Interest Rate Risk Management and Dynamic Portfolio Selections

ABSTRACT

The dynamic portfolio selections in the sense of Markowitz’s mean-variance are addressed in an incomplete market and the effect of interest rate risk on them is discussed. According to Markowitz’s measure risk approach, the interest rate risk is divided into the controllable risk and the uncontrollable risk. The former can be hedged, but the latter cannot. The zero-coupon bond is an efficient tool to avoid the interest rate risk. The optimal payoff resulting from self-financed strategies and the mean-variance efficient frontier are expressed explicitly. The results show that the optimal payoff and the efficient frontier are not affected by the controllable risk of interest rate, but by the uncontrollable risk. The efficient frontier is a part of a hyperbola if there exists the uncontrollable risk. The expected optimal payoff grows with the increase of risk; however, the margin expected optimal payoff lowers. The efficient frontier is a straight line if and only if there is no uncontrollable risk.

The dynamic portfolio selections in the sense of Markowitz’s mean-variance are addressed in an incomplete market and the effect of interest rate risk on them is discussed. According to Markowitz’s measure risk approach, the interest rate risk is divided into the controllable risk and the uncontrollable risk. The former can be hedged, but the latter cannot. The zero-coupon bond is an efficient tool to avoid the interest rate risk. The optimal payoff resulting from self-financed strategies and the mean-variance efficient frontier are expressed explicitly. The results show that the optimal payoff and the efficient frontier are not affected by the controllable risk of interest rate, but by the uncontrollable risk. The efficient frontier is a part of a hyperbola if there exists the uncontrollable risk. The expected optimal payoff grows with the increase of risk; however, the margin expected optimal payoff lowers. The efficient frontier is a straight line if and only if there is no uncontrollable risk.

Cite this paper

nullH. Sun and W. Sun, "Interest Rate Risk Management and Dynamic Portfolio Selections,"*Modern Economy*, Vol. 2 No. 4, 2011, pp. 674-679. doi: 10.4236/me.2011.24075.

nullH. Sun and W. Sun, "Interest Rate Risk Management and Dynamic Portfolio Selections,"

References

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[6] A. E. B. Lim and X. Y. Zhou, “Mean-Variance Portfolio Selection with Random Parameters in a Complete Market,” Applied Mathematics & Optimization, Vol. 27, No. 1, 2002, pp. 101-120.

[7] I. Karatzas, J. Lehoczky, S. E. Shreve and G. Xu, “Martingale and Duality Methods for Utility Maximization in an Incomplete Market,” SIAM Journal of Control and Optimization, Vol. 29, No. 1, 1991, pp. 702–730.

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[11] F. Delbaen and W. Schachermayer, “The Variance-Opti- Mal Martingale Measure for Continuous Processes,” Bernoulli, Vol. 2, No. 1, 1996, pp. 81-105.

[12] C. Goureroux, J. P. Laurent and H. Pham, “Mean-Variance Hedging and Numeraire,” Mathematical Finance, Vol. 8, No. 3, 1998, pp. 179-200.

[13] W. G. Sun and C. F. Wang, “The Mean-Variance Invest- Ment Problem in a Constrained Financial Market,” Journal of Mathematical Economics, Vol. 42, No. 11, 2006, pp. 885-895.

[14] J. P. Laurent and H. Pham, “Dynamic Programming and Mean-Variance Hedging,” Finance and Stochastics, Vol. 3, No. 1, 1999, pp. 83-110.

[1] H. Markowitz, “Portfolio Selection,” Journal of Finance, Vol. 7, No. 1, 1952, pp. 77-91.

[2] R. Merton, “An Intertemporal Capital Asset Pricing Mo- del”, Econometrica, Vol. 41, No. 5, 1971, pp. 867-888.

[3] H. Richardson, “A Minimum Variance Result in Continuous Trading Portfolio Optimization,” Management Science, Vol. 35, No. 3, 1989, pp. 1045-1055.

[4] I. Bajeux-Besnainou and R. Portait, “Dynamic Asset Allocation in a Mean-Variance Framework,” Management Science, Vol. 44, No. 11, 1998, pp. 79-95.

[5] X. Y. Zhou and D. Li, “Continuous-Time Mean-Variance Portfolio Selection: a Stochastic LQ Framework,” Applied Mathematics & Optimization, Vol. 42, No. 1, 2000, pp. 19-33.

[6] A. E. B. Lim and X. Y. Zhou, “Mean-Variance Portfolio Selection with Random Parameters in a Complete Market,” Applied Mathematics & Optimization, Vol. 27, No. 1, 2002, pp. 101-120.

[7] I. Karatzas, J. Lehoczky, S. E. Shreve and G. Xu, “Martingale and Duality Methods for Utility Maximization in an Incomplete Market,” SIAM Journal of Control and Optimization, Vol. 29, No. 1, 1991, pp. 702–730.

[8] H. He and N. D. Pearson, “Consumption and Portfolio Policies with Incomplete Markets and Short-Sale Constraints,” Journal of Economic Theory, Vol. 54, No. 8, 1991, pp. 259–305.

[9] M. Schweizer, “Approximation Pricing and the Variance-Optimal Martingale Measure,” Annals of Probability, Vol. 24, No. 1, 1996, pp. 206-236.

[10] M. Schweizer, “From Actuarial to Financial Valuation Principles,” Insurance: Mathematics & Economics, Vol. 28, No. 1, 2001, pp. 31-47.

[11] F. Delbaen and W. Schachermayer, “The Variance-Opti- Mal Martingale Measure for Continuous Processes,” Bernoulli, Vol. 2, No. 1, 1996, pp. 81-105.

[12] C. Goureroux, J. P. Laurent and H. Pham, “Mean-Variance Hedging and Numeraire,” Mathematical Finance, Vol. 8, No. 3, 1998, pp. 179-200.

[13] W. G. Sun and C. F. Wang, “The Mean-Variance Invest- Ment Problem in a Constrained Financial Market,” Journal of Mathematical Economics, Vol. 42, No. 11, 2006, pp. 885-895.

[14] J. P. Laurent and H. Pham, “Dynamic Programming and Mean-Variance Hedging,” Finance and Stochastics, Vol. 3, No. 1, 1999, pp. 83-110.