Implications Across Portfolios

The risk in a portfolio is measured as the amount of variance that investors can expect based on historical data.

TERMS

• risk

  The potential (conventionally negative) impact of an event, determined by combining the likelihood of the event occurring with the impact, should it occur.

• Co-Variance

  In probability theory and statistics, co-variance is a measure of how much two random variables change together.

• portfolio

  The group of investments and other assets held by an investor.

Portfolio Risk

An investor can reduce portfolio risk by holding combinations of instruments which are not perfectly positively correlated (correlation coefficient). In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification may allow for the same portfolio expected return with reduced risk.

Three assets (apples, bananas, and cherries) can be thought of as a bowl of fruit. The index is a a fruit basket. A full fruit basket probably has 10 or 15 different fruits, but my bowl will be efficient as much as its statistical parameters (risk and return) mimic those of the whole basket. In this unit, we are talking about calculating the risk of a portfolio. In addition, we can extend the implications made by the security market line theory from individual assets to portfolios. How does my bowl of fruit compare to the whole basket and how does that compare to other bowls out there?

Calculating Portfolio Risk

To calculate the risk in my bowl, we need a little more background information on fruit markets. First, we are going to need the variance for each fruit. Remember that the standard deviation answers the question of how far do I expect one individual outcome to deviate from the overall mean. And variance is that number squared. Mathematically, the formula is:

$V(R)=[R−E(R)]^2$

It is the expected value of the difference between the individual return in a given day (R) and the average outcome average return over a year (E(R)).

In order to calculate the variance of a portfolio of three assets, we need to know that figure for apples, bananas, and cherries, and we also need to know the co-variance of each. Co-variances can be thought of as correlations. If every time bananas have a bad day, so do apples, their co-variance will be large. If bananas do great half of the time when cherries do bad and bananas do terrible the other half, their co-variance is zero. If there is zero correlation among all three fruits, we have cut our risk in thirds by owning all three, but if they are perfectly correlated, we haven't diversified away any of our risk.

In reality, they are probably positively correlated, since they are all fruits, but not at all perfectly. Apples and bananas grow in different climates so their performance may be a result of weather patterns in either region. Apples may be a substitute for cherries when cherries are expensive. The overall risk of the portfolio would take into account three individual variances and three co-variances (apples-bananas, apples-cherries, and bananas-cherries) and it would reduce the overall portfolio to the degree that they are uncorrelated.

The formula to compute the co-variance between returns on X and Y:

$Cov(X|Y)=E[(X−E(X))(Y−E(Y)]$

This means what do I expect to see, in a given time period, when I multiply how much X returned off its average performance from how much Y returned off its average. But notice how it could be positive or negative. And if X tends to be up when Y is down, that would make them two good hedges. And it can be shown that:

From co-variance, we get correlation coefficients:

$ρAB=\dfrac{Cov(X|Y)}{σ^2_A⋅σ^2_B}$

In finance and statistics, the Greek letter rho squared represents variance.

So now that we have figures to help us measure the risk and reward of our individual fruit bowl, we can go look at historical figures to determine the expected returns and the risk of the index that comprises the entire fruit basket industry. If our portfolio of investments has diversified away as much risk as is possible given the costs of diversifying, our portfolio will be attractive to investors. If our bowl does not diversify away enough risk, it will not lie on the Security Market Line for those who we are trying to recruit into buying our portfolio.

The same principles that were applied to individual investments in the Understanding the SML section can be applied to the market for portfolio investments. If an institutional investor, such as a city pension fund, looked at two portfolios with identical returns and different risks, they would choose the portfolio that minimized its risk. Thus the only portfolios that are efficient investments are those that effectively diversify the underlying risk away and price their investment efficiently.