On-/Off-the-Run Yield Spread Puzzle: Evidence from Chinese Treasury Markets

Abstract

In this chapter, we document a negative on-/off-the-run yield spread in Chinese Treasury markets. This is in contrast with a positive on-/off-the-run yield spread in most other countries and could be called an “on-/off-the-run yield spread puzzle.” To explain this puzzle, we introduce a latent factor in the pricing of Chinese off-the-run government bonds and use this factor to model the yield difference between Chinese on-the-run and off-the-run issues. We use the nonlinear Kalman filter approach to estimate the model. Regressions results suggest that liquidity difference, market-wide liquidity condition, and disposition effect (unwillingness to sell old bonds) could help explain the dynamics of a latent factor in Chinese Treasury markets. The empirical results of this chapter show evidence of phenomena that are quite specific in emerging markets such as China.

The Kalman filter is a mathematical method named after Rudolf E. Kalman. It is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error. The nonlinear Kalman filter is the nonlinear version of the Kalman filter which linearizes about the current mean and covariance. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown.

We thank the National Natural Science Foundation of China (Grant No. 71101121 and No. 70971114) for financial support.

Appendices

Appendix 1: Nonlinear Kalman Filter

Let y ^on _M,t represent the time t yield of an on-the-run government bond maturing at t _M . Equation 22.3 could be written as

$$ {P}_t^{on}={\displaystyle \sum_{m=1}^M{C}_m{A}_{m,t} \exp \left(-{B}_{m,t}{r}_t\right)}={\displaystyle \sum_{m=1}^M{C}_m \exp \left(-{y}_{M,t}^{on}\left({t}_m-t\right)\right)}. $$

(22.8)

As shown, y ^on _M,t is a nonlinear function of r _t , which is inconsistent with the requirements of the standard Kalman filter that state functions and measurement functions should be linear. So we use the extended (nonlinear) Kalman filter to linearize nonlinear functions. The idea is to employ the Taylor expansions around the estimate at each step. That is, we express y ^on _M,t as

$$ {y}_{M,t}^{on}\left({r}_t\right)\approx {y}_{M,t}^{on}\left({\widehat{r}}_{t|t-1}\right)+\frac{\partial {y}_{M,t}^{on}}{\partial {r}_t}|{}_{r_t={\widehat{r}}_{t|t-1}}\cdot \left({r}_t-{\widehat{r}}_{t|t-1}\right), $$

(22.9)

where ${\widehat{r}}_{t|t-1}$ is the estimate of r _t at time t–1.

To get $\frac{\partial {y}_{M,t}^{on}}{\partial {r}_t}$, we calculate the first-order derivative of P ^on _t with respect to r _t ,

$$ \frac{\partial {P}_t^{on}}{\partial {r}_t}=-{\displaystyle \sum_{m=1}^M{C}_m{A}_{m,t}{B}_{m,t} \exp \left(-{B}_{m,t}{r}_t\right)}=\frac{\partial {P}_t^{on}}{\partial {y}_{M,t}^{on}}\frac{\partial {y}_{M,t}^{on}}{\partial {r}_t}. $$

(22.10)

Thus, we have

$$ \frac{\partial {y}_{M,t}^{on}}{\partial {r}_t}=\frac{{\displaystyle \sum_{m=1}^M{C}_m{A}_{m,t}{B}_{m,t} \exp \left(-{B}_{m,t}{r}_t\right)}}{{\displaystyle \sum_{m=1}^M{C}_m\left({t}_m-t\right) \exp \left(-{y}_{M,t}^{on}\cdot \left({t}_m-t\right)\right)}}. $$

(22.11)

Given ${\widehat{r}}_{t|t-1}$, we can use Eq. 22.8 to calculate ${y}_{M,t}^{on}\left({\widehat{r}}_{t|t-1}\right)$ and then use Eq. 22.11 to get $\frac{\partial {y}_{M,t}^{on}}{\partial {r}_t}$

Finally, the linearized measurement model for the on-the-run issues at time t is

$$ {\mathbf{y}}_{\mathbf{t}}^{\mathbf{on}}={\boldsymbol{\upalpha}}_{\mathbf{t}}^{\mathbf{on}}+{\boldsymbol{\upbeta}}_{\mathbf{t}}^{\mathbf{on}}{r}_t+{\boldsymbol{\upomega}}_{\mathbf{t}}^{\mathbf{on}} $$

