Prepayment risk on callable bonds: theory and test

Abstract

We develop a framework for analyzing prepayment risk on defaultable callable bonds. We argue that prepayment risk emanates from the following informational asymmetry: Callable bond traders cannot determine the issuer’s firm value-maximizing call policy, and their best anticipation is the optimal refinancing policy given by a term structure model. We show that, from the callable bond holder perspective, the issuer’s departure from the optimal refinancing policy translates into an accrued exposure to market risk. The prepayment risk magnitude represents this risk transfer, and we show that callable bond traders can infer it from observable bond characteristics. Tests on callable bond transaction data provide strong evidence for prepayment risk and validate our conjecture that insurance companies trade callable bonds to reduce their exposure to prepayment risk magnitude.

Appendices

Appendix 1: Proof of Proposition 1

By definition, the optimal call threshold $h^{*}$ satisfies

$$\begin{aligned} \left. \frac{dC_{\infty }\left( c,r,h\right) }{dh}\right| _{h=h^{*}}=0. \end{aligned}$$

For a given call policy $h$, the callable bond elasticity is given by

$$\begin{aligned} \mathcal {E}=\frac{dC_{\infty }\left( c,r,h\right) }{dr}\frac{r}{C_{\infty }\left( c,r,h\right) }. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{d\mathcal {E}}{dh}&= d\left( \frac{dC_{\infty }\left( c,r,h\right) }{dr }\frac{r}{C_{\infty }\left( c,r,h\right) }\right) \Bigg /dh, \\&= r\frac{d}{dhdr}\left( \frac{dC_{\infty }\left( c,r,h\right) }{C_{\infty }\left( c,r,h\right) }\right) , \\&= r\frac{\frac{d^{2}C_{\infty }\left( c,r,h\right) }{dhdr}C_{\infty }\left( c,r,h\right) -\frac{dC_{\infty }\left( c,r,h\right) }{dh}\frac{dC_{\infty }\left( c,r,h\right) }{dr}}{C_{\infty }\left( c,r,h\right) ^{2}}. \end{aligned}$$

This derivative at coordinate $h=h^{*}$ can be written as

$$\begin{aligned} \left. \frac{d\mathcal {E}}{dh}\right| _{h=h^{*}}=r\frac{\left. \frac{d^{2}C_{\infty }\left( c,r,h\right) }{dhdr}\right| _{h=h^{*}}C_{\infty }\left( c,r,h\right) -\left. \frac{dC_{\infty }\left( c,r,h\right) }{dh}\right| _{h=h^{*}}\frac{dC_{\infty }\left( c,r,h\right) }{dr}}{C_{\infty }\left( c,r,h\right) ^{2}}=0. \end{aligned}$$

Therefore, the optimal threshold $h^{*}$ that minimizes the callable bond value also minimizes the callable bond elasticity.

Appendix 2: Valuation of $B_{\infty }\left( c,r\right)$

Following Duffie and Singleton (1999) and Barone et al. (1998), the no-arbitrage value of the non-callable consol bond is given by

$$\begin{aligned} B_{\infty }\left( c,r\right) =E_{\mathbb {Q}}\left[ c\int _{0}^{\infty }\exp \left( -\int _{0}^{t}\left( r_{u}+\eta \right) du\right) \mathrm{d}t\right] . \end{aligned}$$

Using Fubini’s theorem,¹⁸

$$\begin{aligned} B_{\infty }\left( c,r\right) =c\int _{0}^{\infty }E_{\mathbb {Q}}\left[ \exp \left( -\int _{0}^{t}\left( r_{u}+\eta \right) du\right) \right] \mathrm{d}t. \end{aligned}$$

