Markowitz Portfolio Theory deals with the **risk and return** of portfolio of investments. Before Markowitz portfolio theory, risk & return concepts are handled by the investors loosely. The investors knew that diversification is best for making investments but **Markowitz** formally built the quantified concept of diversification. He pointed out the way in which the risk of portfolio to an investor is reduced through diversification. The particular measure of portfolio risk was first developed by the Markowitz and the expected risk & return for portfolio are derived on the basis of the covariance relationship.

It is clear fact that the risk of portfolio is not equal to the weighted average of single security risks. According to Markowitz, interrelationships among individual portfolio returns are considered for the purposes of calculation of risk and for the reduction of the portfolio risk to its minimum level for any provided level of return. Following two factors are considered while calculating portfolio risk through variance or standard deviation.

- Weighted single security risks
- Weighted co-movements among returns of securities

**Measuring Co-Movements in Security Returns**

The absolute measure of the co-movements between security returns is referred to as covariance which is used to calculate the portfolio risk. The standard deviation or portfolio variance is calculated by considering the actual covariance between securities in a portfolio. Before calculation of covariance, the relative measure of association is considered which is referred to as correlation coefficient.

**Correlation Coefficient**

The relative co-movements between security returns are measured by the correlation coefficient (ρij). In fact the degree of relatedness of the returns of two securities is measured by the correlation coefficient. Only association is denoted by this measure rather than the causation and the association is limited by +1.0 and 1.0

ρij = +1.0 Perfect Positive Correlation

ρij = -1.0 Perfect Negative Correlation

ρij = 0.0 Zero Correlation

**Covariances**

When there is sufficient amount of correlation among security returns, then actual amount of co-movements must be measured and incorporated into the any measure of portfolio risk, because variance of portfolio are affected by such co-movements. The measure of covariance does this.

The extent of association between the returns for a pair of securities is measured by absolute measure of covariance. Covariance is referred to as the extent to which two random variables move together over time. The covariance can be

- Positive which highlight that the returns on two securities try to move in one direction at the same time. This means that when return of one security increases, the other try to copy the same. The positive covariance results in a positive correlation coefficient.
- Negative which point out that the returns on two securities try to move in opposite direction. This means that when return of one security increases, the other try to decrease. The negative covariance result into negative correlation coefficient.
- Zero which point out that the returns on two securities are independent and have no capacity to move in opposite or same direction together.

Following is the formula for covariance calculation on expected basis.

m

σAB = Σ [RA,i – E ( RA)] [RB,i – E(RB)] pri

i = 1

RA = One possible return on security A

σAB = the covariance between securities A and B

m = no. of probable outcomes for a security for the period

E ( RA) = The expected value of the return on security A

The expected value of product of deviations from the mean is regarded as covariance. The units of variables involved provide the basis for the size of the covariance measure. The main function of covariance is to indicate that whether the association between variables in negative, positive or zero.

**Relating the Covariance and the Correlation Coefficient**

The correlation coefficient and the covariance are related in the following way.

ρAB = σ AB / σA σB

The equation indicates that the covariance standardized by dividing the product of the two standard deviations of returns is simply equal to the correlation coefficient.

In the light of this definition of correlation coefficient, the covariance can be as follow

σ AB = ρAB σA σB

So the covariance can be calculated from the known element of correlation coefficient because the standard deviations of the rates of return of assets will already be provided. Similarly it is easy to calculate correlation coefficient if the covariance is known.

**Calculating Portfolio Risk**

The co-movements in the security returns are characterized by the covariance. After the calculation of covariance, it is easy to calculate the risk of the entire portfolio. There are two conditions for calculating portfolio risk. First is to consider only two securities in the portfolio and second is to consider many securities in which the calculation becomes much complex and difficult.

**The n-Security Case**

The n-security case can be generalized by the two-security case. By combining assets with less than perfect positive correlation, portfolio risk can be reduced. Also the smaller positive correlation is better than the larger positive correlation.

Portfolio risk is the function of risk of every single security and the covariance between the single securities returns. Portfolio risk can be calculated by the following formula in terms of variance

N n n

σ^{2}_{p} = ∑ w_{i}^{2} σ_{i}^{2 }+ ∑ ∑ w_{i} w_{j} σ_{ij}

i = 1 i = 1 i = 1

Where

σ^{2}_{p} = the variance of the return on portfolio

σ_{ij} = the covariance between return for securities i and j

σ_{i} = the variance of return for security

n n

∑ ∑ = double summation sign points out that addition of n^{2} number are done in combination

i = 1 i = 1

w_{i} = the portfolio percentage or weights of investible funds invested in security i.

Two-stock portfolio is dealt exactly with the same stated message. This means that the portfolio risk is a function of

- The weighted risk of every single security
- The weighted covariance among entire security pairs

It should be remembered that portfolio risk is actually determined by the following three pairs

- Covariances
- Variances
- Weights

**Simplifying the Markowitz Portfolio Theory Calculations**

When there are two securities in portfolio, there are two covariances and weighed covariance term is multiplied by the two because the covariance of A with B is similar to the covariance of B with A. In the same sense when there are three securities, there are six covariances. Similarly four securities have 12 covariances and so on. But all these calculations are based on the fact that total no of covariance’s in the Markowitz portfolio theory or Markowitz model is find out as n (n-1), where “n” represents the no. of securities.

In case of two securities total terms in the matrix are n^{2} or four which including two covariance and two variances. Similarly in case of four securities, there are sixteen total terms in the matrix as n^{4 }= 16 including four variances and 12 covariance’s. The diagonal side of the matrix contains the variance terms.

**Efficient Portfolios**

Markowitz Portfolio Theory is helpful in selection of portfolio in such a way that the portfolios should be evaluated by the investor on the basis of their expected return and risk as measured by the standard deviation. The die concept of an efficient portfolio was first derived by Markowitz which is referred to as the portfolio that has the minimum portfolio risk for a certain level of expected return or highest expected return for provided amount of risk. Efficient portfolio can be identified by the investors by setting an expected portfolio return and reducing the portfolio risk at that level of return. An alternative way is to specify certain level of portfolio risk and then expected return of portfolio is tried to maximize at that level of risk. Rational investors prefer efficient portfolios because they include the suitable optimum combination of expected risk & return.

In order to make analysis of efficient portfolios, the first step is to ascertain the risk-return opportunities provided to the investor from a given group of securities. There may develop a large number of portfolios for investor by assigning certain portion of his wealth to each of different considering asset. But the most suitable portfolio for the investor is the efficient portfolio.

Attainable set of portfolios are generated by the available assets or securities. But the risk-averse investors are interest only in those portfolios that offer minimum risk for a given level of return. These efficient portfolios are dominated over all other set of portfolios. The inputs of covariance, variance and expected returns are used to calculate the portfolio with minimum risk against given level of return. The portfolio percentages or weights of investable funds to be invested in every security are utilized to provide solution to the Markowitz model. Because the portfolio weights is the only variable in the Markowitz analysis of selecting efficient portfolios that can be controlled to ascertain the efficient portfolio as all other factors are input like correlation coefficient, expected returns, standard deviations etc. In fact the Markowitz Portfolio Theory is the suitable model for ascertaining the efficient portfolios by the investors in making investments.

Hello everyone! This is Richard Daniels, a full-time passionate researcher & blogger. He holds a Ph.D. degree in Economics. He loves to write about economics, e-commerce, and business-related topics for students to assist them in their studies. That's the sole purpose of Business Study Notes.

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