9.3 Expected Return and Standard Deviation of a Portfolio
Learning Objectives
- Use the expected return formula for a portfolio to determine the weighted average return of multiple stocks within a portfolio.
- Apply the standard deviation formula for a two-stock portfolio to assess the overall risk, considering the weights, individual standard deviations, and correlations between the stocks.
- Evaluate how the correlation between stocks affects the overall risk of a portfolio, demonstrating the benefits of diversification in reducing portfolio volatility.
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]\sigma[/latex] = standard deviation
- [latex]W_{n}[/latex] = the weight (proportion of portfolio) of stock [latex]n[/latex]
- [latex]{\color{purple}{\bar{k_{n}}}}[/latex] = expected return for stock [latex]n[/latex]
- [latex]corr_{x,y}[/latex] = the correlation between the returns of stocks [latex]x[/latex] and [latex]y[/latex]
Formulas Used
- Formula 9.3 – Expected Return (Portfolio)
[latex]\bar{k_{p}}=\sum_{i=1}^{n}W_{i}\bar{k_{i}}[/latex]
OR
[latex]\bar{k_{p}}=W_{1}\bar{k_{1}}+W_{2}\bar{k_{2}}+...+W_{n}\bar{k_{n}}[/latex]
-
Formula 9.4 – Standard Deviation (Two-Stock Portfolio)
[latex]\sigma_{p}=\sqrt{W_{1}^2\sigma_{1}^2+W_{2}^2\sigma_{2}^2+2W_{1}W_{2}\sigma_{1}\sigma_{2}corr_{1,2}}[/latex]
Expected Return
The expected return for a portfolio is simply the weighted average of each stock held in the portfolio. The formula here is
[latex]\boxed{9.3}[/latex] Expected Return (Portfolio)
[latex]\Large{\color{red}{\bar{k_{p}}}}=\sum_{i=1}^{n}{\color{blue}{W_{i}}}{\color{green}{\bar{k_{i}}}}[/latex]
OR
[latex]\Large{\color{red}{\bar{k_{p}}}}={\color{blue}{W_{1}}}{\color{green}{\bar{k_{1}}}}+{\color{blue}{W_{2}}}{\color{green}{\bar{k_{2}}}}+...+{\color{brown}{W_{n}}}{\color{purple}{\bar{k_{n}}}}[/latex]
[latex]{\color{red}{\bar{k_{p}}}}[/latex] represents the expected return for the portfolio
[latex]{\color{blue}{W_{1}}}[/latex] represents the weight (proportion of portfolio) of stock [latex]1[/latex]
[latex]{\color{green}{\bar{k_{1}}}}[/latex] represents the expected return for stock [latex]1[/latex]
[latex]{\color{brown}{W_{n}}}[/latex] represents the weight (proportion of portfolio) of stock [latex]n[/latex]
[latex]{\color{purple}{\bar{k_{n}}}}[/latex] represents the expected return for stock [latex]n[/latex]
Example 9.3.1
What is the expected return of a portfolio made up of [latex]60\%[/latex] Stock A and [latex]40\%[/latex] Stock B when the expected return for Stock A is [latex]10\%[/latex] and the expected return for Stock B is [latex]20\%[/latex]?
Solution
Substitute weights and expected returns for stocks in the portfolio into the portfolio expected return formula (Formula 6.3) and solve:
[latex]\begin{align*}\bar{k_{p}}&=\sum_{i=1}^{n}W_{i}\bar{k_{i}}\\\bar{k_{p}}&=(.60)(10\%)+(.40)(20\%)\\\bar{k_{p}}&=6\%+8\%\\\bar{k_{p}}&=14\%\end{align*}[/latex]
Video: “Expected Return of Two Stock Portfolio” by Kevin Bracker [7:11] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
Standard Deviation
The standard deviation of a portfolio becomes more complicated. It depends not only on the standard deviation and weightings of each stock, but also on the correlation between pairs of stocks.
Correlation
The correlation between a pair of stocks measures how closely the returns for each stock are related. A negative correlation means that the price of one stock tends to fall while the other rises (prices/returns are inversely related). A positive correlation means that the price of one stock tends to rise while the other rises (prices/returns are positively related). Correlations can range from [latex]-1.0[/latex] to [latex]1.0[/latex]. Correlations for real-world variables are almost never at the extremes (perfect positive correlation, no correlation, or perfect negative correlation).
See the Observed Betas, Correlations, and Standard Deviations Table in Appendix B for some standard deviations and correlations from actual companies over the past five years.
Observations from the Table
- The last two rows/columns are for an aggregate US bond fund and the S&P 500 ETF. The purpose of these is to provide a proxy for the US bond market and the US stock market.
- The vast majority ([latex]65[/latex] of [latex]66[/latex]) of correlations between pairs of stocks are positive. This is because all stocks are impacted by general economic factors.
