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Jensen's Alpha

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Summary

One of the basic tenets of finance is that riskier assets should have a higher expected return, as investors would not be willing to take on the additional risk unless they were rewarded appropriately. A common goal of many investors or traders is to generate as high a return as possible with the least amount of risk taken. Many different measures can be used to determine a portfolio’s risk-adjusted performance, including Jensen’s alpha, the Sharpe Ratio, and the Treynor Measure. Jensen’s alpha, or ex-post alpha, is determined by taking the current portfolio return and subtracting the expected return according to the Capital Asset Pricing Model (CAPM).

In this TradeStation Labs Analysis Concepts paper, we will introduce an indicator that applies the concept of Jensen’s alpha to individual equity securities. This allows you to quickly rank and search for stocks that are currently generating positive or negative alpha.

Calculation

The CAPM is a financial model used to calculate the expected return of an asset based on its beta, the average market return, and the risk-free interest rate. Figure 1 below displays the ex-post alpha formula, which includes the CAPM. Alpha represents the excess return generated over what is expected by the CAPM. Beta represents the volatility of a security when compared to the overall market, which is used in the calculation of CAPM.

According to the equation, if an asset has a beta greater than 1, its expected return should be greater than the average market return. If an asset has a beta less than 1, its expected return should be less than the average market return. However, there are times when a portfolio or asset generates a return that is higher than expected based on CAPM. Likewise, there are instances when an asset generates a return that is less than expected based on CAPM. These instances, when the return of an asset differs from its expected return based on CAPM, is when positive or negative ex-post alpha is generated.

Figure 1 – Ex-post Alpha Formula

Figure 1 – Ex-post Alpha Formula
Figure 1 – Ex-post Alpha Formula

In the indicator we are introducing, Jensen’s alpha is applied to an individual security instead of an entire portfolio. The risk-free rate is identified as the 13-week Treasury bill rate ($IRX.X). The beta is calculated solely on the individual security. The average market return is calculated based on the movement of the S&P 500 ($SPX.X). You can choose to calculate the alpha based on the last monthly, quarterly, or yearly return by adjusting an input. The indicator has been formatted specifically for use in both RadarScreen® and Chart Analysis, but can only be used on a daily interval.

Inputs

The first two inputs for the Jensen’s alpha indicator are used to identify the risk-free interest rate and the overall market. The risk-free rate is defined as the 13-week Treasury bill rate, while the overall market is defined as the S&P 500. If you want to change the default risk-free rate and overall market symbol, you can adjust the first two inputs by changing the symbols inside the quotation marks. The third input, “N,” is used to define your beta length.

Figure 2 - Inputs
Name Default Description
iSymbol1 "$IRX.X" The risk-free rate used in the CAPM calculation. We use the 13-week Treasury bill rate to represent the risk-free rate.
iSymbol2 "$SPX.X" The market used in the beta calculation. We use the S&P 500 to represent the overall market.
N 120 Length used for beta calculation.
MonthRet TRUE If you want to compare the average return for the last month against the expected return for the last month, then type true. Otherwise, leave as false.
QuartRet FALSE If you want to compare the average return for the last quarter against the expected return for the last quarter, then type true. Otherwise, leave as false.
YearRet FALSE If you want to compare the average return for the last year against the expected return for the last year, then type true. Otherwise, leave as false.
AvgLen 10 Length used to calculate the average market return and the average return of the security you are applying the indicator to.
AvgAlpha FALSE If you want to average the actual return of the security in order to smooth out alpha, type true. Otherwise, leave as false.
BullishColor Dark Green Color of alpha when alpha is positive.
BearishColor Red Color of alpha when alpha is negative.

The next three inputs are used to determine whether you want to use the last month, quarter, or year to calculate Jensen’s alpha. If you want to use the last month’s alpha, then enter “true” as the input value next to “MonthRet.” Otherwise, enter “false” as the input value. Make sure at least one of the three return-length inputs is defined as “true” while the other two are defined as “false” in order to have the indicator run properly. If you are looking for more short-term trades, you might want to look at the last month’s alpha; for medium term, the last quarter; and for long term, the last year.

The next input, “AvgLen,” is used to smooth out the market return and the return for the last month, quarter, or year for the security you are analyzing. The security’s return is only averaged if the “AvgAlpha” input is defined as “true.” If it is left as “false,” the return for the security will not be averaged and the alpha plotted will be based on the current return for the security for the last month, quarter, or year.

The last two inputs are used to define the color of alpha when alpha is positive or negative. By default, alpha will be dark green when it is positive and red when it is negative.

Analysis

The Jensen’s alpha indicator can be used in both RadarScreen and Chart Analysis. In RadarScreen, the indicator appears as displayed in Figure 3 below. There are four plots for the indicator. The first plot is the alpha. If alpha is positive, the background color of the column will be green. If alpha is negative, the background color of the column will be red. The second column displays the beta value based on the input “N,” which represents the length used to calculate beta. The third plot displays the expected return based on the CAPM formula and the last column shows the actual return realized for the security. The difference between the last two columns is the value that appears in the “Alpha” column.

