# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021113
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## Capital asset pricing model under distribution uncertainty

 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China

* Corresponding author: Peibiao Zhao

Received  March 2021 Revised  April 2021 Early access June 2021

Fund Project: This research was funded by NNSF of China (no.11871275) and by Postgraduate Research & Practice Innovation Program of Jiangsu Province

In this paper, we investigate and demonstrate the capital asset pricing model (CAPM) based on distribution uncertainty (or ambiguity, defined as uncertainty about unknown probability).

We first achieve directly capital asset pricing model based on spectral risk measures (abbreviated as SCAPM) in the case of normal distributions; Then we can characterize SCAPM under the condition of uncertain distributions of returns by solving a robust optimal portfolio model based on spectral measures. Specifically, we do it in the following two folds: 1) Completing first the corresponding effective frontier fitting; 2) Getting the valuation of the market portfolio return $r_m$ and the risk parameters of $\beta_\phi$ in use of the kernel density estimation under the distribution uncertainty of returns.

Finally, by selecting 10 stocks from the constituent stocks of the HS300 Index, and comparing the valuation results from the SCAPM formula with the actual yield in the market, we verify the model proposed in the present paper is reasonable and effective.

Citation: Yu Chen, Zixian Cui, Shihan Di, Peibiao Zhao. Capital asset pricing model under distribution uncertainty. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021113
##### References:
 [1] C. Acerbi, Spectral measures of risk: A coherent representation of subjective risk aversion, Journal of Banking and Finance, 26 (2002), 1505-1518.  doi: 10.1016/S0378-4266(02)00281-9. [2] A. Adam, M. Houkari and J.-P. Laurent, Spectral risk measures and portfolio selection, Journal of Banking and Finance, 32 (2008), 1870-1882.  doi: 10.1016/j.jbankfin.2007.12.032. [3] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068. [4] L. Chen, S. He and S. Zhang, Tight bounds for some risk measures, with applications to robust portfolio selection, Oper. Res., 59 (2011), 847-865.  doi: 10.1287/opre.1110.0950. [5] Q. Chen and Z. Li, Research on CAPM under Chinese liquidity adjustment, Research in Quantitative Economics and Technical Economics, 25 (2008), 66-78. [6] J. Cotter and K. Dowd, Extreme spectral risk measures: an application to futures clearinghouse margin requirements, Journal of Banking and Finance, 30 (2006), 3469-3485. [7] Z. Cui, The Study of Asset Pricing with Spectral Risk Measures, Master's thesis, Nanjing University of Science and Technology, 2021. [8] E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Oper. Res., 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741. [9] X. Deng, Investment Portfolio Problems Based on Spectral Risk Measurement, Master's thesis, Nanjing University of Science and Technology, 2008. [10] X. Diao, B. Tong and C. Wu, Spectrum risk measurement based on EVT and its application in risk management, Journal of Systems Engineering, 30 (2015), 354-369. [11] L. EI Ghaoui, M. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Oper. Res., 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101. [12] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econ., 18 (1989), 141-153.  doi: 10.1016/0304-4068(89)90018-9. [13] Z. He, An empirical study of risk factors in Chinese stock market, Economic Review, 7 (2001), 81-85. [14] D. C. Indro, C. X. Jiang and M. Hu, Mutual fund performance, Journal of Business, 39 (2009), 119-138.  doi: 10.3905/joi.1998.408455. [15] Q. Jia and Z. Chen, An empirical analysis of the effectiveness of Chinese stock market, Financial Research, 7 (2003), 86-92. [16] Z. Kang, X. Li, Z. Li and S. Zhu, Data-driven robust mean-CVaR portfolio selection under distribution ambiguity, Quant. Finance, 19 (2019), 105-121.  doi: 10.1080/14697688.2018.1466057. [17] Y. M. Li, Closed-form solutions for worst-case law invariant risk measures with application to robust portfolio optimization, SSRN Electronic Journal, 66 (2016), 1-16. [18] X. P. Li and X. M. Li, Mean-M effective frontier of risky asset portfolio and an empirical analysis, Chinese Management Science, 13 (2005), 6-11. [19] X. Li, C. Li and J. Wang, Mean-CVaR effective frontier of risky asset portfolio (I), Journal of Management Engineering, 17 (2003), 29-33. [20] X. Li, C. Li and J. Wang, Mean-CVaR effective frontier of risky asset portfolio (II), Journal of Management Engineering, 19 (2005), 1-5. [21] Y. Lian, Super-High-Dimensional Feature Screening Method SEVIS and its Application, Doctoral dissertation, Anhui: University of Science and Technology of China, 2017. [22] J. Lintner, The valuation of risky assets and the selection of risky investments in stock portfolios and capital assets, Stochastic Optimization Models in Finance, 51 (1969), 220-221. [23] H. Markowitz, Portfolio selection, J. of Finance, 7 (1952), 77-91. [24] G. Ch. Pflug, Some remarks on the Value-at-Risk and the Conditional Value-at-Risk, Probabilistic Constrained Optimization, (2000), 272–281. doi: 10.1007/978-1-4757-3150-7_15. [25] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-Atrisk, Journal of Risk, 2 (2000), 21-41. [26] R. T. Rockafellar and S. Uryasev, Conditional Value-at-Risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471. [27] M. Salahi, F. Mehrdoust and F. Piri, CVaR robust mean-CVaR portfolio optimization, ISRN Appl. Math., 2013 (2013), Art. ID 570950, 9 pp. doi: 10.1155/2013/570950. [28] H. Scarf, A min-max solution of an inventory problem, Herbert Scarf's Contributions to Economics, Game Theory and Operations Research, 1 (2005), 19-27. [29] A. Shapiro, On kusuoka representation of law invariant risk measures, Math. Oper. Res., 38 (2013), 142-152.  doi: 10.1287/moor.1120.0563. [30] W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442. [31] G. Szeg, Measures of risk, Journal of Banking and Finance, 26 (2004), 1253-1272. [32] R. H. Tütüncü and M. Koenig, Robust asset allocation, Ann. Oper. Res., 132 (2004), 157-187.  doi: 10.1023/B:ANOR.0000045281.41041.ed. [33] Y. Xiao and E. A. Valdez, A Black-Litterman asset allocation model under elliptical distributions, Quant. Finance, 15 (2015), 509-519.  doi: 10.1080/14697688.2013.836283. [34] S. Zhang, M. Gao and B. Wu, Capital asset pricing model under insufficient diversification of investment: An empirical test based on Chinese A-share market, Management Review, 26 (2014), 24-37. [35] S. Zhu, An empirical test of capital asset pricing model CAPM in Chinese capital market, Statistics and Information Forum, 25 (2010), 95-99. [36] S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Res., 57 (2009), 1155-1168.  doi: 10.1287/opre.1080.0684.

