# Risk and Return Metrics

Risk and Return Metrics

Investing in stocks, bonds, or other financial instruments requires an understanding of the risk and return metrics associated with the investment. These metrics are used to evaluate the potential for financial gain and the potential for loss. Knowing these metrics can help investors make informed decisions about where to put their money.

## Risk Vs. Return Metrics

### What is M-Squared?

Overview: M-Squared, also known as the Modigliani risk-adjusted performance measure, is a method for evaluating the performance of a portfolio or investment strategy. It was developed by Franco Modigliani and Merton Miller, who were awarded the Nobel Prize in Economics for their work in this area.

How it works: The M-Squared measure compares the risk-adjusted return of a portfolio to a benchmark, such as the market index. It is calculated by dividing the excess return of the portfolio (the return above the benchmark) by the portfolio’s volatility (or standard deviation). The result is a ratio that shows how much return the portfolio is generating for each unit of risk.

Possible values: M-Squared typically ranges from negative infinity to positive infinity. A negative M-squared value indicates that the portfolio has underperformed the benchmark, adjusted for the level of risk taken, while a positive M-squared value indicates that the portfolio has outperformed the benchmark, adjusted for the level of risk taken.

Advantages: M-Squared considers both the return and the risk of a portfolio. The M-Squared measure provides a more accurate assessment of a portfolio’s performance.

Limitations: It does not consider other factors, such as the portfolio’s turnover rate or transaction costs, which can also affect the portfolio’s performance. It is important to note that the M-Squared measure is based on the assumption of a normal distribution of returns, which may not always be the case in real-world scenarios.

Usefulness: Here are a few ways in which M-Squared can be useful for evaluating an investment portfolio:

• Comparing portfolios: M-Squared can compare the performance of different portfolios, even if they have different levels of risk. This allows investors to identify which portfolios have the best risk-adjusted returns.
• Benchmarking: M-Squared can compare a portfolio’s performance to a benchmark. RecipeInvesting uses BND, SPY, and 60-40 Balanced Portfolio as benchmarks in all the risk vs. return scatterplots.
• Identifying high-performing assets: M-Squared can identify high performing portfolios.

M-Squared can be used as part of a holistic approach that considers various measures and factors such as downside risk, the portfolio’s objectives and the investor’s risk tolerance.

### What is Alpha?

Overview: Alpha is a measure of a portfolio’s performance that is relative to a benchmark index. It is used to evaluate the performance of an investment manager, and it represents the return on an investment portfolio over the return that a benchmark index has achieved.

How it works: The Capital Asset Pricing Model (CAPM) is a common approach to calculate alpha. Here is the formula for alpha using CAPM: Alpha = Actual Return - (Risk-Free Rate + Beta * (Market Return - Risk-Free Rate)) Where:

• Actual Return: The actual return of the investment
• Risk-Free Rate: The risk-free rate of return, usually the yield on government bonds
• Beta: The systematic risk of the investment, measured relative to the market
• Market Return: The average return of the market

To calculate alpha using CAPM, you first need to calculate the beta of the investment. Beta is a measure of the systematic risk of the investment, which is the risk that cannot be diversified away by holding a well-diversified portfolio. Beta is calculated by regressing the historical returns of the investment against the returns of the market. Once you have the beta, you can use the CAPM formula to calculate the expected return of the investment. The difference between the actual return and the expected return is the alpha. For example, let's say the actual return of a stock was 15%, the risk-free rate was 2%, the market return was 10%, and the beta of the stock was 1.5. The calculation of alpha using CAPM would be: Alpha = 15% - (2% + 1.5 * (10% - 2%)) Alpha = 15% - (2% + 1.5 * 8%) Alpha = 15% - (2% + 12%) Alpha = 1% This means that the stock has an alpha of 1%, which indicates that it has outperformed the market by 1% after adjusting for its risk.

Possible values: The range of alpha values can vary widely depending on the investment and market conditions. A positive alpha indicates that the investment has outperformed its expected returns, while a negative alpha indicates that it has underperformed. The magnitude of alpha can also vary widely, from a small fraction of a percent to several percentage points.

Advantages: The main advantage of alpha in investing is that it provides a measure of the excess return that an investment generates above its expected return, after adjusting for its level of risk. This can help investors to evaluate the performance of investment managers, as well as individual securities, in terms of their ability to generate positive returns over and above what is expected based on their level of risk.

Limitations: The limitations of alpha in investing include its usefulness only for evaluating active investment management, difficulty in accurate calculation, dependence on the choice of benchmark, failure to account for transaction costs and taxes, and not accounting for changes in risk over time.

