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 & 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.  Google Scholar

[2]

A. AdamM. 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.  Google Scholar

[3]

P. ArtznerF. DelbaenJ.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.  Google Scholar

[4]

L. ChenS. 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.  Google Scholar

[5]

Q. Chen and Z. Li, Research on CAPM under Chinese liquidity adjustment, Research in Quantitative Economics and Technical Economics, 25 (2008), 66-78.   Google Scholar

[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.   Google Scholar

[7]

Z. Cui, The Study of Asset Pricing with Spectral Risk Measures, Master's thesis, Nanjing University of Science and Technology, 2021. Google Scholar

[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.  Google Scholar

[9]

X. Deng, Investment Portfolio Problems Based on Spectral Risk Measurement, Master's thesis, Nanjing University of Science and Technology, 2008. Google Scholar

[10]

X. DiaoB. Tong and C. Wu, Spectrum risk measurement based on EVT and its application in risk management, Journal of Systems Engineering, 30 (2015), 354-369.   Google Scholar

[11]

L. EI GhaouiM. 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.  Google Scholar

[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.  Google Scholar

[13]

Z. He, An empirical study of risk factors in Chinese stock market, Economic Review, 7 (2001), 81-85.   Google Scholar

[14]

D. C. IndroC. X. Jiang and M. Hu, Mutual fund performance, Journal of Business, 39 (2009), 119-138.  doi: 10.3905/joi.1998.408455.  Google Scholar

[15]

Q. Jia and Z. Chen, An empirical analysis of the effectiveness of Chinese stock market, Financial Research, 7 (2003), 86-92.   Google Scholar

[16]

Z. KangX. LiZ. 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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[19]

X. LiC. Li and J. Wang, Mean-CVaR effective frontier of risky asset portfolio (I), Journal of Management Engineering, 17 (2003), 29-33.   Google Scholar

[20]

X. LiC. Li and J. Wang, Mean-CVaR effective frontier of risky asset portfolio (II), Journal of Management Engineering, 19 (2005), 1-5.   Google Scholar

[21]

Y. Lian, Super-High-Dimensional Feature Screening Method SEVIS and its Application, Doctoral dissertation, Anhui: University of Science and Technology of China, 2017. Google Scholar

[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.   Google Scholar

[23]

H. Markowitz, Portfolio selection, J. of Finance, 7 (1952), 77-91.   Google Scholar

[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.  Google Scholar

[25]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-Atrisk, Journal of Risk, 2 (2000), 21-41.   Google Scholar

[26]

R. T. Rockafellar and S. Uryasev, Conditional Value-at-Risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[29]

A. Shapiro, On kusuoka representation of law invariant risk measures, Math. Oper. Res., 38 (2013), 142-152.  doi: 10.1287/moor.1120.0563.  Google Scholar

[30]

W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442.   Google Scholar

[31]

G. Szeg, Measures of risk, Journal of Banking and Finance, 26 (2004), 1253-1272.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

S. ZhangM. 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.   Google Scholar

[35]

S. Zhu, An empirical test of capital asset pricing model CAPM in Chinese capital market, Statistics and Information Forum, 25 (2010), 95-99.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. AdamM. 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.  Google Scholar

[3]

P. ArtznerF. DelbaenJ.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.  Google Scholar

[4]

L. ChenS. 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.  Google Scholar

[5]

Q. Chen and Z. Li, Research on CAPM under Chinese liquidity adjustment, Research in Quantitative Economics and Technical Economics, 25 (2008), 66-78.   Google Scholar

[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.   Google Scholar

[7]

Z. Cui, The Study of Asset Pricing with Spectral Risk Measures, Master's thesis, Nanjing University of Science and Technology, 2021. Google Scholar

[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.  Google Scholar

[9]

X. Deng, Investment Portfolio Problems Based on Spectral Risk Measurement, Master's thesis, Nanjing University of Science and Technology, 2008. Google Scholar

[10]

X. DiaoB. Tong and C. Wu, Spectrum risk measurement based on EVT and its application in risk management, Journal of Systems Engineering, 30 (2015), 354-369.   Google Scholar

[11]

L. EI GhaouiM. 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.  Google Scholar

[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.  Google Scholar

[13]

Z. He, An empirical study of risk factors in Chinese stock market, Economic Review, 7 (2001), 81-85.   Google Scholar

[14]

D. C. IndroC. X. Jiang and M. Hu, Mutual fund performance, Journal of Business, 39 (2009), 119-138.  doi: 10.3905/joi.1998.408455.  Google Scholar

[15]

