This is “Standard Deviation”, section 11.4 from the book Finance for Managers (v. 0.1).
This book is licensed under a Creative Commons by-nc-sa 3.0 license. See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it available to everyone else under the same terms.
This content was accessible as of December 29, 2012, and it was downloaded then by Andy Schmitz in an effort to preserve the availability of this book.
Normally, the author and publisher would be credited here. However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed. Additionally, per the publisher's request, their name has been removed in some passages. More information is available on this project's attribution page.
For more information on the source of this book, or why it is available for free, please see the project's home page. You can browse or download additional books there. You may also download a PDF copy of this book (2 MB) or just this chapter (203 KB), suitable for printing or most e-readers, or a .zip file containing this book's HTML files (for use in a web browser offline).
PLEASE NOTE: This book is currently in draft form; material is not final.
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:
Equation 11.4 Standard Deviation of Returns (all outcomes known)
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)
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:
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.