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) return in time period t − 1 tends to be followed by a below-average (above-average) return in time period t. Zero autocorrelation indicates that the returns are linearly independent through time. Positive autocorrelation is seen in trending markets; negative autocorrelation is seen in markets with price reversal tendencies.
We start here by assuming the simplest scenario: The returns on an investment are statistically independent through time, which means there is no autocorrelation. Further, we assume that the return distribution is stationary (i.e., the probability distribution of the return at each point in time is identical). Under these strict assumptions, the distribution of log returns over longer periods of time will tend toward being a normal distribution, even if the very short-term log returns are not normally distributed.
How do we know that log returns will be roughly normally distributed over reasonably long periods of time if the returns have no autocorrelation and if very short-term returns have a stationary distribution? One explanation is that the log return on any asset over a long time period such as a month is the sum of the log returns of the sub-periods. Even if the returns over extremely small units of time are not normally distributed, the central limit theorem indicates that the returns formed over longer periods of time by summing the independent returns of the sub-periods will tend toward being normally distributed.
Why might we think that returns would be uncorrelated through time? If a security trades in a highly transparent, competitive market with low transaction costs, the actions of arbitrageurs and other participants tend to remove pronounced patterns in security returns, such as autocorrelation. If this were not true, then arbitrageurs could make unlimited profits by recognizing and exploiting the patterns at the expense of other traders.
However, markets for securities have transaction costs and other barriers to arbitrage, such as restrictions on short selling. Especially in the case of alternative investments, arbitrage activity may not be sufficient to prevent nontrivial price patterns such as autocorrelation. The extent to which returns reflect nonzero autocorrelation is important because autocorrelation can impact the shape of return distributions. The following material discusses the relationships between the degree of autocorrelation and the shapes of long-period returns relative to short-period returns.
Autocorrelation of returns can be used as a general term to describe possible relationships or as a term to describe a specific correlation measure. Equation 4.20 describes autocorrelation in the context of a return series with constant mean:
