# Sortino Ratio

The Sortino ratio is a means for traders and investors to gauge the risk-adjusted performance of their portfolios or strategies. Namely, they can determine how much return they generate per each unit of risk.

This is very similar to the Sharpe ratio, but the Sortino ratio is designed to improve upon the Sharpe by distinguishing between upside and downside volatility. While Sharpe equally penalizes both upside and downside volatility, the Sortino ratio works to penalize only the downside variety.

Naturally, the aim of risk management and the metrics used to quantify it is to minimize excess downside risk. The Sharpe ratio, given its equal punishment of upside and downside risk, is viewed by many within the financial community as flawed. Despite its simplicity in calculation, its deficiencies can outweigh the positives.

Therefore, Sharpe and Sortino can be used to tell two different stories regarding the reward to risk efficiency of a particular portfolio or strategy.

For portfolios that specialize in capping risk and thus welcome upside volatility – e.g., strategies that buy options or are long convexity / long gamma, or have returns distributions that are positively skewed – the Sortino ratio may be a better measure of risk-adjusted gain than the Sharpe.

A “risk parity” portfolio, which invests in many different asset classes to makes returns largely independent of the economic environment, will also tend to do better when using Sortino over Sharpe because the downside volatility of these portfolios tends to be low.

## Calculation of the Sortino Ratio

The Sortino ratio, S, is defined as:

S = (R – T) / DR

Where:

- R = Portfolio or strategy’s average realized return
- T = Required rate of return
- DR = Is the target downside deviation / “downside risk”

DR is equal to:

DR = √[ ∫ (T – r)^2 * f(r) dr ]

Where:

- T = As above, the required rate of return
- r = Return for the distribution of annual returns, f(r)
- f(r) = Probability distribution of annual returns

The integral calculus function is taken over the area T to -∞ (negative infinity). A continuous, rather than discrete, function is used to avoid the requirement of needing monthly returns to have enough data points to make a statistically accurate calculation. That means that the annual return target needs to be calculated in monthly terms, and this has implications for risk identification. For example, if one requires 10% returns per year, the requirement of needing 0.8% per month will identify higher risk than the annual goal.

## Pros and Cons to the Sortino Ratio

The Sortino ratio can be a better choice by properly rewarding portfolios that have a positive skew in their performance distributions.

The central purpose of risk management and the measures used to describe risk is to avoid large downside risk. Thus, equally penalizing both positive and downside risk is flawed, as is the case of the Sharpe ratio.

The Sortino ratio modifies the Sharpe ratio by using downside deviation rather than standard deviation. In other words, only the returns that fall under a user-specified target or required rate of return are considered risky. Accordingly, if a portfolio or strategy is producing strong upside moves but lower downside moves, the Sortino will view this more favorably in risk-adjusted terms than Sharpe.

On the other hand, for strategies that have negative skews, such as those that sell options – where returns streams are small, positive, and consistent, but can be disrupted by large, negative, and infrequent returns – the Sortino ratio would appropriately penalize those relative to the positive skew strategies. The Sharpe ratio would treat each equally.

## Example Calculation of the Sortino Ratio

Let’s say we have the following set of annual returns:

10%, 4%, 15%, -5%, 20%, -2%, 8%, -6%, 13%, and 23%

### Step One

The average annual return is the sum of these (80%) divided by the number of returns (ten), which is 8%.

Let’s say the target return is the expected annualized returns of the S&P 500, or about 7%.

Therefore, the excess return is 1% (8% – 7%).

### Step Two

Next we calculate the downside deviation. We take into account the negative values, which are -5%, -2%, and -6%. The downside deviations of the others are zero, given they were positive returns.

### Step Three

We square these values:

-5%^2 = .0025

-2%^2 = .0004

-6%^2 = .0036

The average of these downside deviations is the sum divided by the number of returns (ten):

.0025 + .0004 + .0036 = .0065

.0065 / 10 = .00065 = .065%

### Step Four

Next is to calculate the target downside deviation. This is the square root the answer from step 3:

√.00065 = .0255

### Step Five

Finally, calculate the Sortino ratio, which is done by taking the excess return from step one and dividing by the target downside deviation from step four.

.01 / .0255 = 0.392

Considering that most asset classes will produce Sortino ratios of around 0.2 to 0.3 over the long-run, this is a reasonable ratio.