which also explains why Operational Risk does not appear in the correlation matrices.

A numerical example might help, say a firm writes just life protection business and invests solely in equity (with no default risk for simplicity). Let their market risk component of SCR, SCR_m, be £20m and their Life component of SCR, SCR_L, be £10m (which would have been calculated by the firm). The correlation between these two risks is defined by EIOPA to be 0.25.
The basic SCR before diversification is £30m
The diversified SCR is calculated as the sum over all (both) risks as follows:
SCR = SQRT( Corr_m,m * SCR_m * SCR_m + Corr_m,L * SCR_m * SCR_L
+ Corr_L,m * SCR_L * SCR_m + Corr_L,L * SCR_L * SCR_L )
= SQRT( 1 * 20 * 20 + 0.25 * 20 * 10 + 0.25 * 10 * 20 + 1 * 10 * 10 )
= £24.5m
So there is a diversification benefit of £5.5m as the diversified SCR is that much lower than the undiversified SCR.

Intuitively, this is because although each SCR component is the result of a 99.5% VaR (the capital required to cover a "1-in-200 year event") we do not expect all of the 1-in-200 year events to occur at once, i.e. they are not perfectly correlated. So the diversified SCR is lower than the sum of the individual components.

Hope this helps.
Ben