The Sharpe ratio was developed by Nobel laureate William F. Sharpe and is used to help investors understand the return of an investment compared to its risk. The ratio is the average return earned **in excess of the risk-free rate** per unit of volatility or total risk.

Subtracting the risk-free rate from the mean return allows an investor to better isolate the profits associated with risk-taking activities. Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.

## Sharpe Ratio Formula

The Sharpe ratio is calculated by subtracting the risk-free rate from the return of the portfolio and dividing that result by the standard deviation of the portfolio’s excess return.

In 1966, William Sharpe developed this ratio which was originally called it the “reward-to-variability” ratio before it began being called the Sharpe ratio by subsequent academics and financial operators.

Some of the concepts which we require to understand are:

**Returns**– The returns could be of various frequencies such as daily, weekly, monthly or annually as long as the distribution is spread normally since these returns can be annualized to arrive at precise results. Abnormal situations like higher peaks, skewness on the distribution can be a problem area for the ratio as standard deviation does not possess the same effectiveness when these issues exist.

**Risk-Free Rate of Return –**This is used to assess if one is being correctly compensated for the additional risk borne because of the risky asset. Traditionally, the rate of return with no financial loss is the Government securities with the shortest duration (e.g. US Treasury Bill). While such a variant of security has the least amount of volatility, it can be argued that such securities should match with other securities of equivalent duration.

**Standard Deviation –**It is a quantity which expresses how many units from a given set of variables differ from the Mean average of the group. Once this excess return over the risk-free return is computed it has to be divided by the Standard deviation of the risky asset being measured. Greater the number, attractive will the investment appear from a risk/return perspective. However, unless the standard deviation is substantially large, the leverage component may not impact the ratio. Both the numerator (return) and denominator (standard deviation) could be doubled with no problems.

## Understanding The Sharpe Ratio

The Sharpe ratio has become the most widely used method for calculating the risk-adjusted return. Modern Portfolio Theory states that adding assets to a diversified portfolio that have low correlations can decrease portfolio risk without sacrificing return.

Adding diversification should increase the Sharpe ratio compared to similar portfolios with a lower level of diversification. For this to be true, investors must also accept the assumption that risk is equal to volatility which is not unreasonable but may be too narrow to be applied to all investments.

The Sharpe ratio can be used to evaluate a portfolio’s past performance (ex-post) where actual returns are used in the formula. Alternatively, an investor could use expected portfolio performance and the expected risk-free rate to calculate an estimated Sharpe ratio (ex-ante).

The Sharpe ratio can also help explain whether a portfolio’s excess returns are due to smart investment decisions or a result of too much risk. Although one portfolio or fund can enjoy higher returns than its peers, it is only a good investment if those higher returns do not come with an excess of additional risk.

**The greater a portfolio’s Sharpe ratio, the better its risk-adjusted performance**. If the analysis results in a negative Sharpe ratio, it either means the risk-free rate is greater than the portfolio’s return, or the portfolio’s return is expected to be negative. In either case, a negative Sharpe ratio does not convey any useful meaning.

## Sharpe Ratio Example

Client ‘A’ currently is holding a $450,000 invested in a portfolio with an expected return of 12% and a volatility of 10%. The efficient portfolio has an expected return of 17% and a volatility of 12%. The risk free rate of interest is 5%. What is the Sharpe Ratio?

Sharpe Ratio Formula = (Expected Return – Risk-Free rate of return) / Standard Deviation (Volatility)

Sharpe Ratio = (0.12-0.05)/0.10 = 70% or 0.7x

## Advantages of Using Sharpe Ratio

#### #1 – Sharpe Ratio helps in comparing and contrasting new asset addition

It is used to compare the variance of a portfolio’s overall risk-return features whenever a new asset or a class of asset is added to it.

