Value at risk and quantile estimation

Definition Fix a portfolio, a horizon $h$ in years, a reporting currency and a probability measure, usually the physical measure $\mathbb{P}$ or a stated scenario distribution. If $X$ is the profit and loss of the period in currency units, a mark change plus any included cash flows as in Profit and loss attribution and sensitivities, the loss is $L = -X$. With this sign change a larger positive $L$ is worse and a negative $L$ is a gain. The loss distribution is the law of $L$ under the stated measure, described by its distribution function

for real $\ell$, which is nondecreasing and right-continuous, with limit $0$ at $-\infty$ and limit $1$ at $+\infty$. An atom is a value $\ell$ with $\mathbb{P}(L = \ell) > 0$, where $F_L$ jumps by that probability. Losses from discrete events such as defaults, from scenario sets and from empirical samples have atoms, so a definition that works only for continuous distributions is not enough.

Definition For a confidence level $\alpha \in (0, 1)$, the value at risk of the loss $L$ is the lower $\alpha$ quantile of its distribution,

The set is nonempty and bounded below because $F_L$ tends to $1$ at $+\infty$ and to $0$ at $-\infty$ while $0 < \alpha < 1$, so the infimum is finite, and right-continuity makes it attained, $F_L(\operatorname{VaR}_{\alpha}) \ge \alpha$. Hence

and $\operatorname{VaR}_{\alpha}(L) \le \ell$ holds exactly when $F_L(\ell) \ge \alpha$. When $F_L$ is continuous and strictly increasing, VaR is the unique solution of $F_L(\ell) = \alpha$. VaR is an amount in currency units, reported with its horizon, currency, portfolio, loss definition, measure and level. It is neither the largest possible loss nor an average of tail losses, and at an atom it can jump when the probabilities change slightly. If the profit and loss $X = -L$ is stored instead, then $\operatorname{VaR}_{\alpha}(L) = -q^{+}_{1-\alpha}(X)$ with the upper quantile $q^{+}_{\beta}(X) = \inf\{x : F_X(x) > \beta\}$, and the lower quantile of $X$ gives a different answer wherever $F_X$ is flat at level $1 - \alpha$.

Definition For losses $L_1, \dots, L_n$ with equal weights $1/n$, the empirical distribution function is

Write $L_{(1)} \le \dots \le L_{(n)}$ for the sorted sample. The lower inverse of $\widehat F_n$ at level $\alpha$ is the order statistic

Indeed $\widehat F_n(L_{(k)}) \ge k/n \ge \alpha$. Any $\ell < L_{(k)}$ lies below the $n - k + 1$ observations $L_{(k)}, \dots, L_{(n)}$, so $\widehat F_n(\ell) \le (k - 1)/n < \alpha$, because $k$ is the smallest integer with $k \ge n\alpha$. Ties do not affect the argument. This is the lower inverse convention applied to the empirical law, and the losses strictly above $\widehat q_{\alpha}$ carry empirical probability at most $1 - \alpha$. Interpolating between neighboring order statistics defines a different estimator, and numerical libraries differ in their default rules, so two reported VaRs can be compared only after their conventions are matched.

Definition Historical VaR applies the empirical quantile to losses of the current portfolio under past scenarios. Take $n$ historical changes $\Delta x_1, \dots, \Delta x_n$ of the risk factors over windows of the horizon length, apply each to today’s factors $x$ with today’s positions, and revalue, so that

With equal weights, historical VaR at level $\alpha$ is $L_{(\lceil n\alpha \rceil)}$. Relative or logarithmic changes can replace absolute ones, and that choice is part of the method. The number is exact for the empirical law of the scenarios. Treating it as the VaR of the next period assumes that the scenarios are a relevant sample of next-period changes, which is an assumption about the future and not a consequence of a long history. Overlapping windows make the scenario losses dependent, and changes in holdings or in the volatility regime can make older scenarios describe a different distribution. Reweighting the scenarios or revaluing a changed portfolio changes the empirical law, and the quantile has to be recomputed from it.

Definition A risk measure $\rho$ is subadditive on a set of losses if

for all $L_1$ and $L_2$ in the set. VaR is monotone, translation equivariant and positively homogeneous, but it is not subadditive on the set of all losses, and the example below gives two independent losses whose sum has a larger VaR than the sum of their VaRs. Subadditivity does hold in scoped cases. A normal loss with mean $\mu$ and standard deviation $\sigma$ has $\operatorname{VaR}_{\alpha} = \mu + \sigma\,\Phi^{-1}(\alpha)$. If $(L_1, L_2)$ is jointly normal, the sum is normal too, the means add, $\sigma_{L_1 + L_2} \le \sigma_{L_1} + \sigma_{L_2}$, and for $\alpha \ge \tfrac12$ the factor $\Phi^{-1}(\alpha)$ is nonnegative, so VaR is subadditive. A result for one family of distributions is not a theorem about all of them.

Definition Suppose $L_1, \dots, L_n$ are independent with a common distribution function $F$, and $F$ is differentiable at $q_{\alpha} = \operatorname{VaR}_{\alpha}$ with derivative $f(q_{\alpha}) > 0$, for example when $F$ has a density that is continuous and positive at $q_{\alpha}$. Then the empirical quantile $\widehat q_{\alpha} = L_{(\lceil n\alpha \rceil)}$ satisfies

in distribution as $n \to \infty$, a result stated here without proof, which gives the approximate standard error

The reason shows in the empirical distribution function. At the fixed point $q_{\alpha}$, the value $\widehat F_n(q_{\alpha})$ is an average of independent indicators with mean $\alpha$ and variance $\alpha(1 - \alpha)/n$, and near $q_{\alpha}$ a probability error $\delta$ moves the quantile by about $\delta/f(q_{\alpha})$. A thin tail, where $f(q_{\alpha})$ is small, turns a small probability error into a large threshold error. The statement is asymptotic and holds for this sampling law. It is not an exact interval for a given dataset, and atoms at the quantile, dependence such as overlapping windows, changing volatility and sparse tails need other methods.