The Sharpe Ratio is one of the most common performance metrics for both individual stocks and portfolios. However, a more natural choice for a portfolio performance metric is the ratio of the expected return to some Value-at-Risk of the return distribution, which can be characterized as a Generalized Sharpe Ratio i.e., $$ GSR \equiv \lim_{\alpha \to 0} \frac{\mu}{V_\alpha(\mu=0,\sigma,s,k,…)} $$ where \(V_{\alpha}()\) is the Value-at-Risk, \(\alpha\) is the probability threshold and \(\mu\), \(\sigma\), \(s\), etc. are the 1st (mean), 2nd (standard deviation), 3rd (skewness), etc. moments of of the return distribution. \(\mu=0\) indicates that Value-at-Risk calculations assume 0 expected return.

For a standard normal distribution $$V_{0.1587}(\mu=0,\sigma,s,k,…) = V_{0.1587}(\mu=0,\sigma) = \sigma$$ $$V_{0.0250}(\mu=0,\sigma,s,k,…) = V_{0.0250}(\mu=0,\sigma) = 1.96\sigma$$ $$V_{0.0100}(\mu=0,\sigma,s,k,…) = V_{0.0100}(\mu=0,\sigma) = 2.33\sigma$$

Question: If the GSR is our portfolio’s performance metric, what is the decision criteria for including an individual asset in that portfolio?

Now if the portfolio’s performance metric is the standard Sharpe Ratio, then the decision criteria (called a Marginal Sharpe Ratio) is i.e., $$MSR_a = SR_a - \rho SR_m$$

If the MSR is greater than 0, then an asset’s portfolio weight is greater than 0. Is it possible to derive a formula that characterizes a Marginal Generalized Sharpe Ratio (MGSR)?

This is a complicated question mathematically, but it is possible to show that $$GMSR_a = SR_a - (\rho-\gamma) SR_m$$ where \(\gamma\) is related to the second derivative of the asset’s return with respect to the portfolio’s return i.e., $$ \gamma \propto \frac{\partial^2 f(r_m)}{\partial r_m^2} $$ where $$r_a = f(r_m) + \epsilon$$ for some non-linear function f() and error term \(\epsilon\).

(To the best of my knowledge, the proof of this isn’t published. If you’d like to see a proof of this, send me an email.)

This is a pretty surprising result for a few reasons:

  • The ordinary Sharpe Ratio reappears even though it isn’t in our definition of the MGSR
  • Higher order moments of the asset’s distribution and the portfolio’s distribution appear nowhere in this equation

The latter point implies that an individual asset’s skewness in itself doesn’t impact its portfolio weight. The 2nd order factor that is relevant is \(\gamma\).

On further consideration, this is actually just a consequence of the Central Limit Theorem. Imagine a portfolio with an infinite number of uncorrelated assets with arbitrary return distributions. By the Central Limit Theorem, the portfolio’s returns will be normallly distributed!

\(\gamma\) is also related to \(\Gamma\) in traditional optional pricing theory. The “fair” price for an asset corresponds to the discount rate where the MGSR is 0 i.e., $$GMSR_a = SR_a - (\rho-\gamma) SR_m \equiv 0$$ $$SR_a = (\rho-\gamma) SR_m$$ $$\mu_a = (\rho-\gamma) SR_m \sigma_a$$ $$\mu_a = (\rho-\gamma) \mu_m \frac{\sigma_a}{\sigma_m}$$ $$\mu_a = \left(\beta -\gamma\frac{\sigma_a}{\sigma_m}\right)\mu_m$$ which is the classical CAPM formula with an addtional “convexity” term.

While estimating \(\gamma\) isn’t trivial, the qualitative implication is profound: when pricing assets, convexity matters, not skewness.