- For instance, a portfolio manager is considering the addition of a commodities fund allocation to his existing 80/20 investment portfolio of stocks having a Sharpe ratio of 0.81.
- If the new portfolio’s allocation is 40/40/20 stocks, bonds and a debt fund allocation, the Sharpe ratio increases to 0.92.

This is an indication that although the commodities fund investment is volatile as a stand alone exposure, in this case, it actually leads to an improvement of the risk-return characteristic of the combined portfolio, and thus adds a benefit of diversification into another asset class to the existing portfolio. There has to be an involvement of careful analysis that the fund allocation may have to be altered at a later stage if it is having a negative effect on the health of the portfolio. If the addition of the new investment is leading to a reduction in the ratio, it should not be included in the portfolio.

#### #2 – Sharpe Ratio helps in Risk Return Comparision

This ratio can also provide guidance whether the excessive returns of a portfolio are due to careful investment decision making or a result of undue risks taken. Although an individual fund or portfolio can enjoy greater returns than its peers, it is only a reasonable investment if those higher returns do not come with undue risks. The greater the Sharpe ratio of a portfolio, the better its performance has been factoring the risk component. A negative Sharpe ratio indicates that the lesser riskier asset would perform better than the security being analyzed.

Let us take an example for the Risk Return Comparision.

Assume portfolio A had or is expected to have a 12% rate of return with a standard deviation of 0.15. Assuming a benchmark return of about 1.5%, the rate of return (R) would be 0.12, Rf will be 0.015 and ‘s’ will be 0.15. The ratio will be read as (0.12 – 0.015)/0.15 which computes to 0.70. However, this number will make sense when it is compared to another portfolio say Portfolio ‘B’

If portfolio ‘B’ shows more variability than Portfolio ‘A’, but has the same return, it will have a greater standard deviation with the same rate of return from the portfolio. Assuming the standard deviation for Portfolio B is 0.20, the equation would be read as (0.12 – 0.015) / 0.15. The Sharpe ratio for this portfolio will be 0.53 which is lower in comparison to Portfolio ‘A’. This may not be an astonishing result, taking into consideration the fact that both the investments were offering the same return, but ‘B’ had a greater quantum of risk. Obviously, the one which has less risk offering the same return will be a preferred option.

## Criticisms of Sharpe Ratio

The Sharpe ratio utilises the Standard deviation of returns in the denominator as an alternative to the overall portfolio risks, with an assumption that returns are evenly distributed. Past testing has shown that returns from certain financial assets may deviate from a normal distribution, resulting in relevant interpretations of the Sharpe ratio to be misguiding.

This ratio can be improved by various fund managers attempting to boost their apparent risk-adjusted return which can be executed as below:

**Increasing the Time Duration to be measured**: This will result in a lesser probability of volatility. For instance, the annualized standard deviation of daily returns is generally higher than of weekly returns, which in turn is higher than that of the monthly returns. Greater the time duration, clearer picture one has to exclude any one-off factors which can impact the overall performance.**Compounding of the monthly returns**but computing the standard deviation excluding this recently calculated compounded monthly return.**Writing out-of-the money sell and buy decision of a portfolio:**Such strategy can potentially increase the returns by collecting the options premium without paying off for a number of years. Strategies which involve challenging the default risk, liquidity risk or other forms of wide-spreading risks possess the same ability to report an upwardly biased Sharpe ratio.**Smoothening of returns:**Using certain derivative structures, irregular marking to market of less liquid assets or utilizing certain pricing models which underestimate monthly profits or losses, can reduce the expected volatility.**Eliminating Extreme returns:**Too high or too low returns can increase the reported standard deviation of any portfolio since it is distance from the average. In such a case, fund manager may choose to eliminate the extreme ends (best and the worst) monthly returns each year to reduce the standard deviation and affect the results since such a one-off situation can impact the overall average.

Choosing a period for the analysis with the best potential Sharpe ratio, rather than a neutral look-back period, is another way to cherry-pick the data that will distort the risk-adjusted returns.

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