Investment Diversification and Portfolio Risk Analysis - Financial Management

The preceding sections examined the risk and returns associated with investments in single assets —either financial assets (securities) or physical assets. However, most individuals and institutions invest in a portfolio of assets, that is, a collection of two or more assets.

Commercial banks invest in many different types of financial assets when they make loans to consumers and businesses; individuals invest in many different types of financial assets when they buy securities, such as bank certificates of deposit and corporate bonds and stocks; and corporations invest in many different kinds of physical assets when they acquire production and distribution facilities (i.e., plants and equipment).

Consequently, it is important to know how the returns from portfolios of investments behave over time—not just how the returns from individual assets in the portfolio behave. Portfolio risk, the risk associated with collections of financial and physical assets, is considered in this and the following two sections. The questions of importance are as follows:

  • What return can be expected to be earned from the portfolio?
  • What is the risk of the portfolio?

Consider the following example. Suppose that Alcoa (the aluminum industry’s largest producer) is considering diversifying into gold mining and refining. During economic boom periods, aluminum sales tend to be brisk; gold, on the other hand, tends to be most in demand during periods of economic uncertainty. Therefore, let us assume that the returns from the aluminum business and the gold mining business are inversely, or negatively, related.

If Alcoa expands into gold mining and refining, its overall return will tend to be less variable than individual returns from these businesses. This effect is illustrated in Figure. Panel (a) shows the variation of rates of return in the aluminum industry over time; panel (b) shows the corresponding variation of returns from gold mining over the same time frame; and panel (c) shows the combined rate of return for both lines of business.As can be seen from this figure, when the return from aluminum operations is high, the return from gold mining tends to be low, and vice versa. The combined returns are more stable and therefore less risky.

Average Annual Returns and Standard Deviation of Returns of Various

Average Annual Returns and Standard Deviation of Returns of Various

This portfolio effect of reduced variability results because a negative correlation exists between the returns from aluminum operations and the returns from gold mining. The correlation between any two variables—such as rates of return or net cash flows—is a relative statistical measure of the degree to which these variables tend to move together. The correlation coefficient (r) measures the extent to which high (or low) values of one variable are associated with high (or low) values of another. Values of the correlation coefficient can range from +1.0 for perfectly positively correlated variables to –1.0 for perfectly negatively correlated variables. If two variables are unrelated (that is, uncorrelated), the correlation coefficient between these two variables will be 0.

Illustration of Diversification and Risk Reduction:Alcoa

Illustration of Diversification and Risk Reduction:Alcoa

illustrates perfect positive correlation, perfect negative correlation, and zero correlation for different pairs of common stock investments. For perfect positive correlation, panel (a), high rates of return from Stock L are always associated with high rates of return from Stock M; conversely, low rates of return from L are always associated with low rates of return from M. For perfect negative correlation, panel (b), however, the opposite is true; high rates of return from Stock P are associated with low rates of return from Stock Q and vice versa. For zero correlation, panel (c), no perceptible pattern or relationship exists between the rates of return on Stocks V and W.

Illustration of (a) Perfect Positive, (b) Perfect Negative, and (c) Zero

Illustration of (a) Perfect Positive, (b) Perfect Negative, and (c) Zero

In practice, the returns from most investments a firm or individual considers are positively correlated with other investments held by the firm or individual. For example, returns from projects that are closely related to the firm’s primary line of business have a high positive correlation with returns from projects already being carried out and thus provide limited opportunities to reduce risk. In the Alcoa example, if Alcoa were to build a new smelter, it would not realize the risk reduction possibilities that investing in gold mining and refining would produce.

Similarly, the returns from most common stocks are positively correlated because these returns are influenced by such common factors as the general state of the economy, the level of interest rates, and so on. In order to explore further the concepts of diversification and portfolio risk, it is necessary to develop more precise measures of portfolio returns and risk.

Expected Returns from a Portfolio

When two or more securities are combined into a portfolio, the expected return of the portfolio is equal to the weighted average of the expected returns from the individual securities. If a portion, wA, of the available funds (wealth) is invested in Security A, and the remaining portion, wB, is invested in Security B, the expected return of the portfolio, rˆp, is as follows:

ˆrp = wAˆrA + wBˆrB

where rˆA and rˆB are the expected returns for Securities A and B, respectively. Furthermore, wA + wB = 1, indicating that all funds are invested in either Security A or Security B.

