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 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.