Chapter 5 Hedging Interest-Rate Risk with Duration

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

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 HedgingMotivation • 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 HedgingNotation • B(t,T) : price at date t of a unit discount bond paying off $1 at date T (« discount factor ») • Ra(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,

Pricing and Hedging Pricing Certain Cash-Flows • The value at date t (Vt) 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 Interest Rate Risk • Example • Assume today a flat structure of interest rates • Ra(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 Hedging Principles • 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

Duration HedgingDuration • 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 HedgingSensitivity • For small changes, can be approximated by • 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 • Relative variation

Duration Hedging Duration • The absolute sensitivity, is the partial derivative of the bond price with respect to yield • Formally • 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

Duration Hedging Terminology • 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 $107.72 price • The $ Duration of this bond is -809.67 and the modified duration is 7.52. • Interpretation • Rate goes up by 0.1% (10 basis points) • Absolute P&L: -809.67x.0.1% = -$0.80967 • Relative P&L: -7.52x0.1% = -0.752%

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

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

Duration Hedging Properties of Duration • 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 - Example

Duration Hedging Properties of Duration - Linearity • Duration of a portfolio of n bonds where wiis 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

Duration Hedging Hedging • 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 • Hedging instrument • Strategy: hold q units of the hedging instrument so that • Solution

Duration Hedging Hedging • Example: • At date t, a portfolio P has a price $328635, a 5.143% yield and a 6.760 modified duration • Hedging instrument, a bond, has a price $118.786, a 4.779% yield and a 5.486 modified duration • Hedging strategy involves a buying/selling a number of bonds q = -(328635x 6.760)/(118.786x 5.486) = - 3409 • If you hold the portfolio P, you want to sell 3409 units of bonds

Duration Hedging Limits • 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

Chapter 5 Hedging Interest-Rate Risk with Duration