Being able to measure and quantify risk/return is a top priority when trying to evaluate investment portfolios. There are a lot of sophisticated tools that can help investors evaluate risk-adjusted portfolio performance. These tools are also useful to compare investment strategies, indices, or even the historical performance of different money managers and hedge funds.

__Combining Risk and Return into a Single Comparable Value__

There are two main investing goals: (i) get the highest annual payout (Return), by (ii) minimize the chances of losing money (Risk). By combing risk and return into a single value, investors can compare the real performance of different portfolios. This combination is called risk-adjusted portfolio performance. This analysis includes the following risk-adjusted portfolio performance ratios:

(1) Sharpe Ratio

(2) Sortino Ratio

(3) Treynor Measure or Traynor Ratio

(4) Jensen Measure or Jensen Alpha

(5) Calmar Ratio

(6) MAR Ratio

(7) Omega Ratio

(8) Information Ratio or Appraisal Ratio

(-) Key Points of the Analysis

**Three Key Investment Concepts Before Moving On**

Before moving on with the ratios that measure risk-adjusted performance, it is important to mention some basic concepts. These concepts include the risk-free rate of return, the standard deviation (SD), and the maximum drawdown. Most of the following performance ratios are partially based on these three important components.

(i) __The Risk-Free Rate of Return__

The risk-free rate refers to the annual return an investor can earn without taking any risk. Generally, the risk-free rate of return is based on the annualized interest paid on a 3-month Treasury bill. Any investment that is expected to offer lower returns than the risk-free rate of return is an unacceptable investment.

- You can use the annualized return of the 3-month US Treasury Bill in the US or the 3-month Euribor in the EU

(ii) __The Standard Deviation (SD)__

The standard deviation (SD) or Sigma (Greek letter σ) is a statistical tool that measures the amount of variation or dispersion of a set of values. A higher SD shows that the values are spread out. A lower SD shows that the values tend to be close to the mean, this mean value is also called the expected value. Why is this important? Because we want to make sure that the positive historical performance of a portfolio is consistent and can be easily repeated in the future.

By using the standard deviation, investors can normalize the returns of any portfolio. These are some important facts:

- The lower the standard deviation of a portfolio the better
- A lower standard deviation means the return is more consistent and predictable over time
- Standard deviation rises as returns become more volatile, a portfolio with a lower SD is better than a portfolio with a higher SD
- Most commonly, the standard deviation is calculated on a monthly or yearly basis

(iii) __Maximum drawdown__

Maximum drawdown measures the maximum loss of a portfolio from its peak value. It is calculated by subtracting the portfolio's lowest dollar value from its peak dollar value, the result is divided by the peak dollar value.

■ Maximum drawdown (%) = { Peak value($) / Lowest value($) } / Peak value($)

For example, if the peak dollar value of a portfolio is $120,000 and the lower dollar value is $85,000, then the drawdown is:

($120,000 - $85,000) / $120,000 = 37.5%

- The lower the max(drawdown) the better for the portfolio
- Time plays a very important role when calculating the max(drawdown)
- Portfolios with low max(drawdown) over long periods are considered very trustworthy

__The Key Measuring Performance Tools __

As mentioned in the beginning, by combing risk and return into a single value we can reliably compare various portfolios. Keep in mind that a portfolio offering the highest return is not necessarily the best portfolio to invest in.

■ __Use__: All Portfolios | Strategies | Assets

■ __Alternatives__: Sortino Ratio | Omega Ratio

Developed by William F. Sharpe in 1966, the Sharpe ratio, or else the Sharpe index is perhaps the most commonly used portfolio management tool. A Sharpe ratio exceeding 1.0 is considered very good. As an example, the ‘Amundi ETF Nasdaq 100’ between mid-2007 and mid-2022 offered an average annual growth of 16.4% and showed a Sharp ratio of 0.93.

Unlike other ratios, the Sharpe ratio measures the quality of a portfolio not only based on performance but also based on diversification (using the standard deviation).

