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- Explain how standard deviation is a measure of risk.
- Calculate the standard deviation of a set of returns.

There are many potential sources of risk, and not all methods of assessing risk are the same. One popular method (though certainly not the only one) for trying to quantify risk is to examine the standard deviation of the outcome returns. Standard deviation is a term from statistics that helps explain to what degree our results are expected to be different from the averge result. We are, in a sense, finding the “average” size of the “miss”. For a full treatment of standard deviation as a statistical measure, please consult a statistics text; for our purposes, a basic understanding will suffice.

Figure 11.1 Standard Deviation Comparison

Source: JRBrown / Wikimedia Commons / Public Domain

The blue investment has results that are less “clumped” around the mean than in the red investment. The blue investment has a correspondingly higher standard deviation.

Finding standard deviation involves a few steps; if we know all the outcomes and their probabilities, standard deviation is found as follows:

- Find the expected rate of return.
- For each outcome, find the difference between the outcome’s rate of return and the expected rate, and square the difference. So if the expected outcome is 2.5%, but the outcome’s return is −10%, then the difference is (−.10 − (.025) = −.125. This result squared is −.125
^{2}= .015625. Notice that the squaring effectively eliminates the negative numbers (so it doesn’t matter if we miss high or low, just how much we miss by). - Find the weighted mean of these: multiply each by the probability of the outcome and sum.
- Take the square root of this weighted mean to find the standard deviation.

Equation 11.4 Standard Deviation of Returns (all outcomes known)

$$\begin{array}{l}\text{StandardDeviation=}\sqrt{{p}_{1}{({r}_{1}-\overline{r})}^{2}+{p}_{2}{({r}_{2}-\overline{r})}^{2}\dots +{p}_{n}{({r}_{n}-\overline{r})}^{2}}\\ \text{where}\overline{r}\text{is}\text{the}\text{expected}\text{rate}\text{of}\text{return}\text{(weighted}\text{mean)}\end{array}$$If we are using historical returns to estimate our expected return, then we need to make a slight adjustment to the formula. Typically, we assume each past observation (usually daily, weekly, monthly, or yearly returns) to have equal weight. Thus, the expected rate of return is just the arithmetic mean. But since we have just estimated the expected rate of return, we need to adjust our standard deviation calculation (for a detailed reason why, please see any standard statistics text, as it is beyond the scope of this text):

Equation 11.5 Standard Deviation of Returns (estimating expected return from historical)

$$\begin{array}{l}\text{StandardDeviation}=\sqrt{\frac{{({r}_{1}-\overline{r})}^{2}+{({r}_{2}-\overline{r})}^{2}\dots +{({r}_{n}-\overline{r})}^{2}}{n-1}}\\ \text{where}\overline{r}\text{is}\text{the}\text{expected}\text{rate}\text{of}\text{return}\text{(weighted}\text{mean)}\end{array}$$On the bright side, spreadsheet programs can quickly do this calculation for us.

=STDEV.S(outcome1, outcome2, outcome3…)

Thus, if our outcomes for the past 5 years are 5%, 2%, −6%, −2%, and 4%, we can enter:

=STDEV.S(.05,.02,−.06,−.02,.04)

and receive the correct result of 0.0456 or 4.56%.

Most financial calculators can also solve for standard deviation; please consult the instruction manual of your calculator for details.

There are some downsides to using standard deviation as the measure of risk. Standard deviation tells us the most when our outcomes are normally distributed (the “bell curve”). If our outcomes have some extreme outliers (for example, a decent chance for a large positive or negative result), then standard deviation tells us very little about the risk associated with these results. If our results are skewed toward positive or negative outcomes, standard deviation tells us nothing of the effect.

Figuring out the standard deviation of a portfolio can be done using the same procedure as above: just solve for the portfolio returns and then use them to find the standard deviation. What **won’t work** is trying to take the weighted average of the standard deviations of the individual assets! The reason for this is simple, if the assets don’t move in perfect tandem, then the diversification will cause the returns of the portfolio to be less volatile (that is, have a lower standard deviation).

Often, when constructing portfolios, investors will try to use investments that specifically behave differently from each other to try to maximize the benefits of diversification (in turn, minimizing risk). For example, consider the following: asset ABC had returns of 6%, 4%, and 2% over the past 3 years, respectively. DEF had returns of 3%, 5%, and 7%, respectively. The standard deviation of both investments is thus 2%. An equal weighted portfolio (that is, 50% of each) will have returns of 4.5% each year, for a standard deviation of 0%!

In reality, we can rarely find assets that behave so differently from each other (most assets, for example will increase in value as the overall economy improves), so there is a limit to the benefits of diversification. We will explore this further in a following section. Additionally, it seems that in times of crisis (for example, in 2008), many types of assets that typically behave in different ways can all lose value together; in other words, right when we need the benefits of diversification most is when the effect is smallest.

- Standard deviation is one common measure of risk, but not the only one. It does not represent all aspects of risk.
- Standard deviation of a portfolio is not the weighted average of the standard deviations of the components. It must be computed from the portfolio returns.

- An investment has a 10% chance of returning 20%, a 70% chance of returning 10%, and a 20% chance of returning nothing. What is the standard deviation of expected returns?
- An investment has returned 5%, 3%, −4%, 6%, and −7% in each of the last five years, respectively. We have decided that we want to use historical returns as a proxy for expected future returns. What is the standard deviation of these returns?
- Why isn’t a portfolio’s standard deviation the same as its components?