Combining Bond and FX Risk in Normal VaR

Summary

The document asks how to calculate a 10-day, 99% relative normal VaR for a short GBP zero-coupon bond when its value is measured in euros. The central issue is that the position’s euro-denominated profit and loss reflects both changes in the bond’s price and changes in the exchange rate, so bond volatility alone does not capture the full risk.

The answer frames the exposure as two components: the GBP bond and an EUR/GBP currency position. It proposes estimating the FX volatility and combining the two volatilities with a correlation term, with historical analysis offered as one way to estimate that correlation. This provides a general portfolio-risk approach rather than a completed VaR calculation. The answer does not supply an FX volatility, a correlation estimate, or all calculation details, so the stated inputs are insufficient to produce a numerical VaR from the discussion alone.

Key ideas

• A foreign-currency bond position has both bond-price risk and exchange-rate risk when valued in another currency.
• The answer models the exposure as a bond component and an FX component.
• Combined volatility depends on each component’s volatility and their correlation.
• Historical analysis is suggested as one way to estimate the correlation.
• The discussion does not provide enough FX inputs to calculate a numerical VaR.

Tags

# Normal VaR for short bond


# Normal VaR for short bond












So I'm short a GBP denominated zero-coupon bond which has a face value of 1 million pounds and a remaining maturity of 6 months. Furthermore, I have to assume that the daily return of a 6-month zero GBP bond has a volatility of 0.06% (when its price is converted into Euro). The current exchange rate is 0.88 Pound Sterling per Euro and the 6-mont interest rate in GBP is 5% per annum with continuous compounding.

I now have to calculate the 10-day 99% relative normal VaR of this investment and as a hint, I have that I must start by defining the volatility of the P1L of my investment. However, I'm not sure how I can solve this particular exercise as I haven't really worked with the VaR with different currencies.

## Answer by user35980 (score 2, accepted)

https://quant.stackexchange.com/a/55360

I would think that you would treat this as computing the VAR of a two asset case. In your case these assets would be 1/ your GBP bond and 2/ an FX position in EURGBP. You already have a vol measure for asset 1. Once you have a vol measure for the FX, you should be able to obtain the standard deviation of the combined asset via $$\sigma_{X+Y}=\sqrt{\sigma_X^2+\sigma_Y^2 +2\rho\sigma_X\sigma_Y}$$after making an estimate of $\rho$ (the correlation between the GBP bond and EURGBP), say via historical analysis.

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.