2
$\begingroup$

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

| improve this question | |
$\endgroup$
1
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • $\begingroup$ 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 $ES_T > ES_{T+t}$ $\endgroup$ – Brian B Oct 17 '14 at 11:41
  • $\begingroup$ 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 $\endgroup$ – Rohit Arora Oct 18 '14 at 11:45
  • $\begingroup$ Of course empirical is always different. So you should assume a distribution and investigate ES over different times. $\endgroup$ – emcor Oct 18 '14 at 13:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.