VaR by example

1. VaR by example Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

2. Assets NIS TSAMUD $ Yen Deposit 1yr. 6% 4,000 Bonds 10yr. 5% 2,000 Credit 3yr.15%8,000 Liabilities Today L=6% NIS TSAMUD $ Yen Saving 2yr. 4% 1,800 Deposit 1mo. 11% 8,200 Deposit 3mo. L-2% 3,000 Total: (200) 200 4,000 (3,000) VaR example

3. Risk Factors • USD/NIS exchange rate • Yen/NIS exchange rate • Inflation • Real NIS interest rates (IR, 10 yr., 2 yr.) • Nominal NIS IR (1mo., 10 yr.) • USD IR, (1 yr.) • Yen IR, (Libor 3 mo.) VaR example

4. Fair Value For risk measurement we need not only the fair value, but the fair value as a function of risk factors in order to estimate the potential profit/loss. VaR example

5. Fair Value Function VaR example

6. Fair Value Function VaR example

7. Fair Value Function VaR example

8. Biggest market risk Significant risk Significant risk Sensitivity 0.1% 1% 2% 0.5% 0.5% 0.5% 0.5% 0.25% 0.25% -8 40 -60 3 -103 17 -93 -10 2 CPI USD Yen rnominal1mo rnominal3yr rreal2yr rreal10yr rUSD1yr rYen3mo VaR example

9. Real r 2yr 10 yr T Risky Scenario VaR example

10. Are not included into BoI requirements Sensitivity 0.1% 1% 2% 0.5% 0.5% 0.5% 0.5% 0.25% 0.25% -8 40 -60 3 -103 17 -93 -10 2 CPI USD Yen rnominal1mo rnominal3yr rreal2yr rreal10yr rUSD1yr rYen3mo VaR example

11. Gradient Vector Direction of fastest decay (loss). Take the sensitivity vector and divide it by the assumed changes in the risk factors. VaR example

12. What if ... The sensitivity vector allows to estimate quickly an impact of a certain market move on the value of the portfolio. Scalar multiplication of the gradient vector and the hypothetical market change vector gives the predicted loss/gain. VaR example

13. Risk Measurement • The gradient vector describes my exposure to risk factors • The distribution of risk factors allows me to estimate the potential loss together with probability of such an event. • The stress test will describe the response to specific (the most interesting) scenarios. VaR example

14. Risk Management • Swap Dollar Yen • Two forward contracts • Quanto option • FRA (?) • Fixed - floating swap VaR example

15. Duration and IR sensitivity VaR example

16. The Yield to Maturity The yield to maturity of a fixed coupon bond y is given by VaR example

17. Macaulay Duration Definition of duration, assuming t=0. VaR example

18. Macaulay Duration What is the duration of a zero coupon bond? A weighted sum of times to maturities of each coupon. VaR example

19. $ r Meaning of Duration VaR example

20. Proposition 15.12 TS of IR With a term structure of IR (note yi), the duration can be expressed as: VaR example

21. $ r Convexity VaR example

22. FRA Forward Rate Agreement A contract entered at t=0, where the parties (a lender and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S,T]. Assuming continuous compounding we have at time S: -K at time T: KeR*(T-S) Calculate the FRA rate R* which makes PV=0 hint: it is equal to forward rate VaR example

23. Exercise 15.7 Consider a consol bond, i.e. a bond which will forever pay one unit of cash at t=1,2,… Suppose that the market yield is y - flat. Calculate the price of consol. Find its duration. Find an analytical formula for duration. Compute the convexity of the consol. VaR example

24. ALM Duration • Does NOT work! • Wrong units of measurement • Division by a small number VaR example

25. ALM Duration A similar problem with measuring yield VaR example