In recent years, there has been much criticism of the use of the Sharpe Ratio as a tool for measuring investment performance e.g.,
- “It doesn’t account for an asset’s correlation with the overall market”
- “It doesn’t account for skewness or fat tails”
- “Sharpe Ratio estimates don’t properly address account for serial correlation”
and the list goes on.
In this post, I’m going to argue that the Sharpe Ratio is still the best measure of performance quality in the context of a portfolio. The result is a simple heuristic for asset selection:
where SR is shorthand for the Sharpe Ratio, p is the asset being evaluated and m is the rest of one’s portfolio. ϕ is a new type of correlation measure that accounts for non-linear relationships between p and m. If p and m are linearly related, then ϕ is equal to ρ, the standard Pearson correlation coefficient.
Note that the following is an outline and not a formal mathematical proof. If you’re good at math and would like to work on a formal proof, please DM me.
Consider any suitable performance metric as a function of 4 moments: f(μ, σ, s, k). Consider how the performance metric changes if a small amount of some asset p is added to a market portfolio m. An investor will buy this asset if it increases the performance metric i.e., our decision criterion is
where δ is the quantity of p added to our portfolio. Now
Let
Then
Furthermore
Now, by the quotient rule
where
Finally
Again, by the quotient rule
where
Therefore, by the chain rule,
Where the notation ∂f/∂μm, ∂f/∂σm, etc. is shorthand for ∂f/∂μ, ∂f/∂σ, etc. evaluated at (μm, σm, sm, km).
Equivalently,
This equation represents a decision criterion for any suitable performance measure. One obvious choice of the performance measure is the ratio of the expected value to some quantile or Value-at-Risk, which can be characterized as a Generalized Sharpe Ratio i.e.,
where
Q is the quantile function of some distribution with moments μ, σ, s, k and α is some confidence level (typically less than 5%). µ is the expected return of an asset over the risk-free rate. Note that this measure accounts for both skewness and kurtosis.
For example, for a Standard Normal distribution,
or
The functional form of Vα presumably assumes some probability distribution (or a class of distributions). With this form, we can write
Similarly,
Therefore, the decision criterion becomes
which is the result we were looking with
Here, “…” considers higher order moments without describing them explicitly. This is an amazing result. Even though our performance metric is the Generalized Sharpe Ratio, the decision criteria ends up being based on the simple Sharpe Ratio!
To make this expression more intuitive, let’s focus on just the first two terms and assume that
In Appendix A, we show that
where κ is the excess kurtosis, defined as
and γ is defined as
where Γ is the coefficient of a quadratic term in a linear regression (see Appendix A).
Therefore, if we ignore the last term in the parentheses, our decision criteria is
where
The expression ρ - ξγ could be characterized as a convexity adjusted correlation. (It is analogous to Spearman’s rank correlation in that it accounts for non-linear relationships between two random variables.)
Unlike ρ, note that γ is not “invertible” i.e.,
Similarly, we can imagine owning p (instead of m) and consider adding a small quantity of m. In that case, the decision criterion is
If this expression is false, then we’d want to hold none of portfolio m and put 100% into the alternative asset p. In other words,
where w is the optimal proportion of the portfolio held in p. (Note: This simplified analysis does not allow allocation changes within m.)
Finally, we can estimate ξ if we make assumptions about the distribution of m. For example, if we assume that m is Normally distributed (i.e., skew=0, kurtosis=3) and α=0.5% (see Appendix B), then
It’s worth nothing from the tables in Appendix B that ξ appears to converge to 1 as α approaches 0. This suggests that we may be able to derive an estimate of ϕ in the limit i.e.,
Open Questions:
1) How can we formally prove the above result?
2) Does ϕ converge as α approaches 0?
3) How can we derive an estimation algorithm for ϕ?
4) Is there a straightforward way to “invert” ϕ?
5) The above methodology can be described as an extended version of CAPM that contemplates non-linear relationships between assets. Derivative pricing models also consider non-linearity. Are these two approaches consistent?
Appendix A: Quadratic Term of Ordinary Least Squares
Let
Then
We look for coefficients that minimize the sum of square errors i.e.,
Therefore, we have the following system of equations:
or
Finally, we can write
And asymptotically,
Appendix B: Estimating -1/M⋅ ∂M/∂s
The following tables were estimated using the sknor package in Stata:
