Markowitz Theory

Introduction:

Markowitz theory was presented by harry Markowitz in 1952. It’s also known as Modern

portfolio theory.

Modern Portfolio theory:

Modern portfolio theory is a theory of finance that attempts to maximize portfolio

expected return for a given amount of portfolio risk, or equivalently minimize risk for a given

level of expected return, by carefully choosing the proportions of various assets.

Assumptions of portfolio theory

Markowitz theory based on following assumptions

Investors are rational:

Investors are rational and behave in a manner as to maximize their utility with a given level of

income or money.

Free access to Information:

Investors have free access to fair and correct information on the returns and risk. It means that

there will be no hidden information which effects the portfolio selection of investor.

Efficient Market

The markets are efficient and absorb the information quickly and perfectly.

Risk Averse:

Investors are risk averse and try to minimize the risk and maximize return.  A risk

averse investor is an investor who prefers lower returns with known risks rather than higher

returns with unknown risks. So Investors try to maximize return by minimizing risk.

Decision on expected return and Standard deviation:

Investors base decisions on expected returns and variance or standard deviation of these returns

from the mean. Investor decides for portfolio selection on basis of expected return and standard

deviation which is risk.

High return at given level of risk:

Investors prefer higher returns to lower returns for a given level of risk. So if for two portfolio

we have same level of risk then we will select the one with higher return and if we have same

return for two portfolio then we will selected the one with lower risk.

Mathematical Model:

Expected return:

Where   is the return on the portfolio,   is the return on asset i and   is the weighting of

component asset   (that is, the proportion of asset "i" in the portfolio).

Portfolio returns variance:

Where   is the correlation coefficient between the returns on assets i and j. alternatively the

expression can be written as:

,

Where   for i=j.

Portfolio returns volatility (standard deviation):

For a two asset portfolio:

Portfolio

return: 

Portfolio variance: 

For a three asset portfolio:

Portfolio return: 

Portfolio

variance: 

Efficient Frontier:

Markowitz gave idea of efficient frontier which is a set of optimal portfolios that offers the

highest expected return for a defined level of risk or the lowest risk for a given level of expected

return. Portfolios that lie below the efficient frontier are sub-optimal, because they do not

provide enough return for the level of risk. Portfolios that cluster to the right of the efficient

frontier are also sub-optimal, because they have a higher level of risk for the defined rate of

return. According to Markowitz, for every point on the efficient frontier, there is at least one

portfolio that can be constructed from all available investments that has the expected risk and

return corresponding to that point.

The relationship securities have with each other is an important part of the efficient frontier.

Some securities' prices move in the same direction under similar circumstances, while others

move in opposite directions. The more out of sync the securities in the portfolio are (that is, the

lower their covariance), the smaller the risk (standard deviation) of the portfolio that combines

them. The efficient frontier is curved because there is a diminishing marginal return to risk. Each

unit of risk added to a portfolio gains a smaller and smaller amount of return.