Beyond Sharpe: Measuring Risk Adjusted Returns

I’ve been reading Andrew Lo’s In Pursuit of the Perfect Portfolio, an illuminating survey of the history of academic thought in finance. The chapter on Bill Sharpe is superb – Sharpe comes across as both humble and likeable, slightly baffled even at how ubiquitous his name has become in the world of finance. Lo mentions in passing the work of mathematician Benoit Mandelbrot, who over the course of a long and brilliantly wide-ranging career sought to question some of the underpinnings of Sharpe’s work.

As a reminder, the Sharpe ratio is a measure of risk adjusted return calculated very simply as the average return of an investment over the risk-free rate divided by the volatility of that investment (measured as standard deviation). A Sharpe above 1 is generally viewed as decent (i.e. you are getting compensated for the risk you are taking).

It’s a conversation that comes up often in conversations with clients: why do we use Sharpe ratios as a measure of risk, when we know that they are imperfect? Mandelbrot highlighted the major issue with discussions of risk in market contexts – that most definitions of risk assume a normal (Gaussian) distribution of returns, while we know that returns are not normally distributed. Below we show a comparison of two portfolios with the same Sharpe ratio, but where one (red) follows a normal distribution and the other (purple) a real-world distribution with heavier tails and negative skew.

Another way of illustrating this problem is to look at drawdown experience. Two funds with the same Sharpe ratio can have wildly different drawdown profiles.

There’s also the problem that Sharpe uses standard deviation: a measure of volatility that includes both positive and negative volatility in its calculation. Upside volatility is usually not something that investors worry too much about – their focus is on drawdowns.

The Sortino ratio, calculated using only downside volatility, addresses this issue around directionality, but still assumes a normal distribution. In order to capture skewness and the shape of tails, you need to employ measures like Expected Tail Loss (ETL) and Stress Loss. Expected Tail Loss (ETL) (sometimes referred to as Conditional VaR or CvaR) calculates the average of all possible losses given that the specified loss threshold is exceeded.

While ETL is a significant improvement, it still relies on historical data to build its estimates of potential losses, and yet the losses we are worried about didn’t occur in the past – they will occur in the future!

Stress Loss is a qualitative measure of the expected loss in the worst market scenario and so goes some way to addressing this issue. It either uses historical periods of volatility like the 2008 Great Financial Crisis or the 2020 Coronavirus crash, or it submits the portfolio to scenarios that seek to test its resilience to disruptions in liquidity, specific market moves or spikes in consumer or corporate defaults. The problem with Stress Loss is that it is highly subjective and profoundly reliant on the construction and definition of stress scenarios. It also doesn’t convey the likelihood of a loss occurring, just its potential severity.

There are other possible contenders – the Omega ratio, the Calmar ratio, the Sterling ratio, the Burke ratio… All have their benefits and detriments, but none have offered enough insight to supersede the familiarity of Sharpe.

Which sends us back to Sharpe ratios. They are imperfect, but they have become a lingua franca and allow us to compare investments from very different spheres. It feels like there’s a lesson here – that we can spend a long time looking for the perfect answer, but we must not let the perfect become the enemy of the good. It’s important to look beyond Sharpe, but also to recognize that in a fast-moving world, it’s sometimes helpful to have a shorthand for complex ideas.