Using Sharpe Ratios in Practice

Portfolio construction, diversification, high return vs. high Sharpe, and more

In an earlier post, we introduced the Sharpe ratio as a risk-adjusted measure of investment performance. Here we dive deeper into how Sharpe ratios are used in practice. Please read the earlier post before this one.

Sharpes for Portfolio Construction

Previously, we saw that Sharpes are useful for evaluating stand-alone investments.

Perhaps just as important, they let you evaluate combinations of investments (IE. portfolio construction). Below is a stylized example to illustrate this point. RED and BLUE are both equal Sharpe investments (SR=2 each) with the same volatility and risk: which is better?

Trick question. Neither! The best portfolio in this opportunity set can be found by observing that when RED zigs, BLUE zags and vice versa. Their wiggles are “opposite”. So if we combined the two investments, couldn’t we “cancel” the wiggles, reduce risk, and obtain a higher Sharpe portfolio?

Below PURP does exactly this. It’s a 50%/50% portfolio of RED and BLUE. Looks much steadier.

Indeed, while the Sharpe of RED & BLUE were each 2, PURP has a gigantic Sharpe of 5!

So Sharpe ratios also allow us to quantify how well different investments work together in a portfolio. Again, this would be unknowable from returns.

In this way, Sharpes make concrete the often vague term “diversification”: effective diversification should result in the Sharpe of the whole being higher than the Sharpe of the components.

Often, quants will look for strategies which are lowly or negatively correlated and combine them effectively to create high Sharpe models. Even if the individual strategy Sharpes are not massive, when combined properly, the whole can be very strong.

High Return vs. High Sharpe

Below graph shows one investment with a higher Sharpe (HIGH_SR), and another with a higher return (HIGH_RET). Which is better? While the smoothness of HIGH_SR is nice, in the end, “we can’t eat Sharpe” and will have more dollars in our pocket investing in HIGH_RET. So perhaps HIGH_RET is better?

To answer this more clearly, we need to understand leverage. Leverage involves borrowing money to invest more than you have. This is what you do when you take out a mortgage to invest in a house.

Example. Let’s say I have $100 of cash to invest. This is my “equity capital.” I then borrow another $100 to invest a total of $200 in HIGH_SR. Then my leverage is 2x. For every 1% HIGH_SR moves up, I now earn $2 or 2% on my equity capital. Conversely, for every -1% move in HIGH_SR, I lose 2$/2%. Dollar PnL and returns on equity are effectively doubled. As you can probably tell, volatility is also doubled. Since returns and volatility both double, Sharpes remain the same.

So couldn’t we simply lever up the HIGH_SR portfolio to obtain equal or higher return than HIGH_RET while maintaining a high Sharpe?

Below graph shows HIGH_SR levered to 2x (HIGH_SRX2) and 3x (HIGH_SRX3). As you can see, we can maintain the smoothness and high Sharpe of HIGH_SR with leverage AND obtain the same or higher returns than HIGH_RET.

Leverage does bring with it its own risks/costs, however, and users should be cautious and aware of those (see LTCM blow up). Nevertheless, if used carefully, leverage makes high Sharpe strategies very powerful even if they have lower returns initially: you can target higher returns while maintaining the high Sharpe and nice smooth return profile.

Going Deeper

So higher Sharpe = fewer wiggles. “Wiggles” is not exactly a technical term. There are, however, more technical justifications for using the Sharpe ratio. We briefly mention two from statistics and financial theory.

A simple stats approach to evaluating a return series mightseek to determine if the returns are statistically greater than zero. Going back to stat 101, we can compute a t-stat for this. Below is the generic formula for a t-stat:

To determine if a returns series is different from 0, we have

So, playing with the formulas, we can see that the Sharpe ratio as defined above is directly proportional to the t-stat.

If you are evaluating investments with roughly the same number of returns data points, the Sharpe ratio already rank orders them by statistical significance. Kind of useful. And you’re just a scalar away from getting the full t-stat. This is perhaps why, intuitively, we also feel less confident going forward in investments with more wiggles and lower Sharpes.

We don’t like getting too academic, but for what it‘s worth, there is some financial theory which says that “rational” investors who like returns but dislike risk (volatility) should only hold a single portfolio to maximize their happiness/utility. And that portfolio is the highest Sharpe portfolio (IE “tangency portfolio”). So we should always be striving to maximize Sharpe. Interested readers are referred to “Active Portfolio Management” by Grinold and Kahn to learn more.

Conclusion

Returns matter in investing, but so does risk. Sharpe ratios are a risk-adjusted measure of investment performance that tell you how you are being rewarded for the risk you take. Use it to more effectively evaluate investments and portfolios of investments.