extent in which the outcomes differ from the average

 extent in which the outcomes differ from the average

Since modern investors have different types of investment, it is important to make measurements of the risk and the expected return of different types of portfolio. In 1952, the portfolio effective model and the mean variance analysis method has been proposed by Markowitz. Moreover, DeMiguel. et al. (2007) pointed out that although estimated risk was existed, the mean-variance optimal portfolio model could have better performance and it should serve at least as a first obvious benchmark. It is effectively to build asset portfolio for the investors through the modern portfolio theory (MPT). If the volatility of return rate between assets is inconsistent, the stock investors could analyze and calculate the investment portfolio to increase the expect returns at the lowest risk, so as to achieve the purpose of reducing risk and obtaining return.

First, the expected return is. It is essential to understand the extent in which the outcomes differ from the average. Secondly, the formula for calculating variance of the return on the asset becomes. Next, the expected rate of return of an asset portfolio is the average of the expected rate of return of an individual asset, . Then, we obtained the correlation coefficient by standardizing the covariance, and the value of the correlation coefficient fluctuated between -1 and +1. . Next, comparing the variance and covariance parts of the expression for the variance of a portfolio leads to:

or.

Thus the efficient set is reached at by determining that portfolio which has the greatest ratio of excess to standard deviation.

and

When we want to have a portfolio with the same risk and more return or the same return and less risk than a specific point, we could use the efficient frontier. The study of Markowitz (1952) stated that the efficient frontier stretches from the lowest variance portfolio to the highest ratio of excess return portfolio.

However, with the development of this theory in the last 50 years, there are different drawbacks were founded by the mathematicians. Firstly, a number of assumptions need to be met when using this theory to calculate the expected risks and returns based on profit-avoiding psychological method, such as investors are rational and there has the exact information in the market. Secondly, another disadvantage has been provided by Buffett which is inaccurately for the investors when they use historical prices to calculate the volatility. Frazzini et al. (2017) pointed out that Markowitz define risk as the scale of price instability when he used the formula to calculate the variance (Frazzini et al. 2018).Besides, DeMiguel et al. (2018) thought it is too simple to use only two factors(mean and variance) and he added another two factors which are transaction costs, and out-of-sample performance in a specific portfolio perspective .

In 1984, Willian Sharpe proposed a simplified vision. The single-index models calculated the return on a stock, and it is for all stocks I = (1, …, N). In the following part it should derive the standard deviation, expected return, and covariance when the single-index model is applied to represent the joint movement of securities. There were three results. Firstly, the mean return, . Secondly, the variance of a security’s return, . Thirdly, the covariance of returns between securities i and j, .Consequently, the expected return and risk can be estimated for any portfolio, if we have an estimate of , an estimate of , an estimate of , and an estimate of both the expected return and the variance for the market. These are 3N+1 estimats.

In conclusion, the Markowitz theory is available only if these strict assumptions are implemented, and the calculations of these historical data are inaccurate in predicting the future. However, a single index model is simpler and more applicable. The hypothesis of a single index model explains the major forms of stock volatility. Although there are some problems with Markowitz theory, it is still vital because it is the cornerstone of a single index model. The introduction of the above theory provides sufficient evidences for the following analysis.