Definition of Portfolio Analysis

Introduction

Individual securities have risk return characteristics of their own. The future return expected from a security is variable and this variability of returns is termed risk. It is rare to find investors investing their entire wealth in a single security. This is because most investors have an aversion to risk. It is hoped that if money is invested in several securities simultaneously, the loss in one will be compensated by the gain in others. Thus, holding more than one security at a time is an attempt to spread and minimize risk by not putting all our eggs in one basket.

Most investors thus tend to invest in a group of securities rather than a single security. Such a group of securities held together as an investment is what is known as a portfolio.

The process of creating such a portfolio is called diversification. It is an attempt to spread and minimize the risk in investment. This is sought to be achieved by holding different types of securities across different industry groups.

From a given set of securities, any number of portfolios can be constructed. A rational investor attempts to find the most efficient of these portfolios. The efficiency of each portfolio can be evaluated only in terms of the expected return and risk of the portfolio as such. Thus, determining the expected return and risk of different portfolios is a primary step in portfolio management. This step is designated as portfolio analysis.

Expected Return of a Portfolio

As a first step in portfolio analysis, an investor needs to specify the list of securities eligible for selection or inclusion in the portfolio. Next he has to generate the risk-return expectations for these securities. These are typically expressed as the expected rate of return (mean) and the variance or standard deviation of the return.

The expected return of a portfolio of assets is simply the weighted average of the return of the individual securities held in the portfolio. The weight applied to each return is the fraction of the portfolio invested in that security.

Let us consider a portfolio of two equity shares P and Q with expected returns of 15 per cent and 20 per cent respectively.

If 40 per cent of the total funds are invested in share P and the remaining 60 per cent, in share Q, then the expected portfolio return will be:

(0.40 x 15) + (0.60 x 20) = 18 per cent

The formula for the calculation of expected portfolio return may be expressed as shown below:

Where

r_p = Expected return of the portfolio

xi = Proportion of funds invested in security i.

ri = Expected return of security i.

n = Number of securities in the portfolio

Risk of a Portfolio

The variance of return and standard deviation of return are alternative statistical measures that are used for measuring risk in investment. These statistics measure the extent to which returns are expected to vary around an average over time. The calculation of variance of a portfolio is a little more difficult than determining its expected return.

The variance or standard deviation of an individual security measures the riskiness of a security in absolute sense. For calculating the risk of a portfolio of securities, the riskiness of each security within the context of the overall portfolio has to be considered.

This depends on their interactive risk, i.e. how the returns of a security move with the returns of other securities in the portfolio and contribute to the overall risk of the portfolio.

Covariance is the statistical measure that indicates the interactive risk of a security relative to others in a portfolio of securities. In other words, the way security returns vary with each other affects the overall risk of the portfolio.

The covariance between two securities X and Y may be calculated using the following formula:

The calculation of covariance is illustrated below:

Calculation of Covariance

The covariance is a measure of how returns of two securities move together. If the returns of the two securities move in the same direction consistently the covariance would be positive. If the returns of the two securities move in opposite direction consistently the covariance would be negative. If the movements of returns are independent of each other, covariance would be close to zero.

Covariance is an absolute measure of interactive risk between two securities. To facilitate comparison, covariance can be standardized. Dividing the covariance between two securities by product of the standard deviation of each security gives such a standardised measure. This measure is called the coefficient of correlation. This may be expressed as:

It may be noted from the above formula that covariance may be expressed as the product of correlation between the securities and the standard deviation of each of the securities. Thus,

Cov_xy   = r_xyσ_xσ_y

The correlation coefficients may range from - 1 to 1. A value of -1 indicates perfect negative correlation between security returns, while a value of +1 indicates a perfect positive correlation. A value close to zero would indicate that the returns are independent.

The variance (or risk) of a portfolio is not simply a weighted average of the variances of the individual securities in the portfolio. The relationship between each security in the portfolio with every other security as measured by the covariance of return has also to be considered. The variance of a portfolio with only two securities in it may be calculated with the following formula.

