What is the Sharpe Ratio?

Sharpe ratio is used to measure the portfoilo mean excesive return per unit of risk. Risk means the standard deviation of return. Excessive return is the portfolio return minus the risk free rate. Formula looks like this: (mean portfolio return - risk free rate)/standard deviation of returns

Why use excess return in the formula?

1. it makes sense to compare the performance excess of the risk free rate, because if the portfoilio returns can't beat even the risk free rate, why would anybody invest in the portfolio, when they can just invest in the risk free rate bearing instruments like US treasures.
2. For a portfoilio of short term bonds, its Sharpe ratio would approach infinity if we don't account for the risk free rate, For example, a 3-year bond with a yield of 2%. The formula of Sharpe ratio without taking into account of the risk free rate. 2%/0 which is infinity

Sharpe ratio formula explaied

Explained as above. The Sharpre ratio has this formula:

 Sharpre ratio = (expected return - risk free rate) / return standard deviaiton

Here the unit of time is the same for the expected return, the risk free rate and the return standard deviation. For exampe, if the unit of time is year, year should be the unit of time for all three parameters, such as annual expected return, annual risk free rate and annual return standard deviaiton.

expected return - risk free rate is the excess return which means the risk premium. And the risk premium is the return that the investors expect the stock market can generate in excess of the risk free rate. For example, if the 1-year bond yield is 2%, we would want the stock market to give a return higher than 2%, otherwise we would just invest in the bond, because the bond has no risk wihle the stock market does. 

The greater the mean return, the greater the Sharpe ratio.

For a fixed return standard deviation of 1% and a risk free rate of 0%, the greater the mean return, the greater the Sharpe Ratio. a

Mean Return(%)	Sharpe Ratio
0	0
1	0
2	2
3	3
4	4
5	5
6	6

The greater the return standard deviation, the lower the Sharpe ratio.

For a fixed mean return of 2% and a risk free rate of 0%, the greater the return standard deviation, the lower the Sharpe Ratio. The standard deviation can't be 0, or the Shrape ratio would be infinity.

Return Standard Deviation(%)	Sharpe Ratio
1	2
2	1
4	0.5
8	0.25
16	0.125
32	0.0625
64	0.03125

The greater the risk free rate, the lower the Sharpe ratio.

For a fixed mean return of 2% and a standard deviation of 1%, the greater the risk free rate, the lower the Sharpe Ratio. This implies the higher the risk free rate, the higher the return the stock market has to generate in order to acheieve the same Sharpe Ratio.

Risk Free Rate(%)	Sharpe Ratio
0	2
0.1	1.9
0.2	1.8
0.3	1.7
0.4	1.6
0.5	1.5
0.6	1.4
0.7	1.3
0.8	1.2
1	1

Suggested Readings:

• How Sharpe Ratio Affects Retirement Funds
• Visualizing Stock Price Movement With Different Returns Per Unit of Risk (Sharpe Ratio)
• What Are the Mean, Standard Deviation and the Sharpe Ratio of A Leveraged EFT and Comparison to the Non-Leveraged One
• Which Has The HIghest Sharpe Ratio, SPY, SSO or SPXL
  

Conclusion

The Sharpe Ratio is a great way to measure the performance of a investment portfolio. Its indications are the following:

• Portfolio's mean return has a positive linear relationship with the Sharpe Ratio, which means a portfolio with a higher Sharpe Ratio may generate a on average better reutrn.
• Portfolio's return standard deviation has a inverse relationship with the Sharpe Ratio, which means a portfolio with a higher Sharpe Ratio may have a lower return standard deviation. A lower standard deviation means a smaller flucturation in prices and a lower risk.
• The risk free rate has a negative linear relationship with the Sharpe Ratio, which means when the risk free rate is high, the stock market needs to generate an on average better return in order to maintain the same Sharpe Ratio.