Shape of the Yield Curve Under CIR Single Factor Model A Note working paper

Shape of the Yield Curve Under CIR Single Factor Model:A NoteRaymond KanUniversity of TorontoJune,1992AbstractThis note derives the shapes of the yield curve as a function of the current spot rate under Cox,Ingersoll,and Ross(CIR)(1985b)single factor model.Corresponding results reported inCIR are shown to be incorrect.1IntroductionIn Cox,Ingersoll,and Ross(CIR)(1985a,b),they develop a general equilibrium model of the term structure of interest rates.The term structure of interest rates are linked directly to the specifica-tions of preferences,technologies,and the distributions of the underlying sources of uncertainty.As a special case,they derive a single factor model of the term structure.Under this single factor model,the interest rate dynamics can be expressed as:dr=κ(θ−r)dt+σ√rdz(1)where dz is a standard Weiner process,andκ,θ,andσ2are constants,withκθ≥0,andσ2>0.When the current spot rate is r,the yield-to-maturity of aτ-period pure discount bond is then given by:R(r,τ)=rB(τ)−log A(τ)τ(2)whereA(τ)≡2γe(κ+λ+γ)τ/2(κ+λ+γ)(eγτ−1)+2γ2κθ/σ2,(3)B(τ)≡2(eγτ−1)(κ+λ+γ)(eγτ−1)+2γ,(4) 1γ=(κ+λ)2+2σ21/2.(5)From(2),it is easy to seeR(r,0)=r,(6)R(r,∞)=2κθγ+κ+λ.(7)In other words,when the current spot rate is r,the yield curve starts at r and approaches a limit of2κθγ+κ+λas maturity increases to infinity.CIR provide a characterization of the shape of the yield curve as a function of the current spotrate.They claim that when r≤2κθγ+κ+λ,the yield curve is uniformly rising.With r≥κθκ+λ,theyield curve is falling.For2κθγ+κ+λ<r<κθκ+λ,the yield curve is humped.It is easy to see this characterization of the yield curve is incorrect because if the yield curve ishumped when r=2κθγ+κ+λ,then the yield curve cannot be uniformly increasing for all r<2κθγ+κ+λ.Otherwise,R(r,τ)will be a discontinuous function of r for someτ>0,which is contrary to(2)that R(r,τ)is a linear function of r for afixedτ.In fact,for r that is less than but close to2κθγ+κ+λ,the yield curve must also be humped and it has to go above the long-term yield2κθγ+κ+λbefore approaching this limit.Another problem of this characterization is we needκ+λ>0in order for2κθγ+κ+λ<κθκ+λ.However,the model does not always imply this condition.In particular,when theterm premium is positive,λis negative and forκsmall enough,κ+λcould be negative as well.1 In the next section,we will provide the correct characterization of the yield curve as a function of the spot rate under CIR single factor model.2Characterization of the Shape of the Yield CurveBefore we derive the shape of the yield curve as a function of the current spot rate,wefirst note that the yield curve is in fact determined by only three parameters:κ+λ,κθ,andσ2.Therefore, 1In Gibbons and Ramaswamy(1988),they estimate the CIR single factor model andfind thatˆκ+ˆλis−0.7.In the next section,we will show that the yield curve can only be humped or increasing whenκ+λ≤0and r≥0.2by lettingα=κθandβ=κ+λ,we can express the yield curve as follows:R(r,τ)=f(τ)+g(τ)r(8) wheref(τ)≡−2αlog2γe(γ+β)τ/2(γ+β)(eγτ−1)+2γσ2τ=2ασ2τlog(γ+β)(eγτ−1)+2γ2γ−α(γ+β)σ2,(9)g(τ)≡2(eγτ−1)[(γ+β)(eγτ−1)+2γ]τ,(10)γ=β2+2σ21/2.(11)Wefirst prove the following lemmata:Lemma1f(τ)is an increasing function ofτ.It starts out at0with a slope ofα/2whenτ=0 and approaches a limit of2αγ+βasτ→∞.Proof:f (τ)=2ασ2τ2(γ+β)γτeγτ(γ+β)(eγτ−1)+2γ+log2γ(γ+β)(eγτ−1)+2γ=2ασ2τ2F(γτ)(12)where F(x)is defined as:F(x)=xe xe x−1+a+logae x−1+a,a=2γγ+β>1.(13)With L’Hˆo pital’s Rule,one can easily show f(0)=0,f(∞)=2αγ+β,and f (0)=α/2.Note thatF(0)=0and F (x)=xe x(a−1)(e x−1+a)2>0for x>0.Therefore we have F(x)>0for x>0and f (τ)>0 forτ>0.Lemma2g(τ)starts out at1with a slope of−β/2whenτ=0and approaches a limit of0as τ→∞.Ifβ≥0,g(τ)is a uniformly decreasing function ofτ.Ifβ<0,g(τ)first increases to a maximum value and then decreases to0.3g (τ)=−2 β(e γτ−1)2+γ e 2γτ−2γτe γτ−1 [(γ+β)(e γτ−1)+2γ]2τ2=−(γ−β)2σ2τ2G (γτ)(14)where G (x )is defined as:G (x )=(2−a )(e x −1)2+a (e 2x −2xe x −1)(e x −1+a )2,a =2γγ+β>1.