In RadarScreen, you can sort by the “Alpha” column to rank various securities to determine which ones are outperforming the most based on Jensen’s alpha. Notice in the image below, Biogen Idec (BIIB) is the security with the highest alpha within the NASDAQ 100.

Figure 3 –Jensen’s Alpha Indicator in RadarScreen

Figure 3 –Jensen’s Alpha Indicator in RadarScreen

One thing to keep in mind is that even though the return of the security for the last month, quarter, or year may be positive, the overall Jensen’s alpha may be negative. The reason for this is that the return did not compensate enough over the average market return based on the beta of the security. On the other hand, you could see a security with negative return but a positive alpha in a down-trending market. In this case, a security may have fallen less than expected based on its relative risk-adjusted performance to the market.

This indicator allows you to quickly assess whether a security has generated enough return during the last month, quarter, or year based on the risk taken. The Jensen’s alpha indicator can also be applied to sector indexes to determine which sectors are currently generating alpha, as displayed in Figure 4 below. As you can see, the materials and industrial sectors have been outperforming and generating positive alpha based on the last month’s return, whereas energy and consumer staples have been underperforming the most.

Figure 4 – Sector Alpha as of June 24, 2011

Figure 4 – Sector Alpha as of June 24, 2011

The Jensen’s alpha indicator can also be used in a Chart Analysis window. In charts, alpha will plot as a histogram with beta, expected return, and actual return plotting as lines around the histogram. In Figure 5 below is an image of the indicator plotted in a Chart Analysis window. Plotting the indicator in a chart allows you to see the history of alpha for a specific security. In the chart below, we are looking at the last quarter’s alpha. Notice how it lags the security’s underlying price movement. As you increase the length from month to quarter to year, the lag in alpha to the underlying security’s price movement will also increase. Also in the image below, the “AvgAlpha” input is defined as “false,” which is why alpha appears to be choppy over time.

Figure 5 – Jensen’s Alpha in Chart Analysis

Figure 5 – Jensen’s Alpha in Chart Analysis

As of June 24, 2011, Exxon Mobil (XOM) had a negative alpha of 5.91 based on the last quarter’s return. Looking at the chart, you can tell that for the last few years, Exxon Mobil has generated negative alpha more often than positive alpha.

As mentioned earlier, a security can have a negative return with a positive alpha. In Figure 6 below, you can see that Caterpillar Inc, 3M Co, and Verizon Communications all have had negative returns for the last month of -3.17%, -1.94%, and -1.04%, respectively. However, they are all generating slightly positive alpha of 2.25, 1.54, and 1.46, respectively. This means that for the amount of risk taken, which is defined as the beta, the security has not fallen as much as expected. For example, if you had invested $10,000 in Caterpillar Inc, you would have incurred a loss of $314 for the last month. However, if you had invested $10,000 in the S&P Depository Receipts, you would have generated a loss of $425 over the last month.

Figure 6 – Negative Return with Positive Alpha

gure 6 – Negative Return with Positive Alpha

Another method of applying the Jensen’s alpha indicator in TradeStation is to insert the indicator three times into a chart or RadarScreen and adjust the inputs accordingly so you can see the current alpha for the last month, quarter, and year at the same time. In Figure 7 below, we have applied the Jensen’s alpha indicator three times into a chart. The second subgraph is displaying the monthly alpha, the third subgraph is displaying the quarterly alpha, and the last subgraph is displaying the yearly alpha.

Figure 7 – Jensen’s Alpha: Monthly, Quarterly, and Yearly

Figure 7 – Jensen’s Alpha: Monthly, Quarterly, and Yearly

Notice how Dell Inc has generated positive alpha for the last month (6.67) and quarter (9.95). However, for the last year, it has generated slightly negative alpha (-0.01). This might be an indication that the security is transitioning from a period of underperformance to one of outperformance. You may also want to use this indicator as a filter or confirming signal along with other technical or fundamental indicators to help determine your trading decisions.

In the chart above, all of the Jensen’s alpha indicators have been formatted to display the alpha based on the average return of the security by defining the “AvgAlpha” input as “true.” This is why the alpha histogram appears smoother than it did in Figure 5.

Conclusion

Risk-adjusted measures of return are important for both investors and traders. In this TradeStation Labs Analysis Concepts paper, we have taken a common risk-adjusted performance measure and applied it to individual securities to get a better idea of whether the equity security is compensating sufficiently for the amount of risk it has. The risk in this case is defined as beta. Jensen’s alpha helps us determine which sectors or global markets are currently outperforming. At the same time, you can apply it to individual stocks. The indicator also provides you with the expected return of the security based on CAPM.

Depending on the characteristics of the underlying security, the Jensen’s alpha indicator can be used as a basis for a trading strategy. If the security has trending behaviors, positive alpha would generate a bullish signal, whereas negative alpha would generate a bearish signal. For mean-reverting securities, the opposite would be true: positive alpha would generate a bearish signal, while negative alpha would generate a bullish signal. Regardless of the price behavior of the underlying security, it is important to keep track of which equity securities are currently outperforming on a risk-adjusted basis.

Attachments

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