## 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

where **σ(ε _{i})^{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

Using these weights we can construct the active portfolio and calculate the corresponding **α _{A}, β_{A}**, and the

**σ**

_{A}^{2}.## Step 2

Now we determine the size of the optimal portfolio (**w _{A}**) in the overall portfolio. Treynor and Black show that the weight of the active portfolio should be

where **r _{P}** is the return on the passive portfolio,

**r**is the risk-free rate and

_{f }**σ**is the volatility of the passive portfolio.

_{P}^{2}Up to now, we have not yet used the **β _{A}** 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

As such, the overall portfolio consists of an active portfolio with weight w_{A} and a passive portfolio with weight w_{P} = 1-w_{A}.

## 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.

### Treynor-Black model

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