FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton
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
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
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
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 ) 1 1 1 Ra ( t, ) Example: $100 bond wth a 5% coupon F F m cn cn 5% 100 5 N 5% 100 100 105 Therefore, the value s a functon of tme and nterest rates Value changes as nterest rates fluctuate Prcng and Hedgng Prcng Certan Cash-Flows
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: $11.3, or 11.3% Implcatons Prcng and Hedgng Interest Rate Rsk Hedgng nterest rate rsk s economcally mportant Hedgng nterest rate rsk s a complex task: 10 rsk factors n ths example!
Basc prncple: attempt to reduce as much as possble the dmensonalty of the problem Frst step: duraton hedgng Consder only one rsk factor Assume a flat yeld curve Assume only small changes n the rsk factor Beyond duraton Relax the assumpton of small nterest rate changes Relax the assumpton of a flat yeld curve Relax the assumpton of parallel shfts Prcng and Hedgng Hedgng Prncples
Use a proxy for the term structure: the yeld to maturty of the bond It s an average of the whole terms structure If the term structure s flat, t s the term structure We wll study the senstvty of the prce of the bond to changes n yeld: Change n TS means change n yeld Prce of the bond: (actually y/2) m F V y 1 1 Duraton Hedgng Duraton
Interest rate rsk Rates change from y to y+dy dv V dv V '( y) dy V ( y) Duraton Hedgng Senstvty dy s a small varaton, say 1 bass pont (e.g., from 5% to 5.01%) Change n bond value dv followng change n rate value dy V( y dy) V( y) For small changes, can be approxmated by Relatve varaton dv V'( y) dy Sens dy
Duraton Hedgng Duraton The relatve senstvty, denoted as Sens, s the partal dervatve of the bond prce wth respect to yeld, dvded by the bond prce m Formally 1 F V '( y) 1 y 1 1 y Sens / V ( y) V ( y) In plan Englsh: tells you how much relatve change n prce follows a gven small change n yeld mpact It s always a negatve number Bond prce goes down when yeld goes up
Duraton Hedgng Termnology The opposte of the senstvty Sens s referred to as «Modfed Duraton» The absolute senstvty V (y) = Sens x V(y) s referred to as «$ Duraton» Example: Bond wth 10 year maturty Coupon rate: 6% Quoted at 5% yeld or equvalently $107.72 prce The $ Duraton of ths bond s -809.67 and the modfed duraton s 7.52. Interpretaton Rate goes up by 0.1% (10 bass ponts) Absolute P&L: -809.67x.0.1% = -$0.80967 Relatve P&L: -7.52x0.1% = -0.752%
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)
Duraton Hedgng Example Tme of Cash Flow () Cash Flow F w 1 V F 1 y w 1 53.4 0.0506930 0.0506930 Example: m = 10, c = 5.34%, y = 5.34% 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 D m 1 w 8 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
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
Duraton Hedgng Propertes of Duraton - Example Bond Maturty Coupon YTM Prce 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
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
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
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
Example: Duraton Hedgng Hedgng At date t, a portfolo P has a prce $328635, a 5.143% yeld and a 7.108 duraton Hedgng nstrument, a bond, has a prce $118.786, a 4.779% yeld and a 5.748 duraton Hedgng strategy nvolves a buyng/sellng a number of bonds q = -(328635x7.108)/(118.786x5.748) = - 3421 If you hold the portfolo P, you want to sell 3421 unts of bonds
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