Q. Jia and Z. Chen, An empirical analysis of the effectiveness of Chinese stock market, Financial Research, 7 (2003), 86-92.   Google Scholar

[16]

Z. KangX. LiZ. 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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[19]

X. LiC. Li and J. Wang, Mean-CVaR effective frontier of risky asset portfolio (I), Journal of Management Engineering, 17 (2003), 29-33.   Google Scholar

[20]

X. LiC. Li and J. Wang, Mean-CVaR effective frontier of risky asset portfolio (II), Journal of Management Engineering, 19 (2005), 1-5.   Google Scholar

[21]

Y. Lian, Super-High-Dimensional Feature Screening Method SEVIS and its Application, Doctoral dissertation, Anhui: University of Science and Technology of China, 2017. Google Scholar

[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.   Google Scholar

[23]

H. Markowitz, Portfolio selection, J. of Finance, 7 (1952), 77-91.   Google Scholar

[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.  Google Scholar

[25]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-Atrisk, Journal of Risk, 2 (2000), 21-41.   Google Scholar

[26]

R. T. Rockafellar and S. Uryasev, Conditional Value-at-Risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[29]

A. Shapiro, On kusuoka representation of law invariant risk measures, Math. Oper. Res., 38 (2013), 142-152.  doi: 10.1287/moor.1120.0563.  Google Scholar

[30]

W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442.   Google Scholar

[31]

G. Szeg, Measures of risk, Journal of Banking and Finance, 26 (2004), 1253-1272.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

S. ZhangM. 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.   Google Scholar

[35]

S. Zhu, An empirical test of capital asset pricing model CAPM in Chinese capital market, Statistics and Information Forum, 25 (2010), 95-99.   Google Scholar

[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.  Google Scholar

Figure 1.  Closing price chart of Kweichow Moutai (600519) in 2019
Figure 2.  Closing price chart of other nine stocks in 2019
Figure 3.  Fluctuation trend on daily returns of 10 stocks in 2019
Figure 4.  Correlation matrix figure of return on 10 stocks
Figure 5.  Effective frontier of $ (RMS-D_\varTheta(\tau_1, \tau_2)) $ model
Figure 6.  Market portfolio of $ (RMS-D_\varTheta(\tau_1, \tau_2)) $ model
Table 1.  Annualized return rate and covariance matrix of 10 stocks
Stock code 600519 601888 600036 000858 600030 600276 000333 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 000858 600030 600276 000333 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 0.0380 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 0.0490
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 0.1390 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 0.0390 0.0236 0.0410
000333 0.0441 0.0380 0.0387 0.0661 0.0557 0.0444 0.0821 0.0436 0.0397 0.0500
600887 0.0411 0.0355 0.0284 0.0627 0.0416 0.0390 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 0.0490 0.0632 0.0633 0.0410 0.0500 0.0363 0.0458 0.0682
Stock code 600519 601888 600036 000858 600030 600276 000333 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 000858 600030 600276 000333 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 0.0380 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 0.0490
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 0.1390 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 0.0390 0.0236 0.0410
000333 0.0441 0.0380 0.0387 0.0661 0.0557 0.0444 0.0821 0.0436 0.0397 0.0500
600887 0.0411 0.0355 0.0284 0.0627 0.0416 0.0390 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 0.0490 0.0632 0.0633 0.0410 0.0500 0.0363 0.0458 0.0682
Table 2.  Optimal investment ratio of market portfolio in model $ (RMS-D_\varTheta(\tau_1, \tau_2)) $
Stock code 600519 601888 600036 000858 600030 600276 000333 600887 601166 601318
$ x_i $ 0.1788 0.0656 0.1144 0.2543 0.0364 0.1042 0.0970 0.0342 0.0505 0.0638
Stock code 600519 601888 600036 000858 600030 600276 000333 600887 601166 601318
$ x_i $ 0.1788 0.0656 0.1144 0.2543 0.0364 0.1042 0.0970 0.0342 0.0505 0.0638
Table 3.  The spectrum-$ \beta $ value $ \beta_{\phi, i} $ corresponding to each stock
Stock code 600519 601888 600036 000858 600030 600276 000333 600887 601166 601318
$ \beta_{\phi, i} $ 1.0004 0.9990 0.9974 1.0026 0.9987 1.0010 0.9991 0.9968 0.9971 0.9989
Stock code 600519 601888 600036 000858 600030 600276 000333 600887 601166 601318
$ \beta_{\phi, i} $ 1.0004 0.9990 0.9974 1.0026 0.9987 1.0010 0.9991 0.9968 0.9971 0.9989
Table 4.  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%
Table 5.  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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