How do we measure the risk of a financial security?

The modern approach to financial risk dates back to the 1950s and the emergence of the mean-variance approach to portfolio analysis. This approach, developed by Chicago economist Harry Markowitz, takes the view that any investment in financial securities is associated with an uncertainty about its outcome. It is generally impossible to predict with certainty the actual return from an investment, but what we can do is to describe the uncertainty about the likely outcomes in terms of a probability distribution. A probability distribution summarises our degree of belief about the likelihood of all the possible returns. This probability distribution could be based on the past historical performance of the securities, possibly modified to reflect our knowledge of the current market conditions.

On the basis of the probability distribution of returns, we can then compute the mean return, or expected return, on the securities, which is a measure of the centre of the distribution. We can also compute the variance. This is a measure of the spread, or dispersion, of the securities around their mean value. This will be our first measure of the risk of a security. An alternative measure of risk is the square root of the variance, which is called the standard deviation. The main practical difference between the standard deviation and the variance is that the standard deviation is expressed in the same unit of measurement as the returns, and can therefore be directly compared to the expected return.

If we follow the mean-variance approach to portfolio analysis, all we need to know about the portfolio securities are their means and their variances, or equivalently their standard deviations. Investors will have preferences over the mean and the variance of portfolios. Generally, they prefer a greater expected return to a lower expected return, and since they are assumed to be risk averse, less variance to more variance. In principle they may be happy to accept a greater risk, that is, a larger variance for their investment, if this is associated with a sufficiently large increase in the expected return. The portfolio selection problem therefore

goes as follows: investors compute the mean and the variance of all possible investment portfolios, and then select that portfolio which maximises their preferences in terms of the mean-variance combination that it offers. So the mean return from the investment and the variance summarise all the relevant information investors need. Knowing the full distribution of returns would be irrelevant. This clearly simplifies the original problem of choice under uncertainty, since it makes it possible for investors to concentrate on just these two summary measures of the distribution of returns. This approach can’t always be applied. There could be instances, for example, in which investors are concerned with a possible asymmetry of the distribution.

For instance, when large losses are more likely than large gains, and this is not captured by the mean and variance alone. In practice, however, this simplification is usually regarded as satisfactory for a large class of investment problems, and has proven itself to be very useful in many actual applications.