You may not be interested in risk but risk is really interested in you and it will bite you!

Learning objectives:

Introduce the concept of asset volatility and risk measurements.


Too often people concentrate on performances, on return, without evaluating whether the risk they have taken as actually paid of or not. You may not be interested in risk but risk is really interested in you and it will bite you!

As we saw previously return distribution range is one way to illustrate risk, but in the finance world where statistical models are used ad nausea, risk is often translated into standard deviation of rates of return. Standard deviation is a measure of the dispersion of a data set from its mean. In finance, standard deviation is also called volatility. An investment that is said to be volatile is an investment that has or is expected to have periodic returns deviating from its expected average returns. Obviously as investors we are more concerned about the negative deviation (unless you short but that’s another story)!

To compute the standard deviation we need to determine the mean of the numbers set, compute the difference of each number with the mean, sum the squares of each difference, divide the sum by the number of data points minus one, and finally take the square root of the result.

Another important characteristic to remember about standard deviation is that in normal distribution, about two thirds of the observations will be within one standard deviation, 95% within two standard deviations and 99.7% within three standard deviations.
Normal distribution

Enough theory, let’s be a little bit more concrete. The main benefit of using standard deviation as a way to quantify the risk of a security is to be able to have a quick idea of the chances of possible return outcomes. For example if you have annual returns of a security that are characterized by an expected average of 5% and an expected standard deviation of 20%, this means that there is 1-out-of 6 chance that the annual return will be below -15%, or another way to say it is that the security can have a return below -15% every 6 years. If we apply the two standard deviations scenario, it gives a return below -35% every 40 years.

On the table you can see the historical standard deviations of the annual returns for stocks and bonds over the last 10, 20, 30 and 83 years. By showing you the same data over several periods of time we wanted to highlight the fact that the volatility of an investment varies over time.

By selecting the graph showing the annual volatility based on the 20 year rolling period of annual returns you can easily see that the volatility is not constant and can vary significantly. Note that the graph starts at year 1947 because the standard deviation is based on a 20 year period and the available data on returns start in 1927. Note also that the high stock volatility experienced in the late forties and fifties is caused by the significant returns variations that followed the 1929 economic crisis.
Of course you can also notice that for every period, on a yearly basis, bonds were less volatile than stock, but their respective volatility were very close in the second half of the nineties.

In summary, when a financial advisor is recommending you a security, he should be able to provide you with its standard deviation of annual returns or its expected standard deviation if it is a security newly issued. The point here is when we are analysing past performances of a specific investment; we tend to focus on past returns and at the same time neglect the risk characteristics.
Also note that historical standard deviation of annual returns, as often in finance, is not a perfect measure, but it can still help you to get a better idea of the risk associated with an investment.


When one look at the historical standard deviations to compare risks of several assets, he must make sure that he is using the same type of standard deviations. Actually, for a given asset you can find several flavours of standard variations. It all depends on the interval and the past period (length and dates) that have been selected to calculate the standard deviation. As far as intervals are concerned, you can find standard deviations that have been calculated for example from daily, weekly, monthly or yearly returns. You need to understand which types of interval have been used and if you have want to compare assets’ standard deviations that have been based on different intervals you should then normalize the standard deviation on an annual basis for example. As a general note the standard deviation value increases with the square root of the time. If you want to convert a daily based standard deviation to an annualized standard deviation you should multiply the daily based standard deviation by the square root of the number of trading days, 252. Similarly, if you want to convert a monthly based standard deviation to an annualized standard deviation you need to multiply by the square root of 12.

Historical data based standard deviation of an asset returns also depend on the period of time over which it is calculated.

Most of the financial models that use standard deviation as a measure of and asset’s risk must assume that the assets’ returns are normally distributed which is in general not the case. We already illustrated in the previous section showing the CAGR distribution on a histogram that the returns do not follow a bell shape and that they are negatively skewed. Observations of large movements in asset prices in the past are much more common than what we would have been obtained if normal distribution was valid.
For instance the Black Monday crash in October 87 is a 20 sigma event, basically this mean that it should never happen! More recently the S&P500 lost 50% of its value over a 10-month period from May 2008 to March 2009, this is approximately a 3.5 sigma event, meaning that is a 1-in-3000 event.


Here are the formulas of the different dispersion measurement presented in this section:

Standard deviation for a set of sample data:
Standard Deviation formula

S is the symbol used to represent the standard deviation, note that S squared is called the variance. x represents the value of an observation in a sample. The symbol X represents the sample mean. The symbol n represents the total number of data values or observations in a sample.
As a side note, the formula for the sample standard deviation is different than the formula for a population standard deviation. Actually the denominator for a sample standard deviation subtracts 1 from n. Using n instead of n-1 will underestimate the population variance especially for small sample size.

In a previous section we already saw that the range is a good way to illustrate the dispersion of asset returns. The range is a very simple measure of variability and is defined as the difference between the maximum value and the minimum value of a data set.

Another measure is called the mean absolute deviation (MAD), and is the average of the absolute values of the difference between the value of each observation and the arithmetic mean. For instance if asset returns have a mean of 12% and a MAD of 5% we can interpret this result by saying that on average a return will deviate from 12% by plus or minus 5%.
MAD formula

One of the criticisms of the standard variation and its corresponding variance is that they measure both the negative and the positive dispersions, whereas investors may only be interested about negative dispersion. Semivariance of a sample is very similar to variance but it is calculated based only on observations that are below the mean.
Semivariance formula

In investment world, semivariance is also called ‘downside risk’. Also when the distribution is symmetric such as a normal distribution the semivariance is equal to the variance and thus is not very useful. But when a distribution is skewed such as the stock returns distribution, then it can provide additional information.
Semivariance measure can be also be used to quantify the risk of having values below a specific value such as the risk free rate for example. This measure is then called target semivariance. Only observations below the target value are included in the calculation.

One argument often used against dispersion based on historical data is that it is backward looking. Here is where implied volatility comes into play. Actually contrary to the historical standard deviation, the implied volatility is forward looking. Implied volatility can be deduced from derivatives prices and it is based on market consensus. For instance the VIX can be used to estimate the S&P500 volatility anticipated by the market.

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