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

2. Zvi WienerVaR example slide 2 Assets NISTSAMUD$Yen Deposit 1yr. 6%4,000 Bonds 10yr. 5%2,000 Credit 3yr. 15%8,000 Liabilities NISTSAMUD$Yen Saving 2yr. 4% 1,800 Deposit 1mo. 11%8,200 Deposit 3mo. L-2%3,000 Total: (200) 2004,000(3,000) Today L=6%

3. Zvi WienerVaR example slide 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.)

4. Zvi WienerVaR example slide 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.

5. Zvi WienerVaR example slide 5 Fair Value Function

6. Zvi WienerVaR example slide 6 Fair Value Function

7. Zvi WienerVaR example slide 7 Fair Value Function

8. Zvi WienerVaR example slide 8 Sensitivity CPI USD Yen r nominal 1mo r nominal 3yr r real 2yr r real 10yr r USD 1yr r Yen 3mo 0.1% 1% 2% 0.5% 0.25% -8 40 -60 3 -103 17 -93 -10 2 Biggest market riskSignificant risk

9. Zvi WienerVaR example slide 9 Risky Scenario 2yr 10 yrT Real r

10. Zvi WienerVaR example slide 10 Sensitivity CPI USD Yen r nominal 1mo r nominal 3yr r real 2yr r real 10yr r USD 1yr r Yen 3mo 0.1% 1% 2% 0.5% 0.25% -8 40 -60 3 -103 17 -93 -10 2 Are not included into BoI requirements

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

12. Zvi WienerVaR example slide 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.

13. Zvi WienerVaR example slide 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.

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

15. Zvi WienerVaR example slide 15 Duration and IR sensitivity

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

17. Zvi WienerVaR example slide 17 Macaulay Duration Definition of duration, assuming t=0.

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

19. Zvi WienerVaR example slide 19 Meaning of Duration r $

20. Zvi WienerVaR example slide 20 Proposition 15.12 TS of IR With a term structure of IR (note y i ), the duration can be expressed as:

21. Zvi WienerVaR example slide 21 Convexity r $

22. Zvi WienerVaR example slide 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: Ke R*(T-S) Calculate the FRA rate R* which makes PV=0 hint: it is equal to forward rate

23. Zvi WienerVaR example slide 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.

24. Zvi WienerVaR example slide 24 ALM Duration Does NOT work! Wrong units of measurement Division by a small number

25. Zvi WienerVaR example slide 25 ALM Duration A similar problem with measuring yield