When 97.5 is insufficient

Counterparty credit risk on derivative transactions is measured against a 97.5 percentile on a distribution of losses calibrated off historical data. At this percentile, and assuming future market movements would maintain its historical bearings, means that a derivative's value would only surpass our indications 2.5% of the time, or ~6 days in a trading year.

All seemed well until...

• Covid 19 - excessive losses
• Archegos default - excessive losses
• EU power supply strain - excessive losses...

The increasing occurence of excessive losses over the 97.5 percentile highlights a missing statistic in the measurement of credit risk. That is, while one may understand they won't lose more than 10M (for example), 97.5% of the time, there is no indication of how much will be lost in the remaining 2.5%. Hence the need of an alternative statistic to measure tailed losses.

The tailed loss measure analysed here will be the Expected Tail Exposure (ETE).

Expected Tail Exposure

Let $X_t\sim N({\mu }_t,{\sigma }_t^2)$ be a stochastic process over random variables representing a portfolio's value at some time $t$. The ETE is the expected value of $X_t$, conditioned on being past a value of $x(\alpha {)}_t$ with $\alpha$% confidence is given by

$$\begin{array}{r}{\text{ETE}}_{\alpha }(X_t)=\mathbb{E}(X_t|X_t>x(\alpha {)}_t)\end{array}$$

$X_t$ is assumed to be Normal (Gaussian) and may be represented with respect to a standard normal random variable $Z$. That is

$$\begin{array}{r}{X}_t={\mu }_t+{\sigma }_t{Z}_t.\end{array}$$

From the last two equations

$$\begin{array}{rl}{\text{ETE}}_{\alpha }(X_t)& =\mathbb{E}({\mu }_t+{\sigma }_t{Z}_t|{\mu }_t+{\sigma }_t{Z}_t>x(\alpha {)}_t)\\ & =\mathbb{E}({\mu }_t+{\sigma }_t{Z}_t|Z_t>\frac{x(\alpha {)}_t-{\mu }_t}{{\sigma }_t}).\end{array}$$

For clarity, let

$$\begin{array}{r}q=\frac{x(\alpha {)}_t-{\mu }_t}{{\sigma }_t}.\end{array}$$

Expressing conditional expectations with respect to probabilities

$$\begin{array}{rl}{\text{ETE}}_{\alpha }(X_t)& =\frac{{\int }_q^{\mathrm{\infty }}({\mu }_t+{\sigma }_{t}z)f(z)dz}{P(Z>q)}\\ & =\frac{{\int }_q^{\mathrm{\infty }}({\mu }_t+{\sigma }_{t}z)f(z)dz}{1-\alpha }\end{array}$$

where $f$ is the standard normal probability density function. The numerator of this last equation is evalauted below.

$$\begin{array}{rl}{\int }_q^{\mathrm{\infty }}({\mu }_t+{\sigma }_{t}z)f(z)dz& ={\mu }_t{\int }_q^{\mathrm{\infty }}f(z)dz+{\sigma }_t{\int }_q^{\mathrm{\infty }}zf(z)dz\\ & ={\mu }_t(1-\mathrm{\Phi }(q))+\frac{{\sigma }_t}{\sqrt{2\pi }}{\int }_q^{\mathrm{\infty }}z\exp \left(\frac{-z^2}{2}\right)dz\\ & ={\mu }_t(1-\mathrm{\Phi }(q))+\frac{{\sigma }_t}{2\sqrt{2\pi }}{\int }_q^{\mathrm{\infty }}\exp \left(\frac{-z^2}{2}\right)d{z}^2\\ & ={\mu }_t(1-\mathrm{\Phi }(q))-\frac{{\sigma }_t}{\sqrt{2\pi }}{[\exp \left(\frac{-z^2}{2}\right)]}_q^{\mathrm{\infty }}\\ & ={\mu }_t(1-\mathrm{\Phi }(q))-\frac{{\sigma }_t}{\sqrt{2\pi }}[0-\exp \left(\frac{-q^2}{2}\right)]\\ & ={\mu }_t(1-\mathrm{\Phi }(q))+{\sigma }_{t}f(q).\end{array}$$

where $\mathrm{\Phi }$ is the standard normal cumulative distribution function. With the integral evaluated, the conclusion is

$$\begin{array}{r}{\text{ETE}}_{\alpha }(X_t)=\frac{{\mu }_t(1-\mathrm{\Phi }(q))+{\sigma }_{t}f(q)}{1-\alpha }.\end{array}$$

This expression can serve as a complementary statistic to the 97.5 percentile to gauge how much may be lost in the event prices move beyond the 97.5 percentile.