## How Diversification Reduces Risk

Risk of Portfolio (Standard Deviation of Return)

Total risk – Unsystematic Risk – Systematic Risk. Number of Securities in Portfolio.

Both financial theorists and practitioners agree that investors should be compensated for taking one more risk by a higher expected return. Stock prices must therefore adjust to offer higher returns where more risk is perceived, to ensure that all securities are held by someone. Obviously, risk-averse investors wouldn’t buy securities with extra risk without the expectation of extra reward. But not all of the risk of individual securities is relevant in determining the premium for bearing risk. The unsystematic part of the total risk is easily eliminated by adequate diversification. So there is no reason to think that investors will be compensated with a risk premium for bearing unsystematic risk. The only part of total risk that investors will get paid for bearing is systematic risk, the risk that diversification cannot help. Thus, the capital-asset pricing model says that returns (and, therefore, risk premiums) for any stock (or portfolio) will be related to beta, the systematic risk that cannot be diversified away.

The proposition that risk and reward are related is not new. Finance specialists have agreed for years that investors do need to be compensated for taking on more risk. Whait is different about the new investment technology is the definition and measurement of risk. Before the advent of the capital-asset pricing model, it was believed that the return on each security believed that the return from a security varied with the instability of that security’s particular performance, that is, with the variability or standard deviation of the returns it produced. The new theory says that the total risk of each individual security is irrelevant. It is only the systematic component of that total instability that is relevant for valuation.

While the mathematical proof of this proposition would stun even a Yoda, the logic behind it is fairly simple. Consider a case where there are two groups of securities – Group I and Group II – with 20 securities in each. Suppose that the systematic risk (beta) for each security is 1; that is, each of the securities in the two groups tends to move up and down in tandem with the general market. Now suppose that, because of factors peculiar to the individual securities in Group I, the total risk for each of them is substantially higher than the total risk for each security in Group II. Imagine, for example, that in addition to general market factors the securities in Group I are also particularly susceptible to climatic variations, to changes in exchange rates, and to natural disasters. The specific risk for each of the securities in Group I will therefore be very high. The specific risk for each of the securities in Group II, however, is assumed to be very low, and hence the total risk for each of them will be very low. Schematically, this situation appears as follows:

Group I (20 securities) – Group II (20 securities)

Systematic risk (beta) = 1 for each security in both cases.

Specific risk is high for each security is for the Group I and is low for each security in the Group II.

Total risk is high for each security in the Group I and it is low for each security in the Group II.

Now, according to the old theory, commonly accepted before the advent of the capital-asset pricing model, returns should be higher for a portfolio made up of Group I securities than for a portfolio made up of Group II securities, because each security in Group I has a higher total risk, and risk, as we know, has its reward. The advent of the new investment technology changed that sort of thinking. Under the capital-asset pricing model, returns from both portfolios should be equal. Why?

First, as the number of securities in the portfolio approach 20, the total risk of the portfolio is reduced to its systematic level. All of the unsystematic risk is eliminated. As the number of securities in each portfolio is 20, that means that the unsystematic risk has essentially been washed away: An unexpected weather calamity is balanced bu a favorable exchange rate, and so forth. What remains is only the systematic risk of each stock in the portfolio, which is given by its beta of 1. Hence, a portfolio of Group I securities and a portfolio of Group II securities will perform exactly the same with respect to risk (standard deviation) even though the stocks in Group I display higher total risk than the stocks in Group II.

The old and the new views now meet head on. Under the old system of valuation, Group I securities were regarded as offering a higher return because of their greater risk. The capital-asset pricing model says there is no greater risk in holding Groop O securities if they are in a diversified portfolio. Indeed, if the securities of Group I did offer higher returns, then all rational investors would prefer them over Group II securities and would attempt to rearrange their holdings to capture the higher returns from Group I. But by this very process they would bid up the prices of Group I securities and push down the prices of Group II securities until, with the attain,ment of equilibrium (when investors no longer want to switch from security to security), the portfolio for each group had identical returns, related to the systematic component of their risk (beta) rather than to their total risk (including the unsystematic or specific portions).

Because stocks can be combined in portfolios to eliminate specific risk, only the undiversifiable or systematic risk will command a risk premium. Investors will not get paid for bearing risks that can be diversifed away. This is the basic logic behind the capital-asset pricing model.