Usefulness: Alpha is a measure of a portfolio’s excess return, which is the return generated by the portfolio manager’s active decisions and is not explained by the benchmark. It is useful for evaluating an investment portfolio in several ways:

• Measuring manager skill: Alpha can measure the skill of a portfolio manager by comparing the actual return of the portfolio to the benchmark return. A positive Alpha shows the manager has generated returns above the benchmark, and a negative Alpha shows the manager has generated returns below the benchmark.
• Identifying out performance: Alpha can identify which portfolios are outperforming the benchmark. A portfolio with a positive Alpha has performed better than the benchmark and is considered being an attractive investment option.
• Benchmarking: Alpha can compare a portfolio’s performance to a benchmark index. This can help investors determine whether the portfolio is outperforming or underperforming the benchmark.
• Identifying high-performing assets: Alpha can identify the assets in a portfolio that are contributing the most to the portfolio’s overall performance. This can help investors make more informed decisions about which assets to keep in the portfolio and which to sell.
• Risk-adjusted returns: Alpha is a measure of risk-adjusted returns, which allows investors to compare the performance of portfolios with different risk levels.

### What is Sharpe Ratio?

Overview: The Sharpe ratio is a widely used measure of risk-adjusted return that is used to evaluate the performance of an investment portfolio or financial instrument. It compares the return of an investment to its volatility, which is measured by the standard deviation of returns.

How it works: It is calculated by subtracting the risk-free rate of return from the expected return of the investment or portfolio and dividing the result by the standard deviation of the investment or portfolio's returns. More specifically, the formula for the Sharpe ratio is: Sharpe Ratio = (Expected Return - Risk-Free Rate) / Standard Deviation The numerator of the formula represents the excess return that an investor expects to earn above the risk-free rate of return, which is typically the yield on a government bond or another low-risk investment. The denominator of the formula represents the level of risk associated with the investment or portfolio, as measured by the standard deviation of its returns.

Possible values: The Sharpe ratio can take on any positive value, including zero. A higher Sharpe ratio indicates a higher level of risk-adjusted returns, while a lower Sharpe ratio indicates a lower level of risk-adjusted returns. Typically, a Sharpe ratio of 1 or higher is considered good, while a Sharpe ratio of less than 1 is considered less attractive.

Advantages: Sharpe ratio is an important metric for investors and analysts to evaluate the performance of different investment strategies, assets, and portfolios. It can help them determine whether the returns they are earning on their investments are commensurate with the amount of risk they are taking on. A higher Sharpe ratio shows that an investment has provided higher returns for a level of risk. The Sharpe ratio can also compare the performance of different investment managers or funds. It gives a clear picture of the risk-adjusted returns of the manager or fund, which can help investors make more informed decisions when choosing which managers or funds to invest in.

Limitations: It’s important to note that the Sharpe ratio only considers the volatility of returns and not the downside risk. The ratio does not account for the non-normality of returns, which can lead to a distorted picture of risk-adjusted returns. Therefore, it is recommended to use it, along with other metrics, to evaluate the performance of an investment portfolio.

Usefulness: By providing a clear picture of risk-adjusted returns, the Sharpe ratio can help investors and portfolio managers make more informed decisions when evaluating the performance of their investments. A higher Sharpe ratio shows that an investment has provided higher returns for a level of risk, which is more desirable. It helps in identifying whether the returns generated by a portfolio are because of taking on more risk or because of the manager’s skill. It can also compare the performance of different investment managers or funds. It gives a clear picture of the risk-adjusted returns of the manager or fund, which can help investors and portfolio managers make more informed decisions when choosing which managers or funds to invest in.

### What is Sortino Ratio?

Overview: The Sortino ratio is a measure of risk-adjusted return that is like the Sharpe ratio. However, unlike the Sharpe ratio, which uses the standard deviation of returns as a measure of volatility, the Sortino ratio uses the downside deviation of returns. The downside deviation is a measure of the deviation of returns that fall below a certain target or minimum acceptable return.

How it works: To calculate the Sortino ratio, the expected return is subtracted from the actual return, and then the negative returns are squared and averaged. The resulting value is the downside deviation. The Sortino ratio is then calculated by dividing the excess return (i.e., the actual return minus the expected return) by the downside deviation.

Possible values: A higher Sortino ratio indicates that an investment has provided a higher risk-adjusted return and a lower Sortino ratio indicates that the investment has not compensated adequately for the downside risk.

