4.1.1 Value at Risk & Expected Shortfall from Zero

Basic Prerequisites: Probability & Measure Theory (a CDF and a quantile are all you need).

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1. Intuition & Practical Objective

This page builds the why of VaR and Expected Shortfall with no prior risk-management knowledge needed. The objective is one idea: VaR tells you where the tail begins; ES tells you how bad the tail is. VaR reports a single quantile and says nothing about how severe the losses beyond it can be; a portfolio built to satisfy only VaR can therefore hide extreme tail risk, which is exactly what Expected Shortfall captures.

Start with the manager’s sentence everyone has heard: “Our 1-day $99\%$ VaR is $5 million.” Unpack it literally: over the next day, the loss will exceed $5m only about $1$ day in $100$. It says nothing about what happens on that one day. Do you lose $5,000,001 - or $100 million and the firm? VaR is deliberately silent, because it is a quantile, not an average.

Three steps:

1. VaR is a percentile, not a severity. Line up 100 simulated tomorrows, order them worst-to-best, and read off the loss at rank $99$ (for $99\%$ VaR). Everything worse than that line is discarded. That discard is exactly what makes VaR fragile.

2. Expected Shortfall averages what VaR throws away. ES is the mean of the worst $1\%$ - the conditional expectation of the loss given it breached VaR. It is larger than VaR by construction (${\mathrm{ES}}_{\alpha }\ge {\mathrm{VaR}}_{\alpha }$) and it moves when the tail worsens, which VaR does not.

3. The decisive difference is diversification. Two separately-safe bets can be jointly dangerous under VaR, so VaR says merging them created risk - an absurdity that breaks risk limits, capital allocation, and decentralised governance. ES never does this. That is the whole coherence story (03 · Coherent Risk Measures), previewed here with numbers.

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2. Mathematical Ground Truth & Derivations

The two definitions, in words and symbols. Let $L=-\Delta V$ be the loss over the horizon and $\alpha$ the confidence level (e.g. $0.99$):

$${\mathrm{VaR}}_{\alpha }(L)=F_L^{-1}(\alpha )=\inf \{l:\mathbb{P}(L\le l)\ge \alpha \},{\mathrm{ES}}_{\alpha }(L)=\mathbb{E}\left[L\right|L\ge {\mathrm{VaR}}_{\alpha }(L)].$$

VaR is the $\alpha$-quantile of the loss distribution; ES is the mean excess over that quantile. Both are monetary amounts (same units as $L$).

The subadditivity koan (after Artzner §3.3), made discrete. Take two independent bonds. Each loses $100 with probability $4\%$ and loses nothing with probability $96\%$. Ask for the $95\%$ VaR of each alone and then of the merged position:

• Alone: $\mathbb{P}(L>0)=4\%\le 5\%$, so the $95\%$ quantile is ${\mathrm{VaR}}_{95\%}=0$. Each bond looks riskless.
• Merged: $\mathbb{P}(\text{at least one defaults})=1-(0.96{)}^2=7.84\%>5\%$. The $95\%$ quantile is now $100$.

So ${\mathrm{VaR}}_{95\%}(A+B)=100\nleqq {\mathrm{VaR}}_{95\%}(A)+{\mathrm{VaR}}_{95\%}(B)=0$. A merger created measured risk from nothing - the opposite of diversification. ES behaves: ${\mathrm{ES}}_{95\%}(A)={\mathrm{ES}}_{95\%}(B)=80$ and ${\mathrm{ES}}_{95\%}(A+B)=103.2\le 160$.

Why the tail matters in money terms. For a normal loss $L\sim \mathcal{N}(\mu ,{\sigma }^2)$ the closed forms are (Hull eq. 22.1)

$${\mathrm{VaR}}_{\alpha }=\mu +\sigma z_{\alpha },{\mathrm{ES}}_{\alpha }=\mu +\sigma \frac{\phi (z_{\alpha })}{1-\alpha },z_{\alpha }={\Phi }^{-1}(\alpha ).$$

At $\alpha =0.99$ the ratio $\mathrm{ES}/\mathrm{VaR}=1.1457$: the average loss beyond the $99\%$ line is $14.6\%$ larger than the line itself - an amount VaR never reports. Under fat tails the gap explodes.

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3. Computational Implementation - the two-bond koan, by hand and by Monte Carlo

We enumerate the loss distribution exactly (no simulation noise), then confirm it with a Monte Carlo book of 50 bonds. Stdlib only.

Read the first two lines aloud: identical economic exposures, and VaR doubles in extremity purely because the positions were combined. ES gives a sane, sub-additive answer. That is the paper’s whole complaint in one table.

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4. Failure Modes & First-Principles Breakdowns

1. The “it’s just a percentile” trap. A quantile is not a loss; it is a threshold. Reporting only VaR hides the entire loss-given-breach distribution. Always report ES (or the full tail) alongside any VaR number.
2. Mistaking the tail frequency for the tail size. VaR controls how often you breach; it ignores how deep the breach is. A strategy selling far-OTM options maximises small gains while pushing catastrophic losses just past the quantile - profitable until it is fatal (05 · Failure Modes).
3. Confidence-level superstition. "$99\%$" is a convention, not a guarantee. With $250$ trading days a year, a $99\%$ one-day VaR is breached about $2.5$ times per year - the estimator itself must be backtested, not trusted (Kupiec’s POF test; Parametric, Historical & Monte Carlo VaR).
4. Sign and convention confusion. VaR/ES are reported as positive loss numbers; a loss of $L=-100$ has $\mathrm{VaR}=100$. P&L sign errors flip a risk limit into a green light.

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5. References

• Artzner, Delbaen, Eber & Heath, Coherent Measures of Risk, Mathematical Finance 9(3) (1999)
• Hull, Options, Futures, and Other Derivatives
• McNeil, Frey & Embrechts, Quantitative Risk Management (2015)
• Rockafellar & Uryasev, Optimization of Conditional Value-at-Risk, J. Risk 2(3) (2000)

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6. Connected Graph Bridges

• Base: Probability & Measure Theory
• Continue: 02 · VaR Definition & Flaws · Index Hub
• Sibling: Extreme Value Theory & Fat Tails (what happens when the tail is power-law)