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 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 ») • Ra(t,θ) : zero coupon rate – or pure discount rate, – or yield-to-maturity on a zero-coupon bond with maturity date t + θ
1 B(t , t + θ ) = (1 + Ra (t , θ ))θ
• R(t,θ) : continuously compounded pure discount rate with maturity t + θ: B(t , t + θ ) = exp( − θ × R (t , θ )) – Equivalently,
1 R (t , θ ) = − ln( B(t , t + θ )) θ
Pricing and Hedging Pricing Certain Cash-Flows • The value at date t (Vt) of a bond paying cash-flows F(i) is given by: m
m
V (t ) = ∑ Fi B(t , t + i ) = ∑ i =1
[
i =1 1 +
Fi
Ra (t , i )]
i
• Example: $100 bond with a 5% coupon Fi = cN = 5% ×100 = 5 Fm = cN + N = 5% ×100 +100 =105 • 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,θ) = 10% for all θ – 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 Hedging Duration • 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) m Fi V =∑ i ( ) 1 + y i =1
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 dV = V ( y + dy ) − V ( y ) • For small changes, can be approximated by • Relative variation
dV ≈ V ' ( y )dy
dV V ' ( y ) ≈ dy = Sens × dy V V ( y)
Duration Hedging Duration • The relative sensitivity, denoted as Sens, is the partial derivative of the bond price with respect to yield, divided by the bond price 1 m iFi • Formally
− ∑ V ' ( y ) 1 + y i =1 (1 + y ) i Sens = = / V ( y ) V ( y)
• 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 – Bond price goes down when yield goes up
Duration Hedging Terminology • 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 Duration
Fi m i × • Definition of Duration D: i (1 + y ) D= V i =1 • Also known as “Macaulay duration”
∑
• It is a measure of average maturity
• Relationship with sensitivity and modified duration:
D = −Sens × (1 + y ) = MD × (1 + y )
Duration Hedging Example Time of Cash Flow (i) Cash Flow Fi 1 53.4
wi =
1 Fi × V (1+ y) i
i × wi
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
8.0014280
Example: m = 10, c = 5.34%, y = 5.34%
m
D = ∑ i × wi ≅ 8 i =1
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
Bond Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Bond 6 Bond 7 Bond 8 Bond 9 Bond 10
Maturity 1 1 5 5 10 10 20 20 50 50
Coupon 7% 6% 7% 6% 4% 8% 4% 8% 6% 0%
YTM 6% 6% 6% 6% 6% 6% 6% 7% 6% 6%
Price 100.94 100 104.21 100 85.28 114.72 77.06 110.59 100 5.43
Sens -0.94 -0.94 -4.15 -4.21 -7.81 -7.02 -12.47 -10.32 -15.76 -47.17
D 1 1 4.40 4.47 8.28 7.45 13.22 11.05 16.71 50.00
Duration Hedging Properties of Duration - Linearity • Duration of a portfolio of n bonds n
DP = ∑ Di × wi i =1
where wi is the weight of bond i in the portfolio, and:
n
∑w =1 i =1
i
• 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
dP ≈ P ' ( y ) dy
– Hedging instrument
dH ≈ H ' ( y )dy
• Strategy: hold q units of the hedging instrument so that
dP + qdH = ( qH ' ( y ) + P ' ( y ) ) dy = 0
• Solution
P ' ( y ) − P × SensP − P × DurP q=− = = H ' ( y ) H × SensH H × DurH
Duration Hedging Hedging • 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 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