Quantitative Analysis, Risk Management, Modelling, Algo Trading, and Big Data Analysis

## Performance-related Risk Measures

Enterprise Risks Management (ERM) can be described as a discipline by which an organization in any industry assesses, controls, exploits, finances, and monitors risks from all sources for the purpose of increasing the organization’s short- and long-term value to its stakeholders. It is a conceptual framework and when adopted by a company it provides with a set of tools to, inter alia, describe and quantify a risk profile. In general, most of the measures common in the practice of ERM can be broken in two categories: (a) solvency-related measures, and (b) performance-related measures. From a quantitative view point the latter refers to the volatility of the organization’s performance on a going-concern basis.

Performance-related risk measures provide us with a good opportunity to quickly review the fundamental definitions of the tools which concentrate on the mid-refion of the probability distribution, i.e. the region near the mean, and relevant for determination of the volatility around expected results:

Volatility (standard deviation), Variance, Mean
$$Vol(x) = \sqrt{Var(x)} = \left[\frac{\sum_{i=1}^{N}(x_i-\bar{x})^2}{N}\right]^{0.5}$$

Shortfall Risk
$$SFR = \frac{1}{N} \sum_{i=1}^{N} 1_{[x_i\lt T]} \times 100\%$$
where $T$ is the target value for the financial variable $x$. Shortfall Risk measure reflects the improvement over $Vol(x)$ measure by taking into account the fact that most of people are risk averse, i.e. they are more concerned with unfavorable deviations rather than favorable ones. Therefore, $SFR$ can be understood as the probability that the financial variable $x_i$ falls below a specified target level of $T$ (if true, $1_{[x_i\lt T]}$ above takes the value of 1).

Value at Risk (VaR)
In VaR-type measures, the equation is reversed: the shortfall risk is specified first, and the corresponding value at risk ($T$) is solved for.

Downside Volatility (or Downside Standard Deviation)
$$DVol(x) = \left[\frac{\sum_{i=1}^{N}(min[0,(x_i-T)]^2}{N}\right]^{0.5}$$
where again $T$ is the target value for the financial variable $x$. Downside volatility focuses not only on the probability of an unfavorable deviation in a financial vairable (as SFR) but also the extent to which it is favorable. It is usually interpreted as the extend to which the financial variable could deviate below a specified target level.

Below Target Risk
$$BTR = \frac{\sum_{i=1}^{N}(min[0,(x_i-T)]}{N}$$
takes its origin from the definition of the downside volatility but the argument is not squared, and there is no square root taken of the sum.