Treynor-Black model

Unlike the portfolio optimization that an investor can perform using Markowitz’s portfolio selection approach, the Treynor-Black model is a type of active portfolio management. The optimal risky portfolio in the Treynor-Black model consists of a passive (market) portfolio and an active portfolio for which we have alpha forecasts. These alpha forecasts are obtained using a factor model.

At the bottom of this page, we provide an Excel file that implements the Treynor-Black model.

Step 1

First, we determine the weights of the securities in the active portfolio. The weights we assign to the securities should be proportional to

 \dfrac{\alpha_i}{\sigma(\epsilon_i)^2}

where σ(εi)2 is the volatility in the security’s price that is not due to changes in the factors, i.e. unsystematic risk. Hence, the weight assigned to a particular security will be such that

  • the higher the alpha of the security, the higher the weight we assign to the security,
  • the more volatile the security is, due to firm-specific news, the lower the weight.

The ratio of alpha to nonsystematic risk is called the Treynor-Black ratio or appraisal ratio. It measures the value the security would add to our portfolio, on a risk-adjusted basis.

The optimal weights in the Treynor-Black model are

 w_i = \dfrac{\frac{\alpha_i}{\sigma(\epsilon_i)^2}}{\sum_{i=j}^N \frac{\alpha_j}{\sigma(\epsilon_j)^2}}

Using these weights we can construct the active portfolio and calculate the corresponding αA, βA, and the σA2.

\alpha_A = \sum_{i=1}^N w_i \cdot \alpha_i

\beta_A = \sum_{i=1}^N w_i \cdot \beta_i

 \sigma(\epsilon_A)^2 =\sum_{i=1}^N w_i^2 \cdot \sigma(\epsilon_i)^2

Step 2

Now we determine the size of the optimal portfolio (wA) in the overall portfolio. Treynor and Black show that the weight of the active portfolio should be

 w_A = \dfrac{\frac{\alpha_A}{\sigma_A^2}}{\frac{E(r_P)-r_f}{\sigma_P^2}}

where rP is the return on the passive portfolio, rf is the risk-free rate and σP2 is the volatility of the passive portfolio.

Up to now, we have not yet used the βA of the active portfolio. It is possible that the active portfolio we have obtained exhibits a lot of systematic risk, i.e. a high beta. To avoid our overall portfolio from becoming too risky, we make a small correction. That way, we ensure that the portfolio beta of the overall portfolio doesn’t change. The correction looks as follows

 w_A^* = \dfrac{w_A}{1+(1-\beta_A)\cdot w_A}

As such, the overall portfolio consists of an active portfolio with weight wA and a passive portfolio with weight wP = 1-wA.

Benefits

If we have successfully identified securities with positive alphas, the combined portfolio will lie above the efficient frontier using Markowitz’s portfolio selection approach. This will allow us to obtain a steeper capital allocation line and thus better risk-adjusted returns. The extent to which we can improve returns, depends on our stock-picking abilities, i.e. our skill at identifying ‘alpha’.

Summary

The Treynor-Black model is an active investment approach that tries to improve modern portfolio theory. In particular, the approach tries to take into account investors’ stock picks in the optimization process.

FACEBOOK
TWITTER
GOOGLE
http://breakingdownfinance.com/finance-topics/modern-portfolio-theory/treynor-black-model">

Treynor-Black model

Want to apply the Treynor-Black model yourself? Download our Excel file: TreynorBlack.