Quantitative Theory Of Portfolio Theory

Portfolio theory deals with the problem of constructing a collection of assets that reflect the individual needs. When a portfolio is constructed a variety of parameters can be taken into account, such as value, average, the riskiness of the asset.
The financial objectives of the investor determines what types of assets to be used. In this paper a quantitative approach of choosing the portfolio will be discussed.
Modern Portfolio Theory as introduced by Markowitz (1952) frames the time dimension of investing as a single period over which the parameters of the probability distribution of asset returns are both known with certainty and are unchanging. However, neither assumption hold in real life.
The underlying assumption that legitimizes

Markowitz suggested that for each asset that someone wants to include in the portfolio, the asset return and its risk should be measured and he proposed to use mean return for asset return and risk by asset’s variance of return. Apart of variance and return, Markowitz model also takes into account the covariance between all assets. For this reason, the Markowitz framework is commonly referred to as mean-variance portfolio analysis.

Much of the focus has been on mathematical theories behind uncertainty set construction and reformulations resulting in optimization problems that can be solved efficiently; and, as a result, there are many formulations that can be used to build robust equity portfolios.
Since 1990 there have been numerous extensions of the Markowitz’s mean-variance model. (Woo Chang Kim, 2015)

Risk can be described by its two components : Systematic risk, that is common to many assets and is non-diversifiable Non-systematic risk, that is specific to individual assets.
Forming a portfolio of individual assets, the non-systematic risk can be eliminated.
An example of a simple portfolio is a 2 asset portfolio, where the assets can be either stocks, bonds, treasury bills and so son.
The return on the portfolio with two assets is:

The variance: Standard