Definition
Sharpe ratio is used to measure the return of an investment to its risk. It is a mathematical expression that is a measure of the excess return (above the risk-free rate) earned in excess of what would have been earned on a risk vs reward investing investment during the same period. When employed for evaluating a portfolio, the ratio can be computed for any number of time periods. For example, the ratio may be computed from daily price data to evaluate performance over one year, or from monthly returns to monitor results over three years.

History
In 1966, economist William F. Sharpe proposed the ratio as a way of comparing the returns earned by any risky investment with the returns of a benchmark (a security or index of securities designed to serve as a standard for measuring performance). Although Sharpe originally proposed his ratio for measuring the performance of mutual funds, financial analysts today use it to evaluate any investment.

it is also known under other names including: reward-to-variability ratio, Sharpe’s' measure of risk-adjusted return and excess return/standard deviation. The name reward-to-variability ratio is used in India.
It is a popular approach for calculating risk-adjusted relative returns. It contrasts a fund's past or expected returns with those of a benchmark. Given the same amount of risk, a higher ratio indicates that the fund has delivered better returns than the benchmark.
The formula has three elements:
Mean return of investment Standard deviation of investment R-squared. Let's look at each part in turn. In this case, mean represents an average. In other words, we are looking at what is called a time-weighted average return. The numerator of the ratio is the temporal difference between realized or projected returns and a benchmark. The standard deviation of returns over the same time period, a measure of volatility and risk, is the denominator.

Formula:
Sharpe Ratio = (Annualized Return - Risk-Free Rate)/Standard Deviation of Returns

where:
R – average annual return of a portfolio calculated from the beginning to end of the time period. Rf – risk-free rate based on treasury securities with the same maturity as the returns used in the denominator.
Investment metrics such as Sharpe ratio and expected annual return are expressed graphically. Letting:
(1) = R/Rf,

Then, the above formula for the ratio represents point B on a graph. The solid line represents a risk-free investment of par (100% of face value at maturity) paying no interest; the dashed line represents an investment with an expected return of 5 percent compounded annually.

The area between these two lines is the Sharpe ratio from A to A+B on the graph. It measures excess return from the risk-free investment over what would be earned in that period if instead invested in a risky portfolio with expected returns of 5 percent compounded annually.
Pitfalls

The ratio has four important drawbacks:
It is not additive. This means that you cannot add the returns to obtain the ratio for several investments. It does not tell you anything about volatility per se, which limits its usefulness in portfolio management. It is affected by the choice of benchmark. If a strategy performs well versus a poor benchmark, it will get a high ratio. Some think this encourages poor benchmarks! The time frame of the calculation is important, though it is not always obvious how long a timeframe should be chosen when calculating the Ratio.

An Example of Using the
It is sometimes used to determine how adding an investment may influence the portfolio's risk-adjusted returns.
An example; let's say that Roy makes a \$60,000 portfolio investment with an anticipated 15% return and 10% volatility. In addition, the efficient portfolio has an expected return of 18% and 12% volatility with a 5% risk-free interest rate, calculate the Sharpe ratio.
{R (p) – R (f)}/s (p)
here
R (p) = 0.15

R (f) = 0.05

s (p) = 0.10
So,
{0.15 – 0.05}/0.10
= {0.1}/0.10
= 1 / 100%

Mutual Fund
In mutual funds, it represents the approach for measuring performance investing after taking risks into account. The Sharpe ratio applies to an investment in the same way that return on investment [ROI] calculates how well an investment performs relative to its cost.

The ratio’s of a portfolio and balanced funds are used as a means to evaluate their performance relative to the risk they carry and their respective benchmarks. Generally, a Sharpe ratio above 1.00 indicates the fund outperformed its benchmark, while below 1.00 indicates underperformance.
The strength of this ratio is that it can be used in any time frame from daily or monthly returns to annual returns, across any time horizon. Because the it typically measures risk-adjusted returns and not simply return, it is useful for portfolio analysis.

Examples: A fund has a Sharpe ratio of 1.4 and total volatility (greater than the benchmark) of 7%. A balanced portfolio has an expected loss of 3%, and the returns of 30% are within 4% of the benchmark. In these relatively safe portfolios, you can expect to earn more than the return on your principal invested in each fund.