Treynor–Black Portfolio Weights and the Additive Sharpe Ratio

Summary

The document introduces the Treynor–Black model, which combines a market portfolio with an active portfolio that is assumed to offer alpha. It gives formulas for the active allocation using the active portfolio’s alpha and residual variance, along with market return and variance, and raises a question about the model’s squared Sharpe ratio.

The central claim under discussion is that the combined portfolio’s squared Sharpe ratio equals the sum of the market and active portfolios’ squared Sharpe ratios. The text provides no derivation, worked example, or evidence for that identity; instead, it points readers to a longer external treatment and a textbook. Its formulas and question are a useful starting point for studying active portfolio construction, but the assumptions behind the identity and its practical limits are not explained here.

Key ideas

• The Treynor–Black model combines an active portfolio with a market portfolio.
• The active allocation depends on estimated alpha and residual variance relative to market return and variance.
• The document asks why the combined portfolio’s squared Sharpe ratio is expressed as a sum of component squared Sharpe ratios.
• No derivation or empirical evidence is supplied in the document.

Tags

# Derivation Treynor-Black model


# Derivation Treynor-Black model












In the treynor-black model the assumption is that markets are not fully optimal and it is possible to achieve additional alpha on top of the market portfolio. After a mean-variance optimization treynor-black model arrives at optimal weights for the active portfolio $w_A$ and $(1-w_A)$ is the allocation to the market portfolio.

$w_0 = \frac{\frac{\alpha_A}{\sigma_A^2(e)}}{\mu_m/\sigma^2_m}$

$w_A = \frac{w_0}{1+(1-B)w_0}$

I have troubles understanding the statement that the squared sharpe ratio of the full portfolio (containing w_A of the active portfolio and (1-w_A) of the market portfolio) equals the sum of the squared sharpe ratios, i.e.:

$Sp^2 =\{\frac{expectedreturn}{vola} \}^2= \frac{\mu^2}{\sigma_m^2} + \frac{\alpha^2}{\sigma_A^2(e)}$

Below link gives a more detailed description of the problem. But it is also described in identical way in the book of Bodie, Kane, Marcus "investments". https://www.diva-portal.org/smash/get/diva2:1576276/FULLTEXT02.pdf

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.