Portfolio theory is that body of mathematical and business theory that allows us to calculate such things as the expected return of an investment, and the expected return of a portfolio of investments. It is useful to investors who want to perform such calculations. To calculate the expected return for an individual investment in portfolio theory, we first assess the probability of a range of possible outcomes and assign certain expected values to each of the outcomes by considering a range of variables and the varying results they might influence with respect to the asset.
For each possible outcome we multiply the assessed return by its probability and then find the sum of the results. To calculate the expected return on a portfolio, take the weighted average expected return of the individual assets, as previously calculated through an assessment of the various variables and the sum of the probability of the possible outcomes. Then take the sum of the weighted average expected return of the individual assets. To calculate variance for an individual investment, we return to the assessment of the variables that influence particular possible outcomes and then simply calculate the sum of the squares of the difference between the expected return in a particular possible outcome and the mean expected return, not forgetting of course to weight this calculation by the specific probabilities of the particular possible outcomes, given the probability of the variables that feed into the calculation.
As ever, standard deviation is the square root of the variance.
At this stage we might want to calculate the covariance between two assets over a given period. For this you need to calculate the actual return from each given particular period in the overall period that is under consideration. Then we need to establish the difference between the average return and the actual return by subtracting the former from the latter for each of the particular assets we are attempting to assess. In order to assess the covariance of the assets, we now find the sum of these differences and divide by the number of periods we are assessing for the particular assets. Covariance, of course, is a way of measuring the variation between the variance of the variables we are assessing.
At this point, if we want to assess these assets more assiduously the obvious next step is to find the standard deviation for the entire portfolio. If, for instance, the portfolio we are assessing is made up of only two assets, we need to find the proportion of the portfolio that is invested in the first asset and multiply it by the variance of the first asset (as calculated earlier by assessing the various variables that might affect the variance of the first asset). Then we need to find the proportion of the portfolio that is invested in the second asset and multiply it by the variance of the second asset (as calculated earlier by considering the various variables that might affect the variance of the first asset). We add these together then add a third term which is the product of the weighting of the two assets, the standard deviation of the two assets and the correlation coefficient of the two assets (which we should previously have calculated by dividing the covariance of the two assets by the product of the standard deviation of each asset). The standard deviation of the portfolio is the square root of this particular product.
However, this method of calculation is only useful for a portfolio with two assets. For a portfolio with three or more assets, we start by finding the proportion of the portfolio that is invested in the first asset and multiply it by the variance of the first asset (as calculated earlier by assessing the various variables that might affect the variance of the first asset). Then we need to find the proportion of the portfolio that is invested in the second asset and multiply it by the variance of the various variables.