The Integration of Multi-Factor Model of Capital Asset Pricing and Penalty Function for Stock Return Evaluation

Document Type : Research Paper


1 Faculty of Humanities, Islamic Azad University, Qom, Iran

2 Department of Management and Accounting, Qom Branch, Islamic Azad University, Qom, Iran

3 Department of Management and Accounting, Kashan Branch, Islamic Azad University, Kashan, Iran



One of the main concerns of investors is the evaluation of the return on investment, which is conducted using various models such as the CAPM (single-factor model), Fama-French three/five-factor models, and Roy and Shijin’s six-factor model and other models known as multi-factor models. Despite the widespread use of these models, their major drawbacks include sensitivity to unexpected changes, sudden shocks, high turbulence of price bubble, and so on. To eliminate such negatives, the multi-factor model using the penalty function method is used, in which, instead of averaging, the optimization and avoidance of the effects of abnormal changes and other factors affecting the capital market are considered. In order to evaluate stock returns, it is possible to select effective factors, to simulate and develop a model appropriate to the conditions governing the capital market in Iran. In the present study, by forming portfolios of investments and identifying and refining effective factors, the classification and estimation of the hybrid model of penalty and multi-factor (P & PCA) functions were performed based on the functional data during 2007-2017. The results of this study indicated that the extensive use of the simulation algorithm for the penalty function in the form of P & PCA estimation method improves the efficiency of multi-factor methods in stock return evaluation, and that the use of the hybrid algorithm of penalty and multi-factor functions, compared to the exclusive use of multi-factor models, brings a higher accuracy in estimating stock returns.