The Sharpe ratio calculates the excess return of a portfolio over the risk-free rate of the economy (explained above). The result is divided by the standard deviation of the excess return. This is the formula:

■ Sharpe ratio = { (PR − RFR) / SD(p) }

__where__:

PR= Return of the Portfolio

RFR = Risk-Free Rate (such as a US Treasury security)

SD(p) = Standard Deviation of the portfolio’s excess return

__Key Takeaways__

- The Sharpe ratio is an easy way to measure the risk-adjusted returns of a portfolio.
- The Sharpe ratio requires the collection of data over a sufficient period
- The higher the ratio, the greater the investment return, relative to the risk taken
- The ratio illustrates how much excess return is received for the additional risk
- The Sharpe ratio considers unsystematic risk and can be used to compare both well-diversified and less-diversified portfolios

■ __Use__: Risk-Averse Portfolios | Strategies | Assets

■ __Alternatives__: Sharpe ratio

Developed by Frank A. Sortino, the Sortino ratio measures the risk-adjusted return of a portfolio by penalizing returns falling below a user-specified rate of return. On the other hand, the Sharpe ratio equally penalizes both positive and negative returns.

■ Sortino Ratio= ( PR - R(f) ) / SD(d)

__where__:

PR = Portfolio Return (actual or expected)

R(f) = Free Rate of Return

SD(d) = Standard deviation of Negative Portfolio Returns

__Interpreting Sortino Ratio__

A negative Sortino ratio means the portfolio return can’t even beat the risk-free rate.

- Below 0 is completely unacceptable
- Below 1.0 is considered low
- Between 1.0 and 2.0 is considered promising
- Between 2.0 and 3.0 is considered very promising
- Higher than 3.0 is considered excellent

__Key Takeaways__

- The Sortino ratio measures the risk-adjusted return of a portfolio, the higher the better
- The Sortino ratio is similar to the Sharpe ratio, except for the fact that it incorporates a downside deviation for the denominator, instead of a standard deviation
- It penalizes returns below the specified rate of return
- The Sortino ratio focuses on negative returns, and therefore, it is useful for measuring risk-oriented portfolio performance
- The Sortino ratio is considered acceptable above 1.0

(3)** Treynor Measure or Traynor Ratio**

■ __Use__: well-diversified Portfolios / Hedge Funds

■ __Alternatives__: Jensen Alpha

Developed by Jack Treynor in 1965, the Traynor ratio is similar to the Sharpe ratio but instead of a standard deviation in the denominator it uses the relative volatility (beta). The beta coefficient (Greek β) measures the volatility of a portfolio to the general market itself.

■ Treynor Measure = { (PR − RFR) / β }

__where__:

PR= Return of the Portfolio

RFR = Risk-Free Rate

β = beta

__Key Takeaways__

- The Treynor measure is the ratio of the excess return of a portfolio to the systematic risk of that return
- The higher the Treynor measure, the better the portfolio has performed
- It is similar to the Sharpe ratio but is uses beta in the denominator instead of the standard deviation
- The Treynor measure uses the beta (β) to calculate the relative volatility between the portfolio performance and the market performance
- The Treynor Ratio adjusts for systematic risk but ignores portfolio diversification (unsystematic risk), therefore, it should only be used to compare well-diversified portfolios

(4)** Jensen Measure or Jensen Alpha**

■ __Use__: well-diversified Portfolios / Hedge Funds

■ __Alternatives__: Treynor Measure

Developed by Michael C. Jensen in 1968, the Jensen Alpha is similar to the Traynor measure, but it incorporates the Capital Asset Pricing Model (CAPM). The Jensen Alpha shows the portfolio's ability to offer above-average returns, adjusted for market risk.

