1 / 20

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.

lefty
Download Presentation

Chapter 5 Hedging Interest-Rate Risk with Duration

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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

  3. 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

  4. 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,

  5. 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

  6. 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!

  7. 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

  8. 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)

  9. 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

  10. 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

  11. 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%

  12. 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:

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

  14. 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

  15. Duration Hedging Properties of Duration - Example

  16. 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

  17. 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

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

  19. 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

  20. 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

More Related