The expectation in square brackets represents the value of the discount bond that is obtained from the Cox et al. (1985) formula, which yields

$$\begin{aligned} B_{\infty }\left( c,r\right) =c\int _{0}^{\infty }a\left( t\right) \exp \left( -\eta t-b\left( t\right) r\right) \mathrm{d}t, \end{aligned}$$

with

$$\begin{aligned} a\left( t\right)&= \left( \frac{2\gamma e^{0.5\left( \gamma +\kappa +\lambda \right) t}}{\left( \gamma +\kappa +\lambda \right) \left( e^{\gamma t}-1\right) +2\gamma }\right) ^{\frac{2\kappa \theta }{\sigma ^{2}}}, \\ b\left( t\right)&= \frac{2}{\kappa +\lambda +\gamma \coth \left( \frac{ \gamma t}{2}\right) }, \\ \gamma&= \sqrt{\left( \kappa +\lambda \right) ^{2}+2\sigma ^{2}}, \end{aligned}$$

and $\coth \left( x\right) :=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$. The integral in the valuation formula for $B_{\infty }\left( c,r\right)$ is solved numerically with no difficulty.¹⁹

Appendix 3: Proof of Proposition 2

Let $g\left( r,h\right)$ denote the value of the perpetual American call written on the borrower’s financing instrument with call threshold $h$ and strike $K$. This call option entitles the borrower the right to put an end to her current defaultable callable consol with coupon $c$ by paying the strike.

In the one-factor model, using Itô’s lemma and standard arbitrage arguments, $g$ satisfies the following EDP (see Barone et al. 1998)

$$\begin{aligned} \left( \kappa \theta -\left( \kappa +\lambda \right) r\right) g_{r}+\frac{1}{ 2}\sigma ^{2}rg_{rr}=\left( r+\eta \right) g \end{aligned}$$

where $g_{x}$ denotes the partial derivative of $g$ with respect to $x$. The general solution is

$$\begin{aligned} g\left( r,h\right)&=C_{1}\exp \left[ \frac{r}{\sigma ^{2}}\left( \kappa +\lambda -f_{1}\right) \right] r^{\frac{f_{2}}{2}-\frac{\theta \kappa }{ \sigma ^{2}}}M\left( \frac{f_{2}}{2}-\frac{\theta \kappa \left( \kappa +\lambda \right) -\eta \sigma ^{2}}{\sigma ^{2}f_{1}};f_{2};\frac{2rf_{1}}{ \sigma ^{2}}\right) \\&\quad +C_{2}\exp \left[ \frac{r}{\sigma ^{2}}\left( \kappa +\lambda -f_{1}\right) \right] r^{\frac{f_{2}}{2}-\frac{\theta \kappa }{\sigma ^{2}} }L\left( \frac{\theta \kappa \left( \kappa +\lambda \right) -\eta \sigma ^{2} }{\sigma ^{2}f_{1}}-\frac{f_{2}}{2};f_{2}-1;\frac{2rf_{1}}{\sigma ^{2}} \right) \end{aligned}$$

with $C_{1}$ and $C_{2}$ two constants and

$$\begin{aligned} f_{1}&:= f_{1}\left( \kappa ,\lambda ,\sigma \right) =\sqrt{\left( \kappa +\lambda \right) ^{2}+2\sigma ^{2}} \\ f_{2}&:= f_{2}\left( \kappa ,\theta ,\sigma \right) =1+\sqrt{1+\frac{ 4\theta ^{2}\kappa ^{2}}{\sigma ^{4}}-\frac{4\theta \kappa }{\sigma ^{2}}} \end{aligned}$$

and $M\left( .;.;.\right)$ denotes the confluent hypergeometric function, and $L\left( .;.;.\right)$ denotes the generalized Laguerre polynomials.

Boundary condition $g\left( \infty ,h\right) =0$ implies that $C_{2}=0$. The second boundary condition is the payoff upon exercise

$$\begin{aligned} g\left( h,h\right) =B_{\infty }\left( c,h\right) -K. \end{aligned}$$

Substituting into the general solution yields

$$\begin{aligned} g\left( r,h\right) =\left( B_{\infty }\left( c,h\right) -K\right) e^{\frac{\left( r-h\right) }{\sigma ^{2}}\left( \kappa +\lambda -f_{1}\right) }\left( \frac{r}{h}\right) ^{\frac{f_{2}}{2}-\frac{\theta \kappa }{\sigma ^{2}}} \frac{M\left( \frac{f_{2}}{2}-\frac{\theta \kappa \left( \kappa +\lambda \right) -\eta \sigma ^{2}}{\sigma ^{2}f_{1}};f_{2};\frac{2rf_{1}}{\sigma ^{2} }\right) }{M\left( \frac{f_{2}}{2}-\frac{\theta \kappa \left( \kappa +\lambda \right) -\eta \sigma ^{2}}{\sigma ^{2}f_{1}};f_{2};\frac{2hf_{1}}{ \sigma ^{2}}\right) }. \end{aligned}$$

