Introduction to Asset Class Statistics

Elsewhere on this site you will find "green buttons" that provide background information on the different asset classes in which you may invest. In this section, we'll introduce the statistical measures you will encounter on those pages. First, all of the data you'll be seeing will be real, rather than nominal. By that we mean all of the numbers will be net of inflation. We take this approach because one goal of investing is to preserve, and ideally increase your purchasing power over time. Given this, the fact that an asset class earned 12% in nominal terms is meaningless unless you know what inflation was over the same period. If it was 5%, you earned a 7% real return. If it was 14%, you earned a real return of negative (2%).

Second, we will express returns two ways. Arithmetic returns are the simple average of a series of returns earned over a given time period. Geometric returns are the compound annual return earned over the same period. If the same return is earned each year, then the arithmetic and geometric means are the same. However, if different returns are earned in each period, then the two average returns will be different. For example, an investment that returns 15%, (10%), and 10% over three years has an arithmetic average return of 5% per year, but a geometric, or compound average annual return of about 4.42% per year. This example also illustrates a larger point: the greater the volatility of annual returns, the more the geometric average return will tend to be below the arithmetic average return. So why do we use both of these terms? In any given year, the arithmetic return is the best measure of the return you are likely to earn in any given year. This is therefore the return that we use in our asset allocation models. It is also the return we'll use when we develop estimates of the returns different asset classes may earn in the future. However, over a longer holding period (which is a more accurate description of the situation faced by most investors), geometric returns better describe the returns that were actually realized in the past. So we'll use them too, to provide perspective.

Finally, no single statistic provides a complete picture of the riskiness of an asset class. So we'll use four of them. We can hear the groans now! Let us explain. A lot of entry level finance textbooks define risk as the extent to which returns over a given period of time are distributed around the arithmetic average return for the period. In stats-speak, this is known as the variance or the standard deviation of returns (the latter is the square root of the former, but we won't belabor the point...). The problem is, when asked to define financial risk, most people don't talk about the standard deviation of returns. Typically, they'll say something like "risk is the chance I'll lose my money", or "risk is the probability I'll fall short of achieving my goals." In other words, in most people's minds, risk is not the symmetrical concept that textbooks assume it is when they equate it with the standard deviation of returns. So we need to take some other factors into consideration when we talk about the riskiness of an asset class.

The first we've already mentioned. It is the extent to which the returns on a given asset class vary with those on other asset classes. In stats-speak, this is called the covariance, or correlation of returns. If the latter is equal (at one extreme) to 1.0, they two series move in tandem in the same direction (not good for your portfolio). If (at the other extreme), the correlation of returns between two asset classes equals (1.0), they also move in tandem, but in opposite directions (surprisingly, not as good for a portfolio as you might think). And if their correlation equals zero, their returns are completely unrelated. Intuitively, to minimize the downside risk in a portfolio, you'd like to have some asset classes that have low positive or negative correlations with one another.

There is, however, an important catch. When we say that a correlation of zero means that the returns on two asset classes are completely unrelated, we're assuming that both return series are "normal" distributions. These are the familiar bell curve found in every introductory statistics textbook that has ever been written. In fact, once you assume that any set of data is normally distributed, a whole world of different statistical tests opens up to you. For example, you can say with confidence that about 68% of the time, the return on a normally distributed asset will fall within the range defined by the arithmetic average, plus or minus one standard deviation, and about 95% of the time, it will fall between the mean plus or minus two standard deviations. Unfortunately, not many financial assets have returns that are normally distributed. It is for that reason that we have to drag two other statistical terms into our conversation.

The first is called "skewness", and it measures the extent to which a given distribution of returns is "off center" or "tilted" compared to a normal distribution. Specifically, a normal distribution is symmetrical, and has a skewness of zero. A skewness of less than zero indicates that more returns fall below their arithmetic average than above it, while a positive skewness indicates just the opposite. The returns on many asset classes (but not all of them) are negatively skewed, which means that returns below the average are more likely than returns above it. Generally speaking, investors don't like negative skewness (actually, to be technically correct, what we really don't like is co-skweness, or the extent the returns on one asset class tend to skew in the same direction as those on another). Sometimes, however, an investor will accept negative skewness in order to earn the higher returns offered by an asset class. In addition, in a portfolio, positively skewed asset classes can offset negatively skewed ones.

The second statistical measure we need to use to assess asset class risk is called "kurtosis". This measures the extent to which a given distribution of returns is taller or shorter than the normal distribution. Consider the case of two distributions that have the same average return. One of these is normally shaped (that is, it has the form of the typical "bell curve"), while the other is not. If the peak of the non-normal distribution is taller than the normal distribution's peak, it is said to have "excess kurtosis". Because relatively more returns are clustered closer to the average return than in the case of the normal distribution, it must also be the case that relatively more returns also lie in the tails (that is, the extreme ends) of the distribution. In contrast, distributions with less than normal kurtosis have lower peaks than the normal distribution, but also have a lower percentage of returns located in their tails. In both cases, standard deviation will not accurately describe the dispersion of returns around the average. In the case of excess kurtosis, more returns will occur at the extremes than standard deviation would predict, while just the opposite would be the case if kurtosis were less than normal.

Investors' attitude toward kurtosis (again, the technically correct term is co-kurtosis) is harder to generalize about than their attitude toward skewness. Where skewness equals zero or is negative, investors may prefer a low level of kurtosis, as that implies more predictable returns (that is, a series of returns with fewer large extremes). However, where skewness is positive (that is, where above average returns are more likely than below average returns), an investor may prefer a higher level of kurtosis, which implies that relatively large positive returns are more likely than relatively large negative returns.

The key point is that when it comes to judging the riskiness of an asset class, kurtosis and skewness have to be looked at together. The riskiest situation is one with high kurtosis and negative skewness. In this case, you are likely to get more and larger downside surprises than you bargained for (assuming past returns are a reasonable guide, in a statistical sense, to what you can expect in the future). On the other hand, as we have described, high kurtosis also can be a good thing when skewness is positive -- in which case, you'll get more pleasant surprises than you would if the returns were normally distributed.

The final point is that our reviews look at how different asset classes performed during different periods of time. For example, we look at the 70s, 80s, and 90s, which represented periods of high, moderate, and low inflation in many countries. We also look at two shorter crisis period, covering the equity market crash of 1987 and the collapse of Long Term Capital Management in 1998. Finally, we look at the geometric average returns different asset classes delivered over the 1971-2002 period, or the longest set of data we have available. We hope this will help you develop a better intuitive feeling for the real returns different asset classes have delivered under varying historical circumstances. We also hope it will put into better perspective the future return estimates we will present for each asset class.