Main Subjects

[1] Ng, L., Tests of the CAPM with Time-Varying Covariances: A Multivariate GARCH Approach, The Journal of Finance, 2009, 46(4), P.1507-1521, Doi:10.1111/j.1540-6261. 1991.tb04628.x.
[2] Keller, C., Kunzel, P., Souto, M., Measuring Sovereign Risk in Turkey: An Application of the Contingent Claims Approach, IMF Working Paper, 2007, WP/07/233, International Monetary Fund,  Doi:10.5089/9781451 867978.001
[3] Chou, R. Y., Yen, T. J., Yen, Y. M., Risk evaluations with robust approximate factor models, Journal of Banking and Finance, 2017, 82, P.244-264,  Doi:10.1016/j.jbankfin.2016.05.008.
[4] Bai, J., Ng, S., Determining the number of factors in approximate factor models, Econometrica, 2002, 70, P.191–22,  Doi:10.1111/1468-0262.00273.
[5] Alaleh, N. Tamimi, M., Nematpour Dezfuli, A. M., Explanation of Returns Changes in CAPM, TFPM, FFPM in Tehran Stock Exchange, 2013, P. 115-128, Doi:10.2139/ssrn.2252016, (in Persian).
[6] Ebrahimi, M. and Saeidi, A., The effect of accounting variables and company features on stock prices in Tehran stock exchange, Accounting and Auditing, 2010, 17(62), P.1-16, (in Persian).
[7] Markowitz, H. M., Portfolio Selection, Journal of Finance, 1952, 7(1), P.77-91,  Doi:10.1111/j.1540-6261. 1952.tb01525.x.
[8] Markowitz, H. M., Portfolio Selection: Efficient Diversification of Investment, John Wiley and Sons, Inc., 605 Third Avenue, New York, US, 1959,  Doi:10.2307/3006625.
[9] Sharpe, W.F., Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance, 1964, 19(3), P.425-442,  Doi:10.1111/j.1540-6261.1964.tb02865.x.
[10] Lintner, J., The valuation of risk assets and selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 1965, 47, P.13-37,  Doi:10.2307/1924119.
[11] Mossin, J., Equilibrium in a Capital Asset Market, Econometrica, 1966, 34(4),P.768-83,  Doi:10.2307/1910 098.
[12] Black, F., Capital Market Equilibrium with Restricted Borrowing, Journal of Business, 1972, 45(3), P.44-54,  Doi:10.1086/295472.
[13] Mayers, D., Nonmarketable Assets and Capital Market Equilibrium under Uncertainty, Studies in the Theory of Capital Markets, 1972, 3, P.223-248, Praeger Publishers, N.Y., (Cited in Jagannathan, R. and Z. Wang. 1996,  Doi:10.1086/295528.
[14] Breeden, D. T., An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities, Journal of Financial Economics, 1979, 7(3),P. 265-96,  Doi:10.1016/0304-405x(79)90016-3.
[15] Solnik, B. H., An Equilibrium Model of the International Capital Market, Journal of Economic Theory, 1974, 8(4), P.500-24,  Doi:10.1016/0022-0531(74)90024-6.
[16] Adler, M., Dumas, B., International Portfolio Choices and Corporation Finance: A Synthesis, The Journal of Finance, 1983, 38(3),P. 925-984,  Doi:10.1111/j.1540-6261.1983.tb02511.x.
[17] Ross, S., Arbitrage Theory of Capital Asset Pricing, Journal of Economic Theory, 1976, 13(3), 341360,  Doi:10.1016/0022-0531(76)90046-6.
[18] Fama, E. F., French, K. R., Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 1993, 33(1), P.3-56,  Doi:10.1016/0304-405x(93)90023-5.
[19] Hogan, W., Warren, J., Toward the Development of an Equilibrium Capital Market Model Based on Semivariance, Journal of Financial and Quantitative Analysis, 1974, 9(1),P. 1-11,  Doi:10.2307/2329964.
[20] Bawa, V., Lindenberg, E., Capital Market Equilibrium in a mean, Lower Partial Moment Framework, Journal of Financial Economics, 1977, 5(2), P.189-200,  Doi:10.1016/0304-405x(77)90017-4.
[21] Harlow, W.V., Rao, R. K. S., Asset Pricing in a Generalized Mean-Lower Partial Moment Framework: Theory and Evidence, Journal of Financial and Quantitative Analysis, 1989, 24(3), P.285-311,  Doi:10.2307/23 30813.
[22] Rubinstein, M. E., The Fundamental Theorem of Parameter-Preference Security Valuation, Journal of Financial and Quantitative Analysis, 1973, 8(1),P. 61-69,  Doi:10.2307/2329748.
[23] Kraus, A., Litzenberger, R.H., Skewness preference and the valuation of risk assets, Journal of Finance, 1976, 31(4),P.1085-1100,  Doi:10.2307/2326275.
[24] Fang, H., Lai, T., Co-Kurtosis and Capital Asset Pricing, The Financial Review, 1997, 32(2), P.293-307,  Doi:10.1111/j.1540-6288.1997.tb00426.x.
[25] Dittmar. R. R., Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross-Section of Equity Returns, Working Paper, University of North Carolina at Chapel Hill, 2002,  Doi:10.1111/1540-6261.0 0425.