In a big fat nutshell, the proof of the capital-asset pricing model (henceforth to be known as CAPM because we economists love to use letter abbreviations) can be stated as follows:

If investors did get an extra return (a risk premium) for bearing unsystematic risk, it would turn out that diversified portfolios made up of stocks with large amounts of unsystematic risk would give large returns than equally risky portfolios of stocks with less unsystematic risk. Investors would snap at the chance to have these higher returns, bidding up the prices of stocks with less unsystematic risk and selling stocks with equivalent bets but lower unsystematic risk. This process would continue until the prospective returns of stocks with the same betas were equalized and no risk premium could be obtained for bearing unsystematic risk. Any other result would be inconsistent with the existence of an efficient market.

The key relationship of the theory can be shown as follows: As the systematic risk (beta) of an individual stock (or portfolio) increases, so does the return an investor can expect. If an investor’s portfolio has a beta or zero, as might be the case if all his funds were invested in a bank savings certificate (beta would be zero since the returns from the certificate would not vary at all with swings in the stock market), the investor would receive some modest rate of return, which is generally called the risk-free rate of interest. As the indiviual takes on more risk, however, the return should increase. If the investor holds a portfolio with a beta of 1 (as, for example, holding a share in one of the broad stock-market averages) his return will equal the general return from common stocks. This return has over long periods of time exceeded the risk-free rate of interest, but the investment is a risky one. In certain periods the return is much less than the risk-free rate and involves taking substantial losses. This, as we have said, is precisely what is meant by risk.

Risk and Return According to the Capital-Asset Pricing Model

Those who remember their high school algebra will recall that any straight line can be written as an equation. The equation for the model is Rate of Return = Risk-free Rate + Beta (Return from Market – Risk-free Rate).

Alternately, the equation can be written as an expression for the reisj premium, that is, the rate of return on the portfolio or stock over and above the risk-free rate of interest:

Rate of Return – Risk-free Rate = Beta (Return from Market – Risk-free Rate).

The equation says that the risk premium you get on any stock or portfolio increases directly with the beta value you assume. Some may wonder what relationship beta has to the covariance concept that is so critical in the discussion of portfolio theory. The beta for any security is essentially the same thing as the covariance between that security and the market index as measured on the basis of past experience.

A number of different expected returns are possible simply by adjusting the beta of the portfolio. For example, suppose the investor put half of his money in a savings certificate and half in a share of the market averages. In this case, he would receive a return midway between the risk-free return and the return from the market and his portfolio would have an average beta of 0.5 (in general, the beta of a portfolio is simply the weighted average of the betas of its component parts). The CAPM then asserts very simply that to get a higher average long-run rate of return you should just increase the beta of your portfolio. An investor can get a portfolio with a beta larger than 1 either by buying high-beta stocks or by purchasing a portfolio with average volatility of margin. There was an actual fund proposed by a West Coast bank that would have allowed an investor to buy S&P average on margin, thus increasing both his risk and potential reward. Of course, in times of rapidly declining stock prices, such a fund would have enabled an investor to lose his shirt in a hurry. This may explain why the fund found few customers in the 1870s.

Just as stocks had their fads, so beta came into high fashion by the early 1970s. The Institutional Investor, the glossy prestige magazine that spent most of its pages chronicling the accomplishments of professional money managers, put its imprimatur on the movement in 1971 by featuring on its cover the letters BETA on top of a temple and including as its lead story “The Beta Cult!” The New Way to Measure Risk!” The magazine noted that money men whose mathematics hardly went beyond long division were now “tossing betas around with the abandon of Ph.D.s in statistical theory.” Even the Securities and Exchange Commission gave beta its approval as a risk measure in its Institutional Investors Study Report.

In Wall Street the early beta fans boasted that they could earn higher long-run rates of return simply by buying a few high-beta stocks. Those who thought they were able to tine the market thought they had an even better idea. They would buy high-beta stocks when they thought the market was going up, switching to low-beta ones when they feared the market might decline. To accommodate the enthusiasm for this new investment idea, beta measurement services proliferated among brokers, and it was a symbol of progressiveness for an investment house to provide its own beta estimates. Today, you can obtain beta estimates from brokers such as Merrill Lynch and investment advisory services such as Value Line. The beta boosters on the Street oversold their product with an abandon that would have shocked even the most enthusiastic academic scribblers intent on spreading the beta gospel.

Excerpt from 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.