Historical VaR for Foreign Stocks Requires Aligned FX Shocks
Summary
The document explains how to include foreign exchange risk when estimating historical value at risk for a foreign-currency stock position. The position’s home-currency value depends on both the stock price and the exchange rate. Under log returns, the combined shock is the stock return less the FX return, applied to today’s stock price converted at today’s exchange rate.
The response says simple and logarithmic returns can both be used, provided the calculation is consistent throughout the historical VaR procedure. Its central practical point is to pair each historical stock return with the exchange-rate return observed at the same time. Those joint observations preserve the dependence between the two risk factors; independently sampling the returns would discard that relationship. The note gives a general scenario construction rather than a numerical VaR example, and its guidance assumes the FX quotation convention is handled consistently when forming the translated value.
Key ideas
- A foreign stock position has both equity-price and exchange-rate exposure in home-currency terms.
- With log returns, the translated position’s shock combines the stock and FX shocks with opposite signs.
- Simple or log returns can be used if the VaR calculation applies one convention consistently.
- Historical equity and FX shocks should remain paired by observation date to preserve their dependence.
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Full text
# Historical VaR for shares in foreign currency
# 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?
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 by ZRH (score 1, accepted)
https://quant.stackexchange.com/a/44354
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.