show all references

##### References:
 [1] C. Acerbi, Spectral measures of risk: A coherent representation of subjective risk aversion, Journal of Banking and Finance, 26 (2002), 1505-1518.  doi: 10.1016/S0378-4266(02)00281-9. [2] A. Adam, M. Houkari and J.-P. Laurent, Spectral risk measures and portfolio selection, Journal of Banking and Finance, 32 (2008), 1870-1882.  doi: 10.1016/j.jbankfin.2007.12.032. [3] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068. [4] L. Chen, S. He and S. Zhang, Tight bounds for some risk measures, with applications to robust portfolio selection, Oper. Res., 59 (2011), 847-865.  doi: 10.1287/opre.1110.0950. [5] Q. Chen and Z. Li, Research on CAPM under Chinese liquidity adjustment, Research in Quantitative Economics and Technical Economics, 25 (2008), 66-78. [6] J. Cotter and K. Dowd, Extreme spectral risk measures: an application to futures clearinghouse margin requirements, Journal of Banking and Finance, 30 (2006), 3469-3485. [7] Z. Cui, The Study of Asset Pricing with Spectral Risk Measures, Master's thesis, Nanjing University of Science and Technology, 2021. [8] E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Oper. Res., 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741. [9] X. Deng, Investment Portfolio Problems Based on Spectral Risk Measurement, Master's thesis, Nanjing University of Science and Technology, 2008. [10] X. Diao, B. Tong and C. Wu, Spectrum risk measurement based on EVT and its application in risk management, Journal of Systems Engineering, 30 (2015), 354-369. [11] L. EI Ghaoui, M. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Oper. Res., 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101. [12] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econ., 18 (1989), 141-153.  doi: 10.1016/0304-4068(89)90018-9. [13] Z. He, An empirical study of risk factors in Chinese stock market, Economic Review, 7 (2001), 81-85. [14] D. C. Indro, C. X. Jiang and M. Hu, Mutual fund performance, Journal of Business, 39 (2009), 119-138.  doi: 10.3905/joi.1998.408455. [15] Q. Jia and Z. Chen, An empirical analysis of the effectiveness of Chinese stock market, Financial Research, 7 (2003), 86-92. [16] Z. Kang, X. Li, Z. Li and S. Zhu, Data-driven robust mean-CVaR portfolio selection under distribution ambiguity, Quant. Finance, 19 (2019), 105-121.  doi: 10.1080/14697688.2018.1466057. [17] Y. M. Li, Closed-form solutions for worst-case law invariant risk measures with application to robust portfolio optimization, SSRN Electronic Journal, 66 (2016), 1-16. [18] X. P. Li and X. M. Li, Mean-M effective frontier of risky asset portfolio and an empirical analysis, Chinese Management Science, 13 (2005), 6-11. [19] X. Li, C. Li and J. Wang, Mean-CVaR effective frontier of risky asset portfolio (I), Journal of Management Engineering, 17 (2003), 29-33. [20] X. Li, C. Li and J. Wang, Mean-CVaR effective frontier of risky asset portfolio (II), Journal of Management Engineering, 19 (2005), 1-5. [21] Y. Lian, Super-High-Dimensional Feature Screening Method SEVIS and its Application, Doctoral