Expected Shortfall Across Periods and Loss Distributions
Summary
The document asks whether the expected shortfall of a loss series formed from two time periods can be related to the expected shortfalls of the component periods. It explores a simplified distributional case: component losses are treated as normal, while the combined loss distribution is represented as a weighted mixture of normal distributions. The mixture’s quantile and expected shortfall depend on the weights assigned to its components, so the component expected shortfalls alone do not determine the combined measure without further assumptions.
The response gives no general relationship or inequality. It cautions that changing market conditions between periods can make combined expected shortfall either higher or lower. If a return distribution is assumed stable and the periods have equal length, the respondent suggests the distributions should be the same. These points are qualitative; the document offers no empirical analysis, and the mixture formulation relies on assumptions that may not hold in actual loss data.
Key ideas
- The combined expected shortfall depends on how the component distributions are weighted.
- Component expected shortfalls alone do not specify the combined measure in the mixture example.
- Changing market conditions can make combined expected shortfall higher or lower.
- A stable distribution across equal-length periods is suggested as a simplifying assumption.
- The response offers no general inequality or empirical validation.
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Full text
# What are the properties of the Expected Shortall measure when split in multiple time periods?
# What are the properties of the Expected Shortall measure when split in multiple time periods?
Suppose I have a single time series of losses $L$ that consists of two sub-parts $L_1$ and $L_2$.
Is there a relationship that relates the expected shortfall of $L$ to the expected shortfall of $L_1, L_2$
$$ {\rm{L = }}\left[ {\begin{array}{*{20}{c}} {{L_{1,T_1}}}\\ {{L_{2,T}}} \end{array}} \right] $$
We know that $T_1<T$.
Any reference is much appreciated.
Inequalities are fine.
Lets make it simpler. What if I assume that that $L_1, L_2\sim\mathcal{N}(\mu,\sigma^2)$ and $L\sim\mathcal{N}(\mu_1,\sigma_1,\mu_2,\sigma_2,w_1)$ (Bivariate Normal Mixture). Then based on Broda and Paolella (2011), I know that
$$ \begin{array}{c} {w_1} + {w_2} = 1\\ {F_L}\left( x \right) = {w_1}\Phi \left( {\frac{{x - {\mu _1}}}{{{\sigma _1}}}} \right) + {w_2}\Phi \left( {\frac{{x - {\mu _2}}}{{{\sigma _2}}}} \right)\\ {F_L}\left( q \right) = 1 - \alpha \\ E{S_\alpha }\left( L \right) = {w_1}\frac{{\Phi \left( {\frac{{q - {\mu _1}}}{{{\sigma _1}}}} \right)}}{{1 - \alpha }}\left( {{\mu _1} - {\sigma _1}\frac{{\phi \left( {\frac{{q - {\mu _1}}}{{{\sigma _1}}}} \right)}}{{\Phi \left( {\frac{{q - {\mu _1}}}{{{\sigma _1}}}} \right)}}} \right) + {w_2}\frac{{\Phi \left( {\frac{{q - {\mu _2}}}{{{\sigma _2}}}} \right)}}{{1 - \alpha }}\left( {{\mu _2} - {\sigma _2}\frac{{\phi \left( {\frac{{q - {\mu _2}}}{{{\sigma _2}}}} \right)}}{{\Phi \left( {\frac{{q - {\mu _2}}}{{{\sigma _2}}}} \right)}}} \right)\\ = {w^*}_1E{S_\alpha }\left( {{L_1}} \right) + {w^*}_2E{S_\alpha }\left( {{L_2}} \right) \end{array} $$
Even with simplistic assumptions, it depends on the weights in which I combine the 2 distributions in the mixture
## Answer by SRKX (score 1)
https://quant.stackexchange.com/a/15067
I don't think you can say anything general on this type of setup, certainly not from an empirical point of view. Assume the market conditions change between the two periods, then $ES$ could be higher or lower.
If you assume some distribution for you returns, then they should probably be the same if the two periods have the same length.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.