# SHARPE RATIO AND SORTINO RATIO - THE PROFESSIONAL METRICS USED BY HEDGE FUNDS

### Sharpe ratio and sortino ratio

Today, we will take a closer look at professional metrics that measure strategies’ performance – Sharpe Ratio and Sortino Ratio. The Sharpe ratio and the Sortino ratio are both risk-adjusted evaluations of return on investment.

##### WE will cover:
• Definition of Sharpe Ratio and its formula
• Benefits of Sharpe Ratio
• Definition of Sortino Ratio and its formula
• Target Downside Deviation (TDD) and its formula
• Standard Deviation formula
• The difference between TDD and SD

### Sharpe Ratio

Sharpe ratio is one of the most prominent metrics, which takes into account risk and, at the same time, profitability.

Sharpe ratio is an investment performance indicator that is most commonly used in the financial industry. The fundamental goal of this indicator is the possibility of quantifying and comparing the quality of individual investments.

The Sharpe ratio indicator has one primary disadvantage. It penalizes those strategies that tend to be very performance-volatile within the desired returns (or returns above the required risk-free return). But volatility on the return side can be just a natural characteristic of a trading strategy. So it is very questionable whether it should be penalized. Therefore, we will introduce an alternative performance indicator that takes this into account, and that is the Sortino ratio.

But first, let’s look in more detail at the Sharpe ratio before explaining the Sortino ratio, so that we can later compare the two performance indicators.

The Sharpe ratio is calculated as the return of a trading strategy (also investment, an entire mutual fund, or a hedge fund) deducted from the risk-free return divided by the risk, i.e., the volatility (standard deviation) of the return over the risk-free return.The question is, how do we determine “risk-free return”? You don’t have to be scared. The risk-free yield is the classically current achievable interest rate.

##### FORMULA FOR SHARPE RATIO
$\mathrm{Sharpe~Ratio}(x) = \frac{r_x - R_f}{\mathrm{std}(x)}$

$x$  is the given investment (trading strategy),$\large r_{x}$is the average of all returns, $\large R_{f}$is the risk-free return (achievable interest rate of bonds, most common interest rate of 3-month US Treasury Bills is considered as risk-free return),$\mathrm{std}(x)$is the standard deviation of returns.

It logically follows that the larger the value of this indicator, the higher the return that the investment achieved per unit of risk.

$\mathrm{Sharpe~Ratio}(x) = \frac{r_x - R_f}{\mathrm{std}(x)}$

#### BENEFITS OF SHARPE RATIO

Indeed, the Sharpe ratio is the most commonly used indicator in the financial industry, which evaluates our profit level for a given investment against the risk taken. It is an indicator that is well known and is used by many mutual or hedge funds as the fundamental performance metric.

Thanks to the Sharpe ratio, we should compare, for example:

• individual mutual funds,
• hedge funds,

However, this performance indicator also has significant disadvantages.

The Sharpe ratio cannot distinguish between the desired “top” volatility (standard deviation of the desired returns) and the unwanted volatility (the standard deviation of returns/losses below the risk-free return Rf).

Higher and excessive returns may increase the denominator’s value as their primary effect (standard deviations of returns) more than the numerator’s value and thus an overall decrease in the Sharpe ratio. Of course, investors like positive yield volatility. However, they are much more annoyed if the investment has large losses and unwanted volatility.

For a specific example, typical hedge funds focused on long-term trend-following strategies have a Sharpe ratio between 0.5 and 0.9. Contrary, hedge funds using classic convergence strategies (option writing) can have a Sharpe ratio of about 3.

However, you can expect a devastating drawdown at any time for these types of strategies. Sharpe ratio assumes a normal distribution of returns, and that is its weakness.

It tends to give us a too positive rating for trading strategies with a negative skew. It tends to produce many low and consistent returns (it means a very high percentage of profitable trades) but may have a devastating drawdown within a single trade.

