Diversifying investment is a risk management technique that is used to minimize the risk of individual securities by investing in a portfolio of securities. That is, if investors reduce their reliance on particular assets, they can more easily bear a downturn on an individual security. Diversification is the cornerstone of modern portfolio theory that has seen wide applications in financial decision making. Both firms and individuals diversify investments. The former diversify by investing in individual activities, while the latter diversify by investing in portfolios rather than in individual assets.

In portfolio theory, investors are assumed to be risk averse, that is, they prefer lower to higher risk for a given return and will accept a higher risk only if they are compensated with a higher return. This creates indifference curves, lines on which risk and return combinations are offered to investors and as long as investors are on this line, they are indifferent on where to invest. Namely, as risk increases, return increases as well, so that investors are compensated for the increased risk they take on. The expected return of an investment in a single asset is the sum of the returns on that investment conditional on the probability of every return occurring. For example, if there is 60 percent probability that an investment earns a 5 percent return and 40 percent probability that it will earn 15 percent, the expected return on this investment is 9 percent, which is the weighted average of the probability of each event occurring. Furthermore, the expected return on an investment portfolio, i.e., on a combination of investments, is the sum of the weighted average of the returns of the individual holdings of the portfolio.

On the other hand, variance measures the variability of returns and is used as a benchmark for the risk of individual assets. In a single asset, the variance is the square root of the difference between the realized and the expected return on the asset. However, the variance of a portfolio of assets is proportional to the weights that are invested on each asset, to the variance of individual assets, and also to the degree of correlation between the individual assets.

## Correlation

Correlation is a measure of interrelationship between assets and measures the extent to which the returns from two investments move together. If the correlation coefficient is +1 this means that the returns of the two investments always change proportionately in the same direction (perfect positive correlation). A correlation coefficient of –1 means that the returns always move proportionately in the opposite directions (perfect negative correlation). When the correlation coefficient is zero, this means that the returns have no correlation whatsoever and their returns are independent from one another.

The essence of diversification lies on the correlation coefficient of securities. Since the variance of a portfolio has two components, the individual securities’ variance component and the correlation coefficient component, when two assets are negatively correlated, i.e., when the second component is negative, the portfolio variance is greater than portfolio variance. In other words, by investing in securities with negative correlation, the overall risk of the portfolio is reduced. If the correlation is zero, the second term of the portfolio variance is zero, so the variance of the portfolio is simply the sum of the individual variances of each stock included in the portfolio. The benefit of diversification is eliminated when the correlation coefficient is +1. Thus, other things equal, the smaller the correlation between two assets, the smaller the variance (risk) of the portfolio of the two assets. One can create the same risk level and higher expected returns by diversifying one’s investments across a wide range of stocks. The only assumption is that as long as the individual stocks are not perfectly correlated, the risk-return combination of the portfolio will be better than the risk-return combinations of all individual stocks. The benefit of diversification increases as the degree of correlation decreases. This creates the efficient frontier, which represents the set of portfolios that give the highest return at each level of risk, or alternatively, the lowest risk at each level of return.

The above is also extended to portfolios of assets of more than two individual securities. Even though the computation of the correlation of more than two securities becomes very complex, the principle of diversification applies to complex portfolios providing that none of the securities are perfectly positively correlated. Also, as the number of securities that are included in a portfolio increases, the benefit that is derived from diversification of adding one more asset to the portfolio decreases. In other words, the benefit that is derived from the reduction in risk by increasing the number of holdings in a portfolio is outweighed by the additional costs (transaction and monitoring costs) that are associated with increasing the number of securities. For that reason, it has been studied that individual investors do not need to hold more than 1– 15 assets in order to capture 90 percent of the benefits from diversification, while for individual investors the number of shares should not exceed 50.

## Limits

However, there are limits to diversification. These limits are set by the two types of risk that are embedded in asset prices. The first type of risk is called unique or unsystematic risk that refers to the effect that random events may have on individual firms. These events are unique and can be diversified away. The second type of risk is called systematic or market risk and it refers to the risk that is embedded in all securities of a single market, a single industry, or even a single country. For example, the interest rate risk is a type of systematic risk that affects all securities of one country. Thus, as long as portfolios are constructed of assets that include securities from one country, the interest rate risk is no diversifiable and will only be embedded in the total variability of the constructed portfolios.

The way to minimize the systematic risk is to create portfolios of securities that share very few common elements. For example, one can eliminate the country-specific risk by investing in assets in a wide range of countries. Also, the stock-specific risk is diversified away by maintaining a portfolio of assets that is a combination between different types of securities, i.e., bonds, deposits, works of art, real estate. As a rule of thumb, a well-diversified portfolio should not contain more than 40 percent of its value in individual stocks and the remainder should be invested in other forms according to the investor’s liquidity and risk return preferences.

In practice the portfolio theory of diversification has found computational difficulties, since, for example, for a portfolio of 300 securities more than 45,000 need to be calculated. In addition, diversifying investment theory is difficult to apply in physical investment decisions as, in contrast with financial assets, the calculation of correlation coefficients in physical investments is not always accurate. Further, some difficulties become apparent when applying the diversification theory to a multiperiod model. Also, should firms that diversify their individual activities be rewarded for their actions? In other words, should firms evaluate the correlation of each project with its portfolio of existing projects? If so, then the value of a firm’s portfolio of individual projects should be greater than the value of the sum of the parts, thus the present value principle is no longer applicable.

Fortunately, diversification is very difficult to apply in firms in such a manner. This stems from the fact that individual investors are able to diversify their investments more easily. So, investors can invest in a firm this week and pull out next week, whereas a firm would find it extremely difficult to do the same with an investment project.

## Applications

The theory of investment diversification has found wide applications in the construction of indexes that are designed to track the performance of individual securities. Such specific indices are the index funds that contain all stocks that are traded in an index and can be sold as a ready-made investment portfolio to individual investors. The fund’s objectives are to offer simple, diversified solutions at a low cost, yet the main disadvantages of such funds are the tracking error that results from failing to replicate the index accurately and the fact that index funds can only produce the return of the index and not outperform it.

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