Modern Portfolio Theory

Reducing Risk: Modern Portfolio Theory (MPT)

Portfolio theory begins with the premise that all investors are like my wife – they are risk-averse. They want high returns and guaranteed outcomes. The theory tells investors how to combine stocks in their portfolios to give them the least risk possible, consistent with the return they seek. It also gives a rigorous mathematical justification for the time-honored investment maxim that diversification is a sensible strategy for individuals who like to reduce their risks.

The theory was invented in the 1950s by Harry Markowitz. His book, Portfolio Selection, was an outgrowth of his Ph.D. Dissertation at the University of Chicago, Markowitz is a scholarly academic “computenick” type with a most varied background. His experience has ranged from teaching at UCLA to designing a computer language at RAND Corporation and helping General Electric solve manufacturing problems by computer simulations. He has even practiced money management, serving as president of Arbitrage Management Company, which ran a “hedge fund.” (Basically what Markowitz did was to search with the computer for situations where a convertible bond sold at a price that was “out of line” with the underlying common stock. He admitted, however, that it was “no great trick” and that competitors would be joining him in increasing numbers. “Then when we start tripping over each other, buying the same bonds almost simultaneously, the game will be over. Two, three years at most.” Three years later, he admitted that convertible hedges were no longer attractive in the market. Consequently, he had moved on to do hedging operations on the Chicago Board Options Exchange.

What Markowitz discovered was that portfolios of risky (volatile) stocks might be put together in such a way that the portfolio as a whole would actually be less risky than any one of the individual stocks in it.

Manhattan. Macy's. Photo by Elena.

The mathematics of modern portfolio theory (also known as MRT) is recondite and forbidding; it fills the journals and, incidentally, keeps a lot of academics busy. That in itself is no small accomplishment. Fortunately, there is no need to lead you through the labyrinth of quadratic programming for you to understand the core of the theory. A singly illustration will make the whole game clear.

Let’s suppose we have an island economy with only two business. The first is a large resort with beaches, tennis courts, a golf course and the like. The second is a manufacturer of umbrellas. Weather affects the fortunes of both. During sunny seasons the resort does a booming business and umbrella sales pluumet. During rainy seasons the resort owner does very poorly, while the umbrella manufacturer enjoys high sales and large profits. The following table shows some hypothetical returns for the two businesses during the different seasons:

Umbrella Manufacturer – Rainy season – 50%. Sunny season – 25%.

Resort Owner – -25%. Sunny season – 50%.

Suppose that, on average, one-half the seasons are sunny and one-half are rainy (i.e., the probability of a sunny or rainy season is ½). An investor who bought stock in the umbrella manufacturer would find that half the time he earned a 50 percent return and half the time he lost 25 perecent of his investment. On average, he would earn a return of 12 ½ percent. This is what we have called the investor’s expected return. Similarly, investment in the resort would produce the same results. Investing in either one of these businesses would be fairly risky. However, because the results are quite variable and there could be several sunny or rainy seasons in a row.

Suppose, however, that instead of buying only one security an investor with two dollars diversified and put half his money in the umbrella manufacturer’s and half in the resort owner’s business. In sunny seasons, a one-dollar investment in the resort would produce a 50-cent return, while a one-dollar investment in the umbrella manufacturer would lose 25 cents. The investor’s total return would be 25 cents (50 cents minus 25 cents), which is 12 ½ percent of his total investment of two dollars.

Note that during rainy seasons exactly the same thing happens – only the names are changed. Investment in the umbrella manufacturer produces a good 50 percent return while the investment in the resort loses 25 percent. Again, however, the diversified investor makes a 12 1/2 percent return on his total investment.

This simple illustration points out the basic advantage of diversification. Whatever happens to the weather, and thus to the island economy, by diversifying investments over both of the firms the investor is sure of making a 12 1/2% return each year. The trick that make the game work was that while both companies were risky (returns were variable from year to year), the companies were affected differently by weather conditions. (In statistical terms, the two companies had a negative covariance – Statisticians use the term covariance to measure what experts call the degree of parallelism between the return of the two securities. If we let R stand for the actual return from the resort and r be the expected of average return, while U stands for the actual return from the umbrella manufacturer and u is the average return, we define the covariance between U and R (COVUR) as follows:

COVUR = Prob. Rain (U, if rain -u) (R if rain -r) + Prob. Sun (U if sun – u) (R if sun -r).

