# Treynor Black Model

I don’t know how I’m going to summarize this absolute piece of shit here, but I’ll try:

It represents a portfolio management theory that assumes security markets are nearly efficient. The essence of the model:

1. Security analysts analyze a small number of securities in depth to discover any mispricing. all other securities that are not analyzed are assumed to be fair priced.

2. The Market Index portfolio is the passive portfolio (and lies on the CML), and the expected rate of return and the variance will be provided by the macro forecasting unit of the investment management firm.

3. The objective here is to form an active portfolio with limited securities that are supposedly mispriced.

Now, let’s get on with the steps of this model:

1. First of all, we need to know the beta, and the residual standard deviation of each stock that’s being analyzed. Using CAPM, calculate the RRR for each stock.

2. Next, we calculate the expected return of each stock and the expected abnormal return (alpha) is then calculated by subtracting the RRR from the Expected Return. The cost of less than full diversification comes from the nonsystematic risk of the mispriced stock, th variance of the stock’s residual ( variance of the error term e), which offsets the benefit (alpha) of specializing in an underpriced security.

3. After we’re done calculating the alpha and the residual variance (unsystematic risk) for each stock, we need to calculate the weights of the individual stock that should go to make the active portfolio. The weight of each stock is the alpha value divided by the variance of the residual risk.

4. Next, we calculate the alpha value of the active portfolio by adding up (alpha for each stock)(individual weight from #3). In the same way, calculate the Beta (and the residual variance) for the active portfolio, which is the weighted average of the beta (and the residual variance) of each individual stock.

5. From the residual variance, calculate the standard deviation of the error term.

6. Next we need to find w0, which is the weight of the active portfolio in the optimal risky portfolio, using a formulae. This needs to be adjusted for the beta of the active portfolio, and the new weight is w*.

So, the weight of the market index in the Optimal Risky Portfolio is (1-w*).

7. This is getting more interesting, eh? 😉 To calculate the exact make up of the optimal risky portfolio, we will need to calculate the Beta [weighted average of the betas of the market index (it’s beta is 1) and the active port.], the Expected Return [using the Market Model method= alpha of the optimal port. + Beta (Market Risk Premium); Alpha of the Optimal Risky Port. is [W(a)*alpha(a)] and the variance (using Market Model Method= (Beta squared)*(Market Risk Premium squared)+(Market Variance squared)].

8. Next we need to calculate the weight given to the Optimal Risky Portfolio (which is called P and lies on the CAL) and the weight allotted to the Risk Free Rate in the overall Portfolio. For this, we use another equation.

If ‘y’ is the weight of the Optimal Risky Port, then (1-y)= Weight in T Bills.

The weight of individual stocks will be= Y*weight of active port (From #6)*weight of each stock (from #3)

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The prediction of alpha needs to be adjusted depending upon the forecast accuracy of the security analysts. This done by first calculating the correlation between the forecasted alpha and the actual alpha value. Either of them can be the independent variable. Next, square the correlation to get R^2, which is the co-efficient of determination, and is a measure of the % change in the dependent variable that is explained by the % change in the independent variable. The higher the value, the better is the forecast of the security analyst. Then multiply the forecasted alpha with R^2 to get the adjusted Alpha. Then we can use this adjusted alpha to again calculate the weights of the stocks and the optimal risky portfolio.

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I haven’t covered why Active Portfolio Management is helpful even in Efficient Markets. Please go through pages 467-69 for it. I feel too lazy. Sorry!