# 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

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.

• Agree...without an SDE for returns, you cannot relate ES over two time periods. That said it would be a very odd case that would have $ES_T > ES_{T+t}$ – Brian B Oct 17 '14 at 11:41
• Lets make it simpler. What if I assume that that $L_1, L_2\sim\mathcal{N}(\mu,\sigma^2)$ and $L\sim$ Bivariate Normal Mixture – Rohit Arora Oct 18 '14 at 11:45
• Of course empirical is always different. So you should assume a distribution and investigate ES over different times. – emcor Oct 18 '14 at 13:06