Historical VaR for shares in foreign currency

Question

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?

Thanks in advance.

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

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

Answer 1

whether you use simple returns or log returns does not matter at all. If you are using a historical VaR approach, you would take price timeseries (incl for FX), deduce daily returns and the apply them to your position. As long as you are consistently using the same calculation (discrete return/continuous return) on the return timeseries for purposes of VaR calculation, you will get precisely the same results.

Please do note that shocks have to be aligned in order for results to be correct. Rewriting your formula:

$\tilde{P}_{T+1,s}=\frac{R_{T,s}^1}{R_{T,s}^2}e^{r_{t}^1-r_{t}^2}$

So basically you would take today's spot price and convert it to your base currency at today's FX rate. Then you go and take historical log shocks of stock price and FX rate (always measured at coincident times, and not drawn randomly from their respective distributions), and combine them into a series of log shocks that apply to the stock price translated into your base currency. Only if you take the stock & FX log shocks measured at the same time, you get the correlation right. Drawing randomly destroys the correlation.