VaR and Expected Shortfall for Managing the FX Swap Risk

In my recent post on Garman Kohlhagen option values, I presented the probability density (in the risk neutral measure) for the Garman Kohlhagen model.

The above plot shows the probability density of the FX value in 5 years after the start date, when we start at F0=1.65, when the interest rate differenct between CHF and EUR is 1.5% per year and the annual volatility is 2.5%. The y-axes is drawn at the the magic 1.54, meaning that left of the axes the City has to pay additional coupons.

The vertical line (at 1.39194) is chosen in such a way that the filled area measures exactly 0.05. Being a probability density, the complete area (from zero to infinity) has to be 1.
Therefore, the value F=1.39194 (at T=5, with vol=2.5%) is that value for which 5% of the FX random walk results are worse (from the point of view of the city) and 95% are better. This is the definition of the 95% Value at Risk.

In that case, the interest rate (at T=5) the City has to pay is
VaR=(1.54-1.39194)/1.39194 = 10.637%
(plus the basis fee of 0.065%).

The filled area describes those cases that are worse than the 95%VaR, and the 95% expected shortfall is defined as the expectation of losses given that the outcomes are worse than the 95% VaR. Expected shortfall is used to study the tail influence in risk management.

Here we obtain for the expected shortfall (at T=5, vol=2.5%):
ES=13.28% (plus basis fee).

In both cases (VaR and ES) we did not discount the future payments.

Obviously, these extreme outcomes (95% are better) depend heavily on the volatility and on the initial value of F. Remember that in 2011 extreme (historical) volatilities of 16% were observed.

Next: Value at Risk and Expected Shortfall for the combination of 20 coupon payments instead of one isolated coupon.