## Sharpe Ratio Definition

Sharpe ratiois the ratio developed by William F. Sharpe and used by the investors in order to derive the excess average return of the portfolio over the risk-free rate of the return, per unit of the volatility (standard deviation) of the portfolio.

### Explanation

Sharpe Ratio is a critical component for marking the overall returns on a portfolio. It is the average return earned in excess of the risk-free return compared to the total amount of risk borne. It is a way to examine the performance of an investment by adjusting for its risk component. The Sharpe ratio characterizes how well the return of an asset compensates the investor for the risk taken. When comparing two assets versus a common benchmark, the one with a higher Sharpe ratio provides is indicated as a favorable investment opportunity at the same level of risk.

If you look at the table above, you will see that PRWCX has a higher Sharpe Ratio of 1.48 and is the best fund in its group.

Sharpe Ratio, like any other mathematical model, relies on the accuracy of the data which needs to be correct. While examining the investment performance of assets with smoothening of returns, the Sharpe ratio would be derived from the performance of the underlying assets rather than the fund returns. This ratio along with Treynor Ratios and Jeson’s Alphas are often used to rank the performance of various portfolios or Fund managers.

### Formula

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. It was defined in multiple ways till ultimately it was charted as below:

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

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 that 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 calculated 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.

**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

### Calculating Sharpe Ratio in Excel

Now that we know how the formula works, let us calculate Sharpe Ratio in excel.

#### Step 1 – Get the returns in the tabular format

The first step involves arranging for the returns of the portfolio of the mutual fund that you want to analyze. The time period can be monthly, quarterly or annual. The below table provides annual returns of a mutual fund.

#### Step 2 – Get Risk Free Return Details in the table

In this table below, I have made an assumption that the risk-free return is 3.0% throughout the span of 15 years. However, risk-free rate may change each year and you need to put that number here.

#### Step 3 – Find Excess Return

The third step in calculating the Sharpe ratio in excel is to find the excess returns of the portfolio. In our case, the excess return is the Yearly Returns – Risk-Free Return.

#### Step 4 – Find the average of Annual Returns.

The fourth step in calculating the Sharpe ratio in excel is to find the average of the yearly returns. You can use the excel formula AVERAGE to find the average of the portfolio. In our example, we get an average return of 12.09%.

#### Step 5 – Find a Standard Deviation of the Excess Returns

In order to find the standard deviation of excess returns, you can use the excel formula STDEV as given below.

#### Step 6 – Calculate Sharpe Ratio

The final step to calculating the Sharpe ratio in excel is to divide the Average Returns by Standard deviation. We get the ratio = 12.09% / 8.8% = 1.37x

We get the ratio = 12.09% / 8.8% = 1.37x

### 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 on 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 of 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 calculates 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 utilizes 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 decisions of a portfolio:**Such a strategy can potentially increase the returns by collecting the options premium without paying off for a number of years. Strategies that 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 that 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 the distance from the average. In such a case, a 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.

### Ex-Ante and Ex-Post Sharpe Ratio

The Sharpe ratio has been revised multiple times but two general forms that have been used are ex-ante (prediction of future return and variance) and ex-post (analysis of past return variance).

**Ex-ante Sharpe ratio**predictions are simple to**estimate patterns**after observations of the past performance of similar investment activities.**The Ex-post Sharpe Ratio**measures how high the returns were, versus how varied those returns were over a given time period. More specifically, it is the ratio of the differential returns ( the difference between the returns on investment and a benchmark investment) versus the historical variability (standard deviation) of those returns.

### Conclusion

The Sharpe ratio is a standard measure of the performance of the portfolio. Due to its simplicity and ease of interpretation, it is one of the most popular indexes. Unfortunately, most of the users forget the assumptions that result in an inappropriate outcome. You should consider checking the distribution of the returns or validation of the results with equivalent performance measures before arriving at a decision on the market.

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