Since the Sharpe ratio was derived in 1966 by William Sharpe, it has been one of the most referenced risk/return measures used in finance, and much of this popularity can be attributed to its simplicity. The ratio's credibility was boosted further when Professor Sharpe won a Nobel Memorial Prize in Economic Sciences in 1990 for his work on the capital asset pricing model (CAPM).

In this article, we'll break down the Sharpe ratio and the sum of its parts.

### **The Sharpe Ratio ****Defined**

Most people with a financial background can quickly comprehend how the Sharpe ratio is calculated and what it represents. The ratio describes how much excess return you are receiving for the extra volatility that you endure for holding a riskier asset. Remember, you always need to be properly compensated for the additional risk you take for not holding a risk-free asset.

We will give you a better understanding of how this ratio works, starting with its formula:

S (x) = (r_{x} - R_{f}) / StdDev (x)

Where:

- X is the investment
- r
_{x}is the average rate of return of X - R
_{f}is the best available rate of return of a risk-free security (i.e. T-bills) - StdDev(x) is the standard deviation of r
_{x}

**Return (r _{x}**

**)**

The returns measured can be of any frequency (i.e. daily, weekly, monthly or annually), as long as they are normally distributed. Herein lies the underlying weakness of the ratio – not all asset returns are normally distributed.

Kurtosis, fatter tails and higher peaks, or skewness on the distribution can be problematic for the ratio, as standard deviation doesn't have the same effectiveness when these problems exist. Sometimes it can be downright dangerous to use this formula when returns are not normally distributed.

**Risk-Free Rate of Return (r _{f}**

**)**

The risk-free rate of return is used to see if you are being properly compensated for the additional risk you are taking on with the asset. Traditionally, the risk-free rate of return is the shortest-dated government T-bill (i.e. U.S. T-Bill). While this type of security will have the least volatility, some would argue that the risk-free security used should match the duration of the investment it is being compared against.

For example, equities are the longest duration asset available, so shouldn't they be compared with the longest duration risk-free asset available – government issued inflation-protected securities (IPS)?

Using a long-dated IPS would certainly result in a different value for the ratio, because in a normal interest-rate environment, IPS should have a higher real return than T-bills.

For instance, the Barclays U.S. Treasury Inflation-Protected Securities 1-10 Year Index has returned 3.3% for the period ending Sept. 30, 2017, while the S&P 500 Index returned 7.4% over the same timeframe. Although it can be argued that investors are being fairly compensated for the risk of choosing equities over bonds in this period, the bond index's Sharpe ratio of 1.16% versus 0.38% for the equity index would indicate equities are the riskier asset.

**Standard Deviation (StdDev(x)****)**

Now that we have calculated the excess return from subtracting the risk-free rate of return from the return of the risky asset, we need to divide this by the standard deviation of the risky asset being measured. As mentioned above, the higher the number, the better the investment looks from a risk/return perspective.

How the returns are distributed is the Achilles heel of the Sharpe ratio. Bell curves do not take big moves in the market into account. As Benoit Mandelbrot and Nassim Nicholas Taleb note in "How The Finance Gurus Get Risk All Wrong" (*Fortune, *2005*)*, bell curves were adopted for mathematical convenience, not realism.

However, unless the standard deviation is very large, leverage may not affect the ratio. Both the numerator (return) and denominator (standard deviation) could be doubled with no problems. Only if the standard deviation gets too high do we start to see problems. For example, a stock that is leveraged 10-to-1 could easily see a price drop of 10%, which would translate to a 100% drop in the original capital and an early margin call.

### **The Sharpe Ratio and Risk**

Understanding the relationship between the Sharpe ratio and risk often comes down to measuring standard deviation, which is also commonly referred to as the total risk. The square of standard deviation is the variance, as defined by Nobel Laureate Harry Markowitz, who is arguably best known as the pioneer of Modern Portfolio Theory.

So why did Sharpe choose the standard deviation to adjust excess returns for risk and why should we care? We know that Markowitz defined variance, a measure of statistical dispersion or an indication of how far away it is from the expected value, as something undesirable to investors. The square root of variance, or standard deviation, has the same unit form as the data series being analyzed and is such more commonly used to measure risk.

The following example illustrates why investors should care about variance:

An investor has a choice of three portfolios, all with expected returns of 10% for the next 10 years. The average returns in the table below indicates the stated expectation. The returns achieved for the investment horizon is indicated by annualized returns, which takes compounding into account. As the data table and the chart clearly illustrates below, the standard deviation takes returns away from the expected return. If there is no risk – zero standard deviation – your returns will equal your expected returns.

**Expected Average Returns**

Year |
Portfolio A |
Portfolio B |
Portfolio C |

Year 1 | 10.00% | 9.00% | 2.00% |

Year 2 | 10.00% | 15.00% | -2.00% |

Year 3 | 10.00% | 23.00% | 18.00% |

Year 4 | 10.00% | 10.00% | 12.00% |

Year 5 | 10.00% | 11.00% | 15.00% |

Year 6 | 10.00% | 8.00% | 2.00% |

Year 7 | 10.00% | 7.00% | 7.00% |

Year 8 | 10.00% | 6.00% | 21.00% |

Year 9 | 10.00% | 6.00% | 8.00% |

Year 10 | 10.00% | 5.00% | 17.00% |

Average Returns |
10.00% | 10.00% | 10.00% |

Annualized Returns |
10.00% | 9.88% | 9.75% |

Standard Deviation |
0.00% | 5.44% | 7.80% |

### **Using the Sharpe Ratio**

The Sharpe ratio is a measure of return that is often used to compare the performance of investment managers by making an adjustment for risk.

For example, if investment manager A generates a return of 15% while investment manager B generates a return of 12%, it would appear that manager A is a better performer. However, if manager A, who produced the 15% return, took much larger risks than manager B, it may actually be the case that manager B has a better risk-adjusted return.

To continue with the example, say that the risk free-rate is 5%, and manager A's portfolio has a standard deviation of 8%, while manager B's portfolio has a standard deviation of 5%. The Sharpe ratio for manager A would be 1.25, while manager B's ratio would be 1.4, which is better than that of manager A. Based on these calculations, manager B was able to generate a higher return on a risk-adjusted basis.

For some insight: a ratio of 1 or better is considered good; 2 or better is very good; and 3 or better is considered excellent.

### **The Bottom Line**

Risk and reward must be evaluated together when considering investment choices; this is the focal point presented in Modern Portfolio Theory. In a common definition of risk, the standard deviation or variance takes rewards away from the investor. As such, the risk must always be addressed along with the reward when you are looking to choose your investments. The Sharpe ratio can help you determine the investment choice that will deliver the highest returns while considering risk.