For example, consider a portfolio consisting of the common stock of American Electric Power (A), a public utility company, and Boeing (B), an aerospace producer. The expected returns on the two stocks are 12 percent (rˆA) and 16 percent (rˆB), respectively.

A portfolio consisting of 75 percent (wA) invested in American Electric Power and the remainder, or 25 percent (wB), invested in Boeing would yield an expected return, by Equation 6.7, of rˆp = 0.75(12%) + 0.25(16%)= 13.0%

(columns wA and rˆp) and illustrate the relationship between the expected return for a portfolio containing Securities A and B and the proportion of the total portfolio invested in each security. For example, when wA = 1.0 (100%) and wB = 0 (because wA + wB = 1.0), the expected portfolio return is 12 percent, the same as the return for A.When wA = 0.5 (50 percent) and wB = 0.5 (50 percent), the expected portfolio return is 14 percent. As shown earlier, when wA = 0.75 and wB = 0.25, the expected portfolio return is 13 percent. Thus, it can be seen that the expected return from a portfolio of securities is simply equal to the weighted average of the individual security returns, where the weights represent the proportion of the total portfolio invested in each security.

Expected Returns and Portfolio Risk from a Portfolio of the Stocks of

Expected Returns and Portfolio Risk from a Portfolio of the Stocks of

Expected Return from a Portfolio of the Stocks of American Electric

Expected Return from a Portfolio of the Stocks of American Electric

In general, the expected return from any portfolio of n securities or assets is equal to the sum of the expected returns from each security times the proportion of the total portfolio invested in that security:

Expected Returns

Although the expected returns from a portfolio of two or more securities can be computed as a weighted average of the expected returns from the individual securities, it is generally not sufficient merely to calculate a weighted average of the risk of each individual security to arrive at a measure of the portfolio’s risk.Whenever the returns from the individual securities are not perfectly positively correlated, the risk of any portfolio of these securities may be reduced through the effects of diversification.

Thus, diversification can be achieved by investing in a set of securities that have different risk–return characteristics. The amount of risk reduction achieved through diversification depends on the degree of correlation between the returns of the individual securities in the portfolio. The lower the correlations among the individual securities, the greater the possibilities of risk reduction.

The risk for a two -security portfolio, measured by the standard deviation of portfolio returns, is computed as follows:

Portfolio Risk

where wA is the proportion of funds invested in Security A; wB is the proportion of funds invested in Security B; wA + wB = 1; s2 A is the variance of returns from Security A (or the square of the standard deviation for Security A, sA); s2 B is the variance of returns from Security B (or the square of the standard deviation for Security B, sB); and rAB is the correlation coefficient of returns between Securities A and B.

Consider, for example, the portfolio discussed earlier consisting of the common stock of American Electric Power (A) and Boeing (B). The standard deviations of returns for these two securities are 10 percent (sA) and 20 percent (sB), respectively. Furthermore, suppose that the correlation coefficient (rAB) between the returns on these securities is equal to +0.50. Using Equation , a portfolio consisting of 75 percent (wA) invested in American Electric Power and 25 percent (wB) in Boeing would yield a standard deviations of portfolio returns of

Portfolio Risk

With the techniques just described for calculating expected portfolio return and risk, we can now examine in more detail the risk versus return trade-offs associated with investment diversification. The following three special cases illustrate how the correlation coefficient can affect portfolio risk.

Case I: Perfect Positive Correlation (r = + 1.0) Table (columns rˆp and rAB = +1.0) and panel (a) of Figure illustrate the risk–return trade-offs associated with portfolios consisting of various combinations of American Electric Power (A) and Boeing (B) stock when rAB = +1.0. When the returns from the two securities are perfectly positively correlated, the risk of the portfolio is equal to the weighted average of the risk of the individual securities (10 and 20 percent in this example). Therefore, no risk reduction is achieved when perfectly positively correlated securities are combined in a portfolio.

Case II: Zero Correlation (r = 0.0) Table columns rˆp and rAB = 0.0) and panel (b) of Figure 6illustrate the possible trade-offs when rAB = 0.0. In this case, we see that diversification can reduce portfolio risk below the risk of either of the securities that make up the portfolio. For example, an investment consisting of 75 percent in American Electric Power (A) stock and 25 percent in Boeing (B) stock has a portfolio standard deviation of only 9.01 percent, which is less than the standard deviations of either of the two securities (10 and 20 percent, respectively) in the portfolio.

