Hedging Interest-Rate Risk with Duration

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1 FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton

2 Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton Propertes of duraton Hedgng wth duraton

3 Prcng and Hedgng Motvaton Fxed-ncome products can pay ether Fxed cash-flows (e.g., fxed-rate Treasury coupon bond) Random cash-flows: depend on the future evoluton of nterest rates (e.g., floatng rate note) or other varables (prepayment rate on a mortgage pool) Objectve for ths chapter Hedge the value of a portfolo of fxed cash-flows Valuaton and hedgng of random cash-flow s a somewhat more complex task Leave t for later

4 Prcng and Hedgng Notaton B(t,T) : prce at date t of a unt dscount bond payng off $1 at date T («dscount factor») R a (t,) : zero coupon rate or pure dscount rate, or yeld-to-maturty on a zero-coupon bond wth maturty date t + B( t, t θ) 1 (1 R ( t, θ)) R(t,) : contnuously compounded pure dscount rate wth maturty t + : B( t, t θ) exp θ R( t, θ) Equvalently, 1 R( t, θ) lnb( t, t θ) θ a θ

5 The value at date t (V t ) of a bond payng cash-flows F() s gven by: m m F V ( t) F B( t, t ) Ra ( t, ) Example: $100 bond wth a 5% coupon F F m cn cn 5% N 5% Therefore, the value s a functon of tme and nterest rates Value changes as nterest rates fluctuate Prcng and Hedgng Prcng Certan Cash-Flows

6 Example Assume today a flat structure of nterest rates R a (0,) = 10% for all Bond wth 10 years maturty, coupon rate = 10% Prce: $100 If the term structure shfts up to 12% (parallel shft) Bond prce : $88.7 Captal loss: $11.3, or 11.3% Implcatons Prcng and Hedgng Interest Rate Rsk Hedgng nterest rate rsk s economcally mportant Hedgng nterest rate rsk s a complex task: 10 rsk factors n ths example!

7 Basc prncple: attempt to reduce as much as possble the dmensonalty of the problem Frst step: duraton hedgng Consder only one rsk factor Assume a flat yeld curve Assume only small changes n the rsk factor Beyond duraton Relax the assumpton of small nterest rate changes Relax the assumpton of a flat yeld curve Relax the assumpton of parallel shfts Prcng and Hedgng Hedgng Prncples

8 Use a proxy for the term structure: the yeld to maturty of the bond It s an average of the whole terms structure If the term structure s flat, t s the term structure We wll study the senstvty of the prce of the bond to changes n yeld: Change n TS means change n yeld Prce of the bond: (actually y/2) m F V y 1 1 Duraton Hedgng Duraton

9 Interest rate rsk Rates change from y to y+dy dv V dv V '( y) dy V ( y) Duraton Hedgng Senstvty dy s a small varaton, say 1 bass pont (e.g., from 5% to 5.01%) Change n bond value dv followng change n rate value dy V( y dy) V( y) For small changes, can be approxmated by Relatve varaton dv V'( y) dy Sens dy

10 Duraton Hedgng Duraton The relatve senstvty, denoted as Sens, s the partal dervatve of the bond prce wth respect to yeld, dvded by the bond prce m Formally 1 F V '( y) 1 y 1 1 y Sens / V ( y) V ( y) In plan Englsh: tells you how much relatve change n prce follows a gven small change n yeld mpact It s always a negatve number Bond prce goes down when yeld goes up

11 Duraton Hedgng Termnology The opposte of the senstvty Sens s referred to as «Modfed Duraton» The absolute senstvty V (y) = Sens x V(y) s referred to as «$ Duraton» Example: Bond wth 10 year maturty Coupon rate: 6% Quoted at 5% yeld or equvalently $ prce The $ Duraton of ths bond s and the modfed duraton s Interpretaton Rate goes up by 0.1% (10 bass ponts) Absolute P&L: x.0.1% = -$ Relatve P&L: -7.52x0.1% = %

12 Defnton of Duraton D: 1 Also known as Macaulay duraton It s a measure of average maturty Duraton Hedgng Duraton Relatonshp wth senstvty and modfed duraton: D m F (1 V y) D Sens ( 1 y) MD (1 y)

13 Duraton Hedgng Example Tme of Cash Flow () Cash Flow F w 1 V F 1 y w Example: m = 10, c = 5.34%, y = 5.34% D m 1 w Total

14 Duraton of a zero coupon bond s Equal to maturty For a gven maturty and yeld, duraton ncreases as coupon rate Decreases For a gven coupon rate and yeld, duraton ncreases as maturty Increases For a gven maturty and coupon rate, duraton ncreases as yeld rate Decreases Duraton Hedgng Propertes of Duraton

15 Duraton Hedgng Propertes of Duraton - Example Bond Maturty Coupon YTM Prce Sens D Bond 1 1 7% 6% Bond 2 1 6% 6% Bond 3 5 7% 6% Bond 4 5 6% 6% Bond % 6% Bond % 6% Bond % 6% Bond % 7% Bond % 6% Bond % 6%

16 Duraton Hedgng Propertes of Duraton - Lnearty Duraton of a portfolo of n bonds D P n 1 D where w s the weght of bond n the portfolo, and: n w 1 1 Ths s true f and only f all bonds have same yeld,.e., f yeld curve s flat If that s the case, n order to attan a gven duraton we only need two bonds w

17 Prncple: mmunze the value of a bond portfolo wth respect to changes n yeld Denote by P the value of the portfolo Denote by H the value of the hedgng nstrument Hedgng nstrument may be Bond Swap Future Opton Assume a flat yeld curve Duraton Hedgng Hedgng

18 Changes n value Portfolo Hedgng nstrument dp P'( y) dy dp qdh Duraton Hedgng Hedgng dh H'( y) dy Strategy: hold q unts of the hedgng nstrument so that qh' ( y) P'( y) dy 0 Soluton q P'( y) H'( y) P Sens H Sens H P P Dur H Dur H P

19 Example: Duraton Hedgng Hedgng At date t, a portfolo P has a prce $328635, a 5.143% yeld and a duraton Hedgng nstrument, a bond, has a prce $ , a 4.779% yeld and a duraton Hedgng strategy nvolves a buyng/sellng a number of bonds q = -(328635x7.108)/( x5.748) = If you hold the portfolo P, you want to sell 3421 unts of bonds

20 Duraton Hedgng Lmts Duraton hedgng s Very smple Bult on very restrctve assumptons Assumpton 1: small changes n yeld The value of the portfolo could be approxmated by ts frst order Taylor expanson OK when changes n yeld are small, not OK otherwse Ths s why the hedge portfolo should be re-adjusted reasonably often Assumpton 2: the yeld curve s flat at the orgn In partcular we suppose that all bonds have the same yeld rate In other words, the nterest rate rsk s smply consdered as a rsk on the general level of nterest rates Assumpton 3: the yeld curve s flat at each pont n tme In other words, we have assumed that the yeld curve s only affected only by a parallel shft

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