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Expected shortfall and scenario risk

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Start with the idea

Expected shortfall describes the average of the worst tail fraction. Unlike a single VaR threshold, it asks how severe the tail losses are. A recovery calculation then connects any realised drawdown to the gain required to rebuild capital.

Symbols, units & horizon

L: loss random variable, positive for a loss
l: candidate loss threshold
α: confidence level
inf: smallest qualifying threshold
VaRᵤ: loss quantile at probability u
ES: expected shortfall, average upper-tail quantile
∫: integrate over probabilities α to 1
μ_L, σ_L: mean and SD of loss
φ (varphi): standard normal density
Φ: standard normal cumulative probability
Φ⁻¹: standard normal quantile
d: fractional drawdown between 0 and 1

When and why to use this

Use ES beside VaR for tail budgeting and use recovery arithmetic to explain why deep losses constrain future growth. Both feed into liquidity and stress planning.

V a R_{α} (L) = \inf {l : P (L \leq l) \geq α}, E S_{α} = \frac{1}{1 - α} \int_{α}^{1} V a R_{u} (L) d u

Quantile definition + tail averaging

Average tail quantiles

VaR is the smallest loss threshold with cumulative probability at least α. ES integrates quantiles from α to 1 and divides by their interval length 1−α.
In an equally weighted sample with exactly k observations in that tail, ES is their arithmetic mean. Fractional boundary weight is needed if n(1−α) is not an integer.

Work it by hand

Losses 1,2,3,4,10: the worst 40% consists of 4 and 10, so ES at 60%=7. This avoids incorrectly counting all observations tied at a boundary.

Here L is a positive loss, unlike a return or P&L. Expected shortfall averages the worst tail fraction and also handles distributions with mass at the VaR threshold. For a continuous loss distribution it equals the conditional mean beyond VaR. Historical estimates are only as representative as the observations available.

E S_{α} = μ_{L} + σ_{L} \frac{φ (Φ^{- 1} (α))}{1 - α}

Calculus + normal table lookup

Derive normal expected shortfall

For L=μ+σZ and z=Φ⁻¹(α), $E [L | L > V a R] = μ + σ E [Z | Z > z]$ .
The numerator is $\int_{z}^{\infty} u φ (u) d u = φ (z)$ , because $φ' (u) = - u φ (u)$ . Divide by tail probability 1−α.

Work it by hand

α=.975 gives z≈1.96 and φ(z)≈.05844; multiplier=.05844/.025≈2.338. With μ=0 and σ=$1,000, ES≈$2,338.

The second formula assumes normal losses. At 97.5%, the multiplier is about 2.338, greater than the 1.960 VaR multiplier. Neither metric imposes a maximum possible loss. Examine concentration, stressed correlations, jumps and inability to trade.

r e c o v e r y r e t u r n = \frac{d}{1 - d}

Algebra and arithmetic

Solve the required recovery return

After drawdown d, wealth is $E_{0} (1 - d)$ . Recovery r must satisfy $E_{0} (1 - d) (1 + r) = E_{0}$ .
Cancel E₀ and divide by 1−d: $r = \frac{1}{(1 - d)} - 1 = \frac{d}{(1 - d)}$ , for d<1.

Work it by hand

A 30% drawdown needs .3/.7=42.857% growth. At a 100% loss, no finite percentage return on zero capital recovers the account.

A fractional drawdown d of 30% requires 42.86% growth to recover. There is no fixed drawdown-to-volatility ratio that holds across strategies, horizons and return distributions.

Research sources, review dates and limitations

Extend the research question

Stress collateral location as well as total portfolio P&L. Construct a path where a terminally hedged position is liquidated before convergence.

Continue with the connected research module →

Connect the ideas: Constraints and survival

Retrieve: A desired position must fit available capital and explicit limits.

Check the change: Portfolio weights, venue collateral, working orders and redemption obligations impose different constraints.

Portfolio construction → Optimization → Arbitrage → Crypto derivatives → Execution & microstructure → Fund operations & capstone

Explain it yourself: Can an offsetting terminal payoff remove a margin problem today?

Self-assessed. Write your explanation before opening this comparison.

No. Cash may be required before the hedge pays, or in another account. Check the path, collateral location and feasible transfer times.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from statistics import NormalDist
from math import floor

def empirical_es(losses, confidence=.95):
    """Exact integral of empirical quantiles, including fractional boundary mass."""
    if not losses or not 0 < confidence < 1:
        raise ValueError("Nonempty sample, confidence in (0,1)")
    ordered = sorted(losses)
    boundary = len(ordered)*confidence
    index = floor(boundary)
    tail_sum = (index+1-boundary)*ordered[index]+sum(ordered[index+1:])
    return tail_sum/(len(ordered)*(1-confidence))

def normal_es(mean_loss, sd_loss, confidence=.95):
    normal = NormalDist()
    z = normal.inv_cdf(confidence)
    return mean_loss+sd_loss*normal.pdf(z)/(1-confidence)

def recovery_return(drawdown):
    if not 0 <= drawdown < 1:
        raise ValueError("Recovery finite only for drawdown in [0,1)")
    return drawdown/(1-drawdown)

print(normal_es(0, 1), recovery_return(.2))

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