Portfolio standard deviation can be obtained by taking the square root of portfolio variance.

Let us take an example to understand the calculation of portfolio variance and portfolio standard deviation. Two securities P and Q generate the following sets of expected returns, standard deviations and correlation coefficient:

A portfolio is constructed with 40 per cent of funds invested in P and the remaining 60 per cent of funds in Q.

The expected return of the portfolio is given by:

The variance of the portfolio is given by:

The standard deviation of the portfolio is:

s_p  = √ 292 = 17.09 per cent.

The return and risk of a portfolio depends on two sets of factors (a) the returns and risks of individual securities and the covariance between securities in the portfolio, (b) the proportion of investment in each security.

The first set of factors is parametric to the investor in the sense that he has no control over the returns, risks and covariances of individual securities. The second sets of factors are choice variables in the sense that the investor can choose the proportions of each security in the portfolio.

Reduction of Portfolio Risk Through Diversification

The process of combining securities in a portfolio is known as diversification. The aim of diversification is to reduce total risk without sacrificing portfolio return. In the example considered above, diversification has helped to reduce risk. The portfolio standard deviation of 17.09 is lower than the standard deviation of either of the two securities taken separately, which were 50 and 30 respectively.

To understand the mechanism and power of diversification, it is necessary to consider the impact of covariance or correlation on portfolio risk more closely. We shall examine three cases: (a) when security returns are perfectly positively correlated, (b) when security returns are perfectly negatively correlated, and (c) when security returns are not correlated.

Security Returns Perfectly Positively Correlated

When security returns are perfectly positively correlated the correlation coefficient between the two securities will be +1. The returns of the two securities then move up or down together.

The portfolio variance is calculated using the formula:

σ²_p = x₁²σ₁² + x₂²σ₂² + 2x₁x₂(r₁₂σ₁σ₂)

Since r₁₂ = 1, this may be rewritten as:

σ²_p = x₁²σ₁² + x₂²σ₂² + 2x₁x₂σ₁σ₂

The right hand side of the equation has the same form as the expansion of the identity (a + b)2, namely a2 + 2ab + b2. Hence, it may be reduced as

σ²_p = (x₁σ₁+ x₂σ₂ )^2\

The standard 1deviation then becomes

σ_p  = x₁σ₁+ x₂σ₂

This is simply the weighted average of the standard deviations of the individual securities.

Taking the same example that we considered earlier for calculating portfolio variance, we shall calculate the portfolio standard deviation when correlation coefficient is +1.

Portfolio standard deviation may be calculated as:

σ_p = x₁σ₁+ x₂σ₂

 = (0.4) (50) + (0.6) (30)

 = 38

Being the weighted average of the standard deviations of individual securities, the portfolio standard deviation will lie between the standard deviations of the two individual securities. In our example, it will vary between 50 and 30 as the proportion of investment in each security changes.

For example, if the proportion of investment in P and Q are 0.75 and 0.25 respectively, portfolio standard deviation becomes:

σ_p  = (0.75) (50) + (0.25) (30) = 45

Thus, when the security returns are perfectly positively correlated, diversification provides only risk averaging and no risk reduction because the portfolio risk cannot be reduced below the individual security risk. Hence, diversification is not a productive activity when security returns are perfectly positively correlated.

Security Returns Perfectly Negatively Correlated

When security returns are perfectly negatively correlated, the correlation coefficient between them becomes -1. The two returns always move in exactly opposite directions.

The portfolio variance may be calculated as:

σ²_p = x₁²σ₁² + x₂²σ₂² + 2x₁x₂(r₁₂σ₁σ₂)

Since r₁₂ = -1, this may be rewritten as:

σ²_p = x₁²σ₁² + x₂²σ₂² − 2x₁x₂(σ₁σ₂)

The right hand side of the equation has the same form as the expansion of the identity (a - b)2, namely a2 - 2ab + b2. Hence, it may be reduced as:

σ²_p = (x₁σ₁ - x₂σ₂ )²

The standard deviation then becomes:

σ_p = x₁σ₁ - x₂σ₂

For the illustrative portfolio considered above, we can calculate the portfolio standard deviation when the correlation coefficient is —1.