(22.12)

where

$$ {\mathbf{y}}_{\mathbf{t}}^{\mathbf{on}}=\left[\begin{array}{l}{y}_{1,t}^{on}\\ {}{y}_{3,t}^{on}\end{array}\right], $$

$$ {\boldsymbol{\upalpha}}_{\mathbf{t}}^{\mathbf{on}}=\left[\begin{array}{l}{y}_{1,t}^{on}\left({\widehat{r}}_{t|t-1}\right)-{\widehat{r}}_{t|t-1}\cdot \frac{\partial {y}_{1,t}^{on}}{\partial {r}_t}|{}_{r_t={\widehat{r}}_{t|t-1}}\\ {}\kern0.1em {y}_{3,t}^{on}\left({\widehat{r}}_{t|t-1}\right)-{\widehat{r}}_{t|t-1}\cdot \frac{\partial {y}_{3,t}^{on}}{\partial {r}_{\backslash }}|{}_{r_t={\widehat{r}}_{t|t-1}}\end{array}\right], $$

$$ {\boldsymbol{\upbeta}}_{\mathbf{t}}^{\mathbf{on}}=\left[\begin{array}{l}\frac{\partial {y}_{1,t}^{on}}{\partial {r}_t}|{}_{r_t={\widehat{r}}_{t|t-1}}\\ {}\kern0.1em \frac{\partial {y}_{3,t}^{on}}{\partial {r}_t}|{}_{r_t={\widehat{r}}_{t|t-1}}\end{array}\right], $$

and ω ^on _t is the error term,

$$ {\boldsymbol{\upomega}}_{\mathbf{t}}^{\mathbf{on}}=\left[\begin{array}{l}{\omega}_{1,t}^{on}\\ {}{\omega}_{3,t}^{on}\end{array}\right], $$

where the subscript of 1 and 3 represent the 1-year bonds and 3-year bonds, while the superscript on refers to on-the-run bonds.

After we get the measurement function, the third step is to rewrite (22.1) as a discrete state function,

$$ {r}_t=\gamma +\phi {r}_{t-1}+{\varepsilon}_t, $$

(22.13)

where

$$ \gamma =\theta \left(1- \exp \left(-\kappa \cdot \Delta t\right)\right), $$

$$ \phi = \exp \left(-\kappa \cdot \Delta t\right), $$

and ε _t is the error term of r _t and Δt is the size of the time interval in the discrete sample. In our study, Δt = 0.0833. The conditional mean and conditional variance of r _t are

$$ \begin{array}{l}{\widehat{r}}_{t{|t-1}}=\theta \left(1- \exp \left(-k\Delta t\right)\right)+ \exp \left(-k\Delta t\right)\cdot {r}_{t-1}\\ {}Var\left({r}_{t|t-1}\right)={\sigma}_r^2\left(\frac{1- \exp \left(-k\Delta t\right)}{k}\right)\left(\frac{1}{2}\theta \left(1- \exp \left(-k\Delta t\right)+ \exp \left(-k\Delta t\right)\cdot {r}_{t-1}\right)\right).\end{array} $$

(22.14)

Similarly, the state functions of the off-the-run issues are

$$ \begin{array}{l}{r}_t=\gamma +\phi {r}_{t-1}+{\varepsilon}_t\\ {}{l}_t={l}_{t-1}+{\sigma}_l{e}_t.\end{array} $$

(22.15)

The conditional mean and conditional variance of l _t are l _t−1 and σ ² _l Δt, respectively.

The corresponding measurement function is

$$ {\mathbf{y}}_{\mathbf{t}}^{\mathbf{off}}={\boldsymbol{\upalpha}}_{\mathbf{t}}^{\mathbf{off}}+{\boldsymbol{\upbeta}}_{\mathbf{t}}^{\mathbf{off}}{r}_t+{\boldsymbol{\upxi}}_{\mathbf{t}}^{\mathbf{off}}{l}_t+{\boldsymbol{\upomega}}_{\mathbf{t}}^{\mathbf{off}}, $$