In general, when the correlation coefficient between the returns on two securities is less than 1.0, diversification can reduce the risk of a portfolio below the weighted average of the total risk of the individual securities. The less positively correlated the returns from two securities, the greater the portfolio effects of risk reduction. For example, the expected returns from an investment in two firms in different industries, such as ExxonMobil and Delta Airlines, should generally be less positively correlated than the expected returns between two firms in the same industry, such as ExxonMobil and Shell.

Relationship Between Portfolio Expected Return and Risk for the

Relationship Between Portfolio Expected Return and Risk for theRelationship Between Portfolio Expected Return and Risk for the

Case III: Perfect Negative Correlation (r = – 1.0) Table (columns rˆp and rAB = –1.0) and panel (c) of Figure showsthe risk –return relationship when rAB = –1.0. As illustrated, with perfectly negatively correlated returns, portfolio risk can be reduced to zero. In other words, with a perfect negative correlation of returns between two securities, there will always be some proportion of the securities that will result in the complete elimination of portfolio risk. In summary, these three special cases serve to illustrate the effect that the correlation coefficient has on portfolio risk, as measured by the standard deviation. For any given pair of securities, the correlation coefficient is given (or can be estimated), and this number determines how much risk reduction can be achieved with various weighted combinations of the two securities.

Efficient Portfolios and the Capital Market Line

The risk–return relationships just discussed can be extended to analyze portfolios involving more than two securities. For example, consider the graph shown in Figure. Each dot within the shaded area represents the risk (standard deviation) and expected return for an individual security available for possible investment.

The shaded area (or opportunity set) represents all the possible portfolios found by combining the given securities in different proportions. The curved segment from A to B on the boundary of the shaded area represents the set of efficient portfolios, or the efficient frontier. A portfolio is efficient if, for a given standard deviation, there is no other portfolio with a higher expected return, or for a given expected return, there is no other portfolio with a lower standard deviation.

Portfolio Opportunity Set

Portfolio Opportunity Set

Risk -averse investors, in choosing their optimal portfolios, need only consider those portfolios on the efficient frontier. The choice of an optimal portfolio, whether portfolio A that minimizes risk or portfolio B that maximizes expected return or some other portfolio on the efficient frontier, depends on the investor’s attitude toward risk (that is, risk aversion). More conservative investors will tend to choose lower -risk portfolios (closer to A); more aggressive investors will tend to select higher -risk portfolios (closer to B). If investors are able to borrow and lend money at the risk-free rate (rf), they can obtain any combination of risk and expected return on the straight line joining rf and portfolio m as shown in Figure.

When the market is in equilibrium, portfolio m represents the Market Portfolio, which consists of all available securities,weighted by their respective market values. The line joining rf and m is known as the capital market line. The capital market line has an intercept of rf and a slope of (rm rf)/(sm – 0) = (rm rf)/sm. The slope of the capital market line measures the equilibrium market price of risk or the additional expected return that can be obtained by incurring one additional unit of risk (one additional percentage point of standard deviation). Therefore, the equation of the capital market line is and indicates that the expected return for an efficient portfolio is equal to the risk-free rate plus the market price of risk [(rm rf)/sm] times the amount of risk (sp) of the portfolio under consideration.

Efficient Portfolios

Capital Market Line

Capital Market Line

Any risk–return combination on this line between rf and m can be obtained by investing (i.e., lending) part of the initial funds in the risk-free security (such as U.S. Treasury bills) and investing the remainder in portfolio m. Any combination beyond m on this line can be obtained by borrowing money at the risk-free rate and investing the borrowed funds (as well as the initial funds) in portfolio m (that is, purchasing securities on margin).

With the ability to borrow and lend at the risk-free rate, the choice of an optimal portfolio for risk-averse investors involves determining the proportion of funds to invest in the Market Portfolio (m) with the remaining proportion being invested in the risk-free security. More conservative investors will tend to choose investments nearer to the rf point on the capital market line. More aggressive investors will tend to select investments closer to, or possibly beyond, point m on the capital market line.


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