σ_p = (0.4)(50) - (0.6)(30) = 2

The portfolio risk is very low. It may even be reduced to zero. For example, if the proportion of investment in P and Q are 0.375 and 0.625 respectively, portfolio standard deviation becomes:

σ_p = (0.375)(50) - (0.625)(30) = 0

Here, although the portfolio contains two risky assets, the portfolio has no risk at all. Thus, the portfolio may become entirely risk free when security returns are perfectly negatively correlated. Hence, diversification becomes a highly productive activity when securities are perfectly negatively correlated, because portfolio risk can be considerably reduced and sometimes even eliminated. But, in reality, it is rare to find securities that are perfectly negatively correlated.

Security Returns Uncorrelated

When the returns of two securities are entirely uncorrelated, the correlation coefficient would be zero.

The formula for portfolio variance is:

σ²_p = x₁²σ₁² + x₂²σ₂² + 2x₁x₂(r₁₂σ₁σ₂)

Since r₁₂ = 0, the last term in the equation becomes zero; the formula may be rewritten

σ²_p = x₁²s₁² + x₂²σ₂²

The standard deviation then becomes:

σ_p = √x₁σ₁ + x₂σ₂

For the illustrative portfolio considered above the standard deviation can be calculated when the correction coefficient is zero.

σ_p = √(0.4)²(50)²+(.6)²(30)²

= √400+324

= 26.91

The portfolio standard deviation is less than the standard deviations of individual securities in the portfolio. Thus, when security returns are uncorrelated, diversification reduces risk and is a productive activity.

Portfolio Standard Deviations

From the above analysis we may conclude that diversification reduces risk in all cases except when the security returns are perfectly positively correlated. As correlation coefficient declines from +1 to -1, the portfolio standard deviation also declines. But the risk reduction is greater when the security returns are negatively correlated.

Portfolios With More Than Two Securities

So far we have considered a portfolio with only two securities. The benefits from diversification increase as more and more securities with less than perfectly positively correlated returns are included in the portfolio. As the number of securities added to a portfolio increases, the standard deviation of the portfolio becomes smaller and smaller. Hence, an investor can make the portfolio risk arbitrarily small by including a large number of securities with negative or zero correlation in the portfolio.

But, in reality, no securities show negative or even zero correlation. Typically, securities show some positive correlation that is above zero but less than the perfectly positive value (+ 1). As a result, diversification (that is, adding securities to a portfolio) results in some reduction in total portfolio risk but not in complete elimination of risk. Moreover, the effects of diversification are exhausted fairly rapidly. That is, most of the reduction in portfolio standard deviation occurs by the time the portfolio size increases to 25 or 30 securities. Adding securities beyond this size brings about only marginal reduction in portfolio standard deviation.

Adding securities to a portfolio reduces risk because securities are not perfectly positively correlated. But the effects of diversification are exhausted rapidly because the securities are still positively correlated to each other though not perfectly correlated. Had they been negatively correlated, the portfolio risk would have continued to decline as portfolio size increased. Thus, in practice, the benefits of diversification are limited.

The total risk of an individual security comprises two components, the market related risk called systematic risk and the unique risk of that particular security called unsystematic risk. By combining securities into a portfolio the unsystematic risk specific to different securities is cancelled out. Consequently, the risk of the portfolio as a whole is reduced as the size of the portfolio increases. Ultimately when the size of the portfolio reaches a certain limit, it will contain only the systematic risk of securities included in the portfolio. The systematic risk, however, cannot be eliminated. Thus, a fairly large portfolio has only systematic risk and has relatively little unsystematic risk. That is why there is no gain in adding securities to a portfolio beyond a certain portfolio size. Figure depicts the diversification of risk in a portfolio.

