Expected Shortfall: The Loss Beyond the Threshold and the Cost of Exit

A loss percentile does not describe the severity of the losses beyond it. Expected Shortfall addresses that gap, but turning it into a capital measure adds further choices: sample conventions, holding periods, liquidity horizons, stress calibration, diversification constraints and separate charges for risks outside the modelled set. Each layer should retain its own meaning.

REFERENCE · NO CALL

This report consolidates the Expected Shortfall and capital exercises in September 2026 teaching material. Numerical results are illustrative calculations, including several explicitly assumed inputs. Regulatory architecture is explained historically and should not be read as a current implementation statement for a particular jurisdiction.

Define the tail consistently

Let $L$ be a positive loss variable. A robust definition of Expected Shortfall at confidence $\alpha$ is

$$ES_{\alpha}=\frac{1}{1-\alpha}\int_{\alpha}^{1}VaR_u\,du.$$

For a continuous loss distribution this corresponds to average loss beyond the $\alpha$-quantile. The integral definition also handles discrete probability mass at the boundary, where simply averaging observations strictly above VaR can be misleading.

The seven largest losses in the teaching case

Their arithmetic mean is approximately ₹11.20 crore. The source uses that mean as a rounded-tail illustration of 97.5% ES.

However, 2.5% of 250 observations is 6.25, not seven. Under an equal-weight empirical quantile-integral convention, the worst six observations receive full weight and the seventh contributes one quarter. Using the displayed rounded losses gives

$$ES_{97.5\%}\approx\frac{70.18+0.25(8.21)}{6.25} =₹11.5572\text{ crore}.$$

In words: Average the worst six losses and one quarter of the seventh using a total tail weight of six and a quarter.

Both the original ₹11.20 crore teaching headline and the estimator distinction are retained. Neither should be silently substituted for the other. A production process must specify its quantile and fractional-tail convention in advance.

Why a normal benchmark remains useful

For a zero-mean normally distributed loss with standard deviation $\sigma$,

$$ES_{\alpha}=\sigma\frac{\phi(z_{\alpha})}{1-\alpha},$$

where $\phi$ is the standard-normal density and $z_{\alpha}$ its quantile. At 97.5%, the multiplier is approximately 2.3378. The 99% VaR multiplier is approximately 2.3263. They are close under normality, differing by roughly half a percent.

Distinct normal and empirical risk measures in the source case

The roughly ₹2.79 crore gap shows that the tail in the exercise is more severe than the simple normal-linear benchmark suggests.

This comparison does not by itself isolate the reason. Distributional tails, nonlinear payoffs, factor mapping and volatility estimation can all contribute. A controlled model comparison is needed before attributing the whole gap to any single mechanism.

Coherence has a precise meaning

Expected Shortfall is subadditive under its usual integrability conditions:

$$ES(L_A+L_B)\leq ES(L_A)+ES(L_B).$$

This states that the combined risk is no greater than the sum of the component risks. It does not state that adding any position can never increase a portfolio’s risk. That stronger interpretation is false and should not be used to explain diversification.

The teaching example uses two independent bonds, each with a 4% default probability and total loss on default. A ₹100 crore holding in either has 95% VaR of zero and ES of ₹80 crore.

Loss states of the hypothetical diversified portfolio

Its 95% VaR is ₹50 crore, despite the diversification. Its ES is ₹51.6 crore, below the weighted single-name ES of ₹80 crore. The example demonstrates VaR’s possible failure of subadditivity and the importance of treating discrete tail mass correctly. The issuer labels are generic and the entire credit distribution is hypothetical.

Start with the required horizon, not a convenient shortcut

Square-root-of-time scaling follows from restrictive assumptions about independent increments, stable variance and approximately linear exposure. It can be a useful analytical benchmark, but it is not a general identity for nonlinear portfolio tails.

The source exercise reports 10-day ES of ₹27.24 crore from its specified 10-day change construction. Scaling its one-day ₹11.20 crore headline by $\sqrt{10}$ would instead give approximately ₹35.42 crore. The difference is material and demonstrates that a convenient scaling convention can produce another answer.