[26] Merton, R. C., An Intertemporal Capital Assets Pricing Model, Econometrica, 1973, 41(5),P.867-87,  Doi:10.2307/1913811
[27] Lucas, R., Asset Prices in an Exchange Economy, Econometrica, 1978, 46(6), P.1429-1445,  Doi:10.2307/1 913837.
[28] Brock, W. A., Asset Prices in a Production Economy, Social Science Working Paper 275, California Institute of Technology, 1979.
[29] Cochrane, J. H., Production-Based Asset Pricing and the Link Between Stock Returns and Economic Fluctuations, The Journal of Finance, 1991, 46(1), P.209-37,  Doi:10.2307/2328694.
[30] Acharya, V.V., Pedersen, L.H., Asset Pricing with Liquidity Risk. Journal of Financial Economics, 2005, 77 (2), P.375–410,  Doi:10.1016/j.jfineco.2004.06.007.
[31] Jagannathan, R., Wang, Z., The Conditional CAPM and the Cross-Section of Expected Returns, Journal of Finance, 1996, 51(1), P.3-53,  Doi:10.1111/j.1540-6261.1996.tb05201.x.
[32] Carhart, M. M., On persistence in mutual fund performance, The Journal of Finance, 1997, 52(1), P.57-82,  Doi:10.2307/2329556.
[33] Hu, G. X., J. Pan, and J.Wang, Noise as information for illiquidity, Working Paper, 2012,  Doi:10.2139 /ssrn.1689067.
[34] Fama, E. F., K. R. French, A Five-Factor Asset Pricing Model, Working Paper, 2013, Available at SSRN 2287202 (2013).
[35] Racicot, F.-E., and Rentz, W. F., Testing Fama-French’s new five-factor asset pricing model: Evidence from robust instruments, Applied Economics Letters, 2017, 23(6), P.1-5.  Doi:10.1080/13504851.2015.1080798
[36] Roy, R., Shijin, S., Dissecting anomalies and dynamic human capital: The global evidence, Borsa Istanbul Review, 2018, 18(1), P.1-32,  Doi:10.1016/j.bir.2017.08.005.
[37] Shiller, R. J., Human Behavior and the Efficiency of the Financial System, John B., Taylor and Michael Woodford, Editors, Handbook of Macroeconomics, 1998,  Doi:10.1016/s1574-0048(99)10033-8.
[38] Mehra, R., Prescott, E.C., The Equity Premium: A Puzzle, Journal of Monetary Economics, 1985, 15(2), P.145-165,  Doi:10.1016/0304-3932(85)90061-3.
[39] Fama, F., French, K.R., Permanent and Temporary Components of Stock Prices, The Journal of Political Economy, 1988, 96(2), P. 246-273,  Doi:10.1086/261535.
[40] Shiller, R. J., From Efficient Market Theory to Behavioral Finance, The Journal of Economic Perspectives, 2003, 17(1), P. 83-104,  Doi:10.1257/089533003321164967.
[41] Daniel, K. D., Hirshleifer, D., Subrahmanyam, A., Investor Psychology and Security Market Under- and Overreactions, Journal of Finance, 1998, 53(6), P.1839-1885,  Doi:10.1111/0022-1082.00077.
[42] De Bondt, W. F. M., Thaler, R., Does the Stock Market Overreact?, The Journal of Finance, 1985, 40(3), P.793-805,  Doi:10.1111/j.1540-6261.1985.tb05004.x.
[43] Statman, M., Lottery Players/Stock Traders, Financial Analysts Journal, 2002, 58(1), P.14-21, Doi:10.2469 /faj.v58.n1.2506. 
[44] Barberies, N., Thaler R., A Survey OF Behavioral Finance, In G.M., 2003, Doi:10.3386/w9222.  
[45] Tversky, A., Kahneman, D., Judgement under Uncertainity: Heuristics and Biases, Science, 1974, 185 (4157), P.1124-1131,  Doi:10.1126/science.185.4157.1124.
[46] Benartzi, S., Thaler, R.H., Naive Diversification Strategies in Defined Contribution Savings Plan, American Economic Review, 2001, 91(1), P.79-98,  Doi:10.1257/aer.91.1.79.
[47] Saunders, E. M., Stock Prices and Wall Street Weather, American Economic Review, 1993, 83(5), P.1337-1345.
[48] Fama, E.F., French, K.R., The Cross-Section of Expected Returns, Journal of Finance, 1992, 47(2), P.427-465,  Doi:10.2307/2329112.
[49] Fama, E. F., French, K. R., Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics, 1993, 33(1), P.3-56,  Doi:10.1016/0304-405x(93)90023-5.
[50] Jegadeesh, N., Titman, S., Returns to buying winners and selling losers: implications for stock market efficiency, Journal of Finance, 1993, 48, P. 65-91,  Doi:10.1111/j.1540-6261.1993.tb04702.x.
[51] Black, F., Scholes M., The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 1973, 81, P. 637–654,  Doi:10.1086/260062.
[52] Fama, E. F., French, K. R., Incremental variables and the investment opportunity set, Journal of Financial Economics, 2015, 117(3), P.470-488,  Doi:10.1016/j.jfineco.2015.05.001.
[53] Chiah, M., Chai, D., Zhong, A., Li, S., A better model? An empirical investigation of the Fama-French five-factor model in Australia, International Review of Finance, 2016, 16(4), P.595-638,  Doi:10.1111/ irfi.1209 9.