■ Jenson's Alpha = PR – { R(f) + β * ( R(m) - R(f) ) }

__where__:

PR = Return of the Portfolio

R(f) = Risk-Free Rate

β = beta

R(m) = Return of Market Risk

__Key Takeaways__

- The higher the Jensen Alpha, the better the risk-adjusted returns
- The Jensen Alpha shows the excess return of a portfolio compared to its expected return
- The Jensen Alpha is calculated using the CAPM model. The CAPM shows the relationship between systematic risk and expected return for an asset
- Generally, alpha (Greek α) is an investing performance term that describes the ability of a portfolio to beat the market
- A portfolio offering a positive excess return will show a positive Alpha while a portfolio with a negative excess return will show a negative alpha
- A positive Alpha means outperformance, a negative Alpha underperformance, and zero Alpha neutral performance (i.e. Benchmark)
- Jensen's alpha only considers systematic risk, therefore it should be used to compare well-diversified portfolios

■ __Use__: Portfolios | Hedge/Mutual Funds

■ __Alternatives__: MAR Ratio

The Calmar ratio measures the performance of a portfolio or a fund relative to its risk. The performance is calculated by the average annual rate of return minus the annual risk-free rate. Risk is reflected by the maximum drawdown over a period of three years or more.

■ Calmar ratio = { PR – R(f) } / max(drawdown)

__where__:

PR = Average Annual Portfolio Return (3 years or more)

R(f) = Risk-Free Rate

max(drawdown) = the maximum drawdown (3 years or more)

__Example__

As mentioned above, the maximum drawdown measures the maximum loss of a portfolio from its peak value.

For example, the portfolio’s maximum drawdown is 37.5%, the average annual return is 45%, and the risk-free rate is 3.5%. Now, let’s calculate the Calmar ratio:

- Calmar ratio = { 45% – 3.5% } / 37.5% = 1.11

__Key Takeaways__

- The Calmar ratio measures the performance of a portfolio on a risk-adjusted basis, typically, over a three-year period
- The higher the Calmar ratio the better, above 3.0 is considered very good
- Performance is calculated based on the average annual rate of return, while the risk is calculated by the maximum drawdown
- A low Calmar ratio indicates that a portfolio has been at risk of large drawdowns, on the other hand, a high Calmar ratio shows a low drawdown risk
- When comparing two portfolios of similar annual returns, the Calmar ratio is very useful to indicate which portfolio is the best (risk-adjusted)

■ __Use__: Hedge Funds | Strategies

■ __Alternatives__: Calmar Ratio

Developed in 1978 by Leon Rose, the MAR ratio measures the risk-adjusted returns of a hedge fund or an investment strategy. The higher the MAR ratio, the better the risk-adjusted returns.

The ratio is calculated by dividing the compound annual growth rate (CAGR) by the maximum drawdown of a particular period.

■ MAR Ratio = CAGR / max(drawdown)

__where__:

CAGR is the Compound Annual Growth Rate (It measures the annual growth rate of an investment assuming all profits are reinvested at the end of each period)

max(drawdown) is the maximum drawdown of the portfolio for a particular period

__Key Takeaways__

- MAR stands for ‘Managed Account Reports Ratio’
- The MAR ratio is useful for the calculation of the risk-adjusted returns of hedge funds and investment strategies
- Return is determined by using the Compound Annual Growth Rate (explained above) and risk as the maximum drawdown
- As the analysis period gets wider, the MAR ratio will offer significantly lower numbers, as the ratio is very sensitive to the maximum drawdown of the portfolio
- The MAR ratio is similar to the Calmar ratio, however, the two ratios may provide completely different results over a long period of analysis

■ Best Use: Complex Portfolios | Hedge Funds | Strategies | Assets

■ Alternatives: Sharpe ratio | Information Ratio

Developed by Con Keating and William Shadwick in an article titled “A Universal Performance Measure”, in 2002, the Omega ratio is a risk-return performance measure. The Omega ratio is considered a good alternative for the Sharpe ratio.

