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Market Risk Metrics – Jensen’s Alpha

Jensen’s Alpha is the risk-adjusted performance metric that measures a portfolio manager’s returns against those of a benchmark. For asset allocation, the portfolio consists of the instrument (e.g. equity stock) being analyzed and the benchmark is a broad market index (e.g. S&P 500) – in other words we are assessing the instrument’s returns against those of a broad market index. We are measuring the instrument’s performance relative to the respective broad market index where the return represented by alpha is the unique or excess return of that instrument over the market.

If the alpha is not statistically different from zero there is no unique return over the return earned by the benchmark, i.e. the instrument’s performance is in line with that of the market. A statistically positive alpha means that the instrument has performed better than the market index while a negative value for alpha means that the instrument underperformed the market index.

The alpha is estimated using Regression Analysis. In particular, a least squares estimation approach is employed whereby we minimize the sum of squared differences between the instrument’s actual returns and those estimated using the following equation:

Where

RIt = Daily rate of return of instrument, I, at time t

Rf = Daily risk free rate of return. The risk free rate of return is usually specified in annualized terms. In order to convert the annualized figure to a daily number we will employ the following formula (we assume 252 trading days in a year):

Daily risk free rate of return = (1+ annual risk free rate of return)1/252-1

RMt = Daily rate of return of the market index, M, at time t

?I = Beta of the instrument with respect to the market index

?I = Daily unique rate of return of instrument I.

Annualized ?I = Annualized unique rate of return of instrument I over the market rate of return, i.e. Jensen’s Alpha.

Let us now review the step-by-step calculation methodology for Jensen’s Alpha in EXCEL.

First obtain the time series price data over the given period of analysis for the given instruments as well as the broad market index.

Next, calculate the daily return time series from the price data. The daily returns are calculated by taking the natural logarithm of the ratio of successive (consecutive) prices:

The resulting actual return series for the stocks and the market index are given below:

We have calculated the beta for stock ABC and XYZ with respect to the broad market index as 1.20 and 2.16 respectively. The method for calculating beta has been discussed in our earlier post:

Market Risk Metrics – Beta with respect to market risk indices

The annual risk free rate is given as 12%. The daily risk free rate as discussed above works out to:

Daily risk free rate = (1+12%)1/252-1=0.045%.

Using the formula for estimated returns given above and assuming initial dummy values for daily alpha returns for the stocks of 0.5% each we arrive at the estimated return series as follows:

Note that the estimated returns given above are not the final estimated values as they are dependent on the initial dummy values input-ed for the daily alpha. Note also that the daily alpha is assumed to be constant over the dataset, i.e. it does not vary with time.

Next, at each data point we will calculate the squared differences between the actual returns and the estimated returns, i.e. Square Difference = (Actual return –Estimated return)2. We then calculate the sum of the squared differences across all data points. The results of these two steps are illustrated below for our example:

We will now set up EXCEL’s Solver function so that the sums of squared differences are minimized by changing the values of daily alphas.

For each stock we will follow the process mentioned below:

Step 1: Data Tab>Solver

Step 2: “Set Target Cell” to the “Sum of squared difference” cell for the stock

Step 3: Equal to: Min or Value =0

Step 4: By Changing Cells: “Daily est. alpha” cell for the stock

Step 5: Click on the Solve button to solve.

Follow the above mentioned procedure for each instrument on the portfolio being assessed.

The resulting daily and annualized alphas after Solver converged to a solution are as follows:

The annualized estimated alphas for stocks ABC and XYZ work out to 93.32% and -35.86% respectively. We can see that stock ABC has over performed the market whereas stock XYZ has underperformed the market.

Significance Testing of Alphas

It is important to test the alphas for statistical significance. One way of doing this is to calculate the information ratio or t-statistic which is the ratio of the estimated annualized alpha to the standard error (annualized) of the regression. If the absolute value of this ratio exceeds 1.96 (which is critical value at the 5% significance level) then we can say that the alpha is different from zero and that it is statistically significant.

The annualized standard error, i.e. the denominator of the above mentioned ratio is calculated as follows:

First calculate the difference between the actual and estimated returns:

Next, for each stock, calculate the daily volatility of these differences, i.e. the standard deviation of the error terms. This can be done in EXCEL using the STDEV() function applied to the array of differences. For stocks ABC and XYZ this works out to 1.60% and 4.51% respectively.

Annualized volatility, i.e. the annualized standard error term is calculated using the following formula:

They work out to 25.45% and 71.54% for stocks ABC and XYZ respectively.

Now for each stock we calculate the statistic for the significance test. For stock ABC this is calculated as follows:

Test Statistic = absolute value of {Annualized Alpha/ Annualized standard error} = 93.32%/25.45%=3.67.

In a similar manner the test statistic for stock XYZ works out to 0.50.

As mentioned earlier if the value of the test statistic exceeds 1.96, i.e. the critical value for the t-test at the 5% significance level, then we can say that the alpha is different from zero and that it is statistically significant. Using this test, we see that only the Jensen’s alpha for stock ABC is statistically significant given the dataset used, as the test statistic for this instrument, 3.67, exceeds the critical value for the test, 1.96.

 

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