dissertation, Anhui: University of Science and Technology of China, 2017. [22] J. Lintner, The valuation of risky assets and the selection of risky investments in stock portfolios and capital assets, Stochastic Optimization Models in Finance, 51 (1969), 220-221. [23] H. Markowitz, Portfolio selection, J. of Finance, 7 (1952), 77-91. [24] G. Ch. Pflug, Some remarks on the Value-at-Risk and the Conditional Value-at-Risk, Probabilistic Constrained Optimization, (2000), 272–281. doi: 10.1007/978-1-4757-3150-7_15. [25] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-Atrisk, Journal of Risk, 2 (2000), 21-41. [26] R. T. Rockafellar and S. Uryasev, Conditional Value-at-Risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471. [27] M. Salahi, F. Mehrdoust and F. Piri, CVaR robust mean-CVaR portfolio optimization, ISRN Appl. Math., 2013 (2013), Art. ID 570950, 9 pp. doi: 10.1155/2013/570950. [28] H. Scarf, A min-max solution of an inventory problem, Herbert Scarf's Contributions to Economics, Game Theory and Operations Research, 1 (2005), 19-27. [29] A. Shapiro, On kusuoka representation of law invariant risk measures, Math. Oper. Res., 38 (2013), 142-152.  doi: 10.1287/moor.1120.0563. [30] W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442. [31] G. Szeg, Measures of risk, Journal of Banking and Finance, 26 (2004), 1253-1272. [32] R. H. Tütüncü and M. Koenig, Robust asset allocation, Ann. Oper. Res., 132 (2004), 157-187.  doi: 10.1023/B:ANOR.0000045281.41041.ed. [33] Y. Xiao and E. A. Valdez, A Black-Litterman asset allocation model under elliptical distributions, Quant. Finance, 15 (2015), 509-519.  doi: 10.1080/14697688.2013.836283. [34] S. Zhang, M. Gao and B. Wu, Capital asset pricing model under insufficient diversification of investment: An empirical test based on Chinese A-share market, Management Review, 26 (2014), 24-37. [35] S. Zhu, An empirical test of capital asset pricing model CAPM in Chinese capital market, Statistics and Information Forum, 25 (2010), 95-99. [36] S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Res., 57 (2009), 1155-1168.  doi: 10.1287/opre.1080.0684.
Closing price chart of Kweichow Moutai (600519) in 2019
Closing price chart of other nine stocks in 2019
Fluctuation trend on daily returns of 10 stocks in 2019
Correlation matrix figure of return on 10 stocks
Effective frontier of $(RMS-D_\varTheta(\tau_1, \tau_2))$ model
Market portfolio of $(RMS-D_\varTheta(\tau_1, \tau_2))$ model
Annualized return rate and covariance matrix of 10 stocks
 Stock code 600519 601888 600036 858 600030 600276 333 600887 601166 601318 $\mu_i$ 0.0887 0.0462 0.0391 0.0905 0.0473 0.0481 0.0441 0.0411 0.0337 0.0518 $\hat\sigma_{ij}$ 600519 601888 600036 858 600030 600276 333 600887 601166 601318 600519 0.0887 0.0462 0.0391 0.0905 0.0473 0.0481 0.0441 0.0411 0.0337 0.0518 601888 0.0462 0.0929 0.0274 0.0621 0.0344 0.0429 0.038 0.0355 0.0227 0.0373 600036 0.0391 0.0274 0.0578 0.0417 0.0475 0.0298 0.0387 0.0284 0.0442 0.049 000858 0.0905 0.0621 0.0417 0.1541 0.0685 0.0574 0.0661 0.0627 0.0364 0.0632 600030 0.0473 0.0344 0.0475 0.0685 0.139 0.0416 0.0557 0.0416 0.0497 0.0633 600276 0.0481 0.0429 0.0298 0.0574 0.0416 0.1192 0.0444 0.039 0.0236 0.041 000333 0.0441 0.038 0.0387 0.0661 0.0557 0.0444 0.0821 0.0436 0.0397 0.05 600887 0.0411 0.0355 0.0284 0.0627 0.0416 0.039 0.0436 0.0772 0.0241 0.0363 601166 0.0337 0.0227 0.0442 0.0364 0.0497 0.0236 0.0397 0.0241 0.0617 0.0458 601318 0.0518 0.0373 0.049 0.0632 0.0633 0.041 0.05 0.0363 0.0458 0.0682
