The Treynor-Black Model

Investment Theory

Lorenzo Naranjo

Fall 2024

Model Setup

We have n risky assets with excess returns following a single-index model R_{i} = \alpha_{i} + \beta_{i} R_{m} + e_{i}, where \operatorname{E}(e_{i}) = 0, \operatorname{Cov}(R_{m}, e_{i}) = 0 and \operatorname{Cov}(e_{i}, e_{j}) = 0 for i \neq j, for all i, j = 1, \ldots, n.
Suppose we form a portfolio P with the n risky assets and the market portfolio M.
Denote by w_{i} the weights of the n risky assets and w_{M} the weight in the market portfolio so that \sum_{i = 1}^{n} w_{i} + w_{M} = 1.

We can write the excess returns of this portfolio over the risk-free asset as R_{P} = \alpha_{P} + \beta_{P} R_{M} + e_{P}.
- The portfolio alpha includes the alpha of all the securities except the market that has no alpha.
Therefore, \alpha_{P} = \sum_{i = 1}^{n} w_{i} \alpha_{i}.
We will see later that the alpha of the optimal portfolio is positive since its weights are positive for securities with positive alpha, and negative otherwise.

The beta of this portfolio is a weighted average of the beta of all securities and the beta of the market, which is one.
Thus, \beta_{P} = \sum_{i = 1}^{n} w_{i} \beta_{i} + w_{M}.

The idiosyncratic risk includes the firm-specific risks of all securities except the market that has no idiosyncratic risk.
Because the firm-specific risks are uncorrelated with each other, we have that \sigma^{2}(e_{P}) = \sum_{i = 1}^{n} w_{i}^{2} \sigma^{2}(e_{i}).

We can compute the portfolio expected return and standard deviation as \operatorname{E}(R_{P}) = \sum_{i = 1}^{n} w_{i} \alpha_{i} + \beta_{P} \operatorname{E}(R_{M}), \tag{1} and \sigma_{P}^{2} = \beta_{P}^{2} \sigma_{M}^{2} + \sum_{i = 1}^{n} w_{i}^{2} \sigma^{2}(e_{i}). \tag{2}

The expected return and variance in (1) and (2) depend on the alphas generated by the active securities and the systematic exposure of the resulting portfolio.
- This is the essence of the Treynor-Black model.
- The active part of portfolio, which might behave like a zero-cost portfolio, generates an alpha but might increases the total risk by adding diversifiable risk.
- It might also increase the expected return by loading on systematic risk, and therefore increase the systematic risk of the resulting portfolio.

Determine a list of securities that you believe are mispriced.
Use your market research to determine \alpha_{i} for each mispriced security.
- Remember that \alpha_{i} is positive if you think that the security is undervalued and negative otherwise.
Run regressions for all securities in your active portfolio and get \beta_{i} and \sigma^{2}(e_{i}) = \sigma_{i}^{2} - \beta_{i}^{2} \sigma_{M}^{2}.
Obtain an estimate for the market risk-premium \operatorname{E}(R_{M}) = \mu_{M} - r_{f}.
Compute the variance of market returns \sigma_{M}^{2}.

The objective is to find the portfolio weights w_{1}, w_{2}, \ldots, w_{n} and w_{M} that maximize its Sharpe ratio.
Since \beta_{P} = \sum_{i = 1}^{n} w_{i} \beta_{i} + w_{M}, this is equivalent to finding w_{1}, w_{2}, \ldots, w_{n} and the beta of the final portfolio that maximize the Sharpe ratio of the target portfolio.
Maximizing the Sharpe ratio is equivalent to minimizing the variance of the portfolio for a given expected return.
- Only one particular expected return satisfies that the sum of the weights is one.
- For other targets of the expected return, we would have to borrow or invest the difference at the risk-free rate.

If we denote by \mu_{P} a possible target expected return, the minimization problem is as follows: \begin{aligned} \min_{\{w_{1}, w_{2}, \ldots, w_{n}, \beta_{P}\}} \quad & \frac{1}{2} \sigma_{P}^{2} \\ \textrm{s.t.} \quad & \operatorname{E}(R_{P}) = \mu_{P} - r_{f}. \end{aligned}
If you are interested in how to solve the model see the notes.
- You need to use some standard constrained optimization calculus techniques, i.e. form a Lagrangian and compute first-order conditions.

Compute \lambda = \frac{1}{\frac{\operatorname{E}{R_{M}}}{\sigma^{2}_{M}} + \sum_{i = 1}^{n} \frac{\alpha_{i}}{\sigma^{2}(e_{i})} (1 - \beta_{i})}.
The portfolio weights are given by \begin{aligned} w_{i} & = \lambda \frac{\alpha_{i}}{\sigma^{2}(e_{i})}, \\ w_{M} & = 1 - \sum_{i = 1}^{n} w_{i}. \end{aligned} \tag{3}

We can prove the following relationship between the Sharpe ratio of the optimal portfolio and the market Sharpe ratio, \left( \frac{\operatorname{E}(R_{P})}{\sigma_{P}} \right)^{2} = \sum_{i = 1}^{n} \left(\frac{\alpha_{i}}{\sigma(e_{i})}\right)^{2} + \left( \frac{\operatorname{E}(R_{M})}{\sigma_{M}} \right)^{2}.
The equation shows that the Sharpe ratio of the optimal portfolio is higher than the Sharpe ratio of the market alone.
- Each security contributes positively in increasing the Sharpe ratio of the optimal portfolio, regardless of whether the alpha is positive or negative.

For a given security i, the term \alpha_{i} / \sigma(e_{i}) is called its information ratio, and measures how good its alpha is.
- If the volatility of the firm-specific risk is small, then it will be easier to diversify that risk away and hence extract the alpha of the security.
- Each security contributes to increasing the squared value of the Sharpe ratio of the optimal portfolio by the square of its information ratio.