Markowitz Portfolio Theory: Optimizing Portfolio Weights for Acceptable Baseline and Minimum Volatility
Modern portfolio theory can be defined as the mathematical framework for assembling a portfolio for assembling the portfolio of assets in such a manner that the anticipated return is maximised for a given level of risk (Barberis et al. 2015). The essay aims to provide a key insight where an asset risk and return must not be assessed by itself but by the process it contributes to the portfolios overall risk and return.
As defined in the Markowitz portfolio theory, it defines models as the rate of return on the assets in the form of random samples. In the later stages, the goal is to select the portfolio weighting factors optimally. In context to the Markowitz theory, the most favourable set of weights is one where the portfolio attains an acceptable baseline and the expected rate of return is with minimum volatility (Bodie 2013). The variance concerning the rate of return of an instrument is considered as the surrogate for its volatility.
Markowitz portfolio theory assumes that the investors are risk averse, which represents those two portfolios that provide the identical expected return the investors, will favour the less risky portfolio. Hence, investors will only take on increased risk given that compensation is offered with higher amount of risk (Zabarankin, Pavlikov and Uryasev 2014). Conversely, investors in desire of higher return should accept more amount of risk. The exact amount of trade off will be similar for all investors but efficient investors will assess the trade off in a different way depending upon the individual risk aversion characteristics.
The implication is that rational investors may not make an investment in a portfolio given that a second a portfolio is present with a more favourable risk-expected return portfolio (Fama and French, 2017). The efficient market line appears in the capital asset pricing model to portray the rates of return concerning the efficient portfolios subject to the risk level standard deviation for a market portfolio along with the risk free rate of return. The efficient market line is established by sketching down the tangent line from the interception point of the efficient frontier to the place where the anticipated return on holdings is equivalent to the risk-free rate of return (Ai, Croce and Li 2013). However, the capital market line is better than the efficient frontier since it takes into the considerations the infusion of risk free asset in the market portfolio.
Efficient Market Line and Capital Asset Pricing Model: Determining Fair Prices of Investments
The capital asset pricing model establishes that the market portfolio is the efficient front line. The capital asset pricing model determines the fair price of investment. Once the fair value is ascertained, it is compared with the market price. A stock can be considered as the good buy if the estimated price is higher than the market price (Kuehn, Simutin and Wang 2016). However, if the price is lower than the market price then the stock will not be considered as the good buy. The variance portfolio is where the Cov measures the correlation being the covariance between the securities. It is assumed that the random return of securities is normally distributed hence; the random vector of return is drawn from the multivariate normal distribution (Tsuji, 2017). As consequences to this, the anticipated return and variance of any security is finite. The monetary model of mean variance analysis is developed by the Harry Markowitz in 1952, which assumes that investors generally take in to the considerations the greater amount of return with lower risk. The model treats portfolio in the form of single point.
The model originally incorporated iso-mean and iso-variance acting as a surface for proving portfolio optimisation. Similarly, iso-mean surfaces generally form the shape that has the similar expected return (Fernandez 2017). In addition to this, Tobin’s Separation Theory postulates an investor can exercise control on the risk of a basket of risky investments either by borrowing at a risk free rate and leveraging the portfolio or alternatively lending the risk free rate and tempering risk. Therefore, large number of investors is risk unenthusiastic and the ultimate preference of most investors is to combine the risky basket of securities with the risk free bonds as this helps in lowering the downside risk of the portfolio. In common parlance, it is termed as stock or bond asset allocation decision (Prat 2016). The risk free bond allocation is generally termed as the short-term treasury securities or else it is expanded into the short-term investment grade corporate paper.
The classical Markowitz portfolio optimization theory illustrates that portfolio risk acts as the variance of the portfolio return and seeks allocation that minimizes the risk that is subjected to a target anticipated return. In actual practice, the anticipated returns and the covariance matrix of return is generally unknown and they are estimated from the historical data. This leads to the introduction of numerous problems, which results that Markowitz theory is impracticable in the applications of actual portfolio management (Kou, Peng and Zhong 2017). Over the long period, the financial data is characteristically non-stationary. This restricts the amount of information that can be used to derive the meaningful estimation of the mean and covariance of the asset return vector. In addition to this, sample covariance has several parameters and requires huge amount of estimation of information.
