So, in general terms, how do we derive (model) an optimum, efficient diversified portfolio of investments?

To begin with, let us develop the "two asset case" where a company have funds to invest in two profitable projects, A and B. One proportion x is invested in A and (1-x) is invested in B.

We know from Activity 1 that the expected return from a portfolio R(P) is simply a weighted average of the expected returns from two projects, R(A) and R(B), where the weights are the proportional funds invested in each. Mathematically, this is given by:

But, what about the likelihood (probability) of the portfolio return R(P) occurring? Markowitz defines the proportionate risk of a two-asset investment as the portfolio variance:

Percentage risk is then measured by the portfolio standard deviation (i.e. the square root of the variance):

Unlike the risk of a single random variable, the variance (or standard deviation) of a two-asset portfolio exhibits three separable characteristics:

1) The risk of the constituent investments measured by their respective variances,

2) The squared proportion of available funds invested in each,

3) The relationship between the constituents measured by twice the covariance.

The covariance represents the variability of the combined returns of individual investments around their mean. So, if A and B represent two investments, the degree to which their returns (ri A and r B) vary together is defined as:

For each observation i, we multiply three terms together: the deviation of r.(A) from its mean R(A), the deviation of ri(B) from its mean R(B) and the probability of occurrence pi. We then add the results for each observation.

Returning to Equations (2) and (3), the covariance enters into our portfolio risk calculation twice and is weighted because the proportional returns on A vary with B and vice versa.

Depending on the state of the world, the logic of the covariance itself is equally simple.

- If the returns from two investments are independent there is no observable relationship between the variables and knowledge of one is of no use for predicting the other. The variance of the two investments combined will equal the sum of the individual variances, i.e. the covariance is zero.

- If returns are dependent a relationship exists between the two and the covariance can take on either a positive or negative value that affects portfolio risk.

1) When each paired deviation around the mean is negative, their product is positive and so too, is the covariance.

2) When each paired deviation is positive, the covariance is still positive.

3) When one of the paired deviations is negative their covariance is negative.

Thus, in a state of the world where individual returns are independent and whatever happens to one affects the other to opposite effect, we can reduce risk by diversification without impairing overall return.

Under condition (iii) the portfolio variance will obviously be less than the sum of its constituent variances. Less obvious, is that when returns are dependent, risk reduction is still possible.

To demonstrate the application of the statistical formulae for a two-asset portfolio let us consider an equal investment in two corporate capital projects (A and B) with an equal probability of producing the following paired cash returns.

We already know that the expected return on each investment is calculated as follows:

R(A) = (0.5 x 8) + (0.5 x 12) = 10%

R(B) = (0.5 x 14) + (0.5 x 6) = 10%

Using Equation (1), the portfolio return is then given by:

R(P) = (0.5 x 10) + (0.5 x 10) = 10%

Since the portfolio return equals the expected returns of its constituents, the question management must now ask is whether the decision to place funds in both projects in equal proportions, rather than A or B exclusively, reduces risk?

To answer this question, let us first calculate the variance of A, then the variance of B and finally, the covariance of A and B. The data is summarised in Table 2.1 below.

With a negative covariance value of minus 8, combining the projects in equal proportions can obviously reduce risk. The question is by how much?

Table 2.1: The Variances of Two Investments and their Covariance

Using Equation (2), let us now calculate the portfolio variance:

And finally, the percentage risk given by Equation (3), the portfolio standard deviation:

Activity 3

Unlike our original example, which underpinned Activities 1 and 2, the current statistics reveal that this portfolio is not riskless (i.e. the percentage risk represented by the standard deviation a Is not zero). But given that our investment criteria remain the same (either minimise a, given R; or maximise R given a ) the next question to consider is how the portfolio's risk-return profile compares with those for the individual projects. In other words is diversification beneficial to the company?

If we compare the standard deviations for the portfolio, investment A and investment B with their respective expected returns, the following relationships emerge.

These confirm that our decision to place funds in both projects in equal proportions, rather than either A or B exclusively, is the correct one. You can verify this by deriving the standard deviations for the portfolio and each project from the variances in the Table 2.1.

Found a mistake? Please highlight the word and press Shift + Enter