(15)With L’Hˆo pital’s Rule,one can easily show g (0)=1,g (∞)=0,and g (0)=−β/2.Consider the numerator of G (x ).By Taylor series expansion,we have(2−a )(e x −1)2+a (e 2x −2xe x −1)=∞ i =2a i x i i !(16)wherea i =(2i −2)(2−a )+(2i −2i )a,i ≥2.(17)If β≥0,we have a ≤2and a i ≥0.Therefore G (x )≥0for x ≥0and g (τ)is a decreasing function of τfor τ≥0.When β<0,we have a 2<0and note that whenever a i >0,it implies a i +1>0.Therefore,a i changes sign only once and by a simple extension of Descartes’Rule of Signs,G (x )has only one positive root.2Hence,when β<0,g (τ)is an increasing function when τ=0and it reaches a maximum value before decreasing to 0as τ→∞.We now describe the yield curve by the following proposition:Proposition 1If κ+λ>0,the yield curve is uniformly falling when r ≥κθκ+λand it is uniformly rising when 0≤r ≤κθγ.For κθγ<r <κθκ+λ,the yield curve first increases to a maximum value andthen decreases to the long-term yield2κθγ+κ+λ.If κ+λ≤0,the yield curve is uniformly rising when 0≤r ≤κθγ.For r >κθγ,the yield curve first increases to a maximum value and then decreases to the long-term yield2κθγ+κ+λ.2Let Z be the number of positive zeros of a power series with radius of convergence ρ=∞and let the number of changes of signs in the sequence of coefficients be C .Then Z ≤C and C −Z is a non-negative even number.In our case,C =1and we must have Z =1.See,for example,P´o lya and Szeg¨o (1976,Part V)for a proof of this result.4R2(r,τ)=∂R(r,τ)∂τ=f (τ)+g (τ)r=2ασ2τ2F(γτ)−r(γ−β)4αG(γτ)=2ασ2τ2H(r,γτ)(18)where H(r,x)is defined as:H(r,x)=F(x)−r(γ−β)4αG(x).(19)Wefirst show that for afixed r,H(r,x)can have at most one positive root.H2(r,x)=∂H(r,x)∂x=F (x)−r(γ−β)4αG (x)=xe x(a−1)(e x−1+a)2−r(γ−β)4α2axe x(e x+1−a)(e x−1+a)3=xe x(e x−1+a)3∞i=0b i x ii!,(20)where b i are obtained by Taylor series expansion and they are:b0=a(a−1)1−rβα,(21)b i=(a−1)1−rγα,i≥1.(22)Therefore,b i can change sign at most once and H2(r,x)can have at most one positive root.Since H(r,0)=0,H(r,x)can also have at most one positive root by Rolle’s theorem.Hence,the yield curve can change direction at most once.Ifβ>0,b i will change sign if and only ifαγ<r<αβ.If β≤0,b i will change sign if and only if r>αγ.In other words,forβ>0,the yield curve is humped if and only ifαγ<r<αβand forβ≤0,the yield curve is humped if and only if r>αγ.By lemmata1and2,we have R2(r,0)=f (0)+g (0)r=α−βr2.Ifβ>0,R2(r,0)>0when r<αβand R2(r,0)≤0when r≥αβ.Therefore,the yield curve will be uniformly falling if r≥αβand it will be uniformly increasing if r≤αγ.Forαγ<r<αβ,the yield curve is humped.Ifβ≤0,5R2(r,0)is always positive and the yield curve can only be upward sloping or humped.For r≤αγ, the yield curve is uniformly increasing,and for r>αγ,the yield curve is humped.3SummaryWe derive the correct characterization of the yield curve under CIR single factor model.For 0≤r≤κθγ,the yield curve is uniformly increasing.Forκ+λ>0,the yield curve is uniformly .For all the other cases,the yield curve is humped.decreasing if r≥κθκ+λReferences[1]Cox,John C.,Ingersoll,Jonathan E.,Jr.,and Ross,Stephen A.(1985a),“An IntertemporalGeneral Equilibrium Model of Asset Prices,”Econometrica,53,363–384.[2]Cox,John C.,Ingersoll,Jonathan E.,Jr.,and Ross,Stephen A.(1985a),“A Theory of theTerm Structure of Interest Rates,”Econometrica,53,385–407.[3]Gibbons,Michael R.,and Ramaswamy,Krishna(1988),“The Term Structure of Interest Rates:Empirical Evidence,”Working Paper,Stanford University and University of Pennsylvania. [4]P´o lya,George,and Szeg¨o,Gabor(1976),“Problems and Theorems in Analysis,”Volume II,Translated by Claude Elias Billigheimer,Springer-Verlag,Berlin,Heidelberg,New York.6。