Treynor-Black Model

Treynor-Black Model

Analyst		X	Y
Alpha		4%	7%
Residual risk		30%	40%
Correlation between forecasted and realized alphas		0.85	0.60

Using the Treynor-Black model and alphas adjusted for each analyst's forecast accuracy, what is the optimal allocation of the stock recommended by X to the active portfolio?

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Answer:

W_X=(4%*0.85^2)/30% / ((4%*0.85^2)/30% + (7%*0.60^2)/40%)
= 0.6046... = 60%

W_Y=1-W_X=40%

Note that as the alphas are adjusted downward, not only does the allocation within the active portfolio change, but the allocation between the active portfolio and the market portfolio changes as well. The allocation to the active portfolio will decline and the allocation to the market portfolio will increase as a result of adjusting for forecast accuracy.

Correlation^2=R^2