Advantages: The Sortino ratio only considers downside volatility, which provides a more accurate measure of risk-adjusted return in situations where investors are more concerned with avoiding losses than achieving higher returns. This makes the Sortino ratio especially useful for evaluating the performance of investments that have highly asymmetric return distributions, such as hedge funds, private equity, or other alternative investments. Sortino ratio is also more sensitive to the risk of larger losses, as it penalizes downside risk more heavily than upside risk. This makes it more useful for evaluating investments with non-normal or highly skewed return distributions, and for identifying investment strategies that have a high risk of large losses.

Limitations: It is important to note that the Sortino ratio only considers the downside volatility of returns and not the upside volatility. The ratio does not account for the non-normality of returns, which can lead to a distorted picture of risk-adjusted returns. It is also recommended to use it along with other metrics to evaluate the performance of an investment portfolio.

Usefulness: The Sortino ratio is useful for evaluating the performance of an investment portfolio because it focuses specifically on downside risk, which is the risk of losing money. It can help investors and portfolio managers identify investments that have a high risk of losing money and make more informed decisions about which investments to keep and which to sell.

## Risk Metrics

### What is Maximum Drawdown?

Overview: Maximum drawdown (MDD) is a measure of the risk in a portfolio that calculates the largest single drop from peak to bottom in the value of an investment. Investors and portfolio managers widely used it as a metric to evaluate the risk of a portfolio. It is a measure of the risk of loss, rather than volatility. MDD is widely used because it not only reflects the worst-case scenario but also the time required to recover from that scenario.

How it works: When evaluating a portfolio, the maximum drawdown is the difference between the highest peak value and the lowest trough value, usually measured as a percentage. For example, if an investment is worth \$100 at its peak, and then drops to \$80, the maximum drawdown is 20%.

Possible values: The maximum drawdown is a measure of the largest percentage decline that an investment or portfolio has experienced from its peak value to its subsequent trough. The possible values of maximum drawdown can range from zero to 100%, with higher values indicating a greater loss.

Advantages: We consider maximum drawdown an important metric because it shows how much capital an investor can expect to lose during the worst period of a portfolio’s performance. It also shows how long it would take to recover from that loss. A portfolio with a lower maximum drawdown is less risky than one with a higher maximum drawdown, as it would take less time to recover from a loss.

Limitations: It is important to note that maximum drawdown is a backward-looking metric, meaning it only considers past performance and not future performance. Therefore, it should not be the only metric used to evaluate a portfolio. Other important risk metrics include volatility, Value at Risk (VaR), and expected shortfall (ES).

Usefulness: Maximum drawdown can evaluate the performance of a portfolio over time, and it can compare the performance of different portfolios or investment strategies. For example, we may consider a portfolio with a lower maximum drawdown to be less risky than a portfolio with a higher maximum drawdown, even if the returns of both portfolios are the same. Maximum drawdown can also identify trends in portfolio performance over time. For example, if a portfolio has a consistently high maximum drawdown over a period, it may show that the portfolio is not well diversified, and that we expose it to a high level of risk. Besides evaluating the portfolio’s performance, maximum drawdown can also help investors set realistic expectations for the portfolio’s performance. For example, if an investor knows a portfolio has a maximum drawdown of 30%, they can expect that the portfolio’s value may drop by up to 30% in time. This can help investors make more informed decisions about when to buy or sell investments in the portfolio. It’s important to note that we usually use the maximum drawdown as a relative measure, comparing with other portfolio’s max-drawdown or benchmark. Also, it’s important to consider the time frame the max-drawdown was calculated, as well as other factors like the investor’s goals and risk tolerance.

### What is Standard Deviation?

Overview: Standard deviation measures the deviation of the portfolio’s returns from its mean return over a period. We often used it as a proxy for risk, with higher standard deviation showing a higher level of risk and lower standard deviation showing a lower level of risk.

How it works: To calculate the standard deviation of an investment portfolio, we first need to calculate the portfolio’s returns over a period. We can do this by taking the difference between the end-of-period value of the portfolio and the beginning-of-period value, divided by the beginning-of-period value. Once we have the portfolio’s returns, we can then calculate the mean return by taking the average of the returns. Next, we calculate the deviation of each return from the mean return. This is done by subtracting the mean return from each individual return. We then square each deviation and take the average of the squared deviations. Finally, we take the square root of this average to get the standard deviation.

Possible values: Here are some examples to illustrate the possible values of standard deviation:

• A standard deviation of 0 indicates that all the data points are identical, and there is no variability or dispersion around the mean.
• A low standard deviation indicates that the data points are close to the mean and that there is less variability or dispersion.
• A high standard deviation indicates that the data points are far from the mean and that there is greater variability or dispersion.
• The standard deviation can be equal to the mean, which means that the data points are widely dispersed around the mean.
• The standard deviation can also be much larger than the mean, which indicates that the data points are highly dispersed and spread out.

