# Historical VaR for shares in foreign currency

I'm currently studying John Hull's [1] example on historical value at risk for portfolio consisting of four stock indices.

In this example Hull converts the prices of the stock indices to the home currency first and then calculates the daily returns of the portfolio in home currency. Which he later uses to calculate the VaR.

What made my question, is that e. g. Auer [2] suggests using lognormal returns for both exchange rates and stock prices. Therefore I'd assume, that a foreign stock is dependent from two risk factors and calculate the tomorrows scenario price $$\tilde { P } _ { T + 1 , s }$$ as follows: $$\tilde { P } _ { T + 1 , s } = \frac{{ R } _ { T , s } ^ { 1 } e ^ {r _ { t } ^ { 1 }}}{{ R } _ { T , s } ^ { 2 } e ^ {r _ { t } ^ { 2 }}},$$

where $${ R } _ { T , s } ^ { 1 }$$ is todays ($$T$$) stock price in foreign currency, $$r _ { t } ^ { 1 }$$ the scenario log return of the stock, $${ R } _ { T , s } ^ { 2 }$$ is todays exchange rate and $$r _ { t } ^ { 2 }$$ the scenario logarithmic return of the exchange rate.

Is there a reason why Hull suggests using simple returns instead of log returns? Is it a fair assumption to calculate the value of a foreign stock, as written above?

[1] Options, futures and other derivatives; 2018; p. 519 ff.

[2] Hands-On Value-at-Risk and Expected Shortfall; 2018; p. 22

$$\tilde{P}_{T+1,s}=\frac{R_{T,s}^1}{R_{T,s}^2}e^{r_{t}^1-r_{t}^2}$$
• OK i see, thanks for describing precisely where you have the issue. If you convert today's stock price to your base currency, and then apply some historical return (as determined from stock prices), that would be wrong. Likewise, the other approach you describe is wrong. Both approaches do not factor in the correlation of FX and stock price. If you use $r_t^1$, then you have to use $r_t^2$ for the same t. I will amend my answer accordingly. – ZRH Mar 3 at 10:09