MBA (Finance) – IV Semester, Investment and Portfolio Management, Unit 4.1
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:
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:
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:
indicates that n2numbers of values are to be summedup. These
values are obtained by substituting the values of xi, xj and σij 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.
The portfolio standard deviation is:
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,
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:
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.
Each portfolio is characterized by its expected return and risk.
Determining the expected return and risk (variance or standard deviation) of
each portfolio that can be created from a set of selected securities is the
first step in portfolio management and is called portfolio analysis.
