## Pages

### VaR and Expected Shortfall - the City Swap Continued

For the exposure analysis of THE swap, I recommend to re-read
We valuate THE swap on February 5 (describes the payoff of the swap),
The future development of exchange rates (describes the Garman Kohlhagen model),
Garman Kohlhagen analyzed and FX option values underGarman Kohlhagen (distribution properties and option values), and
VaR and expected shortfall for managing the FX swap risk (that covers the risk measures for a single swaplet, i.e. the coupon payment at one coupon date).

For the exposure analysis of the swap in total, the risk numbers (VaR or expected shortfall) can not just be added: The termsheet of the swap states that each coupon to be paid by the City depends on the EUR/CHF exchange rate. Thus, there is significant path dependence (in some Asian style) contained. For the calculation of the VaR or the expected shortfall, we therefore use Monte Carlo simulation.

 Mathematica code for the Monte Carlo simulation. An initial rate of 1.65 and a constant (low) annual volatility of 2.5% is used. Note that the "0.5" in In[4] reflects the fact that coupons are paid semi-annually.
We calculated 100 000 paths of FX rates. The 95% VaR is that the City has to pay 104% of the notional amount of 195 mio CHF (over the lifetime, not discounted, no basis fee included) and the 95% expected shortfall is 127%.

This VaR and expected shortfall is extremely sensitive to volatility changes. The 2.5% in the above example are rather low. For the last 14 years (Feb. 2000 thru Feb.2014), the average historical volatility (in the sense of realized variance) was 6.32%. If we use this value, we obtain a 95% VaR of 276% (over the lifetime) and 95% ES of 347%.

The possible gain (Bank pays CHF Libor to City) in this simple Garman-Kohlhagen model was always 30%.