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Ciao, I'm working on VaR and Expected Shortfall and this question came out. For a given portfolio VaR can be computed w.r.t. all the risk factors or just for a subset. Infact you can decide to 'freeze' some of them in order to isolate the impact of certains risk factors rather then others.

Let me formalise a little bit. For simplicity we will consider a portfolio $P_t$ composed by one stock $FX_t \cdot S_t$. It will depend on the risk factor $r_t$, the interest rate; $FX_t$ is the FX rate. We can use these very classical models (it is not so important I tink):

\begin{align} dS_t & = r_tS_t dt + \sigma_S S_t dW^S_t \\ dr_t &= (\theta_r - \gamma_r r_t) dt + \sigma_r dW^r_t \\ dFX_t &= (\theta_{FX} - \gamma_{FX} FX_t) dt + \sigma_{FX} dW^{FX}_t \\ \end{align} (suppose all the Brownian motion have $0$ correlation...for simplicity again)

By standard integration: \begin{align} S_t & = S_0 \exp \left[ \left(r_t - \frac{1}{2}\sigma_S^2 \right)dt + \sigma_S W_t \right] \\ r_t & = r_0e^{-\gamma_r t} + e^{-\gamma_r t}\frac{\theta_r}{\gamma_r}(e^{r_t t}-1) + \sigma_r e^{-\gamma_r t}\frac{e^{2\gamma_r t}-1}{2\gamma_r t}W^r_t \\ FX_t & = FX_0e^{-\gamma_{FX} t} + e^{-\gamma_{FX} t}\frac{\theta_{FX}}{\gamma_{FX}}(e^{FX_t t}-1) + \sigma_{FX} e^{-\gamma_{FX} t}\frac{e^{2\gamma_{FX} t}-1}{2\gamma_{FX} t}W^{FX}_t \end{align}

With $VaR(FX_t, r_t)$ I mean the VaR of $P_t$ computed w.r.t both the risk factors, while $VaR(r_t)$ is the VaR computed with the $FX_t$ freezed (i.e. considered as a constant).

The question is then:

Is there a relation between and $VaR(FX_t, r_t)$ and $VaR(r_t)$ such as $$VaR(FX_t, r_t) > VaR(r_t)$$ Is there an idea (even totally euristic one) which could show which of the two quantities is in general the greater?

Right now I 'm trying to do the formal computation following the models I've imposed but I think the work would be more efficient if, more or less, I have an idea about what to look for.

Thank you! Ciao!

AM

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