 Stock code 600519 601888 600036 858 600030 600276 333 600887 601166 601318 $\mu_i$ 0.0887 0.0462 0.0391 0.0905 0.0473 0.0481 0.0441 0.0411 0.0337 0.0518 $\hat\sigma_{ij}$ 600519 601888 600036 858 600030 600276 333 600887 601166 601318 600519 0.0887 0.0462 0.0391 0.0905 0.0473 0.0481 0.0441 0.0411 0.0337 0.0518 601888 0.0462 0.0929 0.0274 0.0621 0.0344 0.0429 0.038 0.0355 0.0227 0.0373 600036 0.0391 0.0274 0.0578 0.0417 0.0475 0.0298 0.0387 0.0284 0.0442 0.049 000858 0.0905 0.0621 0.0417 0.1541 0.0685 0.0574 0.0661 0.0627 0.0364 0.0632 600030 0.0473 0.0344 0.0475 0.0685 0.139 0.0416 0.0557 0.0416 0.0497 0.0633 600276 0.0481 0.0429 0.0298 0.0574 0.0416 0.1192 0.0444 0.039 0.0236 0.041 000333 0.0441 0.038 0.0387 0.0661 0.0557 0.0444 0.0821 0.0436 0.0397 0.05 600887 0.0411 0.0355 0.0284 0.0627 0.0416 0.039 0.0436 0.0772 0.0241 0.0363 601166 0.0337 0.0227 0.0442 0.0364 0.0497 0.0236 0.0397 0.0241 0.0617 0.0458 601318 0.0518 0.0373 0.049 0.0632 0.0633 0.041 0.05 0.0363 0.0458 0.0682
Optimal investment ratio of market portfolio in model $(RMS-D_\varTheta(\tau_1, \tau_2))$
 Stock code 600519 601888 600036 858 600030 600276 333 600887 601166 601318 $x_i$ 0.1788 0.0656 0.1144 0.2543 0.0364 0.1042 0.097 0.0342 0.0505 0.0638
 Stock code 600519 601888 600036 858 600030 600276 333 600887 601166 601318 $x_i$ 0.1788 0.0656 0.1144 0.2543 0.0364 0.1042 0.097 0.0342 0.0505 0.0638
The spectrum-$\beta$ value $\beta_{\phi, i}$ corresponding to each stock
 Stock code 600519 601888 600036 858 600030 600276 333 600887 601166 601318 $\beta_{\phi, i}$ 1.0004 0.999 0.9974 1.0026 0.9987 1.001 0.9991 0.9968 0.9971 0.9989
 Stock code 600519 601888 600036 858 600030 600276 333 600887 601166 601318 $\beta_{\phi, i}$ 1.0004 0.999 0.9974 1.0026 0.9987 1.001 0.9991 0.9968 0.9971 0.9989
Results of absolute difference between SCAPM valuation and actual rate of return
 Stock code 600519 601888 600036 000858 600030 600276 000333 600887 601166 601318 $\Delta_i$ -21.18% 7.11% 4.25% -50.98% 3.23% -3.31% -0.09% 14.77% 17.22% 3.07%
 Stock code 600519 601888 600036 000858 600030 600276 000333 600887 601166 601318 $\Delta_i$ -21.18% 7.11% 4.25% -50.98% 3.23% -3.31% -0.09% 14.77% 17.22% 3.07%
Results of relative difference between SCAPM valuation and actual rate of return
 Stock code 600519 601888 600036 000858 600030 600276 000333 600887 601166 601318 $\delta_i$ -31.13% 17.90% 10.00% -52.05% 7.41% -6.60% -0.19% 46.25% 58.34% 7.01%
 Stock code 600519 601888 600036 000858 600030 600276 000333 600887 601166 601318 $\delta_i$ -31.13% 17.90% 10.00% -52.05% 7.41% -6.60% -0.19% 46.25% 58.34% 7.01%
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