# The Sharpe Ratio As An Efficiency Metric

Ratios and normalized metrics are used regularly in the hard sciences, particularly when it comes to comparing scenarios and outcomes. The efficiency of a vehicle, for instance, is typically measured in miles per gallon, or the distance traveled per unit of energy. A Toyota Prius at about 50 MPG is without a doubt substantially more efficient compared to say a top fuel dragster.

The financial world has its equivalent of miles per gallon: the Sharpe Ratio, which combines both return and volatility into a single metric

In it’s original form, presented here, the ratio quantifies an assets return in excess of a risk-free rate (the risk premium) per unit of volatility. In his 1966 paper Nobel Laureate William Sharpe derived this measure of investment efficiency from a linear relationship between return and volatility (risk). [1] The plot below shows this linear relationship between US Large Company Stocks (S&P 500) and the thirty day Treasury Bill. The volatility of the thirty day T-Bill was set to zero to satisfy the “risk-free” criteria assumed in Sharpe’s work.

In the absolute sense risk-free returns don’t really exist. The thirty day Treasury Bill does indeed have a small amount of volatility associated with it. From 1926 through 2015 the annualized standard deviation was approximately 0.9%. To build a more accurate model this volatility should be accounted for, and as Harry Markowitz demonstrated, the standard deviation of a portfolio is not simply the weighted sum of the constituent standard deviations. Using Markowitz’s formulas results in the following efficient frontier

Admittedly I’m splitting hairs. Sharpe’s model is in fact a very close approximation of the Markowitz efficient frontier when comparing various mixes of an asset and the 30 day Treasury Bill. The linearity only begins to break down as the portfolio approaches a 100% allocation to Treasury Bills–it’s rather trivial.

But there’s a bigger issue with the Sharpe Ratio, and it isn’t the fault of the ratio itself, rather how the ratio is interpreted. At first glance it would appear that higher Sharpe Ratios indicate more desirable assets. Is a higher Sharpe Ratio always better? Consider the performance of the 5 year Treasury Note and Large Company US stocks

 Asset Performance (1926 – 2012) US Large Co. Stocks 5 Year Treasury Note Annualized Total Return 9.8% 5.4% Standard Deviation 19.1% 4.4% Sharpe Ratio 0.33 0.42 Source: SBBI [3]

If one were to use the Sharpe Ratio as the sole criteria for making investment decisions then Treasury Notes have clearly been superior to US stocks. The comparison here isn’t all that different from the Prius and the drag racer I mentioned previously. Efficiency and absolute performance are very different things. One option may be more efficient, but is it really sufficient to accomplish the task at hand? It probably depends on what the objective is in the first place. Sharpe himself recognized that using only mean return and variance (or standard deviation) was too simple to fully capture the needs of every investment decision

Clearly, comparisons based on the first two moments of a distribution do not take into account possible differences among portfolios in other moments or in distributions of outcomes across states of nature that may be associated with different levels of investor utility.

When such considerations are especially important, return mean and variance may not suffice, requiring the use of additional or substitute measures. [2]

Efficiency may be one useful measure to consider, but it doesn’t necessarily equate to “better.”

Offered without comment…

26 September 2016 8:38 pm CDT

When I originally put this post together I struggled with the topic of leverage as it does play an important role in the overall theory that Sharpe presented. The perspective I shared above was viewed through the lens that leverage was not a choice available to investors. This was a conscious decision as I felt it distracted from the bigger message I was trying to communicate. Consequently some of the feedback I received revolved around this omission.

The manner in which leverage makes higher Sharpe ratios more appealing can be demonstrated in the following two scenarios. Both were constructed using the numbers above for the 5 year Treasury Note and US stocks.

Scenario 1
A portfolio of Treasury Notes is levered to the same level of volatility as US stocks. The resulting return would have been 3.5% + 0.42*19.8% = 11.8%. This would require a leverage ratio of 11.8%/5.4% = 2.19.

Scenario 2
Again, a portfolio of Treasury Notes is levered, but this time to achieve the same return as US stocks. The resulting volatility would be (9.8% – 3.5%)/0.42 = 15.0% with a required leverage ratio of 9.8%/5.4% = 1.81. Clearly less volatile than US stocks.

The use of leverage is key to understanding why a higher Sharpe Ratio–the risk adjusted return–is considered the preferred choice. But this thought pattern assumes that investors are comfortable with borrowing to finance investment, and can do so at a reasonable interest rate. In practice the use of leverage should not be taken lightly, and is not something that I believe is best for the do-it-yourself set. Hence my hesitation to bring it up in the initial discussion. My original intent, in a pragmatic way, was to demonstrate the necessity to look beyond simple metrics that appear to be an all-in-one solution to investment decisions.

References
1. Sharpe, William F. Mutual Fund Performance. Journal of Business. January 1966. pp. 119-138.
2. Sharpe, William F. The Sharpe Ratio. The Journal of Portfolio Management. Fall 1994. https://web.stanford.edu/~wfsharpe/art/sr/sr.htm
3. 2013 Ibbotson SBBI Classic Yearbook. Morningstar Inc. Chicago, IL. pp. 184-211.