September 8, 2016

MRM | ||
---|---|---|

1 | < 0.5% | |

2 | 0.5% - 5% | |

3 | 5.0% - 12% | |

4 | 12% - 20% | |

5 | 20% -30% | |

6 | 30% -80% | |

7 | > 80% |

Since the level of the VEV is derived from the VaR, a logical first step is to calculate the value at risk (VaR) for a particular financial instrument. Here the regulators have opted for the $97.5\%$-VaR. The probability of a loss exceeding this amount is $2.5\%$. On a sample of 200 closing prices, the 2.5% VaR level corresponds to the sixth smallest return. A logical starting point when calculating the VaR-number of a particular holding is the assumption that the price returns are normally distributed. One can estimate the mean and variance of the underlying normal distribution using historical price observations. The $2.5\%$-percentile gets us the estimation of the VaR.

The disadvantage of this method is the assumption of normality. Stock price returns are often right or left skewed, which makes the assumption of normality invalid.
A solution is to make use of the **Cornish-Fisher expansion**. The Cornishâ€“Fisher expansion is an asymptotic expansion used to approximate the quantiles of a probability distribution based on its moments. The Cornish-Fisher expansion delivers us a closed-form formula to obtain percentiles for non-normal distributions when limiting ourselves for example to only the first four moments of the distribution: the mean, the variance, the skewness and the excess kurtosis. The Cornish-Fisher expansion hence gives us a method to calculate the VaR directly from these four moments:

We have the following formula to derive the VaR of the price returns at the 97.5% confidence level: \begin{equation}VaR_{0.975}=\sigma(-1.96+0.474s-0.0687k + 0.146s^2)-\frac{1}{2}\sigma^2 \end{equation} with $\sigma$ the standard deviation, $s$ the skewness and $k$ the excess kurtosis.

- The historical $97.5\%$-VaR corresponds with the sixth smallest return, which is in this particular example $-3.45\%$. So only in $2.5\%$ of the business days a log return smaller than this number is expected to be observed.
- To obtain the Cornish-Fisher VaR (limiting ourselves to the first four moments), we first estimate the standard deviation, skew and kurtosis. For this example the parameter estimates are
$$\begin{equation}
\begin{array}{lll}
\sigma &= &0.0166\\
s &=& 1.1247\\
k &= &10.4444\\
\end{array}
\end{equation}$$

Using the above equation for the Cornish-Fisher VaR, we obtain a $97.5\%$-VaR of $-3.26\%$.

The annualised VEV numbers are respectively $48.19\%$ and $83.7\%$, using the square root of time rule with 250 trading days.

The advantage of using Cornish-Fisher to obtain the VEV is that we have a closed form formula. But we have to remark that the formula is not always valid. There are limits on the skew and kurtosis of a distribution under which the Cornish-Fisher holds. Very often practitionars also mix up the skewness and kurtosis