Chapter 5 Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES

Outline Pricing and Hedging –Pricing certain cash-flows –Interest rate risk –Hedging principles Duration-Based Hedging Techniques –Definition of duration –Properties of duration –Hedging with duration

Pricing and Hedging Motivation Fixed-income products can pay either –Fixed cash-flows (e.g., fixed-rate Treasury coupon bond) –Random cash-flows: depend on the future evolution of interest rates (e.g., floating rate note) or other variables (prepayment rate on a mortgage pool) Objective for this chapter –Hedge the value of a portfolio of fixed cash-flows Valuation and hedging of random cash-flow is a somewhat more complex task –Leave it for later

Pricing and Hedging Notation B(t,T) : price at date t of a unit discount bond paying off $1 at date T (« discount factor ») R a (t,  ) : zero coupon rate –or pure discount rate, –or yield-to-maturity on a zero-coupon bond with maturity date t +  R(t,  ) : continuously compounded pure discount rate with maturity t +  : –Equivalently,

The value at date t (V t ) of a bond paying cash-flows F(i) is given by: Example: $100 bond with a 5% coupon Therefore, the value is a function of time and interest rates –Value changes as interest rates fluctuate Pricing and Hedging Pricing Certain Cash-Flows

Example –Assume today a flat structure of interest rates –R a (0,  ) = 10% for all  –Bond with 10 years maturity, coupon rate = 10% –Price: $100 If the term structure shifts up to 12% (parallel shift) –Bond price : $88.7 –Capital loss: $11.3, or 11.3% Implications –Hedging interest rate risk is economically important –Hedging interest rate risk is a complex task: 10 risk factors in this example! Pricing and Hedging Interest Rate Risk

Basic principle: attempt to reduce as much as possible the dimensionality of the problem First step: duration hedging –Consider only one risk factor –Assume a flat yield curve –Assume only small changes in the risk factor Beyond duration –Relax the assumption of small interest rate changes –Relax the assumption of a flat yield curve –Relax the assumption of parallel shifts Pricing and Hedging Hedging Principles

Use a “proxy” for the term structure: the yield to maturity of the bond –It is an average of the whole terms structure –If the term structure is flat, it is the term structure We will study the sensitivity of the price of the bond to changes in yield: –Change in TS means change in yield Price of the bond: (actually y/2) Duration Hedging Duration

Duration Hedging Sensitivity Interest rate risk –Rates change from y to y+dy –dy is a small variation, say 1 basis point (e.g., from 5% to 5.01%) Change in bond value dV following change in rate value dy For small changes, can be approximated by Relative variation

The absolute sensitivity, is the partial derivative of the bond price with respect to yield Formally Duration Hedging Duration In plain English: tells you how much absolute change in price follows a given small change in yield impact It is always a negative number –Bond price goes down when yield goes up

The relative sensitivity $Sens / V(y) with the opposite sign, or -V’(y) / V(y) is referred to as « Modified Duration » The absolute sensitivity V’(y) = Sens is referred to as « $Duration » Example: –Bond with 10 year maturity –Coupon rate: 6% –Quoted at 5% yield or equivalently $ price –The $ Duration of this bond is and the modified duration is Interpretation –Rate goes up by 0.1% (10 basis points) –Absolute P&L: x.0.1% = -$ –Relative P&L: -7.52x0.1% = % Duration Hedging Terminology

Definition of Duration D: Also known as “Macaulay duration” It is a measure of average maturity Relationship with sensitivity and modified duration: Duration Hedging Duration

Example: m = 10, c = 5.34%, y = 5.34% Duration Hedging Example

Duration of a zero coupon bond is –Equal to maturity For a given maturity and yield, duration increases as coupon rate –Decreases For a given coupon rate and yield, duration increases as maturity –Increases For a given maturity and coupon rate, duration increases as yield rate –Decreases Duration Hedging Properties of Duration

Duration Hedging Properties of Duration - Example

Duration Hedging Properties of Duration - Linearity Duration of a portfolio of n bonds where w i is the weight of bond i in the portfolio, and: This is true if and only if all bonds have same yield, i.e., if yield curve is flat If that is the case, in order to attain a given duration we only need two bonds

Principle: immunize the value of a bond portfolio with respect to changes in yield –Denote by P the value of the portfolio –Denote by H the value of the hedging instrument Hedging instrument may be –Bond –Swap –Future –Option Assume a flat yield curve Duration Hedging Hedging

Changes in value –Portfolio Duration Hedging Hedging –Hedging instrument Strategy: hold q units of the hedging instrument so that Solution

Example: –At date t, a portfolio P has a price $328635, a 5.143% yield and a modified duration –Hedging instrument, a bond, has a price $ , a 4.779% yield and a modified duration Hedging strategy involves a buying/selling a number of bonds q = -(328635x 6.760)/( x 5.486) = If you hold the portfolio P, you want to sell 3409 units of bonds Duration Hedging Hedging

Duration hedging is –Very simple –Built on very restrictive assumptions Assumption 1: small changes in yield –The value of the portfolio could be approximated by its first order Taylor expansion –OK when changes in yield are small, not OK otherwise –This is why the hedge portfolio should be re-adjusted reasonably often Assumption 2: the yield curve is flat at the origin –In particular we suppose that all bonds have the same yield rate –In other words, the interest rate risk is simply considered as a risk on the general level of interest rates Assumption 3: the yield curve is flat at each point in time –In other words, we have assumed that the yield curve is only affected only by a parallel shift Duration Hedging Limits