9.2 Expected Return and Standard Deviation of a Single Security
Learning Objectives
- Apply the formula for expected return to determine the anticipated rate of return for a single security based on the probability distribution of possible outcomes.
- Use the standard deviation formula to measure the variability of returns for a single security, interpreting the implications of higher or lower standard deviation on investment risk.
- Analyze probability distributions to understand the likelihood of various states of nature and their corresponding returns, enhancing the ability to predict and manage financial risks.
Formula & Symbol Hub
Symbols Used
- [latex]\sum_{i=1}^{n}[/latex] = repeatedly sum the expression from [latex]i[/latex] to [latex]n[/latex]
- [latex]\bar{k}[/latex] = expected return of a stock
- [latex]P_i[/latex] = the probability of the ith possible outcome (state of nature)
- [latex]k_i[/latex] = the return under the ith outcome (state of nature)
- [latex]P_n[/latex] = the probability of the nth possible outcome (state of nature)
- [latex]k_n[/latex] = the return under the nth outcome (state of nature)
- [latex]\sigma[/latex] = standard deviation
Formulas Used
-
Formula 9.1 – Expected Return (Single Security)
[latex]\bar{k}=\sum_{i=1}^{n}P_{i}k_{i}[/latex]
OR
[latex]\bar{k}=P_{1}k_{1}+P_{2}k_{2}+...+P_{n}k_{n}[/latex]
-
Formula 9.2 – Standard Deviation (Single Security)
[latex]\sigma=\sqrt{\sum_{i=1}^{n}P_{i}\left(k_{i}-\overline{k}\right)^2}[/latex]
OR
[latex]\sigma=\sqrt{P_{1}\left(k_{1}-\overline{k}\right)^2+P_{2}\left(k_{2}-\overline{k}\right)^2+...+P_{n}\left(k_{n}-\overline{k}\right)^2}[/latex]
Expected Return
The expected return of a security is based on the probability distribution of returns. Before we get into the details of the expected return, let’s briefly introduce the concept of a probability distribution. A probability distribution is a representation of possible outcomes (states of nature) that may occur and the likelihood (probability) of each outcome. If you think of a coin flip, the probability distribution has two possible outcomes (heads or tails) and each outcome has a [latex]50\%[/latex] chance of occurring (technically, this is not true as even in a “fair” coin flip, the side that starts up has about a 51\% chance of occurring). When dealing with financial securities, the number of possible outcomes is nearly infinite and it is not possible to know the exact outcomes or probabilities of those outcomes. Therefore, we are really only approximating the true probability distribution.
Video: “Probability Distribution” by Kevin Bracker [6:35] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
Specifically, the expected return is the probability of a specific state of nature occurring times the return under that state of nature summed across all possible states of nature.
[latex]\boxed{9.1}[/latex] Expected Return (Single Security)
[latex]\Large{\color{red}{\bar{k}}}=\sum_{i=1}^{n}{\color{blue}{P_{i}}}{\color{green}{k_{i}}}[/latex]
OR
[latex]\Large{\color{red}{\bar{k}}}={\color{blue}{P_{1}}}{\color{green}{k_{1}}}+{\color{blue}{P_{2}}}{\color{green}{k_{2}}}+...+{\color{purple}{P_{n}}}{\color{brown}{k_{n}}}[/latex]
[latex]{\color{red}{\bar{k}}}[/latex] represents the expected return of the stock
[latex]\sum_{i=1}^{n}[/latex] means “sum the following expression repeatedly from [latex]i[/latex] to [latex]n[/latex]”
[latex]{\color{blue}{P_i}}[/latex] represents the probability of the [latex]i[/latex]th possible outcome (state of nature)
[latex]{\color{green}{k_i}}[/latex] represents the return under the [latex]i[/latex]th outcome (state of nature)
[latex]{\color{purple}{P_n}}[/latex] represents the probability of the [latex]n[/latex]th possible outcome (state of nature)
[latex]{\color{purple}{k_n}}[/latex] represents the return under the [latex]n[/latex]th outcome (state of nature)
Example 9.2.1
After researching Stock A we have determined that there are [latex]3[/latex] possible outcomes for the next year ([latex]3[/latex] states of nature). The first possibility is the economy enters a recession causing the stock to have a return of [latex]-15\%[/latex]. The probability of this occurring is [latex]20\%[/latex]. The second possibility is that the economy goes smoothly, but does not experience rapid growth causing the stock to rise and offer a [latex]10\%[/latex] return. The probability of this occurring is [latex]50\%[/latex]. The third possibility is that the economy booms, causing the stock to provide a [latex]35\%[/latex] rate of return. The probability of the economy booming is [latex]30\%[/latex] (note that the probabilities must sum to [latex]1.0[/latex] and the states of nature should be mutually exclusive).
| State of Nature | Probability | Return |
|---|---|---|
| Recession | [latex]0.20[/latex] | [latex]-15\%[/latex] |
| Normal | [latex]0.50[/latex] | [latex]10\%[/latex] |
| Boom | [latex]0.30[/latex] | [latex]35\%[/latex] |
What is the expected rate of return?