Advantages: One of the primary benefits of standard deviation is that it is widely used and understood, making it a common language for investment professionals. It is also relatively easy to calculate and can be applied to a wide range of investment types, including stocks, bonds, mutual funds, and other investment vehicles. Another advantage of standard deviation is that it incorporates all deviations from the mean, both positive and negative, which provides a comprehensive measure of risk. This makes it particularly useful for comparing the risk of different investments with similar returns, as investors can use the standard deviation to assess the level of risk in each investment and make informed decisions about which investments align with their risk tolerance and investment objectives.

Limitations: Despite being a widely used and valuable tool for assessing risk, standard deviation has several limitations that investors and financial professionals should be aware of. One of the most significant limitations of standard deviation is that it assumes a normal distribution of returns. This assumption can be problematic since some investments may have fat tails, which means there is a higher probability of extreme returns, both positive and negative, than would be expected in a normal distribution. Another limitation of standard deviation is that it treats positive and negative deviations from the mean as equal. However, investors may view losses differently than gains, which means standard deviation may not provide a complete picture of the risk associated with an investment.

Usefulness: Standard deviation is a widely used measure of risk in investment portfolios. It provides investors with a way to quantify the volatility of their portfolio’s returns over a period. We can use this measure to evaluate a portfolio and make informed investment decisions like:

• To measure the risk of a portfolio. A higher standard deviation shows that the returns of the portfolio are more spread out and are more likely to deviate from the mean, showing a higher level of risk. Conversely, a lower standard deviation shows that the returns of the portfolio are more consistent and are less likely to deviate from the mean, showing a lower level of risk.
• Compare the risk of different portfolios. For example, if one portfolio has a higher standard deviation than another, it shows that the first portfolio is riskier than the second. This can be useful for investors who are trying to balance risk and return in their portfolio. By comparing the standard deviation of different portfolios, investors can make informed decisions about which portfolio is the best fit for their risk tolerance and investment goals.
• Evaluate the diversification of a portfolio. Diversification is spreading investments across different asset classes and sectors to reduce the overall risk of the portfolio. By comparing the standard deviation of unique assets within a portfolio, investors can determine which assets are contributing the most to the portfolio’s risk and make adjustments accordingly.

In combination with other measures we can also use standard deviation in combination with other measures of risk, such as Value at Risk (VaR) and Conditional Value at Risk (CVaR), to provide a more comprehensive assessment of the risk of an investment portfolio. These measures can provide a more detailed picture of the portfolio’s risk by considering different scenarios and probabilities. ​​​

### What is Downside Deviation?

Overview: Downside deviation is a measure of risk that quantifies the amount of variation or dispersion of negative returns in an investment portfolio. It is calculated by measuring the deviation of negative returns from a portfolio’s mean return, and is often used as a proxy for risk, with higher downside deviation showing a higher level of risk.

How it works: To calculate downside deviation, we first need to calculate the portfolio’s negative returns over a period. We can do this by taking the difference between the end-of-period value of the portfolio and the beginning-of-period value, divided by the beginning-of-period value. Then we can select only the negative returns, which are the returns that are below the mean return. Once we have the portfolio’s negative returns, we can then calculate the mean return by taking the average of the negative returns. Next, we calculate the deviation of each negative return from the mean return. This is done by subtracting the mean return from each individual negative return. We then square each deviation and take the average of the squared deviations. Finally, we take the square root of this average to get the downside deviation.

Possible values: The possible values of downside deviation will depend on the specific data set and the minimum acceptable return (MAR) used in the calculation. In general, downside deviation can range from 0 to infinity, with a lower value indicating lower risk and a higher value indicating higher risk. A value of 0 would indicate that there are no returns that fall below the MAR, which would imply no downside risk in the portfolio. A higher value would indicate that there are significant fluctuations in the returns that fall below the MAR, which would imply a higher level of downside risk in the portfolio.

Advantages: One of the key benefits of downside deviation is that it focuses on downside risk, which provides a more accurate picture of the potential losses an investor might experience in a portfolio. By considering only the volatility of returns that fall below a certain threshold or minimum acceptable return (MAR), downside deviation helps investors identify the specific sources of risk in a portfolio. Another advantage of downside deviation is that it is more sensitive to extreme events than other risk measures. This is important because extreme events can have a disproportionate impact on a portfolio, and downside deviation takes this into account. By giving greater weight to large negative returns than to large positive returns, downside deviation provides a more nuanced assessment of risk.