Finally, the value of the defaultable callable coupon-bearing perpetuity is that of the equivalent non-callable perpetuity less the value of the call option

$$\begin{aligned} C_{\infty }\left( c,r,h\right) =B_{\infty }\left( c,r\right) -g\left( r,h\right) \!. \end{aligned}$$

Appendix 4: Estimation of term structure model parameters

Consistent with our theoretical framework, we estimate a one-factor Cox–Ingersoll–Ross term structure model for the instantaneous risk-free rate. The risk-neutral dynamics of the instantaneous risk-free rate are

$$\begin{aligned} dr_{t}=\left( \kappa \theta -\left( \kappa +\lambda \right) r_{t}\right) \mathrm{d}t+\sigma \sqrt{r_{t}}\mathrm{d}W_{t}. \end{aligned}$$

Following Duffee (1999) and Duan and Simonato (1999), our estimation method relies on the approximate linear Kalman filter applied to exponential-affine term structure models.

At time $t$, the yield of the discount bond maturing in $\tau$ units of time is given by

$$\begin{aligned} y_{t}\left( \tau \right) =-\frac{1}{\tau }\ln \left( a\left( \tau \right) \right) +\frac{1}{\tau }b\left( \tau \right) r_{t}, \end{aligned}$$

where the functions $a\left( .\right)$ and $b\left( .\right)$, defined in “Appendix 2,” contain all four parameters to estimate.

The filtering approach assumes that yields for different maturities are observed with errors of unknown magnitude. Specifically, observing $N$ bonds with different maturities, we write the measurement equation as follows

$$\begin{aligned} \left[ \begin{array}{c} y_{t}\left( \tau _{1}\right) \\ y_{t}\left( \tau _{2}\right) \\ \vdots \\ y_{t}\left( \tau _{N}\right) \end{array} \right] =\left[ \begin{array}{c} -\ln \left( a\left( \tau _{1}\right) \right) /\tau _{1} \\ -\ln \left( a\left( \tau _{2}\right) \right) /\tau _{2} \\ \vdots \\ -\ln \left( a\left( \tau _{N}\right) \right) /\tau _{N} \end{array} \right] +\left[ \begin{array}{c} b\left( \tau _{1}\right) /\tau _{1} \\ b\left( \tau _{2}\right) /\tau _{2} \\ \vdots \\ b\left( \tau _{N}\right) /\tau _{N} \end{array} \right] r_{t}+\left[ \begin{array}{c} \varepsilon _{t,1} \\ \varepsilon _{t,2} \\ \vdots \\ \varepsilon _{t,N} \end{array} \right] , \end{aligned}$$

where $\varepsilon _{t,j}$ is a normally distributed error term with zero mean and standard deviation $\sigma _{j}$.

Consider the discretized version of the stochastic process for $r_{t}$ and denote by $\Delta t$ the time step. The transition equation is given by

$$\begin{aligned} r_{t+\Delta t}=\theta \left( 1-e^{-\kappa \Delta t}\right) +e^{-\kappa \Delta t}r_{t}+\eta _{t+\Delta t}, \end{aligned}$$

where $\eta _{t+\Delta t}$ is an error term with zero mean and variance equal to

$$\begin{aligned} \frac{\sigma ^{2}}{\kappa }\left[ r_{t}\left( e^{-\kappa \Delta t}-e^{-2\kappa \Delta t}\right) +\frac{\theta }{2}\left( 1-e^{-\kappa \Delta t}\right) ^{2}\right] . \end{aligned}$$

The measurement and transition equations characterize the state space on which we apply the standard Kalman filter recursion. Since the terms in the measurement equation are nonlinear and the error term in the transition equation is not normal, the Kalman filter methodology is only approximate in this case. However, Duan and Simonato (1999) provide Monte Carlo evidence that this procedure remains adequate for the Cox–Ingersoll–Ross model.

