Lecture 13: Hedging with duration and convexity and review Finance 688: Investment Administration Professor John Chalmers Read Chapter 12 problems 1-5

Duration and Convexity are risk management tools  Basic ideas are applicable to all assets  Often not analytically tractable, make heroic assumptions  Primary uses  Asset liability management (managing the firm’s exposure)  Bank managers manage loan portfolio risk  insurance company portfolios, pension fund portfolios  Portfolio selection (in which bonds do we invest)  risk aversion of investors  matching particular liabilities (a retirement plan)  Security selection (how to best implement a trading strategy)  how to best play information about interest rates  e.g. if you know rates are coming down long maturities? MBS?

Convexity helps the Estimates

Three portfolios Duration increases as coupons decrease Convexity increases as coupons decrease Suppose your liabilities look like the 5% bond, what can we do to hedge with the other two portfolios?

Ten year 10%, 5% and 0% bonds

Neutral hedge  The objective of a neutral hedge is to desensitize portfolio value from changes in interest rates.  In general, any hedging problem solves for the amount to buy of various instruments that you can use to hedge. The number of assets required to hedge with will be equal to the number of dimensions on which you wish to hedge.  If D is zero this implies that changes in interest rates will have no impact on the value of your portfolio. This is portfolio immunization. Depends on parallel shift assumption.  Suppose liability is 10% bond. Duration hedge with zero:  Remember the duration of portfolio is weighted average of the duration of the assets in the portfolio

Duration Hedge Suppose liability is 10% bond. Duration hedge with zero:

Duration and Convexity Hedge  Match the duration of your portfolio along with the convexity of the portfolios

Bullet versus Barbell Hedge  Bullet effectively matches duration with assets of maturity similar to the asset that is being hedged. For example hedge a bond with 6 year duration with 6 year zero.  Barbell matches duration with bonds with very different maturities. For example, hedge a 6 year duration bond with a 1 year zero and a 13 year zero.  Bullet hedges will come closer to matching duration and convexity than a barbell hedge. The barbell will have higher convexity, which is fine if rates are a changing.

Summary Hedging with duration and convexity This is useful in many contexts, including the corporate managers, portfolio managers and business line people. Duration and PVBP are the crudest but most often encountered measures of price sensitivity The topics on the exam.