Recalculating Portfolio VaR Across Currencies

# recalculate VaR from one currency to another


# recalculate VaR from one currency to another












I have a strange question. For example, I have 95% VaR (1Y, delta-normal) for portfolio with one stock in same currency (for example, USD). What minimal information should I have for recalculation this VaR from USD to RUB? USD_RUB and this portfolio stock are correlated.

## Answer by Dimitri Vulis (score 2, accepted)

https://quant.stackexchange.com/a/65942

It's not a strange question at all. It's actualy quite common for firms doing business in different countries to need to calculate VaRs in multiple currencies.

Case 1: your exposure to the price of the stock is linear (i.e. no options). You simply own some shares. It does not matter whether you own U.S. shares (e.g. IBM US) or Russian shares (e.g. Lukoil in Moscow) or American Depositary Receipts (ADRs) of Russian shares. At any point in time, you have a USD-denominated share price, and a RUB-denominated share price (USD price * RUB/USD exchange rate). (The USD price is RUB price / RUB/USD exchange rate).

To calculate the USD VaR, it may be acceptable to make some assumptions about distributions and to multiply the exposure by the (historical) volatility of the USD price and normsinv of the confidence interval. Likewise, to calculate the RUB VaR it may be acceptable to multiply the exposure by the volatility of the RUB price instead. (Note that this may not be the same as mutiplying the vector of your exposures (stock and fx) by the covariance mutrix and the transpose of exposures.) However, because the correlation $\rho$ is very "information-lossy", there is no way (that would be acceptable to model validators / auditors / regulators) even to estimate one currency VaR from the other currency VaR and the $\rho$ of the stock returm and the exchange rate return, because the price and the exchange rate may move together in complicated ways that $\rho$ does not "know".

Case 2: some of your exposures are non-linear, or you're not allowed to make assumptions necessary for case 1, and so you calculate the VaR by consiering the P&L impact under a large number of scenarios (historical and/or from Monte Carlo) for the stock price and for the exchange rate, rank the resulting P&Ls, and look at the worst 1% (for example). As before, you have to run this calculation separately for each currency.

Edit:

> Could you explain your Case 1 a little bit deeper?

Sure. Suppose you own 1 share of IBM common equity. Your accounting is in USD. What's the most money you can lose over $d$-day horizon with 95% probability ?

The calculation that you learn in an undergraduate course is:

You find the daily time series of IBM stock prices (USD-denonminated, of course), adjusted for splits and dividends.

You calculate the series of total returns: $\mathrm{return}_i=\frac{\mathrm{price}_i-\mathrm{price}_{i-1}+\mathrm{dividend}_i}{\mathrm{price}_{i-1}}$ from day $i-1$ to day $i$.

You calculate the annualized standard deviation of the returns and denote it $\sigma_{IBM,USD}$.

Your exposure to IBM's price, scaled to 100%, is just the mark to market of your position. Multiply $\sigma_{IBM,USD}$, the absolute value of your exposure $|\delta|$, sqrt($d$ days horizon/trading days per year), and normsinv(95%)$\approx$ 1.645 standard deviations to obtain the USD VaR.

This methodology would have been fine in 1990. This might not fly with model validation and regulators in 2020 because you assume that this $\sigma$ describes all the features of your time series, and that you need no information beyond this one number. (Further, if you had more than one stock in your portfolio, then you'd want more complicated correlations.)

But assuming that you are allowed to calculate your VaR this way, what could you do to calculate a RUB VaR if your accounting is in RUB? Now you have not one, but 2 bets: you bet the IBM's USD price will go up, and you also bet than USD will appreciate against RUB. So you have two exposures, each equal to your position's value. Let $\Delta=\left(\matrix{\delta & \delta}\right)$ denote the vector of your 2 exposures - same value twice.

You can actually do two things here.

You find the time series of the RUB/USD exchange rates. Let $\sigma_{RUB}$ denote the historical volatility of its returns, $\rho$ denote the correlation of RUB returns and IBM USD returns, and $c=\rho\sigma_{RUB}\sigma_{IBM,USD}$ denote their covariance. You'd need to deal with the practical problems of one series having data on days when the other does not, as well of rates observed at different time zones. You again assume that these few numbers tell you all you need to know about the features of your time series, which is likely to be problem in 2021. (In particular, you assume that, despite the large discrepancy between USD and RUB nominal interest rates, the average change in their exchange rate is 0. Is that realistic?)

The variance-covariance matrix $C=\left(\matrix{\sigma_{IBM,USD}^2 & c\\ c & \sigma_{RUB}^2}\right)$.

Mutliply $\mathrm{sqrt}(\mathrm{mmult}(\mathrm{mmult}(\Delta,C),\mathrm{transpose}(\Delta)))$ and the same ratio of days and normsinv as above to obtain RUB VaR.

One advantage is that you can calculate "component VaR" to see how much of your RUB VaR comes from your currency bet, and how much comes from your equity bet.

If you want to play forensic accountant, you could start out from your USD VaR (calculated from the product of $|\delta|$, $\sigma_{IBM,USD}$, et al ; divide the VaR by $\sigma_{IBM,USD}$, ratio of days, and normsinv to back out the stock exposure, and include the RUB exposure and $\rho$ in the mix, and do the above matrix multiplication.

Note that, even if you know for sure that all the risk comes from the exposure to a single stock, you can recover the absolute value of the delta, but you don't know the sign of the delta. You have the same VaR whether you are long 1 share or short 1 share. But the sign does not matter - the RUB VaR is the same whether you are long IBM (and USD) or short IBM (and USD).

However if your USD VaR came from more than one exposure (e.g. 2 stocks rather than 1), then you would not be able to able to back out the exposures.

Also if the USD VaR had been calculated by ranking the P&L impacts of historical scenarios, rather than by multiplying $|\delta|$ and $\sigma$, then dividing the VaR by $\sigma$ is unlikely to be very close to the original $|\delta|$, illustrating the skepticism of normality assumptions.

Another approach is to calculate the time series of IBM stock price denominated in RUB, and calculate their returns, and the volatility of these returns $\sigma_{IBM,RUB}$, and to multiply the absolute value of your sensitivity to IBM RUB price by $\sigma_{IBM,RUB}$. Arguably, it's not much worse than the matrix multiplication above.

## Answer by emot (score 1)

https://quant.stackexchange.com/a/65944

Consider stock $S$ and $X$ the currency rate. You want to calculate VaR for $SX$.Assume that $S$ and $X$ follows: $$S_{t}=S_{t-1}*exp(Z_{1,t}*\sqrt \Delta t*\sigma_1)$$ $$X_{t}=X_{t-1}*exp(Z_{2,t}*\sqrt \Delta t*\sigma_2)$$ where $Z_1$ and $Z_2$ are correlated standard Gaussians with correlation $\rho$ and mean 0.

Then $$ln(\frac{S_{t}*X_{t}}{S_{t-1}*X_{t-1}})=Z_1*\sqrt \Delta t*\sigma_1+Z_2*\sqrt \Delta t*\sigma_2$$ Then variance of this is $Var(Z_1*\sqrt \Delta t*\sigma_1+Z_2*\sqrt \Delta t*\sigma_2)=\sigma_1^2*\Delta t+\sigma_2^2*\Delta t+2*\sigma_1*\sigma_2*\Delta t*Cov(Z_1,Z_2)$ $$=\sigma_1^2*\Delta t+\sigma_2^2*\Delta t+2*\sigma_1*\sigma_2*\Delta t *\rho$$ VaR for quantile $\alpha$ is $Y=\Phi^{-1}(1-\alpha)*\sqrt {\sigma_1^2*\Delta t+\sigma_2^2*\Delta t+2*\sigma_1*\sigma_2*\Delta t *\rho}$

To transfer it to price space $$S_{t}*X_{t}*(exp(-Y)-1)$$

To sum up, you need volatiltiy of the Stock $S$, FX Rate $X$ and correlation $\rho$ between returns of the Stock and FX rate.