The risk-free yield curve is obtained from the St-Louis Federal Reserve Bank database. We use weekly observations of the 6-month, 1-, 2-, 3-, 5-, 10-, and 30-year US Treasury constant maturity rates²⁰ from January 5, 1990, to December 26, 2003.²¹ Estimation results are presented in the table below (standard errors are reported below in parentheses).

 Kalman filter estimates of the Cox--Ingersoll--Ross model of US Treasury bond yields, January 1990--December 2003

$\theta$	$\kappa$	$\sigma$	$\lambda$
$\underset{\left( 0.005889\right) }{0.046581}$	$\underset{\left( 0.038419\right) }{0.287423}$	$\underset{\left( 0.000973\right) }{0.024373}$	$\underset{\left( 0.036529\right) }{-0.116064}$

Bond maturity	Measurement error
6 months	$\underset{\left( 0.000059\right) }{0.001918}$
1 year	$\underset{\left( 0.014744\right) }{0.000001}$
2 years	$\underset{\left( 0.000069\right) }{0.003166}$
3 years	$\underset{\left( 0.000102\right) }{0.004368}$
5 years	$\underset{\left( 0.000146\right) }{0.006188}$
10 years	$\underset{\left( 0.000143\right) }{0.007571}$
30 years	$\underset{\left( 0.000169\right) }{0.010519}$

Appendix 5: Dynamics of the CIR process in a binomial tree

We outline the procedure developed by Nawalkha and Beliaeva (2007). The CIR process for the short rate has non-constant volatility. As a consequence, its discretized version in a binomial tree entails that the tree is not recombining. Nelson and Ramaswamy (1990) use the following change of variable to fix this issue

$$\begin{aligned} r_{t}=\frac{x_{t}^{2}\sigma ^{2}}{4}, \end{aligned}$$

where $x_{t}$ denotes the variable that is used in the recombining binomial tree. By construction, the process $x_{t}$ has unit volatility, and the up and down nodes can therefore be defined as $x_{u}=x+\sqrt{\Delta t}$ and $x_{d}=x-\sqrt{\Delta t}$ where $\Delta t$ denotes the time step. However, the tree must be adjusted for the drift of the $x_{t}$ process and for the constraint that the CIR process $r_{t}$ must remain positive. Nawalkha and Beliaeva (2007) propose a truncation of the $x-$binomial tree at zero. They distinguish two cases.

• If $x>0$ then $r>0$, the up and down nodes are

  $$\begin{aligned} x_{u}&= x+\left( J+1\right) \sqrt{\Delta t} \\ x_{d}&= \max \left( 0,x+\left( J-1\right) \sqrt{\Delta t}\right) \!, \end{aligned}$$

  where $J$ is an integer such that

  $$\begin{aligned} J=\left\{ \begin{array}{l@{\quad }l} Z &{} \text {if}\,\,Z\,\,\text { is even} \\ Z+1 &{} \text {if }\,\,Z\,\,\text { is odd} \end{array} \right. \end{aligned}$$

  and

  $$\begin{aligned} Z=Floor\left( \frac{\sqrt{x^{2}\sigma ^{2}\left( 1-\left( \kappa +\lambda \right) \Delta t\right) +4\kappa \theta \Delta t}-x\sigma }{\sigma \sqrt{ \Delta t}}\right) , \end{aligned}$$

  where $Floor\left( u\right)$ is the integer value of $u$, rounded down. The probabilities of up and down moves ensure that the drift of the short rate is correctly matched. They are given by

  $$\begin{aligned} p_{u}&= \frac{\left( \kappa \theta -\left( \kappa +\lambda \right) r\right) \Delta t+r-r_{d}}{r_{u}-r_{d}} \\ p_{d}&= 1-p_{u}. \end{aligned}$$

• If $x=0$ then $r=0$, the down node simply stays at zero $x_{d}=0$. The up node is the node closest to the truncation line of zero which verifies

  $$\begin{aligned} x_{u}\ge \frac{2}{\sigma }\sqrt{\kappa \theta \Delta t}. \end{aligned}$$

  The probabilities of up and down moves are given by

  $$\begin{aligned} p_{u}&= \frac{\kappa \theta \Delta t}{r_{u}} \\ p_{d}&= 1-p_{u}. \end{aligned}$$