It is hard to predict price variations of financial assets so it is usual to assume that successive returns are relatively independent of each other. This means that uncertainty will increase as the holding period increases, the distribution will become more dispersed and its variance will increase. Put another way, the variance of -day returns will increase with n. Therefore it is not possible to compare w-day variance with w-day variance on the same scale. It is standard to assume statistically independent returns4 and to express a standard deviation in annual terms. Thus in financial markets we define

Annual volatility = (100o\A4)%, (1.2)

where A is an annualizing factor, the number of returns per year.5 In this way volatilities of returns of different frequencies may be compared on the same scale in a volatility term structure (§2.2.2, §3.3 and §4.4.1).

To understand what correlation is, consider a joint density of two random variables.6 A joint density may be visualized as a mountain: the more symmetric this mountain is about both the axes representing the two variables, the less information can be gained about the value of one variable by knowing the value of the other; that is, the lower the correlation between the two variables. For highly correlated variables the joint density will have more of a ridge in a direction between the axes of the two variables.

Figure 1.2 shows three scatter plots, where synchronous observations on each of the returns are plotted as horizontal and vertical coordinates. A scatter plot is a sample from the joint density of the two returns series, and so if the returns have no correlation their scatter plot will be symmetrically dispersed, like the one in Figure 1.2a; a high value on one axis will be no indication that the corresponding value on the other axis will be high or low. But if they have a high positive correlation the joint density will have a ridge sloping upwards, as in Figure 1.2b; when one variable has a high value the other will also tend to have a high value. If they have negative correlation the joint density will have a downwards sloping ridge as in Figure 1.2c; when one variable has a high value the other will tend to have a low value, and vice versa.

Correlation is a measure of co-movements between two returns series. Strong positive correlation indicates that upward movements in one returns series tend to be accompanied by upward movements in the other, and similarly

4 Two random variables X and Y are independent if and only if their joint density function h(x, y) is simply the product of the two marginal densities. That is. if X has density ) and Y has density g(y) then X and Y are independent if and only if h(x, y)=f(x)g(y).

5 The annualizing factor is a normalizing constant: the variance increases with the holding period but the annualizing factor decreases. The number of trading days (or risk days) per year is usually taken for the conversion of a daily standard deviation into an annualized percentage; that is. often =250 or 252 in (1.2). Note the continuation of this footnote in §2.1.1 (footnote 5).

6 The joint density . , ) of two random variables X and Y is a real-valued function of the two variables where the total area underneath the surface is one: Jy\x, y)dxdy=\. The joint probability that X takes values in one range and Y takes values in another range is the area under the function defined by these two ranges.

Uncertainty will increase as the holding period increases, the distribution will become more dispersed and its variance will increase

• • • •

• • • •

Figure 1.2 (a) Zero correlation; (b) positive correlation; (c) negative correlation.

downward movements of the two series tend to go together. If there is a strong negative correlation then upward movements in one series are associated with downward movements in the other.

A simple statistical measure of co-movements between two random variables is covariance, the first product moment about the mean of the joint density function. That is, cov(X, Y) = E[(X-\ix){Y- uy)], where \ix=E(X) and =£( ). Covariance is determined not only by the degree of co-movement but also by the size of the returns. For example, monthly returns are of a much

greater order of magnitude than daily returns, so the covariance of monthly returns will normally be greater than the covariance of any daily returns in the same market.

Since covariance is not independent of the units of measurement, it is a difficult measure to use for comparisons. It is better to use the correlation, which is a standardized form of covariance that is independent of the units of measurement. For two random variables X and Y the correlation is just the covariance divided by the product of the standard deviations, that is:

corr(Z,7) = cov(X,Y)/y/[ V(X)V(Y)]. (1.3)

Equivalently, using parameter notation rather than operator notation:7

Pxy = / (1-4)

Correlation does not need to be annualized like volatility because it is already in a standardized form. Normalizing the covariance as we have in (1.3) and (1.4) will always give a number that lies between -1 and + 1. High positive correlation indicates that the returns are strongly associated, or co-dependent, because they tend to move together in the same direction. High negative correlation indicates that the returns are still highly co-dependent, but they tend to move in opposite directions.

The greater the absolute value of correlation, the greater the association or co-dependency between the series. If two random variables are statistically independent then a good estimate of their correlation should be insignificantly different from zero. We use the term orthogonal to describe such variables. However, the converse is not true. That is, orthogonality (zero correlation) does not imply independence, because two variables could have zero covariance and still be related (the higher moments of their joint density function could be different from zero).

Orthogonality does not imply independence; two variables could have zero covariance and still be related

In financial markets, where there is often a non-linear dependence between returns, correlation may not be an appropriate measure of co-dependency. Correlation is related to the slope parameter of a linear regression model (§A. 1.1). Comparison of the ordinary least squares (OLS) formula with (1.3) shows that the correlation p and the slope coefficient (3 are related as

P=pv,

(1.5)

where v denotes the relative volatility of Y (the dependent variable) with respect to X (the independent variable). That is, v = aY/®x- Thus correlation is only a linear measure of association. If a regression line were fitted to the data in Figure 1.3a as illustrated, the estimate of the slope coefficient (3 will be highly

7 Equations (1.3) and (1.4) say exactly the same thing. We tend to use parameter notation because it is concise, but often prefer operator notation for algebraic manipulation. For example V(X+ Y)=V(X)+ V{ ) + 2cov( V.))

is perhaps easier to write than ax+Y

+ rsY2 + 2ov

Correlation is only a linear measure of association