2.8 Value-at-Risk: Normal versus Student-t · M02 Statistics of returns

Motivation

Brokers, clearing corporations and regulators turn the distribution of returns into a single loss number. NSE Clearing's margin system (SPAN, see M07) and every bank's market-risk capital rest on Value-at-Risk and its relatives. The choice of distribution matters most exactly where the number is used.

VaR and Expected Shortfall#

Setup

For a position with one-period return $r$ , the Value-at-Risk at confidence level $c$ is the loss not exceeded with probability $c$ : $\text{VaR}_c = -q_{1-c}(r)$ , where $q_p$ is the $p$ -quantile. Expected Shortfall is $\text{ES}_c = -E[r \mid r \le q_{1-c}(r)]$ , the average loss given that VaR is exceeded.

Proposition 2.14

(a) If $r \sim N(\mu, \sigma^2)$ : $\text{VaR}_c = -\mu + \sigma\Phi^{-1}(c)$ and $\text{ES}_c = -\mu + \sigma\,\phi(\Phi^{-1}(c))/(1-c)$ . (b) If $r = \mu + s T_\nu$ with $T_\nu$ standard Student- $t$ , $\nu > 2$ , and $s = \sigma\sqrt{(\nu-2)/\nu}$ (so $\operatorname{sd}(r) = \sigma$ ), then $\text{VaR}_c = -\mu + \sigma\sqrt{(\nu-2)/\nu}\; t_\nu^{-1}(c)$ .

Proof

(a) $q_{1-c} = \mu + \sigma\Phi^{-1}(1-c) = \mu - \sigma\Phi^{-1}(c)$ . For ES, $E[Z \mid Z \le -a] = -\phi(a)/\Phi(-a)$ for standard normal $Z$ , using $\int_{-\infty}^{-a} z\phi(z)\,dz = -\phi(a)$ ; with $a = \Phi^{-1}(c)$ and $\Phi(-a) = 1 - c$ the result follows. (b) $\operatorname{Var}(T_\nu) = \nu/(\nu-2)$ , so the scale $s$ gives variance $\sigma^2$ ; quantiles transform linearly and $t_\nu$ is symmetric.∎

Comparing the two models#

Worked example 2.8.1

Take daily $\sigma = 1.2\%$ , $\mu = 0$ .

Level	Normal	Student- $t$ , $\nu = 4$	Student- $t$ , $\nu = 3$
99% VaR	2.79%	3.18%	3.15%
99.9% VaR	3.71%	6.09%	7.08%

At 99% the distributions roughly agree, because matching variance forces the $t$ to be narrower in the middle; the extra tail mass shows up further out. At 99.9% the $t$ VaR is 1.6–1.9 times the normal one. On a ₹50 lakh Nifty position the 99.9% one-day VaR is ₹1.86 lakh under the normal model and ₹3.04–3.54 lakh under the $t$ models.

Worked example 2.8.2Nifty sample, in-sample check

With the full-sample daily standard deviation (1.29%), the normal 99% VaR (2.97%) was exceeded on 1.79% of days, against 1% expected; the normal 99.9% VaR (3.96%) was exceeded on 0.81% of days, eight times the nominal 0.1%. The $t_4$ 99.9% VaR (6.52%) was exceeded on 0.15% of days. The empirical 0.1% quantile of daily returns was $-7.2\%$ .

Practitioner notes

VaR says nothing about the size of losses beyond it. Basel's Fundamental Review of the Trading Book moved bank capital from 99% VaR to 97.5% Expected Shortfall for this reason.
Unconditional VaR with full-sample volatility is wrong in both directions: too high in calm periods and too low in stressed ones. Conditional VaR uses a volatility forecast (2.7), and that matters more than the choice of distribution.
Historical-simulation VaR (use the empirical quantile of the last 250 or 500 days) needs no distributional assumption but cannot produce a loss larger than the worst day in the window.

2.8 Value-at-Risk: Normal versus Student- $t$

VaR and Expected Shortfall#

Comparing the two models#

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