Portfolio Theory

Suppose a certain amount of money is to be invested in a portfolio of some selected securities (stocks and/or stock funds).  By varying the amount invested in each security, we can create an infinite number of portfolio possibilities - each with a unique risk-return attributes - from which to choose.  Some of these portfolios are efficient portfolios - they yield maximum returns with minimum risk levels.  Modern Portfolio Theory (MPT) is about identifying these efficient portfolios from a universe of portfolio possibilities.  MPT is the mathematical tool that enables an investor to allocate optimally securities to either maximize return for the level of risk he is willing to accept, or, minimize risk for the level of return he seeks.  Note that a collection of all efficient portfolios is called the efficient frontier.

Components Composition Portfolio A  Composition Portfolio B Composition Portfolio C Security 1 20.00% 2.26% 4.22% Security 2 15.00 17.02 25.16 Security 3 5.00 5.99 6.09 Security 4 60.00 74.73 64.53   Total = 100.00% Total = 100.00% Total = 100.00%           Portfolio A Portfolio B Portfolio C Expected Return 7.10% 7.38% 7.10% Ret Std Deviation 22.50% 22.50% 19.94%

For example, let us examine three portfolios: A, B and C.  B and C are efficient portfolios, while A is not.  Consider A and B. Portfolio A has the same standard deviation as portfolio B, and hence both are of the same risk level. Nevertheless, they differ in their returns - portfolio B yields higher average return than portfolio A.  An investor who is comfortable with this risk level will choose portfolio B to maximize her return.  Now, let us compare A and C. Portfolios A and C yield equal returns with different risk levels.  Portfolio A has a higher standard deviation than Portfolio C, thus A is riskier than C.  An investor who is happy with this return will choose portfolio C to minimize his risk.

With the passage of time, the initial compositions of an efficient portfolio may be out of balance.  Several factors may cause these changes, such as dividend reinvestments, capital gain reinvestments, differing growth rates among securities, etc.  To keep a portfolio efficient, an investor should periodically rebalance it by reducing the amount invested in some securities and increasing others in accordance to MPT.  At times, rebalancing calls for selling off some well-performed securities and buying more of the under-performed.  To most of us, this is counter intuitive.  We incline to do just the opposite - sell non-performers and buy good performers to improve portfolio average return.  It is important to remember, however, that in an investment portfolio, what matters is the risk and return of the portfolio as a whole - not the risk and return of the individual security.  A well-diversified portfolio will have some securities not performing well, and some performing well.  If we do the opposite of what MPT recommends, we may reduce diversification and increase risk.  Remember that the currently under-performed securities are the insurance for when the well-performed are not doing so well in the future.

Portfolio Risk & Return.  A portfolio average return is measured in percentage.  Each security contributes its average return to the portfolio, and the sum of all the contributions is the portfolio average return.  The size of each security's contribution is proportional to the size of that security in the portfolio.  For example, if a security has a mean return of 10%, and the amount invested in that security accounted for 25% of the total portfolio, the contribution of that security's mean return to the portfolio's average return is 2.5%. The sum of these individual contributions - the portfolio average return - is called the weighted average of the security mean returns.

The risk of a portfolio - the return variance or its square root (the return standard deviation) - is not so straightforward.  Unlike portfolio average return, portfolio variance is not simply the weighted average of all security variances; it is a combination of security return variances and covariances, and the proportion of the securities in the portfolio.  Security returns covariance measures how the combined returns of two securities deviate from their individual averages.  The following illustrates the mechanics of calculating this:

Columns (1) and (2) show the monthly returns of securities A and B.  Deviations of the monthly returns from their average returns are in column (3) for security A, and column (4) for security B. Column (5), labeled "Covary AB," is the product of "Deviation A" [column (3)] multiplied by "Deviation B" [column (4)].  The covariance of securities A and B is the average of "Covary AB" [column (5)], which is just the sum of the numbers in column(5), divided by 12 (number of months).

Recall that portfolio variance is the combination of security return variances and covariances, and the size of the securities relative to the overall portfolio.  In the above example, if we invest equal amount in both securities, the portfolio variance would be 0.1574%. Clearly, this is less than the risk (the variance) of the individual securities, and the reason is that the two securities do not move in tandem as indicated by the negative covariance.  Mathematically, as the number of securities in a portfolio goes up, the contribution of an individual security's variance to portfolio variance decreases.  This is the basis of diversification - that the individual risk of securities can be reduced. As the number of securities gets very large, the contribution of the individual securities' variance to the portfolio variance diminishes, and the portfolio variance approaches the average covariance.  These terms, the covariances, cannot be diversified away, and these are the only sources of risk in a portfolio containing a very large number of securities.

Assumption & Usage.  The basis of MPT is the characteristics of the securities that make up a portfolio.  These characteristics - the average return, the return variance, and the return covariance - are all computed from historical data. MPT assumes that the characteristics of the securities in the future will be the same as those in the past.  This assumption - that history repeats itself - merits a further examination, and the best way to do so is through a personal everyday anecdote.  From our own experience, we know that friends can tell each other's future behavior based upon past associations.  And the more diverse situations friends went through together, the better understanding they have of each other's characters, hence the more comfortable they are about presuming each other's future behavior.  Likewise, in securities, we need to know how securities responded to a wide range of industrial situations and business environments.  The wider the range, the more meaningful the historical data, and the better the characterizations of the securities, thus the surer the investors can be about MPT assumption.  In friendship, it is not the years of friendship that necessarily matter, but rather the diversity of situations that friends encountered together. In securities, it is not the years of available data that necessarily matter, but rather the breadth of relevant economic situations the data covers.

When we assume, we expect.  When we expect, we do not predict.  Therefore, MPT is an "expectation" - not a "predictive" model.  MPT does not forecast a specific portfolio return; it merely outlines possible returns.  Instead of a forecasted return, MPT gives an expected return and the variance associated with it.  The expected return, together with the variance, defines return distributions.  To illustrate the difference between something "forecasted" and something "expected," consider the following example.  Suppose we know from the past, that a friend's mood at any given morning may range from depressed to jubilant, where most mornings he is calm and serious.  If we assume that his mood attributes are the same in the future as they were in the past, we can expect that tomorrow morning he will be calm and serious - calm and serious is the expected mood.  If, however, we learned that he was just fired from his work, we can predict his mood to be depressed tomorrow morning - depressed is the forecasted mood.  In the preceding example, forecasting was trivial because the factor affecting our friend's mood was singular, namely: the involuntary termination of his employment.  When there are multiple causing factors, such as the case in a security's return, forecasting becomes very complex.  Even with two factors, forecasting becomes quite complex.  For example, if we also learned that on the day our friend was fired he won US$25,000 in the state lottery, predicting his mood the next morning would not be so easy.  This is because determining the net effect of simultaneously winning the lottery and being fired is not simple.  For securities, there are so many economics and non-economics factors affecting their returns that forecasting them accurately in a consistent way is almost impossible.  Because of this, in the long run, investors are better off with an "expectation" model like MPT.