The direction of error is not guaranteed. Serial dependence, changing volatility, jumps, nonlinear pricing and the construction of overlapping observations all matter. A conservative-looking result in one case does not establish that the shortcut is conservative for every book.

Overlapping multi-day windows can increase the number of usable scenarios, but the resulting observations are dependent. They do not provide the same independent information as an equally large sample of nonoverlapping periods. Estimation and uncertainty analysis should preserve that distinction.

Liquidity horizons add risk from slower-moving exits

The historical liquidity horizons in days

The calculation uses nested sets of factors with increasingly long assigned horizons, rather than multiplying each instrument’s total risk by an arbitrary exit-time factor.

A schematic form is

$$ES_{LH}=\sqrt{ES_T(P)^2+ \sum_{j=2}^{m}ES_T(P,j)^2\frac{LH_j-LH_{j-1}}{T}},$$

In words: Combine the squared base-period risk with additional squared risks weighted for their longer holding periods, then take the square root.

where $T$ is the base horizon, $P$ the portfolio, $LH_j$ successive liquidity horizons, and $ES_T(P,j)$ the base-horizon ES with only the factors in the relevant longer-horizon set varying.

The case uses ₹27.24 crore for all-factor ten-day ES and ₹4.41 crore for the volatility-only component extending from ten to twenty days. It obtains

$$\sqrt{27.24^2+4.41^2}\approx₹27.60\text{ crore}.$$

The modest uplift reflects this particular factor mix. It is not evidence that liquidity adjustments are generally small. A book concentrated in longer-horizon spread or volatility factors can behave very differently.

Keep stress calibration separate from current-period measurement

The historical teaching framework applies a reduced-set stress calibration because some risk factors have longer observable histories than others. Its illustrative scaling is

$$ES_{\mathrm{stress}}=ES_{R,S}\max\left(\frac{ES_{F,C}}{ES_{R,C}},1\right),$$

In words: Scale reduced-set stressed risk by the current full-to-reduced risk ratio, without allowing that ratio to reduce it.

where $R$ and $F$ identify reduced and full factor sets, and $S$ and $C$ stress and current periods. The source explicitly assumes ₹27.40 crore, ₹12.90 crore and ₹10.55 crore for these three inputs, giving approximately ₹33.50 crore.

Those assumed figures are a separate calibration illustration. They should not be represented as observed outputs from the one-year desk sample or spliced into the later capital arithmetic as if all figures formed one fully recomputed production chain. The existence of a formula does not create the missing long-history data.

The reduced-set coverage condition also concerns the prescribed measurement through time, not merely whether one displayed ratio exceeds a threshold. A production implementation needs the full dated evidence for that test.

Aggregate capital with an explicit input ledger

Another source exhibit takes ₹27.60 crore of diversified capital and ₹29.94 crore as the sum of separate risk-class amounts. A half-weight to each produces ₹28.77 crore. It then uses an assumed non-modellable-factor charge of ₹1.08 crore and a historical multiplier of 1.83:

Capital teaching inputs, outputs and evidence status

The source also assumes a ₹76.29 crore standardised benchmark and illustrates an additional surcharge. Default-risk and other applicable components remain separate. These exhibits teach the capital architecture; they are not enough to certify a complete regulatory return.

ES still needs outcomes testing

Capital input governance should also distinguish sampling uncertainty from uncertainty about the model. More observations can improve estimation of a chosen distribution, but they do not establish that omitted factors, recovery assumptions or liquidity behaviour are correct. Conversely, changing the distribution after every difficult observation can make the calculation unstable. The review should identify which uncertainty is being measured and which control is intended to address it.

ES is not singly elicitable in the same way as a quantile, but it is jointly elicitable with VaR and can be assessed using specialised backtests. Saying that ES cannot be backtested is incorrect. Counting VaR exceptions and testing realised tail severity are complementary exercises.

Small tail samples, noisy estimates and model-dependent critical values remain constraints. A strong ES report therefore combines a precise estimator definition, historical calibration status, horizon construction, tail-severity evidence and clear separation of assumed from calculated capital inputs. That is what makes the final number interpretable.