correlation.
Exhibit 4.3 Diversification between Two Assets
The highest possible correlation and least diversification potential is when the assets' correlation coefficient is positive: perfect positive correlation. The straight line to the lower right between points A and B in Exhibit 4.3 plots the possible standard deviations and mean returns achievable by combining asset A and asset B under perfect positive correlation. The line is straight, meaning that the portfolio risk is a weighted average of the individual risks. This illustrates that there are no benefits to diversification when perfectly correlated assets are combined. The idea is that diversification occurs when the risks of unusual returns of assets tend to cancel each other out. This does not happen in the case of perfect positive correlation, because the assets always move in the same direction and by the same proportion.
The greatest risk reduction occurs when the assets' correlation coefficient is −1: perfect negative correlation. The two upper-left line segments connecting points A and B in Exhibit 4.3 plot the possible standard deviations and mean returns that would be achieved by combining asset A and asset B under perfect negative correlation. Notice that the line between A and B moves directly to the vertical axis, the point at which the standard deviation is zero. This illustrates ultimate diversification, in which two assets always move in opposite directions; therefore, combining them into a portfolio results in rapid risk reduction, or even total risk reduction. This zero-risk portfolio illustrates the concept of a perfect two-asset hedge and occurs when the weight of the investment in asset A is equal to the standard deviation of asset B divided by the sums of the standard deviations of A and B.
But the most realistic possibility is represented by the curve in the center of Exhibit 4.3. This is the more common scenario, in which the assets are neither perfectly positively nor perfectly negatively correlated; rather, they have some degree of dependent movement. The key point to this middle line in Exhibit 4.3 is that when imperfectly correlated assets are combined into a portfolio, a portion of the portfolio's risk is diversified away. The risk that can be removed through diversification is called diversifiable, nonsystematic, unique, or idiosyncratic risk.
In alternative investments, the concept of correlation is central to the discussion of portfolio implications. Further, graphs with standard deviation on the horizontal axis and expected return on the vertical axis are used as a primary method of illustrating diversification benefits. Assets that generate diversification benefits are shown to shift the attainable combinations of risk and return toward the benefit of the investor, meaning less risk for the same amount of return. The key point is that imperfect correlation leads to diversification that bends portfolio risk to the left, representing the improved investment opportunities afforded by diversification.
In the case of asset returns, true future correlations can only be estimated. Past estimated correlation coefficients not only are subject to estimation error but also are typically estimates of a moving target, since true correlations should be expected to change through time, as fundamental economic relationships change. Further, correlation coefficients tend to increase (offer less diversification across investments and asset classes) in times of market stress, just when an investor needs diversification the most.
4.3.5 Beta
The beta of an asset is defined as the covariance between the asset's returns and a return such as the market index, divided by the variance of the index's return, or, equivalently, as the correlation coefficient multiplied by the ratio of the asset volatility to market volatility:
where βi is the beta of the returns of asset i (Ri) with respect to a market index of returns, Rm. The numerator of the middle expression in Equation 4.19 measures the amount of risk that an individual stock brings into an already diversified portfolio. The denominator represents the total risk of the market portfolio. Beta therefore measures added systematic risk as a proportion of the risk of the index.
In the context of the capital asset pricing model (CAPM) and other single-factor market models, Rm is the return of the market portfolio, and the beta indicates the responsiveness of asset i to fluctuations in the value of the market portfolio. In the context of a single-factor benchmark, Rm would be the return of the benchmark portfolio, and the beta would indicate the responsiveness of asset i to fluctuations in the benchmark. In a multifactor asset pricing model, the beta indicates the responsiveness of asset i to fluctuations in the given risk factor, as is discussed in Chapter 6.
Exhibit 4.2 illustrates the computation of beta using a market index's return as a proxy for the market portfolio. Beta is similar to a correlation coefficient, but it is not bounded by +1 on the upside and −1 on the downside.
There are several important features of beta. First, it can be easily interpreted. The beta of an asset may be viewed as the percentage return response that an asset will have on average to a one-percentage-point movement in the related risk factor, such as the overall market. For example, if the market were to suddenly rise by 1 % in response to particular news, a fund with a market beta of 0.95 would be expected on average to rise 0.95 %, and a fund with a beta of 2.0 would be expected to rise 2 %. If the market falls 2 %, then a fund with a beta of 1.5 would have an expected decline of 3 %. But actual returns deviate from these expected returns due to any idiosyncratic risk. The risk-free asset has a beta of zero, and its return would therefore not be expected to change with movements in the overall market. The beta of the market portfolio is 1.0.
The second feature of beta is that it is the slope coefficient in a linear regression of the returns of an asset (as the Y, or dependent variable) against the returns of the related index or market portfolio (as the X, or independent variable). Thus, the computation of beta in Exhibit 4.2 using Equation 4.19 may be viewed as having identified the slope coefficient of the previously discussed linear regression. Chapter 9 discusses linear regression.
Third, because beta is a linear measure, the beta of a portfolio is a weighted average of the betas of the constituent assets. This is true even though the total risk of a portfolio is not the weighted average of the total risk of the constituent assets. This is because beta reflects the correlation between an asset's return and the return of the market (or a specified risk factor) and because the correlation to the market does not diversify away as assets are combined into a portfolio.
Similar to the correlation coefficient between the returns of two assets, the beta between an asset and an index is estimated rather than observed. An estimate of beta formed with historical returns may differ substantially from an asset's true future beta for a couple of reasons. First, historical measures such as beta are estimated with error. Second, the beta of most assets should be expected to change through time as market values change and as fundamental economic relationships change. In fact, beta estimations based on historical data are often quite unreliable, although the most reasonable estimates of beta that are available may be based at least in part on historical betas.
4.3.6 Autocorrelation
The autocorrelation of a time series of returns from an investment refers to the possible correlation of the returns with one another through time. For example, first-order autocorrelation refers to the correlation between the return in time period t and the return in the previous time period (t − 1). Positive first-order autocorrelation is when an above-average (below-average) return in time period t − 1 tends to be followed by an above-average (below-average) return in time period t. Conversely, negative first-order autocorrelation is when an above-average (below-average)