(22.16)

where off refers to off-the-run bonds,

$$ {\mathbf{y}}_{\mathbf{t}}^{\mathbf{off}}=\left[\begin{array}{l}{y}_{1,t}^{off}\\ {}{y}_{3,t}^{off}\end{array}\right], $$

$$ {\boldsymbol{\upalpha}}_{\mathbf{t}}^{\mathbf{off}}=\left[\begin{array}{l}{y}_{1,t}^{off}\left({\widehat{r}}_{t|t-1},{\widehat{l}}_{t|t-1}\right)-{\widehat{r}}_{t|t-1}\cdot \frac{\partial {y}_{1,t}^{off}}{\partial {r}_t}|{}_{r_t={\widehat{r}}_{t|t-1}}-{\widehat{l}}_{t|t-1}\cdot \frac{\partial {y}_{1,t}^{off}}{\partial {l}_t}|{}_{l_t={\widehat{l}}_{t|t-1}}\\ {}{y}_{3,t}^{off}\left({\widehat{r}}_{t|t-1},{\widehat{l}}_{t|t-1}\right)-{\widehat{r}}_{t|t-1}\cdot \frac{\partial {y}_{3,t}^{off}}{\partial {r}_t}|{}_{r_t={\widehat{r}}_{t|t-1}}-{\widehat{l}}_{t|t-1}\cdot \frac{\partial {y}_{3,t}^{off}}{\partial {l}_t}|{}_{l_t={\widehat{l}}_{t|t-1}}\end{array}\right], $$

$$ {\boldsymbol{\upbeta}}_{\mathbf{t}}^{\mathbf{off}}=\left[\begin{array}{l}\frac{\partial {y}_{1,t}^{off}}{\partial {r}_t}|{}_{r_t={\widehat{r}}_{t|t-1}}\\ {}\kern0.1em \frac{\partial {y}_{3,t}^{off}}{\partial {r}_t}|{}_{r_t={\widehat{r}}_{t|t-1}}\end{array}\right], $$

$$ {\boldsymbol{\upxi}}_{\mathbf{t}}^{\mathbf{off}}=\left[\begin{array}{l}\frac{\partial {y}_{1,t}^{off}}{\partial {l}_t}|{}_{l_t={\widehat{l}}_{t|t-1}}\\ {}\frac{\partial {y}_{3,t}^{off}}{\partial {\mathrm{l}}_t}|{}_{l_t={\widehat{l}}_{t|t-1}}\end{array}\right], $$

$$ {\boldsymbol{\upomega}}_{\mathbf{t}}^{\mathbf{off}}=\left[\begin{array}{l}{\omega}_{1,t}^{off}\\ {}{\omega}_{3,t}^{off}\end{array}\right], $$

$$ \frac{\partial {y}_{M,t}^{off}}{\partial {r}_t}=\frac{{\displaystyle \sum_{m=1}^M{C}_m{A}_{m,t}{B}_{m,t} \exp \left({D}_{m,t}-{B}_{m,t}{r}_t-\left({t}_m-t\right){l}_t\right)}}{{\displaystyle \sum_{m=1}^M{C}_m\left({t}_m-t\right) \exp \left(-{y}_{M,t}^{off}\cdot \left({t}_m-t\right)\right)}}, $$

$$ \frac{\partial {y}_{M,t}^{off}}{\partial {l}_t}=\frac{{\displaystyle \sum_{m=1}^M{C}_m{A}_m\left({t}_m-t\right) \exp \left({D}_{m,t}-{B}_{m,t}{r}_t-\left({t}_m-t\right){l}_t\right)}}{{\displaystyle \sum_{m=1}^M{C}_m\left({t}_m-t\right) \exp \left(-{y}_{M,t}^{off}\cdot \left({t}_m-t\right)\right)}}, $$

and ${\widehat{l}}_{t|t-1}$ is the estimate of l _t at time t–1.

Once we get the state functions and the measurement functions, we employ the regular iterative prediction-update procedure and the method of quasi-maximum likelihood to estimate the parameters. When estimating the parameters of the off-the-run issues, we use just the parameters γ and ϕ estimated from the on-the-run issues to identify σ _l .

