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 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
Pricing and Hedging
Notation
• 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 +
θ a t θ R θ t t B )) , ( 1 ( 1 ) , (
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with maturity t + :
• The value at date t (Vt) of a bond paying cash-flows
F(i) is given by:
105 100 100 % 5 5 100 % 5 N cN F cN F m i
• Example: $100 bond with a 5% coupon
• Therefore, the value is a function of time and interest rates
– Value changes as interest rates fluctuate
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Pricing and Hedging
• 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
• 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
• 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
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Sensitivity
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dy Sens dy y V y V V dV ) ( ) ( '• 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 relative sensitivity, denoted as Sens, is the partial derivative of the bond price with respect to yield, divided by the bond price
• Formally
Duration Hedging
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• In plain English: tells you how much relative change in price follows a given small change in yield impact • It is always a negative number
• The opposite of the sensitivity Sens is referred to as « Modified Duration »
• The absolute sensitivity V’(y) = Sens x V(y) 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
• Definition of Duration D:
• Also known as “Macaulay duration” • It is a measure of average maturity
• Relationship with sensitivity and modified duration:
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Time of
Cash Flow (i) Cash Flow
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1 53.4 0.0506930 0.0506930 2 53.4 0.0481232 0.0962464 3 53.4 0.0456837 0.1370511 4 53.4 0.0433679 0.1734714 5 53.4 0.0411694 0.2058471 6 53.4 0.0390824 0.2344945 7 53.4 0.0371012 0.2597085 8 53.4 0.0352204 0.2817635 9 53.4 0.0334350 0.3009151 10 1053.4 0.6261237 6.2612374 Total 8.00142808
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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
Duration Hedging
Properties of Duration - Example
Bond Maturity Coupon YTM Price Sens D
Bond 1 1 7% 6% 100.94 -0.94 1 Bond 2 1 6% 6% 100 -0.94 1 Bond 3 5 7% 6% 104.21 -4.15 4.40 Bond 4 5 6% 6% 100 -4.21 4.47 Bond 5 10 4% 6% 85.28 -7.81 8.28 Bond 6 10 8% 6% 114.72 -7.02 7.45 Bond 7 20 4% 6% 77.06 -12.47 13.22 Bond 8 20 8% 7% 110.59 -10.32 11.05 Bond 9 50 6% 6% 100 -15.76 16.71 Bond 10 50 0% 6% 5.43 -47.17 50.00
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
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• 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
• Changes in value – Portfolio
Duration Hedging
Hedging
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– Hedging instrument• Strategy: hold q units of the hedging instrument so that
• Example:
– At date t, a portfolio P has a price $328635, a 5.143% yield and a 7.108 duration
– Hedging instrument, a bond, has a price $118.786, a 4.779% yield and a 5.748 duration
• Hedging strategy involves a buying/selling a number of bonds
q = -(328635x7.108)/(118.786x5.748) = - 3421
• If you hold the portfolio P, you want to sell 3421 units of bonds
Duration 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
