# Risk Measures

A risk measure is used to determine the amount of an asset or set of assets to be kept in reserve. The purpose of this reserve is to make the downside risk taken by financial institutions, such as banks and insurance companies, acceptable to regulators.

## Mathematical Definition

A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents portfolio returns. The common notation for a risk measure associated with a random variable $X$ is $\rho(X)$. A risk measure should have certain properties:

Normalized

[$]\rho(0) = 0[$]

That is, the risk of holding no assets is zero.

Translative

[$]\mathrm{If}\; a \in \mathbb{R},\;\mathrm{then}\; \rho(Z + a) = \rho(Z) - a[$]

The portofolio $A$ is just adding cash $a$ to your portfolio $Z$. In particular, if $a=\rho(Z)$ then $\rho(Z+A)=0$. In financial risk management, translation invariance implies that the addition of a sure amount of capital reduces the risk by the same amount.

Monotone

[$]\mathrm{If} \; Z_1 \leq Z_2 ,\; \mathrm{then} \; \rho(Z_2) \leq \rho(Z_1)[$]

That is, if portfolio $Z_2$ always has better values than portfolio $Z_1$ under almost all scenarios then the risk of $Z_2$ should be less than the risk of $Z_1$. E.g. If $Z_1$ is an in the money call option (or otherwise) on a stock, and $Z_2$ is also an in the money call option with a lower strike price. In financial risk management, monotonicity implies a portfolio with greater future returns has less risk.

## Coherent Risk Measures

A coherent risk measure is a risk measure that satisfies properties of sub-additivity and homogeneity.

[$] \rho(Z_1 + Z_2) \leq \rho(Z_1) + \rho(Z_2)[$]

The risk of two portfolios together cannot get any worse than adding the two risks separately: this is the diversification principle. In financial risk management, sub-additivity implies diversification is beneficial.

Positive homogeneity

[$]\mathrm{If}\; \alpha \ge 0,\; \mathrm{then} \; \rho(\alpha Z) = \alpha \rho(Z)[$]

Loosely speaking, if you double your portfolio then you double your risk. In financial risk management, positive homogeneity implies the risk of a position is proportional to its size.

### Convex risk measures

The notion of coherence has been subsequently relaxed. Indeed, the notions of sub-additivity and positive homogeneity can be replaced by the notion of convexity:

Convexity

[$]\text{If } \lambda \in [0,1] \text{ then } \rho(\lambda Z_1 + (1-\lambda) Z_2) \leq \lambda \rho(Z_1) + (1-\lambda) \rho(Z_2)[$]

## VaR and TVaR

Value at Risk (VaR) is a measure of the risk of investments. It estimates how much a set of investments might lose, given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses.

In financial mathematics and financial risk management, VaR is defined as: for a given portfolio, time horizon, and probability $p$, the $p$ VaR is defined as a threshold loss value, such that the probability that the loss on the portfolio over the given time horizon exceeds this value is $p$.

### Mathematical definitions

Given a confidence level $\alpha \in (0,1)$, the VaR of the portfolio at the confidence level $\alpha$ is given by the smallest number $l$ such that the probability that the loss $L$ exceeds $l$ is at most $(1-\alpha)$.Mathematically, if $L$ is the loss of a portfolio, then $\operatorname{VaR}_{\alpha}(L)$ is the level $\alpha$-quantile, i.e.

[$]\operatorname{VaR}_\alpha(L)=\inf\{l \in \mathbb{R}:\operatorname{P}(L \gt l) \le 1-\alpha\}=\inf\{l\in \mathbb{R}:F_L(l) \ge \alpha\}.[$]

Given a random variable $L$ which is the loss of a portfolio at some future time and given a parameter $0 \lt \alpha \lt 1$ then the tail value at risk is defined by 

[$] \operatorname{TVaR}_{\alpha}(L) = \frac{\left(F_L(v_{\alpha}) - \alpha \right)v_{\alpha} + \left(1- F_L(v_{\alpha})\right)\operatorname{E}[L \mid L \gt v_{\alpha} ]}{1 - \alpha},\hspace{5 pt} v_{\alpha} = \operatorname{VaR}_{\alpha}(L). [$]

Note the following:

• The VaR is not a coherent risk measure since it violates the sub-additivity property.
• $v_{\alpha} \leq \operatorname{TVaR}_{\alpha}(L)$ with equality if and only if $\operatorname{P}(L \gt v_{\alpha}) = 0$, i.e., the two risk measures are equal whenever the loss is bounded (almost surely) by the VaR.
• When the distribution of the losses, $F_{L}$, is continuous at $v_{\alpha}$, then $\operatorname{TVaR}$ reduces to $\operatorname{E}\left[L \mid L \gt v_{\alpha}\right]$.

If the loss distribution is continuous then the formula for the TVaR can be expressed in terms of limited expected values:

[] \begin{align*} \operatorname{TVaR}_{\alpha}(L) &= \operatorname{E}\left[L \mid L \gt v_{\alpha}\right] \\ &= v_{\alpha} + \frac{1}{1 - \alpha} \left (\operatorname{E}[L] - \operatorname{E}[L \wedge v_{\alpha}]. \right ) \end{align*} []