- The average correlation across pairs is a low, positive value ([latex]0.31[/latex] in our sample). While general economic factors cause a tendency for stock to move together, firm-specific factors push correlations towards zero.
- Stocks in similar industries tend to have higher correlations than the average stock ([latex]0.50[/latex] vs. [latex]0.31[/latex]).
- Each stock has a positive correlation with the overall stock market.
- The stock and bond market have a negative (although essentially zero) correlation during this time-period.
- The bond market has a much lower standard deviation than the stock market, but also generated much lower returns during this time-period.
- Most individual stocks have a standard deviation that is higher than the overall stock market. This is because the stock market represents a diversified portfolio which has eliminated most of the firm-specific risk. The exception is Pepsi in our sample which is essentially the same standard deviation.
- The historical annualized returns are not the same as the expected returns. It is unrealistic to expect [latex]30\%+[/latex] annual returns for Deere, Boston Beer or Amazon going forward (which is not the same as saying that these stocks can’t generate those returns). Also, it is unreasonable to expect people to buy stock in Molson Coors and Exxon with the anticipation of losing nearly [latex]12\%[/latex] or [latex]8\%[/latex] each year.
- While standard deviations will vary depending on overall market conditions, large cap stocks over this [latex]5-[/latex]year window had standard deviations of approximately [latex]15-42\%[/latex]. Note that the standard deviations will be bigger or smaller depending on both the stock and the time period and that Boston Beer’s [latex]42.5\%[/latex] is noticeably larger than the next closest company.
Standard Deviation for a Two-Stock Portfolio
In order to calculate the standard deviation of a two-stock portfolio, we will use the following formula:
[latex]\boxed{9.4}[/latex] Standard Deviation (Two-Stock Portfolio
[latex]\Large{\color{red}{\sigma_{p}}}=\sqrt{{\color{blue}{W_{1}}}^2{\color{BurntOrange}{\sigma_{1}}}^2+{\color{NavyBlue}{W_{2}}}^2{\color{Bittersweet}{\sigma_{2}}}^2+2{\color{green}{W_{1}}}{\color{Aquamarine}{W_{2}}}{\color{BurntOrange}{\sigma_{1}}}{\color{Bittersweet}{\sigma_{2}}}{\color{brown}{corr_{1,2}}}}[/latex]
[latex]{\color{red}{\sigma_{p}}}[/latex] represents the standard deviation of the portfolio
[latex]{\color{blue}{W_{1}}}[/latex] represents the weight (proportion of portfolio) of stock [latex]1[/latex]
[latex]{\color{BurntOrange}{\sigma_{1}}}[/latex] represent the standard deviation of stock [latex]1[/latex]
[latex]{\color{Aquamarine}{W_{2}}}[/latex] represents the weight (proportion of portfolio) of stock [latex]2[/latex]
[latex]{\color{Bittersweet}{\sigma_{2}}}[/latex] represent the standard deviation of stock [latex]2[/latex]
[latex]{\color{brown}{corr_{1,2}}}[/latex] represents the correlation between the returns of stocks [latex]1[/latex] and [latex]2[/latex]
Example 9.3.2
Consider a two-stock portfolio in which [latex]60\%[/latex] of your money is invested in stock A and [latex]40\%[/latex] of your money is invested in stock B. Stock A has a standard deviation of [latex]50\%[/latex] and stock B has a standard deviation of [latex]70\%[/latex]. The correlation between the returns for stock A and stock B are [latex]0.30[/latex]. You want to find the standard deviation of this portfolio.
Solution
Substitute values into Formula 6.4 for calculating the standard deviation of a two-stock portfolio and solve:
[latex]\begin{align*}\sigma_{p}&=\sqrt{W_{1}^2\sigma_{1}^2+W_{2}^2\sigma_{2}^2+2W_{1}W_{2}\sigma_{1}\sigma_{2}corr_{1,2}}\\[1ex]\sigma_{p}&=\sqrt{(0.6)^2(50)^2+(0.4)^2(70)^2+2(0.6)(0.4)(50)(70)(0.3)}\\[1ex]\sigma_{p}&=\sqrt{(0.36)(2500)+(0.16)(4900)+(504)}\\[1ex]\sigma_{p}&=\sqrt{900+784+504}\\[1ex]\sigma_{p}&=\sqrt{2188}\\[1ex]\sigma_{p}&=46.78\%\end{align*}[/latex]
Note that in Example 9.3.2, the standard deviation of the portfolio is less than the standard deviation of either stock separately. This illustrates the concept of diversification. As long as the correlation is less than [latex]1.0[/latex] (which it will be for any two stocks), the risk of the portfolio is less than the weighted average risk of the two securities which make up the portfolio (and sometimes — like here — even less than the lowest risk stock in the portfolio). While we will not cover the process of calculating the expected return and standard deviation for larger portfolios in this class, in practice, most portfolios hold far more securities.
Video: “Standard Deviation of a Two Stock Portfolio” by Kevin Bracker [7:06] 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.