[54] Kubota, K., Takehara, H., Does the Fama and French five-factor model work well in Japan?, International Review of Finance, 2018, 18(1), P.137-146,  Doi:10.1111/irfi.12126.
[55] Campbell, J. Y., Understanding risk and return, Journal of Political Economy, 1996, 104(2), P.298-345,  Doi:10.1086/262026.
[56] Kim, D., Kim, T. S., Min, B. K., Future labor income growth and the cross-section of equity returns. Journal of Banking and Finance, 2011, 35(1), P.67-81,  Doi:10.1016/j.jbankfin.2010.07.014.
[57] Kuehn, L.-A., Simutin, M., Wang, J. J., A labor capital asset pricing model, The Journal of Finance, 2017, 72(5), P.2131-2178,  Doi:10. 1111/jofi.12504.
[58] Racicot, F.-E., and Rentz, W. F., Testing FamaeFrench’s new five-factor asset pricing model: Evidence from robust instruments, Applied Economics Letters, 2017, 23(6), P.1-5,  Doi:10.1080/13504851.2015.1080798
[59] Khani Farahani, A., Sheshmani, M., Mohades, A., Studying the Expected Returns Based on Carhart Model Com-pared to CAPM Model and Implicit Capital Cost Model Based on Cash and Capital Flow of Growth and Value Stocks, Advances in Mathematical Finance and Applications, 2017, 2(4), P. 61-79. Doi:10.22034/AMFA. 2017.536267.
[60] Matevž, S., Lončarski, I., Multi-factor asset pricing models: Factor construction choices and the revisit of pricing factors, Journal of International Financial Markets, Institutions and Money, 2018, 55, P.65-80,  Doi:10. 1016/j.intfin.2018.02.006.
[61] Cao, V. N., Gray, P., Zhong, A., Investment-related anomalies in Australia: Evidence and explanations, International Review of Financial Analysis, 2019, 61(C) P.97-109. Doi:10.1016/j.irfa.2018.10.007.
[62] Ramzi Radchobeh, Z., Rezazadeh, J., Kazemi, H., Ambiguity Theory and Asset Pricing: Empirical Evidence from Tehran Stock Exchange, Advances in Mathematical Finance and Applications, 3(4), P.101-114, 2018, Doi: 10.22034/AMFA.2019.579558.1143.
[63] Ando, T., Bai, J., Asset pricing with a general multifactor structure, Journal of Financial Econometrics, 2014, 13(3), P.556-604,  Doi:10.1093/jjfinec/nbu026.
[64] Takács, G., Gas Lift Manual, Pennwell Books, 2005.
[65] Francis, B., Low-Dimensional Geometry: from Euclidean Surfaces to Hyperbolic Knots, AMS Bookstore, 2009,  Doi:10.1090/stml/049.
[66] Stock, J.H., Watson, M.W., Forecasting using principal components from a large number of predictors, Journal of the American Statistical Association, 2002, 97, P.1167–1179,  Doi:10.1198/016214502388618960.
[67] Donoho, D.L., Jonhstone, I.M., Ideal spatial adaptation by wavelet shrinkage, Biometrika, 1994, 81, P. 425–455,  Doi:10.1093/biomet/81.3.425.
[68] Bai, J., Panel data models with interactive fixed effects, Econometrica, 2009, 77, P.1229–1279, Doi:10. 398 2/ecta6135.
[69] Morovati Sharifabadi, A., Golshan, M., Prediction of stock returns using artificial neural networks and multivariate regression (Case study: Companies accepted in Tehran Stock Exchange), Second National Conference on Accounting, Financial Management and Investment, Gorgan, Scientific and Professional Association of Golestan Managers and Accountants, 2013, (in Persian).
[70] Kashanipour, M., Salehnejad, H., Rezaei, A., Yousefi Manesh, D., The effect of variations of the stock price fluctuation rate on stock returns and trade volume of companies accepted in Tehran Stock Exchange, Journal of Accounting Knowledge, 2013, 2(12), P.155-174, (in Persian).
[71] Mashayekhi, B., Panahi, D., The Investigation of Relationship between Financial Ratios and Stock Return: Income Smoother Firms Versus Non-Income Smoother ones, Financial Research Journal, 2008, 9(3), (in Persian).
[72] Makian, N., Mousavi, F. S., Prediction of stock prices in Pars Oil Products Company using Neural Networks and Regression Method, Economic Modeling, 2012, 6(18), P.105-121, (in Persian).
[73] Monajemi, A. H., Abzari, M. and Rayati, A. R., Prediction of stock prices in the stock market using fuzzy neural network and genetic algorithms and its comparison with Artificial Neural Network, Quantitative Economics Quarterly, 2014, 6(3), 26-41, (in Persian).
[74] Mehrara, A., Atf, Z., Asgari, Z., Data mining techniques in stock price prediction, National Conference on Accounting, Finance, and Investment Management, Gorgan, Golestan University of Applied Sciences, 2012, (in Persian).