# American Institute of Mathematical Sciences

January  2016, 21(1): 185-203. doi: 10.3934/dcdsb.2016.21.185

## The optimal mean variance problem with inflation

 1 School of Insurance, Central University of Finance and Economics, Beijing 10086, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, China 3 Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon Tong, Hong Kong, China

Received  December 2013 Revised  September 2014 Published  November 2015

The risk of inflation is looming under the current low interest rate environment. Assuming that the investment includes a fixed interest asset and $n$ risky assets under inflation, we consider two scenarios: inflation rate can be observed directly or through a noisy observation. Since the inflation rate is random, all assets become risky. Under this circumstance, we formulate the portfolio selection problem and derive the efficient frontier by solving the associated HJB equation. We find that for a given expected portfolio return, investment at time $t$ is linearly proportional to the price index level. Moreover, the risk for the real value of the portfolio is no longer minimal when all the wealth is put into the fixed interest asset. Finally, for the mutual fund theorem, two funds are needed now instead of the traditional single fund. If an inflation linked bond can be included in the portfolio, the problem is reduced to the traditional mean variance problem with a risk-free and $n+1$ risky assets with real returns.
Citation: Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. The optimal mean variance problem with inflation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 185-203. doi: 10.3934/dcdsb.2016.21.185
##### References:
 [1] A. Bensoussan, J. Keppo and S. P. Sethi, Optimal consumption and portfolio decisions with partially observed real prices, Mathematical Finance, 19 (2009), 215-236. doi: 10.1111/j.1467-9965.2009.00362.x. [2] M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238. [3] J. Cea, Lectures on Optimization - Theory and Algorithm, Tata Institute of Fundamental Research, Bombay, 1978. [4] S. N. Chen and W. T. Moore, Uncertain inflation and optimal portfolio selection: A simplified approach, The Financial Review, 20 (1985), 343-352. doi: 10.1111/j.1540-6288.1985.tb00312.x. [5] C. H. Chiu and X. Y. Zhou, The premium of dynamic trading, Quantitative Finance, 11 (2011), 115-123. doi: 10.1080/14697681003685589. [6] W. S. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993. [7] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100. [8] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [9] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555. doi: 10.1137/S0363012900378504. [10] J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a value-at-risk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531-547. doi: 10.3934/jimo.2012.8.531. [11] J. Z. Liu and K. F. C. Yiu, Optimal stochastic differential games with VaR constraint, Discrete & Continuous Dynamical Systems - Series B, 18 (2013), 1889-1907. doi: 10.3934/dcdsb.2013.18.1889. [12] J. Z. Liu, K. F. C. Yiu and T. K. Siu, Optimal investment of an insurer with regime-switching and risk constraint, Scandinavian Actuarial Journal, 2014 (2014), 583-601. doi: 10.1080/03461238.2012.750621. [13] S. Manaster, Real and nominal efficient sets, Journal of Finance, 34 (1979), 93-102. doi: 10.1111/j.1540-6261.1979.tb02073.x. [14] C. Munk, C. Sorensen and T. N. Vinther, Dynamic asset allocation under mean-reverting returns, stochastic interest rates and inflation uncertainty, International Review of Economics and Finance, 13 (2004), 141-166. doi: 10.1016/j.iref.2003.08.001. [15] T. K. Siu, Long-term strategic asset allocation with inflation risk and regime switching, Quantitative Finance, 11 (2011), 1565-1580. doi: 10.1080/14697680903055588. [16] B. H. Solnik, Inflation and optimal portfolio choice, Journal of Financial and Quantitative analysis, 13 (1978), 903-925. doi: 10.2307/2330634. [17] A. Zhang, Stochastic Optimization in Finance and Life Insurance: Applications of the Martingale Method, Ph.D thesis, University of Kaiserslautern, 2008. [18] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33. doi: 10.1007/s002450010003. [19] K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 979-989. doi: 10.1016/j.automatica.2010.02.027.

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##### References:
 [1] A. Bensoussan, J. Keppo and S. P. Sethi, Optimal consumption and portfolio decisions with partially observed real prices, Mathematical Finance, 19 (2009), 215-236. doi: 10.1111/j.1467-9965.2009.00362.x. [2] M. J. Brennan and Y. Xia, Dynamic asset allocation under inflation, Journal of Finance, 57 (2002), 1201-1238. [3] J. Cea, Lectures on Optimization - Theory and Algorithm, Tata Institute of Fundamental Research, Bombay, 1978. [4] S. N. Chen and W. T. Moore, Uncertain inflation and optimal portfolio selection: A simplified approach, The Financial Review, 20 (1985), 343-352. doi: 10.1111/j.1540-6288.1985.tb00312.x. [5] C. H. Chiu and X. Y. Zhou, The premium of dynamic trading, Quantitative Finance, 11 (2011), 115-123. doi: 10.1080/14697681003685589. [6] W. S. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993. [7] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100. [8] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [9] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555. doi: 10.1137/S0363012900378504. [10] J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a value-at-risk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531-547. doi: 10.3934/jimo.2012.8.531. [11] J. Z. Liu and K. F. C. Yiu, Optimal stochastic differential games with VaR constraint, Discrete & Continuous Dynamical Systems - Series B, 18 (2013), 1889-1907. doi: 10.3934/dcdsb.2013.18.1889. [12] J. Z. Liu, K. F. C. Yiu and T. K. Siu, Optimal investment of an insurer with regime-switching and risk constraint, Scandinavian Actuarial Journal, 2014 (2014), 583-601. doi: 10.1080/03461238.2012.750621. [13] S. Manaster, Real and nominal efficient sets, Journal of Finance, 34 (1979), 93-102. doi: 10.1111/j.1540-6261.1979.tb02073.x. [14] C. Munk, C. Sorensen and T. N. Vinther, Dynamic asset allocation under mean-reverting returns, stochastic interest rates and inflation uncertainty, International Review of Economics and Finance, 13 (2004), 141-166. doi: 10.1016/j.iref.2003.08.001. [15] T. K. Siu, Long-term strategic asset allocation with inflation risk and regime switching, Quantitative Finance, 11 (2011), 1565-1580. doi: 10.1080/14697680903055588. [16] B. H. Solnik, Inflation and optimal portfolio choice, Journal of Financial and Quantitative analysis, 13 (1978), 903-925. doi: 10.2307/2330634. [17] A. Zhang, Stochastic Optimization in Finance and Life Insurance: Applications of the Martingale Method, Ph.D thesis, University of Kaiserslautern, 2008. [18] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33. doi: 10.1007/s002450010003. [19] K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 979-989. doi: 10.1016/j.automatica.2010.02.027.
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