Another Partial Solution: Conditional Sharpe Ratio

“What this country needs is a good five-cent cigar,” Thomas Marshall (Woodrow Wilson’s vice president) once remarked. Updating the quip for 21st century finance might run as follows: Investors need a good risk metric. Alas, what’s needed and what’s available isn’t usually, if ever, one and the same in the money game. The next best thing is tapping several flawed metrics that are flawed in different ways.

Last week I discussed one “partial solution” in the search for an upgrade to the nearly 50-year-old Sharpe ratio, the widely used but flawed measure of quoting risk premiums (return less a risk-free rate) per unit of performance volatility (standard deviation). As many critics have charged over the years, standard deviation falls well short of the ideal definition of investment risk. So, too, does everything else.
One big challenge is modeling what’s known as tail risk, or the possibility—the virtual inevitability—that investment losses will exceed expectations implied by a normal distribution. Last week’s look at one attempt at trying to anticipate non-normality was the modified Sharpe ratio, which incorporates skewness and kurtosis into the calculation. Another possibility is the so-called conditional Sharpe ratio (CSR), which attempts to quantify the risk that an asset or portfolio will experience extreme losses.
To understand CSR it’s necessary to start with so-called value at risk (VaR), the much maligned metric that was (and still is) widely used and widely abused. At its core, VaR tries to tell us what the possibility of loss is up to some confidence level, usually 95%. So, for instance, one might say that a certain portfolio is at risk of losing X% for 95% of the time. What about the remaining 5%? That’s where the trouble lies, of course, and trying to model the last 5% (or 1% for a 99% confidence level) is devilishly hard. Some analysts say it’s simply impossible. Misinformed or not, conditional VaR, or CVaR, dares to tread into this black hole of fat taildom. For the conditional Sharpe ratio, CVaR replaces standard deviation in the metric’s denominator.
As a recent research paper from Ibbotson Associates explains,

CVaR is a comprehensive measure of the entire part of the tail that is being observed, and for many, the preferred measurement of downside risk. In contrast with CVaR, VaR is only a statement about one particular point on the distribution. Intuitively, CVaR is a more complete measure of risk relative to VaR and previous studies have shown that CVaR has more attractive properties (see for example, Rockafellar and Uryasev (2000) and Pflug (2000)).

The main question with CVaR is one of choosing a methodology for calculation. The standard approach—the parametric method—is to assume a normal distribution, in which case CVaR can be estimated with only three inputs: average (or expected) return, volatility, and a conventional assumption about distributions. Easy but fraught with caveats.
Fortunately, there are several alternative approaches for estimating CVaR. One is using Monte Carlo simulations, which come in a variety of flavors and assumptions. The basic version can be easily run in Excel. You can also estimate CVaR based on an historical sample. If, for instance, you were calculating tail risk for the S&P 500 through a CVaR prism, you would identify the extreme return outliers in the past and factor the data into your CVaR modeling. WIth a little bit of work in a spreadsheet, you can come up with a rough estimate fairly easily. More sophisticated analytics require programming in Matlab or R.
Readers at this point are probably wondering how much insight is offered in CVaR, and by extension the conditional Sharpe ratio. Probably less than its strongest advocates argue. The basic message in conditional Sharpe ratio, like that of its modified counterpart, is that investors underestimate risk by roughly a third (or more?) when looking only at standard deviation and related metrics. That’s a critical message. The details get fuzzy once you move beyond that reasonable conclusion. One reason is that quantifying expected risk in the tail is a serious challenge and open to a fair amount of debate.
Indeed, you can’t model uncertainty per se. Even when it comes to modeling the known unknowns it’s still best to regularly repeat the following: no one risk measure can profile the true nature of risk in all its fury and variation. But we can try, if only to develop a deeper understanding of risk analytics’ strengths and weaknesses. At some point, however, you’re still flying blind.

2 thoughts on “Another Partial Solution: Conditional Sharpe Ratio

  1. Roberto

    I have a question about using CVaR or MVaR in portfolio optimization. If the covariance matrix is normally populated with Correl(1,2) * Std(1) * Std(2) would a CVaR substitution look like this? Correl(1,2) * CVaR(1) * CVaR(2). Or is there more to it?

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