Incorporating Convexity in Bond Portfolio Immunization Using Multifactor Model: A Semidefinite Programming Approach
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Incorporating Convexity in Bond Portfolio Immunization Using Multifactor Model: A Semidefinite Programming Approach Wei Zhu1 · Cai-Hong Zhang2 · Qian Liu1 · Shu-Shang Zhu1 Received: 26 July 2017 / Revised: 16 January 2018 / Accepted: 29 January 2018 / Published online: 20 February 2018 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2018
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
This paper is dedicated to Professor Duan Li in celebration of his 65th birthday. This research is partially supported by the National Natural Science Foundation of China (Nos. 71471180 and 71571062).
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Shu-Shang Zhu [email protected] Wei Zhu [email protected] Cai-Hong Zhang [email protected] Qian Liu [email protected]
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Department of Finance and Investment, Sun Yat-Sen Business School, Sun Yat-Sen University, Guangzhou 510275, China
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Ping An Securities Company Ltd, Beijing 100033, China
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W. Zhu et al.
strategies exhibit more accurate approximation to the value change of bonds and thus result in better immunization performance. Keywords Immunization · Duration · Convexity · Multifactor model · Semidefinite programming Mathematics Subject Classification 90c25 · 91B28
1 Introduction Bond portfolio immunization derives from the following simple question: Suppose an investor has a fixed amount of debt, e.g., $100, to repay after 10 years, then how should he/she invest his/her money now to guarantee that the return after 10 years can afford the debt? The answer is trivial if the interest rate does not change in the following 10 years. However, if we consider changes of interest rate, the answer is not so simple. The safest way is to buy a zero coupon bond, which will pay $100 after 10 years; however, it is usually impossible to find such a bond. A compromise is to invest a coupon bond portfolio to approximate the payoff of zero coupon bond. Since coupon bond pays coupons during the period, a 10-year coupon bond is actually equivalent to a zero coupon bond whose term is less than 10 years, and a natural measure of the “actual term” is the weighted average of time when coupons are paid. This is the original idea of duration. The concept of
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