Limitations: Downside deviation only considers negative returns, ignoring positive deviations from the target return. As a result, it may provide an incomplete view of an investment's overall risk and reward potential. It may not accurately reflect the volatility of an investment, as it only measures downside risk. Downside deviation does not account for the frequency of negative returns. An investment with a high downside deviation could have a few large negative returns, or it could have many small negative returns. These two scenarios have different implications for an investor's overall risk, but downside deviation does not differentiate between them. Lastly, downside deviation assumes that negative returns are equally bad, regardless of their magnitude. In reality, some negative returns may be more harmful to an investor than others. For example, a large negative return can have a more significant impact on an investor's portfolio than a small negative return.

Usefulness: Downside deviation is useful for investors who are concerned about the potential for loss in their portfolio. It provides a more accurate measure of risk by focusing specifically on the deviation of negative returns. A higher downside deviation shows that the negative returns of the portfolio are more spread out and are more likely to deviate from the mean, showing a higher level of risk. Conversely, a lower downside deviation shows that the negative returns of the portfolio are more consistent and are less likely to deviate from the mean, showing a lower level of risk. Investors can also use downside deviation to compare the risk of different portfolios. For example, if one portfolio has a higher downside deviation than another, it shows that the first portfolio is riskier than the second, especially in terms of potential losses. This can be useful for investors who are trying to balance risk and return to their portfolio. We can also use downside deviation in combination with other measures of risk, such as Value at Risk (VaR) and Conditional Value at Risk (CVaR) to provide a more comprehensive assessment of the risk of an investment portfolio.

### What is Beta?

Overview: Beta is a measure of the volatility of a stock or portfolio compared to the market. It is a coefficient that indicates how much the price of an asset moves in relation to the movement of the overall market.

How it works: It is calculated by comparing the returns of the investment to the returns of a benchmark index, such as the S&P 500. To calculate beta, you need to first collect historical returns data for the investment and the benchmark index over a period of time. Then, you can use statistical tools such as regression analysis to calculate the relationship between the investment's returns and the benchmark's returns. The beta coefficient represents the slope of the line that describes the relationship between the investment's returns and the benchmark's returns.

Possible values: The beta of a stock or portfolio can range from a negative one to positive infinity. A negative beta shows that the stock or portfolio is likely to move in the opposite direction of the market. A stock or portfolio with a beta of one is likely to move in the same direction as the market, while a stock or portfolio with a beta of two or more is likely to move more than the market. The beta of a portfolio is the weighted average of the betas of the individual stocks or assets in the portfolio. The weight of each stock or asset is determined by its relative size or weight in the portfolio. For example, if a portfolio comprises three stocks with betas of 0.5, 1.0, and 2.0, the portfolio beta will be 1.4. It’s important to note that the beta of a portfolio is not necessarily the same as the beta of its individual components. For example, a portfolio of stocks with an average beta of 1.0 might have a portfolio beta of 1.5 if the stocks are not equally weighted.

Advantages: Beta is a relatively simple and easy-to-understand measure. It can be calculated with just a few inputs, making it accessible to a wide range of investors, from casual investors to professional money managers. It is also widely used and accepted by investors and analysts, which can facilitate communication and understanding among market participants. The use of a common measure like beta can make it easier for investors to compare and evaluate different investments, as well as for analysts to make recommendations and share insights. Lastly, beta can be a useful tool in constructing a well-diversified investment portfolio. By combining assets with different betas, investors can create a portfolio that balances risk and return potential. This can help investors achieve their financial goals while minimizing the impact of market volatility.

Limitations: While beta can be a useful tool for evaluating an investment's risk and return potential, it also has some limitations that investors should be aware of. Beta is a historical measure, which means that it is based on past data and may not reflect future market conditions. This can make it difficult to use beta as a predictive tool for evaluating an investment's future risk and return potential. It also does not take into account all the factors that can influence an investment's performance, such as company-specific events, management decisions, and macroeconomic factors. Therefore, beta may not provide a complete picture of an investment's risk and return potential. Beta does not account for the impact of fees, taxes, and other transaction costs on an investment's returns. These costs can have a significant impact on an investment's performance and should be taken into account when evaluating an investment's risk and return potential.

Usefulness: Beta is a powerful tool that can help investors identify and manage risk in their portfolios. By understanding the beta of a portfolio, investors can assess the potential rewards and risks of their investments and make informed decisions. The beta of a portfolio can also compare portfolios and assess the risk-reward balance of different strategies. For example, an investor with a high-risk tolerance may choose to invest in a portfolio with a higher beta than an investor with a low-risk tolerance. Beta can also identify potential opportunities in the market. By understanding the beta of a portfolio, investors can identify stocks or assets that may offer higher returns than the market. For example, an investor may choose to invest in a stock with a beta of two if they believe it will outperform the market.