Sortino Ratio Calculation

The Sortino Ratio is a financial metric that differentiates between good and bad volatility in the assessment of an investment’s risk-adjusted returns. While the widely-used Sharpe Ratio considers all variability in returns as risk, the Sortino Ratio focuses specifically on downside risk, providing a more nuanced view of an investment’s performance. This is particularly useful for investors who are more concerned about downside events than upside variability.

Components of the Sortino Ratio

To understand the Sortino Ratio, one needs to be familiar with its primary components:

1. Expected Return (( R )): This is the mean return of an investment over a specified period.

2. Risk-Free Rate (( R_f )): Typically, this is the return of an investment considered “risk-free,” such as U.S. Treasury bonds.

3. Downside Deviation (( DD )): Unlike standard deviation that considers all deviations, downside deviation only considers returns that fall below a targeted or minimum acceptable return (MAR).

Formula

The Sortino Ratio is calculated using the following formula:

[ \text{Sortino Ratio} = \frac{R - R_f}{DD} ]

Where:

• ( R ) is the expected return of the investment.
• ( R_f ) is the risk-free rate.
• ( DD ) is the downside deviation.

Calculation Steps

1. Define the Time Period

Choose the period over which you want to analyze the investment returns, such as daily, monthly, or yearly returns. The chosen period should align with your investment horizon and risk preferences.

2. Calculate the Mean Return (( R ))

Sum the periodic returns and divide by the number of periods to find the average return.

[ R = \frac{\sum_{i=1}^{n} R_i}{n} ]

Where:

• ( \sum ) denotes the summation of returns.
• ( R_i ) represents the return in each period.
• ( n ) is the number of periods.

3. Determine the Minimum Acceptable Return (MAR)

The MAR is often set at the risk-free rate (( R_f )), but it can be adjusted based on the investor’s target return or other benchmarks.

4. Calculate the Downside Deviation (( DD ))

Downside Deviation is a measure of downside risk and is calculated as follows:

1. Subtract the MAR from each periodic return to determine the deviations.
2. Replace all positive deviations with zero, since the focus is on downside risk.
3. Square the negative deviations.
4. Sum these squared deviations.
5. Divide by the number of periods.
6. Take the square root of the result.

Mathematically:

[ DD = \sqrt{\frac{1}{n} \sum_{i=1}^{n} \left( \min(0, R_i - \text{MAR}) \right)^2} ]

Where:

• ( R_i ) is the return in period ( i ).
• ( \text{MAR} ) is the minimum acceptable return.
• ( n ) is the number of periods.

5. Insert Variables into the Formula

Finally, use the calculated values of ( R ), ( R_f ), and ( DD ) in the Sortino Ratio formula.

Example Calculation

Consider the following example where an investment’s annual returns over 5 years are: 10%, 5%, -2%, 12%, 8%. Assume the risk-free rate is 3%.

1. Mean Return (( R )): [ R = \frac{10 + 5 - 2 + 12 + 8}{5} = \frac{33}{5} = 6.6\% ]

2. MAR: Let’s set MAR to the risk-free rate, ( R_f ) = 3%.

3. Calculate Deviations:
   • Year 1: 10% - 3% = 7% (positive, set to 0 for downside deviation)
   • Year 2: 5% - 3% = 2% (positive, set to 0 for downside deviation)
   • Year 3: -2% - 3% = -5% (negative, include in calculation)
   • Year 4: 12% - 3% = 9% (positive, set to 0 for downside deviation)
   • Year 5: 8% - 3% = 5% (positive, set to 0 for downside deviation)

   Negative deviations: 0, 0, -5%, 0, 0

4. Square Negative Deviations:
   • ( (-5\%)^2 = 0.0025 )
5. Sum Squares and Calculate Downside Deviation:
   • Sum: 0.0025
   • Divide by number of periods: ( \frac{0.0025}{5} = 0.0005 )
   • Take the square root: ( \sqrt{0.0005} \approx 0.02236 ) or 2.236%
6. Sortino Ratio: [ \text{Sortino Ratio} = \frac{6.6\% - 3\%}{2.236\%} = \frac{3.6\%}{2.236\%} \approx 1.61 ]

Interpretation

A higher Sortino Ratio indicates a lower probability of downside risk relative to the expected return. Generally, a Sortino Ratio above 1 is considered good, indicating that the investment’s return is high compared to its downside risk. A higher Sortino Ratio also suggests that the investment is more efficient in generating returns per unit of downside risk.

Practical Considerations

Use Cases

1. Risk Management: Investors and financial analysts use the Sortino Ratio to understand better the risk-adjusted returns considering only the downside risk, which is especially critical for retirement funds, conservative portfolios, and risk-averse investors.

2. Performance Benchmarking: Financial performance metrics often compare the Sortino Ratio of different investments, funds, or portfolios to benchmark them against peers or market averages.

Limitations

1. Subjectivity in MAR: The selection of the Minimum Acceptable Return can significantly impact the ratio. Different investors may have different MARs, leading to inconsistent comparisons.
2. Not Considering All Risks: The Sortino Ratio does not account for all kinds of risks, including macroeconomic risks, market risk, and specific investment risks.
3. Only Historical Data: Like many financial metrics, the Sortino Ratio relies on past performance, which may not necessarily predict future results.

Tools for Calculation

Several online tools and financial software can help compute the Sortino Ratio efficiently:

• Portfolio Visualizer: An online tool that provides detailed portfolio analysis, including the Sortino Ratio (https://www.portfoliovisualizer.com/).
• Financial Planning Software: Many platforms like Morningstar, Bloomberg, and Mint offer risk-adjusted performance metrics, including the Sortino Ratio (https://www.morningstar.com/, https://www.bloomberg.com/, https://www.mint.com/).

Conclusion

The Sortino Ratio is a valuable tool for investors seeking to evaluate the risk-adjusted returns of their investments while focusing specifically on downside risk. By filtering out the upside volatility, it offers a clearer picture of how well an investment compensates for the risk of undesirable returns. Employing the Sortino Ratio in conjunction with other financial metrics can lead to more informed and effective investment decisions.