13.3 Term structure models

Transcription

1 13.3 Term srucure models Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +) (1) =δ + φ ( δ) f (N) = E ( +N 1 )=δ + φn 1 ( δ) i E (y +1)+ (1) h δ + φ( δ)+ = 1 y () δ = 1+φ ³ δ... y (N) δ = 1+φ + φ +.. N i δ = 1 φn δ N(1 φ) 1Differen shapes, upward and downward sloping. Up when shor raes expeced o rise. More complex shapes? Move pas AR(1)!. No average slope E(y (i) ) all he same. Well, we imposed expecaions! 3. Yields move in locksep a "1 facor model Term srucure model in general Ingrediens: 1) Wrie a ime series model for he discoun facor in discree or coninuous ime. a) Solve M forward, M,+n = M +1 M +...M +n.then = E [M,+n ] b) Solve differenial / difference equaion, i.e. solve backward from P () =1, h i = E M,+1 P (n 1) +1 Resul: = funcion of sae variables ha drive M. This is easies for logs. Translaion from levels o logs: eiher coninuous ime or lognormal disribuions. 118

2 Discree-ime single-facor Vasicek Here: A sandard single facor model discree-ime Vasicek. The end resul: +1 δ = φ δ + v +1. f () = δ + φ 1 δ f (3) = δ + φ δ... y () (1 + φ) = δ + δ 1 1 y (3) = δ + (1 + φ + φ ) 3... δ λ σ ε 1 (1 + φ) + λ(1 + φ) + λ ½ 1 σ ε σ ε 1+(1+φ) ¾ + λ [1 + (1 + φ)] Inuiion for now: he firs equaion ells you where ineres raes are going over ime. The second and hird ses of equaions ell you where each forward rae is a any dae, depending only on where he shor rae is on ha dae; a single facor model. Derivaion: σ ε Suppose m follows he ime series model. x +1 δ = φ (x δ)+ε +1 m +1 = lnm +1 = 1 λ σ ε x λε +1 This is jus a model for m, wih a convenien sae variable x. Inuiion: x shifs he mean of ln m +1 around. If you remember ha R f =1/E(m), you can see ha specifying a model for he mean of m is he key o hinking abou ineres raes. The 1/σ erm jus shifs he mean of ln m down, and offses a 1/σ erm which will pop up laer. To be specific, ε and v are iid Normal wih σ ε,σ v,σ εv. δ,ρ,λ are free parameers; we ll pick hese o make he model fi as well as possible. Bond prices. = E (M +1 M +...M +n ) This is easier o do recursively, P () = 1 = E m +1 P (n 1). +1 Here we go. P (1) = E (M +1 )=E e m +1 P (1) = e 1 λ σ ε x + 1 λ σ ε = e x 119

3 p (1) = lne (m +1 )= x = x Now you see why I se up he problem wih he 1/λ σ ε o begin wih! The one year ineres rae reveals he laen sae variable x I could have wrien he model as +1 δ = φ δ + ε +1 m +1 = 1 λ σ ε λε +1 a shor rae process plus a marke price of risk. Then, by aking ln E (M +1 ) I would have checked ha he he model produces is he same I sared wih. Take your pick. Which is more confusing: a) saring wih an x you can see and hen showing ha i urns ou o be? b) saring wih an assumed process and hen showing ha i s in fac he one period rae, ha he model is self-consisen (in he language of CP appendix.) On o he nex price. P () = E m +1 P (1) +1 = E ³e m +1+p (1) +1 ³ = E e 1 λ σ ε x λε +1 x +1 P () = E ³ e 1 λ σ ε x λε +1 δ φ(x δ) ε +1 P () = E ³ e δ (1+φ)(x δ) 1 λ σ ε (1+λ)ε +1 p () = δ (1 + φ)(x δ) 1 λ σ ε + 1 (1 + λ) σ ε p () = δ (1 + φ)(x δ)+( 1 + λ)σ ε From prices, we find yields and forwards, y () (1 + φ) = δ + (x δ) 1 f () = p (1) p () 1 + λ σ ε = δ (x δ)+δ +(1+φ)(x δ) ( 1 + λ)σ ε = δ + φ (x δ) ( 1 + λ)σ ε Now he res of he mauriies. You can solve he discoun rae forward and inegrae p (3) = loge (M +1 M + M +3 ) = loge e 1 λ σ ε x λε +1 1 λ σ ε x +1 λε + 1 λ σ ε x + λε +3 = loge e 3δ 3 λ σ ε (1+φ+φ )(x δ) λε +1 λε + λε +3 (1+φ)ε +1 ε + This will work afer much algebra 1

4 Insead, le s do i recursively derive a differenial equaion for price as a funcion of sae variables. Guess = A n B n (x δ) hen = E M +1 P (n 1) +1 A n B n (x δ) = loge exp 1 λ σ ε x λε +1 exp (A n 1 B n 1 (x +1 δ)) = loge exp 1 λ σ ε x λε +1 + A n 1 B n 1 φ (x δ) B n 1 ε +1 = loge exp δ + A n 1 (1 + B n 1 φ)(x δ) 1 λ σ ε λε +1 B n 1 ε +1 A n B n (x δ) = δ + A n 1 (1 + B n 1 φ)(x δ)+ B n 1 λ + 1 B n 1 σ ε The consan and he erm muliplying x mus separaely be equal. Thus, B n =1+B n 1 φ A n = δ + A n 1 + B n 1 λ + 1 B n 1 We have ransformed he soluion of a sochasic differenial equaion plus inegral o he soluion of an ordinary differenial equqaion. σ ε Tha s easy o solve B = B 1 = 1 B = 1+φ B 3 = 1+φ + φ B n = Xn 1 φ j = 1 φn 1 φ j= A = A 1 = δ A = δ + λ + 1 σ ε A 3 = 3δ + (1 + φ) λ + A 4 = 4δ + You see he paern from here Yields, forwards, reurns, ec. follow. y (n) f (n) = p (n 1) p (n) 1 (1 + φ) σ ε 1+φ + φ λ φ + φ = 1/n p (n). Forwards are even simpler, = (A n 1 A n ) (B n 1 B n )(x δ) = δ + B n 1 λ + 1 B n 1 σ ε + φ n 1 (x δ) σ ε 11

