The Marginal Sharpe Ratio: A New Heuristic for Asset Selection
Portfolio optimization strategies typically involve an unintuitive procedure that combines individual assets’ expected returns, individual assets’ risks (i.e., standard deviations of expected returns), and the correlations amongst all assets to determine an optimal portfolio.
The following post explores a simple heuristic that greatly simplifies the standard portfolio optimization algorithm: the Marginal Sharpe Ratio.
Consider an asset p being added to an existing portfolio m. Moreover, say that an investor’s goal is to choose a quantity of p and m that maximize the portfolio’s Sharpe Ratio.
Let μₚ be the expected return of p and μₘ be the expected return of m. Moreover, let σₚ and σₘ be the expected standard deviation of the returns of p and m, respectively.
The standard result from Portfolio Theory is that an asset has a place in our portfolio if its “alpha” is greater than 0 i.e.,
Here, βₚ,ₘ is the standard definition used in Portfolio Theory.
Note that this heuristic doesn’t give us the exact optimal quantity of p. It just tells us that the optimal quantity is greater than 0.
It can be shown that
Note that beta is not invertible i.e.,
We can also consider the case where we currently own p and want to add some amount of m to our portfolio. In this case, Portfolio Theory suggests that we want some amount p if
(Of course, to do this division, we’re assuming that beta (and hence rho) is greater than 0.)
Now we can say the following about the optimal weight (w) of p:
While this heuristic works, it’s not very intuitive because the betas aren’t invertible. Instead, note that
can be rewritten as
Note that μ/σ is the Ordinary Sharpe Ratio (SR). Now we can define the Marginal Sharpe Ratio (ξ) as
Now we can rewrite our heuristic as
which is arguably much easier to interpret than the earlier version (primarily since ρ is invertible while β is not). This result suggests that the Marginal Sharpe Ratio is a convenient - and semantically consistent - alternative to alpha. It also provides better intuition about the relationship between the Orindary Sharpe Ratio and an asset’s value in the context of a portfolio.
Example: Say two hedge funds both report a 1.00 Sharpe Ratio at a time period when the (estimated) Sharpe Ratio of the S&P 500 was 1.50. Moreover, say fund 1 has a 0.25 correlation with the S&P 500, whereas fund 2 has a 0.75 correlation. In this case, fund 1 has a Marginal Sharpe Ratio of 0.625, whereas fund 2 has a Marginal Sharpe Ratio of just -0.125. In other words, fund 1 would be deemed investment-worthy, while fund 2 would not.