From the preceding table of returns and assumed probabilities we can fill in the relevant numbers:

COVUR = 1/2 (0.50 – 0.125) (-0.25 – 0.125) + 1/2 (0.25 – 0.125) (0.50 – 0.125). = -0.141

Whenever the return from two securities moves in tandem – when un goes up the other always goes up – the covariance number will be a large positive number. If the returns are completely out of phase, as in the present example, the two securities are said to have negative covariance).

As long as there is some lack of parallelism in the fortunes of the individual companies in the economy, diversification will always reduce risk. In the present case, where there is a perfect negative relationship between the companies’ fortunes (one always does well when the other does poorly), diversification can totally eliminate risk.

Of course, there is always a rub, and the rub in this case is that the fortunes of most companies move pretty much in tandem. When there is a recession, and people are unemployed, they may buy neither summer vacations nor umbrellas. Therefore, one should not expect in practice to get the neat kind of total risk elimination just shown. Nevertheless, since companies’ fortunes don’t always move completely in parallel, investment in a diversified portfolio of stocks is likely to be less risky than investment in one or two single securities.

It is easy to carry the lessons of this illustration to actual portfolio construction. Suppose you were considering combining General Motors and its major supplier of new tires in a stock portfolio. Would diversification be likely to give you much risk reduction? Probably not. It may not be true that “as General Motors goes, so goes the nation” but it surely does follow that if General Motors’ sales slump, G.M. will be buying fewer new tires from the tire manufacturer. In general, diversification will not help much if there is a high covariance between the returns of the two companies.

On the other hand, if General Motors were combined with a government contractor in a depressed area, diversification might reduce risk substantially. It usually has been true that as the nation goes, so goes General Motors. If consumer spending is down (or if an oil crisis comes close to paralyzing the nation) General Motors” sales and earnings are likely to be down and the nation’s level of unemployment up. Now, if the government makes a habit during times of high unemployment of giving out contacts to the depressed area (to alleviate some of the unemployment miseries there) it could well be that the returns of General Motors and those of the contractor do not move in phase. The two stocks might have very little covariance or, better still, negative covariance.

The example may seem a bit strained, and most investors will realize that when the market gets clobbered just about all stocks go down. Still, at least at certain times, some stocks do move against the market. Gold stocks are often given as an example of securities that do not typically move in the same direction as the geeral market. The point to realize in setting up a portfolio is that while the variability (variance) of the returns from individual stocks in important, even more important in judging the risk of a portfolio is covariance, the extent to which the securities move in parallel. It is this covariance that plays the critical role in Markowitz’s portfolio theory.

True diversification depends on having stocks in your portfolio that are not all dependent on the same economic variables (consumer spending, business, investment, housing construction, etc.) Wise investors will diversify their portfolios not by names of industries but by the determinants that influence the fluctuations of various securities.

The following chart illustrates the theory quite nicely. Looking first at the top line of the figure, marked “U.S. Stocks,” we see that as the number of securities in the portfolio contains close to 20 equal-sized and well-diversified issues, the total risk (standard deviation of returns) of the portfolio is reduced by about 70 percent. Further increase in the number of holdings does not produce and significant further risk reduction. Of course, we are assuming that the stocks in the portfolio utilities would not produce an equivalent amount of risk reduction.

Having learned the twin lessons that diversification reduces risk and the diversification is most helpful if one can find securities that don’t move in tandem with the general market, investors in the 1980s have sought to apply these principles on the international scene. Since the movement of foreign economies is not always synchronous with that of the U.S. economy, we should expect some additional benefits from including foreign companies in the portfolio. The potential benefits of international diversification are illustrated in the bottom line of the figure. Here, the stocks are drawn not simply from France, Germany, Italy, Belgium, the Netherlands, and Switzerland. As expected, the international diversified portfolio tends to be less risky than the one of corresponding size drawn purely from stocks directly traded on the NYSE. Even further benefits would be achieved by including securities from the Pacific Rim countries, such as Japan and Australia.

Burton G. Malkiel. A Random Walk Down Wall Street, including a life-cycle guide to personal investing. First edition, 1973, by W.W. Norton and company, Inc.