Incorporating Convexity in Bond Portfolio Immunization Using Multifactor Model: A Semidefinite Programming Approach

Abstract

Bond portfolio immunization is a classical issue in finance. Since Macaulay gave the concept of duration in 1938, many scholars proposed different kinds of duration immunization models. In the literature of bond portfolio immunization using multifactor model, to the best of our knowledge, researchers only use the first-order immunization, which is usually called as duration immunization, and no one has considered second-order effects in immunization, which is well known as “convexity” in the case of single-factor model. In this paper, we introduce the second-order information associated with multifactor model into bond portfolio immunization and reformulate the corresponding problems as tractable semidefinite programs. Both simulation analysis and empirical study show that the second-order immunization strategies exhibit more accurate approximation to the value change of bonds and thus result in better immunization performance.

Appendix: Calculation of Duration Vector and Convexity Matrix

The expressions of the elements of duration vector $\varvec{d}=\left( \frac{\partial p}{\partial \beta _0}, \frac{\partial p}{\partial \beta _1}, \frac{\partial p}{\partial \beta _2}\right) '$ can be calculated via the first-order derivation as

$$\begin{aligned} \frac{\partial p}{\partial \beta _0}= & {} -\sum \limits _t t c_t \mathrm{e}^{-R(t)t},\\ \frac{\partial p}{\partial \beta _1}= & {} -\sum \limits _t \tau c_t \mathrm{e}^{-R(t)t} \left( 1-\mathrm{e}^{-\frac{t}{\tau }}\right) ,\\ \frac{\partial p}{\partial \beta _2}= & {} -\sum \limits _t c_t \mathrm{e}^{-R(t)t} \left( \tau -\tau \mathrm{e}^{-\frac{t}{\tau }}-t \mathrm{e}^{-\frac{t}{\tau }}\right) . \end{aligned}$$

The expressions of the elements of convexity matrix

$$\begin{aligned} H=\left( \begin{array}{ccc} \frac{\partial ^2 p}{\partial \beta _0^2} &{}\quad \frac{\partial ^2 p}{\partial \beta _0\partial \beta _1} &{}\quad \frac{\partial ^2 p}{\partial \beta _0\partial \beta _2} \\ \frac{\partial ^2 p}{\partial \beta _0\partial \beta _1} &{}\quad \frac{\partial ^2 p}{\partial \beta _1^2} &{}\quad \frac{\partial ^2 p}{\partial \beta _1\partial \beta _2} \\ \frac{\partial ^2 p}{\partial \beta _0\partial \beta _2} &{}\quad \frac{\partial ^2 p}{\partial \beta _1\partial \beta _2} &{}\quad \frac{\partial ^2 p}{\partial \beta _2^2} \\ \end{array} \right) \end{aligned}$$

can be calculated via the second-order derivation as

$$\begin{aligned} \frac{\partial ^2 p}{\partial \beta _0^2}= & {} \sum \limits _t t^2 c_t \mathrm{e}^{-R(t)t},\\ \frac{\partial ^2 p}{\partial \beta _0\partial \beta _1}= & {} \sum \limits _t t \tau c_t \mathrm{e}^{-R(t)t} \left( 1-\mathrm{e}^{-\frac{t}{\tau }}\right) ,\\ \frac{\partial ^2 p}{\partial \beta _0\partial \beta _2}= & {} \sum \limits _t t c_t \mathrm{e}^{-R(t)t} \left( \tau -\tau \mathrm{e}^{-\frac{t}{\tau }}-t \mathrm{e}^{-\frac{t}{\tau }}\right) ,\\ \frac{\partial ^2 p}{\partial \beta _1^2}= & {} \sum \limits _t \tau ^2 c_t \mathrm{e}^{-R(t)t} \left( 1-\mathrm{e}^{-\frac{t}{\tau }}\right) ^2,\\ \frac{\partial ^2 p}{\partial \beta _1\partial \beta _2}= & {} \sum \limits _t \tau c_t \mathrm{e}^{-R(t)t} (1-\mathrm{e}^{-\frac{t}{\tau }})\left( \tau -\tau \mathrm{e}^{-\frac{t}{\tau }}-t \mathrm{e}^{-\frac{t}{\tau }}\right) ,\\ \frac{\partial ^2 p}{\partial \beta _2^2}= & {} \sum \limits _t c_t \mathrm{e}^{-R(t)t} \left( \tau -\tau \mathrm{e}^{-\frac{t}{\tau }}-t \mathrm{e}^{-\frac{t}{\tau }}\right) ^2. \end{aligned}$$