The ratio is calculated as the ratio of probability-weighted profits and losses. To calculate the ratio, we need to know the cumulative excess return of the portfolio:

■ Omega Ratio = { Σ(Winning) – Benchmarking } / { Σ(Benchmarking) – Losing }

__Key Takeaways__

- The Omega ratio is a multi-purpose risk-return performance tool, an alternative to the Sharpe ratio
- The higher the Omega ratio the better the probabilities of winning in comparison to loosing
- In contrast to the Sharpe ratio, the Omega considers excess returns and not absolute returns
- The ratio incorporates all distribution moments and all risk-return attributes (Standard Deviation, Mean, Kurtosis, and Skewness)
- The Omega ratio is particularly useful for the evaluation portfolios or funds with abnormal distributions over time

(8) **The Information Ratio or Appraisal Ratio**

■ __Use__: Portfolio Relative Returns (benchmark-based)

■ __Alternatives__: Sharpe Ratio | Omega Ratio

The information ratio or else appraisal ratio measures the risk-adjusted return of a portfolio relative to a benchmark index. In other words, the information ratio calculates the excess return a manager generated relative to a benchmark index, divided by the extra risk that this manager was forced to accept in order to beat the benchmark index. This excess return is called the ‘active return’.

■ Information ratio = { R(P) -R(b) } / SD

__where__:

R(P) = Portfolio returns

R(b) = Benchmark Index returns (i.e. S&P 500)

SD = Standard Deviation of return

The information ratio works similarly to the Sharpe ratio, however, the information ratio uses an index as the benchmark and not the risk-free return as the Sharpe ratio.

__Key Takeaways__

- The information ratio is the ratio of the active return of a portfolio divided by the tracking error of its return (standard deviation)
- Active return refers to the difference between the returns of a portfolio and the returns of a benchmark index (i.e. S&P 500)
- The information ratio works similarly to the Sharpe ratio
- The information ratio is useful to investors for measuring the relative returns of a portfolio against a major index

**The Key Points of Our Analysis**

__These are some useful points__:

- Risk matters as much as return, therefore, we need to evaluate the performance of any investment on a risk-adjusted basis
- A portfolio offering the highest return is not necessarily the best portfolio to invest in
- By combing risk and return into a single value, investors can easily compare the performance of different portfolios
- The lower the standard deviation of a portfolio the better, as a lower standard deviation means the returns of the portfolio are more consistent and predictable
- A portfolio with a lower standard deviation is better than a portfolio with a higher standard deviation
- The risk-free rate of return refers to the annual return an investor can get without taking any risk. Any investment that is expected to offer lower returns than the risk-free rate of return is an unacceptable investment
- Generally, the risk-free rate of return is based on the annualized interest paid on a 3-month Treasury bill
- Maximum drawdown measures the maximum loss of a portfolio from its peak value. It is calculated by subtracting the portfolio's lowest dollar value from its peak dollar value, the result is divided by the peak dollar value
- CAGR is the Compound Annual Growth Rate and measures the annual growth rate of an investment assuming all profits are reinvested at the end of each period
- The Sharpe ratio is perhaps the most commonly used portfolio management tool. The Sharpe ratio considers unsystematic risk and can be used to compare both well-diversified and less-diversified portfolios
- The Sortino ratio is similar to the Sharpe ratio, however, it focuses on negative returns, and therefore, it is useful for measuring risk-oriented portfolio performance
- The Treynor Ratio adjusts for systematic risk but ignores portfolio diversification (unsystematic risk), therefore, it should only be used to compare well-diversified portfolios
- Jensen's alpha also considers systematic risk, therefore, it should also be used to compare well-diversified portfolios
- When comparing two portfolios of similar annual returns, the Calmar ratio is very useful to indicate which portfolio is the best (risk-adjusted). A high Calmar ratio shows a low drawdown risk
- The MAR ratio is similar to the Calmar ratio, however, the two ratios may provide completely different results over a long period of analysis
- The Omega ratio is particularly useful for the evaluation of complex portfolios or funds with abnormal distributions over time. Based on a benchmark (i.e. index) the Omega ratio considers excess portfolio returns and not absolute returns like the Sharpe ratio
- The information ratio works similarly to the Sharpe ratio, however, it uses an index as the benchmark and not the risk-free return. The active return of the information ratio refers to the difference between the returns of a portfolio and the returns of a benchmark index (i.e. S&P 500)

■ __Essential Tools for Measuring Portfolio Performance__

Giorgos Protonotarios for Tradingcenter.org (c) -July 2022

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