Appendix 2: Matlab Codes

22.2.1 Codes for the On-the-Run Bonds

%*************define the likelihood function***************%

function [logfun v1 zz QQ RR rr]=kalfun(param)

k=param(1);

theta=param(2);

sigm=param(3);

sigm2=param(4);

v1=zeros(63,2);

v=zeros(2,1);

rr=zeros(63,1);

zz=zeros(63,2);

RR=0;

QQ=0;

load data.mat

z1=data(:,1); % YTMs of one-year bonds

z3=data(:,3); % YTMs of three-year bonds

c1=data(:,2); % cash flows of one-year bonds

c3=data(:,4:6); % cash flows of three-year bonds

couponrate=data(:,7); % the coupon rate of three-year bonds

z=[z1,z3];

%gam=sqrt(k^2+2*theta^2);

gam=sqrt(k^2+2*sigm^2);

tao=[1 2 3]';

dt=1/12;

a=zeros(3,1);

b=zeros(3,1);

for j=1:3

     a(j)=log((2*gam*exp(k*j/2+gam*j/2))/((k+gam)*(exp(gam*j)-1)+2*gam)^(2*k*theta/sigm^2));

b(j)=(2*exp(gam*j)-2)/((k+gam)*(exp(gam*j)-1)+2*gam);

end

r_(1)=theta; %the initial value of r

A= exp(-k*dt);

P_=(1/(1-A^2))*

(sigm^2*(1-exp(-k*dt))/k)*(theta*(1-exp(-k*dt))/2+r_(1)*exp(-k*dt)); %the initial value of P

C=theta*(1-exp(-k*dt)); %r(i)=C+A*r(i-1)

zm=zeros(63,2);          %the prediction of YTM

R=sigm2*[1 0;0 sqrt(1/3)]; %the covariance of measurement functions

logfun=0;

for i=1:63

Q=(sigm^2*(1-exp(-k*dt))/k)*(theta*(1-exp(-k*dt))/2+r_(i)*exp(-k*dt)); %the conditional variance of state functions

pz1(i)=(c1(i)*b(1)*exp(a(1)-b(1)*r_(i)))/c1(i)*exp(-z1(i)*1);           %the partial derivative of one year z against r

pz3(i)=sum(c3(i,:)'.*b.*exp(a-b*r_(i)))/sum(c3(i,:)'.*tao.*exp(-z3(i)*tao)); %the partial derivative of three year z against r

P1=c1(i)*exp(a(1)-b(1)*r_(i));     %the prediction price of one-year bond

P3=sum(c3(i,:)'.*exp(a-b*r_(i))); % the prediction price of three-year bond

%zm1=bndyield(P1,c1(i),'20-Jan-1997','20-Jan-1998',1);

zm1=-log(P1/c1(i));   %nonlinear measurement function for one-year bonds

zm3=bndyield(P3,couponrate(i),'20-Jan-1997','20-Jan-2000',1); %nonlinear measurement function for three-year bonds

H=[pz1(i) pz3(i)]';

%C1=[zm1 zm3]'-H*r_(i);

%zm(i,:)= C1+H*r_(i);

%zm(i,:)= C1+H*r(i);

zm(i,:)=[zm1 zm3]';     %the prediction of YTMs

v=z(i,:)'-zm(i,:)'; %the error of measurement functions

v1(i,:)=v';        % the error between the prediction and the real value

F=H*P_*H'+R;          %the kalman gain

if det(F)<=0

          logfun=0;

          return

end

rr(i,:)=r_(i);

zz(i,:)=zm(i,:);

r(i)=r_(i)+P_*H'*inv(F)*v; %update r

P=P_-P_*H'*inv(F)*H*P_;     %update P

ll=-0.5*log(det(F))-0.5*v'*inv(F)*v; %likelihood function

logfun=logfun+ll;

     r_(i+1)=A*r(i)+C;     %predict r

     P_=A*P*A'+Q;               %predict P

end

QQ=Q;

RR=R;

logfun=-logfun;

function covv=covirance(param)

covv=zeros(4,4);

for i=1:4

     for j=1:4

     parama=param;

     paramb=param;

     paramab=param;

     parama(i)=param(i)*1.01;

     paramb(j)=param(j)*0.99;

     paramab(i)=param(i)*1.01;

     paramab(j)=paramab(j)*0.99;

     ua=kalfun(parama);

     db=kalfun(paramb);

     udab=kalfun(paramab);

     kk=kalfun(param);

     covv(i,j)=(ua+db-kk-udab)/((0.01*param(i))*(0.01*param(j)));

end

end

22.2.2 Codes for the Off-the-Run Bonds

%************define the likelihood function *************%

function [logfun LL zz v1 QQ RR]=kalfunL(paramL)

sigm3=paramL(1);