Sharpe Single Index Model: Lowering Input Requirements for Mean Variance Setting
The optimal theory of Markowitz generally leads to amplify large amount of errors of anticipation in certain directions. This stems from the fact that if the variance of the asset is underestimated in significant amount and hence, appears to be small and the optimal portfolio will assign a large amount of weight to it. Identically, a large amount of weight will be assigned if the mean return generated from an asset or a sub-portfolio appears to be large as consequences of being significantly overestimated (Prat 2016). Consequently, to this, the risk of estimated optimal portfolio is typically under predicted and its return is over predicted. It is noteworthy to denote that in several stocks practical circumstances the number of stocks is larger than the number of historical return per stock, leading to single sample covariance matrix.
Although the model of Markowitz is understood as the classic attempt to create a comprehensive technique to introduce the theory of diversification of investment in a portfolio in the form of risk lowering mechanism it also consists of several limitations. Sharpe further expanded the model of Markowitz when he introduced the theory of capital asset pricing model in order to solve the problem relating to the determination of correct arbitrage-free fair or equal price of an asset (Carpenter and Whitelaw 2016). Sharpe single index model proposes that relationship between each pair of securities can be indirectly measured by facilitating comparison of each security under a common factor market index performance, which is shared among all the securities. Therefore, the model can lower the burden of large input requirements with different computations in the mean variance setting. The model needs (3n+2) inputs of data and estimates the alpha and beta for each security. It also estimates the unsystematic risk for each of the security with anticipated return on the market index and estimates the variance of return on the market index. Because of such kind of simplicity, the single model index of Sharpe has attained much popularity in the area of investment finance in comparison to the Markowitz model.
One thing, which the newest investors understands or may have heard of, is the portfolio diversification. The portfolio diversification blends a variety of class of asset in order to reduce the exposure of risk. However, a well-diversified stock portfolio is just one component of putting together the best possible portfolio (Fernandez 2017). Diversification is not just about different types of stocks but among the different assets is how investors can mitigate risk. Even with the well diversified stocks portfolio a person is still exposed to the market risk that cannot be diversified away by making addition to the stocks. Diversification among the class of assets functions by spreading the investments among several assets having correlation with each other. This allows the investors to lower their volatility in portfolio because different assets move up and down in price at different times and at different rates. Therefore, diversification of the portfolio among different class of assets results in more consistency and helps in improving the overall portfolio performance.
The assumptions stated in the CAPM model are unrealistic as this makes the real world application difficult (Tsuji 2017). The unlimited risk free borrowing and the assumptions related to lending are unrealistic. However, such kinds of assumptions are dropped in the black CAPM. Neutral taxes do not exist since the investments do not face the same tax rates.
Conclusion:
To conclude with, the theoretical standpoint beta is not yet proven dead. In addition, the lack of sufficient market portfolio proxy must not lead to disregarding of the model. Hence, the present real world data must be used with caution. Conceivably a model that is more easily transferable in the real world may provide a better guide to the investors. The ultimate intentions of this model are to provide a guide for the investment through an asset requiring risk premium.
Reference list:
Ai, H., Croce, M.M. and Li, K., 2013. Toward a quantitative general equilibrium asset pricing model with intangible capital. Review of Financial Studies, 26(2), pp.491-530.
Barberis, N., Greenwood, R., Jin, L. and Shleifer, A., 2015. X-CAPM: An extrapolative capital asset pricing model. Journal of Financial Economics, 115(1), pp.1-24.
Bodie, Z., 2013. Investments. McGraw-Hill.
Carpenter, J.N. and Whitelaw, R.F., 2016. The Development of China’s Stock Market and Stakes for the Global Economy?.
Fama, E.F. and French, K.R., 2017. International tests of a five-factor asset pricing model. Journal of Financial Economics, 123(3), pp.441-463.
Fernandez, P., 2017. The Capital Asset Pricing Model. In Economic Ideas You Should Forget (pp. 47-49). Springer International Publishing.
Giannetti, M. and Wang, T.Y., 2016. Corporate scandals and household stock market participation. The Journal of Finance.
Kou, S., Peng, X. and Zhong, H., 2017. Asset pricing with spatial interaction. Management Science.
Kuehn, L.A., Simutin, M. and Wang, J.J., 2016. A labor capital asset pricing model.
Prat, R., 2016. Five Objections Against Using a Size Premium When Estimating the Required Return of Capital with the Capital Asset Pricing Model.
Tsuji, C., 2017. A Non-linear Estimation of the Capital Asset Pricing Model: The Case of Japanese Automobile Industry Firms. Applied Finance and Accounting, 3(2), pp.20-26.
Zabarankin, M., Pavlikov, K. and Uryasev, S., 2014. Capital asset pricing model (CAPM) with drawdown measure. European Journal of Operational Research, 234(2), pp.508-517.
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