past behavior is a reasonable indication of its future behavior. However, many alternative investments are especially problematic in this context. For example, historical data may not exist for venture capital investment in new firms or may be difficult to observe or to obtain in cases such as private equity, where most or all trades are not publicly observable. Especially in alternative investments such as hedge funds, return distributions are expected to change as the fund's investment strategies and use of leverage change through time. In these cases and many others, ex ante return distributions may need to be based on economic analysis and modeling rather than simply projected from ex post data.
Nevertheless, whether based on prior observations or on economic analysis, the return distribution is a central tool for understanding the characteristics of an investment. The normal distribution is the starting point for most statistical applications in investments.
4.1.2 The Normal Distribution
The normal distribution is the familiar bell-shaped distribution, also known as the Gaussian distribution. The normal distribution is symmetric, meaning that the left and right sides are mirror images of each other. Also, the normal distribution clusters or peaks near the center, with decreasing probabilities of extreme events.
Why is the normal distribution so central to statistical analysis in general and the analysis of investment returns in particular? One reason is empirical: The normal distribution tends to approximate many distributions observed in nature or generated as the result of human actions and interactions, including financial return distributions. Another reason is theoretical: The more a variable's change results from the summation of a large number of independent causes, the more that variable tends to behave like a normally distributed variable. Thus, the more competitively traded an asset's price is, the more we would expect that the price change over a small unit of time would be the result of hundreds or thousands of independent financial events and/or trading decisions. Therefore, the probability distribution of the resulting price change should resemble the normal distribution. The formal statistical explanation for the idea that a variable will tend toward a normal distribution as the number of independent influences becomes larger is known as the central limit theorem. Practically speaking, the normal distribution is relatively easy to use, which may explain some of its popularity.
4.1.3 Log Returns and the Lognormal Distribution
For simplicity, funds often report returns based on discrete compounding. However, log returns offer a distinct advantage, especially for modeling a return probability distribution. In a nutshell, the use of log returns allows for the modeling of different time intervals in a manner that is simple and internally consistent. Specifically, if daily log returns are normally distributed and independent through time, then the log returns of other time intervals, such as months and years, will also be normally distributed. The same cannot be said of simple returns. Let's take a closer look at why log returns have this property.
The normal distribution replicates when variables are added but not when they are multiplied. This means that if two variables, x and y, are normally distributed, then the sum of the two variables, x + y, will also be normally distributed. But because the normal distribution does not replicate multiplicatively, x × y would not be normally distributed. Aggregation of discretely compounded returns is multiplicative. Thus, if R1, R2, and R3 represent the returns for months 1, 2, and 3 using discrete compounding, then the product [(1 + R1)(1 + R2)(1 + R3)] − 1 represents the return for the calendar quarter that contains the three months. If the monthly returns are normally distributed, then the quarterly return is not normally distributed, and vice versa, since the normal distribution does not replicate multiplicatively. Therefore, modeling the distribution of discretely compounded returns as being normally distributed over a particular time interval (e.g., monthly) technically means that the model will not be valid for any other choice of time interval (e.g., daily, weekly, annually).
However, the use of log returns, discussed in Chapter 3, solves this problem. If Rm = ∞1, Rm = ∞2, and Rm = ∞3 are monthly log returns, then the quarterly log return is simply the sum of the three monthly log returns. The normal distribution replicates additively; thus, if the log returns over one time interval can be modeled as being normally distributed, then the log returns over all time intervals will be lognormal as long as they are statistically independent through time.
Further, log returns have another highly desirable property. The highest possible simple (non-annualized) return is theoretically + ∞, while the lowest possible simple return for a cash investment is a loss of −100 %, which occurs if the investment becomes worthless. However, the normal distribution spans from − ∞ to + ∞, meaning that simple returns, theoretically speaking, cannot truly be normally distributed; a simple return of −200 % is not possible. Thus, the normal distribution may be a poor approximation of the actual probability distribution of simple returns. However, log returns, like the normal distribution itself, can span from − ∞ to + ∞.
There are two equivalent approaches to model returns that address these problems: (1) use log returns and assume that they are normally distributed, or (2) add 1 to the simple returns and assume that it has a lognormal distribution. A variable has a lognormal distribution if the distribution of the logarithm of the variable is normally distributed. The two approaches are identical, since the lognormal distribution assumes that the logarithms of the specified variable (in this case, 1 + R) are normally distributed.
In summary, it is possible for returns to be normally distributed over a variety of time intervals if those returns are expressed as log returns (and are independent through time). If the log returns are normally distributed, then the simple returns (in the form 1 + R) are said to be lognormally distributed. However, if discretely compounded returns (R) are assumed to be normally distributed, they can only be normally distributed over one time interval, such as daily, since returns computed over other time intervals would not be normally distributed due to compounding.
4.2 Moments of the Distribution: Mean, Variance, Skewness, and Kurtosis
Random variables, such as an asset's return or the timing of uncertain cash flows, can be viewed as forming a probability distribution. Probability distributions have an infinite number of possible shapes, only some of which represent well-known shapes, such as a normal distribution.
The moments of a return distribution are measures that describe the shape of a distribution. As an analogy, in mathematics, researchers often use various parameters to describe the shape of a function, such as its intercept, its slope, and its curvature. Statisticians often use either the raw moments or the central moments of a distribution to describe its shape. Generally, the first four moments are referred to as mean, variance, skewness, and kurtosis. The formulas of these four moments are somewhat similar, differing primarily by the power to which the observations are raised: mean uses the first power, variance squares the terms, skewness cubes the terms, and kurtosis raises the terms to the fourth power.
4.2.1 The Formulas of the First Four Raw Moments
Statistical moments can be raw moments or central moments. Further, the moments are sometimes standardized or scaled to provide more intuitive measures, as will be discussed later. We begin with raw moments, discussing the raw moments of an investment's return, R. Raw moments have the simplest formulas, wherein each moment is simply the expected value of the variable raised to a particular power:
(4.1)
The most common raw moment is the first raw moment and is known as the mean, or expected value, and is an indication of the central tendency