Diversification of risk

The figure shows the portfolio risk declining as the number of securities in the portfolio increases, but the risk reduction ceases when the unsystematic risk is eliminated.

Risk-Return Calculations of Portfolios With More Than Two Securities

The expected return of a portfolio is the weighted average of the returns of individual securities in the portfolio, the weights being the proportion of investment in each security. The formula for calculation of expected portfolio return is the same for a portfolio with two securities and for portfolios with more than two securities. The formula is:

Where

r_p  = Expected return of portfolio.

x_i = Proportion of funds invested in each security.

r_i  = Expected return of each security.

n = Number of securities in the portfolio.

Let us consider a portfolio with four securities having the following characteristics

The expected return of this portfolio may be calculated using the formula:

The portfolio variance and standard deviation depend on the proportion of investment in each security, as also the variance and covariance of each security included in the portfolio.

The formula for portfolio variance of a portfolio with more than two securities is as follows:

Where

indicates that n2numbers of values are to be summedup. These values are obtained by substituting the values of x_i, x_j and σ_ij for each possible pair of securities.

The method of calculation can be illustrated through an example.

A convenient way to obtain the result is to set up the data required for calculation in the form of a variance-covariance matrix. Let us consider a portfolio with three securities A, B and C. The proportions of investment in each of these securities are 0.20, 0.30 and 0.50 respectively. The variance of each security and the covariance of each possible pair of securities may be set up as a matrix as follows:

Variance-Covariance Matrix

The entries along the diagonal of the matrix represent the variances of securities A, B and C. The other entries in the matrix represent the covariance of the respective pairs of securities such as A and B, A and C, B and C.

Once the variance-covariance matrix is set up, the computation of portfolio variance is a comparatively simple operation.

Each cell in the matrix represents a pair of two securities. For example, the first cell in the first row of the matrix represents A and A; the second cell in the first row represents securities A and B, and so on. The variance or covariance in each cell has to be multiplied by the weights of the respective securities represented by that cell. These weights are available in the matrix at the left side of the row and the top of the column containing the cell. This process may be started from the first cell in the first row and continued for all the cells till the last cell of the last row is reached. When all these products are summed up, the resulting figure is the portfolio variance. The square root of this figure gives the portfolio standard deviation.

The variance of the illustrative portfolio given above can now be calculated.

σ_P2  =   (0.2 x 0.2 x 52) + (0.2 x 0.3 x 63) + (0.2 x 0.5 x 36)

+(0.3 x 0.2 x 63) + (0.3 x 0.3 x 38) + (0.3 x 0.5 x 74)

+(0.5 x 0.2 x 36) + (0.5 x 0.3 x 74) ÷ (0.5 x 0.5 x 45)

=53.71.

The portfolio standard deviation is:

σ_P = √53.71 = 7.3287

We have seen earlier that covariance between two securities may be expressed as the product of correlation coefficient between the two securities and standard deviations of the two securities.

Thus,

^σij^=rij^σi^σj

Where

Hence, the formula for computing portfolio variance may also be stated in the following form:

To illustrate the use of this formula let us calculate the portfolio variance and standard deviation for a portfolio with the following characteristics.

It may be noted that correlation coefficient between P and P, Q and Q, R and R is 1.

The variance-covariance matrix may be set up as follows:

The portfolio variance can now be calculated using this variance-covariance matrix as shown below:

The portfolio standard deviation is:

σ_p  = .√65.4945 = 8.09

A portfolio is a combination of assets. From a given set of n securities, any number of portfolios can be created. The portfolios may comprise of two securities, three securities, all the way up to ‘n’ securities. A portfolio may contain the same securities as another portfolio but with different weights. Thus, new portfolios can be created either by changing the securities in the portfolio or by changing the proportion of investment in the existing securities.