Capital asset pricing model under distribution uncertainty

Yu Chen; Zixian Cui; Shihan Di; Peibiao Zhao

doi:10.3934/jimo.2021113

Article Contents

2022, Volume 18, Issue 5: 3283-3313. 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
^* Corresponding author: Peibiao Zhao

Received: February 28, 2021

Revised: March 31, 2021

Early access: June 2021

Published: November 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 91B24, 91B28; Secondary: 90C90.

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Figure 1. Closing price chart of Kweichow Moutai (600519) in 2019

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Figure 2. Closing price chart of other nine stocks in 2019

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Figure 3. Fluctuation trend on daily returns of 10 stocks in 2019

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Figure 4. Correlation matrix figure of return on 10 stocks

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Figure 5. Effective frontier of $ (RMS-D_\varTheta(\tau_1, \tau_2)) $ model

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Figure 6. Market portfolio of $ (RMS-D_\varTheta(\tau_1, \tau_2)) $ model

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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

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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

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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

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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%

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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%

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