5 Resul: +1 δ = φ δ + ε +1. f () = δ + φ δ f (3) = δ + φ δ f (4) = δ + φ 3 δ... y () (1 + φ) = δ + δ 1 1 y (3) = δ + (1 + φ + φ ) 3... ( 1 + λ)σ ε 1 (1 + φ) + λ(1 + φ) σ ε 1 1+φ + φ + λ(1 + φ + φ ) δ λ ½ 1 σ ε σ ε 1+(1+φ) ¾ + λ [1 + (1 + φ)] 1. Jus like EH bu now a risk premium!. Shapes: A seady decline from σ erms, (risk premim) + exponenial decay from E( ). (expecaions hypohesis) 3. Shor rae process plus one facor model. All yields move in locksep indexed by (or any oher yield). The shape is ied o he level. I looks like you can price oher bonds by arbirage bu ha is only because we resric our model o have one facor. 4. is also sufficien o forecas all yields. 5. Risk premium comes from cov(m, )=λσ ε marke price of ineres rae risk. If here were a securiy whose payoff were ε +1 is price would be driven by cov(m, ε +1 ). 6. The premium can go eiher way depending on he sign of λ. My guess: lower y +1means (1) higher m (bad sae) means + (m =.. ε) sign and negaive premium. This is a ypical resul. The real erm srucure ough o slope down, as long erm bonds are safer for long-erm invesors. However, as long as we separae marke prices of risk from consumpion and ineres daa (as we did wih he CAPM!) we can incorporae an upward sloping yield curve wih λ< 7. The risk premium is consan over ime hough as we ll see no in daa. 8. Risk neuraliy λ =does no mean expecaions since here is anoher erm. This is Anoher force for ypical downward slope. However, i s quaniaively very small, since σ ε.1 9. Anoher way o see risk premia is o look a reurns, r (n) +1 = p (n 1) +1 p (n) =(A n 1 A n ) B n 1 (x +1 δ)+b n (x δ) = δ B n 1 λ + 1 B n 1 σ ε B n 1 (x +1 δ)+b n (x δ) σ ε 1

6 Expeced reurns E r (n) +1 = δ B n 1 λ + 1 B n 1 σ ε +(B n B n 1 φ)(x δ) E r (n) +1 = δ B n 1 λ + 1 B n 1 σ ε +(x δ) E rx (n) +1 = B n 1 λ + 1 B n 1 σ ε You see he expeced reurns differ by mauriy, bu he risk premium is consan over ime no wha Fama-Bliss find. 1. The limiing yield and forward rae are consans. There is no rue level shif. We ll see his is quie general level shifs imply an arbirage opporuniy a he long end of he yield curve. 11. Yields can be negaive hey are normally disribued here. M>means P> no P<1. TheCIRmodelfixeshisup. Le s see an example. 1. I chose some parameers o fi he FB zero coupon bond daa. I ran a regression of +1 on o ge ρ; I ook ³ he variance of errors from ha regression o ge σ ε ; I ook he mean δ = E. Finally, I picked he marke price of risk λ o fi he average 5 year forward spread: f (5) = δ + ρ 4 1 δ 1+ρ + ρ + ρ 3 + λ(1 + ρ + ρ + ρ 3 ) σ ε E f (5) δ λ = 1 1+ρ + ρ + ρ 3 (1 + ρ + ρ + ρ 3 )σ ε. I plo y (n) for a bunch of. The dashed lines in he righ hand graph give he expecaions hypohesis erms from above, so you can see he disorion from risk aversion λ and he Jensen s inequaliy σ ε erm. 13

7 percen percen percen percen forwards, λ = -14.8, ρ =.83, 1 x σ = forwards and expecaions mauriy mauriy.1 yields.1 yields and expecaions mauriy mauriy Cool! This capures some basic paerns; yields are upward sloping when lower, downward sloping when higher. The subsanial risk premium I esimaed o mach he average upward slope does inroduce a subsanial deviaion of he model from expecaions a he long end. 3. Noe already: he parameers φ, λ can be chosen o mach he cross secion of yields he shapes of hese curves or he ime series hear(1)coefficien of he shor rae and he expeced bond reurns. These do no necessarily give he same answer, a sign of model misspecificaion. 4. Take he hisory of. Find he model-implied y (n) : compare wih daa. 14

8 yield spread yield spread yields yields Simulaion of Vasicek yield model year zero coupon yields You can see a decen fi upwward sloping yields when he ineres rae is low. Bu you can see yields are going up o a consan long-erm value, raher han some sor of local mean. This is clearer if we plo spreads, Simulaion of Vasicek yield spreads year zero coupon yield spreads Answer: we need a wo-facor model... 15