## Mean large number of securities can present

Mean variance analysis focuses on investors looking to maximise the risk-return trade-off i.e. given a certain level of risk (i.e. standard deviation of returns), investors will look to maximise their return for that level of risk. An investor who believes in mean-variance analysis will only choose to invest in portfolios that lie on the efficient frontier (Joshi & Paterson, 2013), a subset of the total opportunity set of portfolios available to the investor. Portfolios on this frontier are said to be efficient in the sense that: for any given return level, no other portfolio has a lower level of risk and for a given level of risk, no other portfolio has a greater return.

Conducting mean-variance analysis for a large number of securities can present a problem in the sense that we will be required to calculate a vast number of points in order to perform our analysis. While under traditional mean-variance analysis, asset returns and variances are known (Joshi & Paterson, 2013), this is not true in practice. In order to reduce the number of calculations we need to make, we use factor models, with one or more factors, each of which representing a certain market index, macroeconomic factors or various sectors.

A single factor model states that the return of a security is determined by one single factor representing the market, whereas a multi-factor model will state that the return is now dependent on multiple indices. In both models, there is an error term that is uncorrelated for different assets. The simplicity in such models arises from the fact that all the securities in a given portfolio will owe their correlation to the factors that they all have in common (Joshi & Paterson, 2013).