Solution
Substitute probabilities and returns into the Expected Return formula (Formula 6.1) and solve:
[latex]\begin{align*}\bar{k}&=\sum_{i=1}^{n}P_{i}k_{i}\\\bar{k}&=(.20)(-15\%)+(.50)(10\%)+(.30)(35\%)\\\bar{k}&=-3\%+5\%+10.5\%\\\bar{k}&=12.5\%\end{align*}[/latex]
The expected return is [latex]12.5\%[/latex].
Video: “Expected Return of Single Security” by Kevin Bracker [5:24] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
Standard Deviation
The standard deviation measures the variability of possible returns and is represented by the lower-case Greek symbol sigma ([latex]\sigma[/latex]). The smaller the standard deviation, the more likely we are going to earn something “close” to our expected return. The greater the standard deviation, the greater the chance that we may earn something far more (good) or far less (bad) than our expected return. The formula for this is (remember that [latex]\bar{k}[/latex] is our symbol for expected return):
[latex]\boxed{9.2}[/latex] Standard Deviation (Single Security)
[latex]\Large{\color{red}{\sigma}}=\sqrt{\sum_{i=1}^{n}{\color{blue}{P_{i}}}\left({\color{green}{k_{i}}}-{\color{purple}{\overline{k}}}\right)^2}[/latex]
OR
[latex]\Large{\color{red}{\sigma}}=\sqrt{{\color{blue}{P_{1}}}\left({\color{green}{k_{1}}}-{\color{purple}{\overline{k}}}\right)^2+{\color{blue}{P_{2}}}\left({\color{green}{k_{2}}}-{\color{purple}{\overline{k}}}\right)^2+...+{\color{Bittersweet}{P_{n}}}\left({\color{brown}{k_{n}}}-{\color{purple}{\overline{k}}}\right)^2}[/latex]
[latex]{\color{red}{\sigma}}[/latex] (sigma) represents the standard deviation
[latex]{\color{blue}{P_{i}}}[/latex] represents the probability of the [latex]i[/latex]th outcome (state of nature)
[latex]{\color{green}{k_{i}}}[/latex] represents the return under the [latex]i[/latex]th outcome (state of nature)
[latex]{\color{purple}{\overline{k}}}[/latex] represent the expected return for the stock
[latex]{\color{Bittersweet}{P_{n}}}[/latex] represents the probability of the [latex]n[/latex]th outcome (state of nature)
[latex]{\color{brown}{k_{n}}}[/latex] represents the return under the [latex]n[/latex]th outcome (state of nature)
Things to Watch Out For
- It is easy to get confused with decimals and percentages. The best way to do these calculations is to always leave the weights as decimals and the returns as a regular number. For instance, if you have a probability of [latex]0.10[/latex] and a return of [latex]15\%[/latex], you would put the probability into your calculator as [latex]0.10[/latex] and the return as [latex]15[/latex].
- Be careful with your order of operations.
- Do ([latex]k_1-\bar{k}[/latex]) first
- Then square that
- Then multiply by [latex]P_1[/latex]
- Repeat for all [latex]n[/latex] states of nature
- Add them up
- Finally, take the square root
Example 9.2.2
Using the data and calculated expected return from Example 9.2.1 above, calculate the standard deviation for stock A?
Solution
Substitute probability, return, and expected return into the formula for standard deviation (Formula 9.2) and solve:
[latex]\begin{eqnarray*}\sigma&=&\sqrt{\sum_{i=1}^{n}P_{i}\left(k_{i}-\overline{k}\right)^2}\\[1ex]\sigma&=&\sqrt{0.2\left(-15-12.5\right)^2+0.5\left(10-12.5\right)^2+0.3\left(35-12.5\right)^2}\\[1ex]\sigma&=&\sqrt{0.2\left(756.25\right)+0.5\left(6.25\right)+0.3\left(506.25\right)}\\[1ex]\sigma&=&\sqrt{151.25+3.13+151.88}\\[1ex]\sigma&=&\sqrt{306.25}\\[1ex]\sigma&=&17.50\%\end{eqnarray*}[/latex]
The standard deviation for Stock A is [latex]17.50\%[/latex].
Video: “Standard Deviation Single Security” by Kevin Bracker [7:15] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
Attribution
“Chapter 7 – Risk Analysis” from Business Finance Essentials by Dr. Kevin Bracker; Dr. Fang Lin; and Jennifer Pursley is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.