sigm4=paramL(2);

load dataL.mat

z1=dataL(:,1); %YTMs of one-year bonds

z3=dataL(:,3); % YTMs of three-year bonds

c1=dataL(:,2); % cash flows of one-year bonds

c3=dataL(:,4:6); % cash flows of three-year bonds

couponrate=dataL(:,7); %the coupon rate of three-year bonds

r=dataL(:,8); %the estimated r in the CIR model

% the estimated parameters in the CIR model

k=0.08899;

theta=0.022659;

sigm=0.063329;

gam=sqrt(k^2+2*sigm^2);

z=[z1,z3];

tao=[1 2 3]';

dt=1/12;

a=zeros(3,1);

b=zeros(3,1);

e=zeros(3,1);

zz=zeros(63,2);

v1=zeros(63,2);

v=zeros(2,1);

RR=0;

QQ=0;

for j=1:3

     a(j)=log((2*gam*exp(k*j/2+gam*j/2))/((k+gam)*(exp(gam*j)-1)+2*gam)^(2*k*theta/sigm^2));

     b(j)=(2*exp(gam*j)-2)/((k+gam)*(exp(gam*j)-1)+2*gam);

     e(j)=(sigm3^2*tao(j)^3)/6;

end

L_(1)=0;  %the initial value of L

v1=zeros(63,2);

v=zeros(2,1);

P_=0;

zm=zeros(63,2);          %the prediction of YTM

R=sigm4*[1 0;0 sqrt(1/3)];    %the covariance of measurement functions

logfun=0;

for i=1:63

Q=sigm3^2*dt;          %the conditional variance of state functions

pz1(i)=(c1(i)*exp(a(1)-b(1)*r(i)+e(1)-L_(i)*1))/c1(i)*exp(-z1(i)*1); %the partial derivative of one year z against r

pz3(i)=sum(c3(i,:)'.*tao.*exp(a-b*r(i)+e-L_(i)*tao))/sum(c3(i,:)'.*tao.*exp(-z3(i)*tao));         %the partial derivative of three year z against r

P1=c1(i)*exp(a(1)-b(1)*r(i)+e(1)-L_(i)*1);     %the prediction price of one-year bond

P3=sum(c3(i,:)'.*exp(a-b*r(i)+e-L_(i)*tao));     % the prediction price of three-year bond

%zm1=bndyield(P1,c1(i),'20-Jan-1997','20-Jan-1998',1);

zm1=-log(P1/c1(i));         %nonlinear measurement function for one-year bonds

zm3=bndyield(P3,couponrate(i),'20-Jan-1997','20-Jan-2000',1);   %nonlinear measurement function for three-year bonds

H=[pz1(i) pz3(i)]';

%C1=[zm1 zm3]'-H*r_(i);

%zm(i,:)= C1+H*r_(i);

%zm(i,:)= C1+H*r(i);

zm(i,:)=[zm1 zm3]';     %the prediction of YTMs

v=z(i,:)'-zm(i,:)';     %the error of measurement functions

v1(i,:)=v';               % the error between the prediction and the real value

F=H*P_*H'+R;             %the kalman gain

if det(F)<=0

          logfun=0;

          return

end

LL(i,:)=L_(i);

zz(i,:)=zm(i,:);

L(i)=L_(i)+P_*H'*inv(F)*v;   %update r

P=P_-P_*H'*inv(F)*H*P_;         %update P

ll=-0.5*log(det(F))-0.5*v'*inv(F)*v;

   logfun=logfun+ll;

   L_(i+1)=L(i);     %predict r

     P_=P+Q;             %predict P

end

QQ=Q;

RR=R;

logfun=-logfun;

function covvL=coviranceL(paramL)

covvL=zeros(2,2);

for i=1:2

     for j=1:2

     parama=paramL;

     paramb=paramL;

     paramab=paramL;

     parama(i)=paramL(i)*1.0000001;

     paramb(j)=paramL(j)*0.9999999;

     paramab(i)=paramL(i)*1.0000001;

     paramab(j)=paramab(j)*0.9999999;

     ua=kalfunL(parama);

     db=kalfunL(paramb);

     udab=kalfunL(paramab);

     kk=kalfunL(paramL);

    covvL(i,j)=(ua+db-kk-udab)/((0.0000001*paramL(i))*(0.0000001*paramL(j)));

end

end