In picture below, you can see an illustrative comparison of two trading approaches:

• Positive skew yields a trend-following strategy (many small losses and a few very profitable trades that cover small losses and add extra profitable trades. Classically trading strategies with about 30-40% profitable trades and a Risk Reward Ratio between 2.5-3.5)
• Negative skew yields an option strategy (many small profits and a few very drastic loss trades). Classically, these are systems around 90% of profitable trades with a Risk Reward Ratio from 0 to 1.
• Also previously mentioned problem – when the profit occurred only in one year, and other years the equity was flat – Sharpe ratio is not able to detect that and gives very positive value.

### SORTINO RATIO

Contrary to the Sharpe Ratio, only those returns with a lower value than the investor-defined target (in the so-called Desired Target Return) are considered risky. To clearly illustrate the difference between the Sharpe ratio and Sortino ratio in the context of a defined target – Desired Target return, you can see a picture below:

Large fluctuations in returns are a sign of volatility and risk. But if an investment or a trading strategy, by its very nature, has more significant profitable returns with smaller loss movements, these strategies should not be penalized. A classic example of such a strategy is trend-following systems.

##### FORMULA FOR SORTINO RATIO

$\mathrm{Sortino~Ratio}(x) = \frac{r_x-R_t}{TDD}$

, where $\small R$ is the average of the values of the yield periods. $\small T$ is the desired target return of a strategy (or the minimum required return). $\small TDD$ is the targeted “bottom” deviation.

Large fluctuations in returns are a sign of volatility and risk. But if an investment or a trading strategy, by its very nature, has more significant profitable returns with smaller loss movements, these strategies should not be penalized. A classic example of such a strategy is trend-following systems.

##### Target Downside Deviation (TDD) And ITS FORMULA

Target Downside Deviation is defined as the square root of the averages of squares (in Root Mean Square) of deviations of realized underperforming returns and losses derived from the target value of revenue, where all returns above the target value are equal to 0. Mathematically, this relationship we calculate as:

$\mathrm{Sharpe~Ratio}(x) = \frac{r_x - R_f}{\mathrm{std}(x)}$

$\small Xi$ = i-th

$\small N$ = total number of returns

$\small T$ = target yield value

You can see that the TDD calculation is very similar to the standard deviation (SD) measure to calculate the Sharpe ratio.

##### STANDARD Deviation FORMULA
$\large SD=\sqrt{\frac{1}{N}}\sum_{i=1}^{N}(Xi-u)^{2}$

$\small Xi$ = net yield

$\small N$ = total number of revenues

$u$ = average of all Xi yields

##### The difference between TDD and SD

The differences between TDD and standard deviation are then:

1. In TDD, deviations are calculated as Xi from the target yield value. In SD, the deviations are calculated from the average of all Xi.
2. In TDD, all Xi above the target yield value is equal to 0, but they are still included. The Standard Deviation calculation then does not have the Min () function included in the analysis.

To sum it up: the Sharpe ratio cannot distinguish between desired “positive” volatility (standard deviation of desired yields) and negative volatility (standard deviation losses below the value of risk-free yield, Rf ). Higher and excessive returns may have the primary effect of increasing the denominator value (standard deviations of returns) more than the numerator value and, thus, the overall decrease in the Sharpe ratio.

However, investors like the positive volatility of returns. The solution to this issue can be the performance indicator Sortino ratio. Due to the nature of its calculation, it doesn’t penalize strategies with variability required positive returns. It only penalizes those strategies that have a high negative variability of insufficient income and losses.

So what investors consider a fair value of the Sharpe and Sortino ratio that signs a reasonable degree of expected return for an acceptable low amount of risk?

Usually, any Sharpe and Sortino ratio greater than 1.0 is valued as a good investment, higher than 2.0 is as outstanding, and a 3.0 or higher ratio is excellent. A ratio under 1.0 is considered as not sufficient. Every time you get a very high value of these ratios – be careful. The risk of overfitting of strategy and other biases are very high.

If you are looking for practical examples of calculating the Sharpe and the Sortino ratio on concrete strategy, you can find it in our FREE EBOOK.