1. Interest Rate Risk Risk Management Prof. Ali Nejadmalayeri, Dr N a.k.a. “Dr N”

2. Asset Liability Structure Imagine a bank with A assets and L liabilities. The traditional measures of asset and liability sensitivity to interest rates were GAPs: –Dollar Maturity Gap Amount of assets repriced over the period minus the liabilities repriced –Percentage Maturity Gap Dollar maturity gap as a percentage of assets Downside is that Gap is static!Downside is that Gap is static!

3. Basics of Bond Pricing In discrete time, the value of bond is the sum of all cash flow present values: Interest RateCouponN = 4 yrsN = 24 yrs y = 3%6%$111.15$150.81 y = 6%6%$100.00 y = 9%6%$90.28$70.88 y = 3%9%$122.30$201.61 y = 6%9%$110.40$137.65 y = 9%9%$100.00 y = 12%9%$90.89$76.65

4. Bond Prices & Yields

5. Duration Time to break even or more precisely, the sensitivity to small interest rate movements: Interest RateCouponN = 4 yrsN = 24 yrs y = 3%6%2.522 11.158 11.158 y = 6%6%2.485 9.842 9.842 y = 9%6%2.449 8.672 8.672 y = 3%9%2.503 11.137 11.137 y = 6%9%2.466 9.826 9.826 y = 9%9%2.431 8.661 8.661 y = 12%9%2.396 7.656 7.656

6. Price & Duration For small interest rate changes, Δy, then price change is defined by the following: –By redefining the duration as modified duration, the ratio of duration to yield, then:

7. Duration & Pricing

8. Duration & A-L Risks In an asset-liability risk analysis, then duration of both assets and liabilities could matter: The modified duration of net worth (capital) is then given by:

9. Zero-Coupon Bonds Price of any coupon bond is a weighted sum of zero-coupon bonds: In continuous time, with non-flat yield curve, then:

10. Duration and Zeros The duration with a non-flat yield curve and continuous time pricing, also known as Fisher- Weil duration, D F, is then: The percentage price change of a bond, ΔB/B, then is given by ΔB/B = − D F  ΔThe percentage price change of a bond, ΔB/B, then is given by ΔB/B = − D F  Δ

11. Convexity & Pricing

12. Convexity and Bond Prices ii 2A zero-coupon bond maturing at t + i with a price of P t (t + i) and continuous compounded interest rate of r t (t + i) has duration of i and convexity of i 2. For a coupon bond, then change in price due to Δ change in the yield-to-maturity is: − Modified Duration  Δ + Convexity  (Δ) 2The percentage in price is then equal to: − Modified Duration  Δ + Convexity  (Δ) 2

13. Convexity & Bonds’ Prices We have seen that duration of a coupon bond is: The convexity of the coupon bond then is:

14. Cash Flow Portfolios The value, W, of any portfolio of cash flows, C t (t + i), in year t + 1 can be priced using the price of zero-coupon bonds associated with payment period, P t (t + i): The change if the value of the portfolio then is:

15. Forward Curve Model Under pure expectation hypothesis, the price of one-year zero coupon bond for delivery of one year from today is governed by one-year forward rate, f t (t + i). Using continuous compounding, f t (t + i) is: So value of any zero-coupon bond is a function of all future forward rate, i.e., forward curve: