Understanding Jensen's alpha: The geometry of risk adjustments

What is it about?

Investors seeking alpha aim to improve the performance of an existing portfolio by augmenting the opportunity set (i.e. by trading in new assets). Jensen showed how to mix a new asset with the existing portfolio such that the resulting portfolio was orthogonal to the existing one. The additional expected return, obtained with this approach, was subsequently called Jensen's alpha. However, by leveraging Jensen's orthogonal portfolio any expected return can be targeted, thus rendering the whole concept meaningless. We re-formulate the problem as: how much can an investor improve his Sharpe ratio by augmenting the opportunity set - and we show that this problem has a well defined solution.

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Photo by Micaela Parente on Unsplash

Photo by Micaela Parente on Unsplash

Why is it important?

Dynamically trading a portfolio with too high leverage ultimately leads to bankruptcy (as first shown by Kelly). Moreover, the expected return of any portfolio scales linearly with the leverage factor, but so does the volatility. For this reason, we propose to look at orthogonal Sharpe ratios rather than so-called risk adjusted returns.