Common portfolio metrics

Everybody wants their investment to produce a maximum possible return so the absolute result is important. However, how you get to the final destination is also critical. In fact, most of the popular portfolio performance metrics focus on the path your investment takes.

Introduction

Take a look at the 2 portfolio options above. Which one would you pick and why? Portfolio B should be preferable to Portfolio A, all else being equal. Moreover, many people will prefer Portfolio B even if it produces a lower absolute return than portfolio A.

This makes sense for two fundamental reasons.

First, having your investment results fully depend on market timing is a difficult sale to make. Predicting future market environment (or even adequately assessing current market conditions) is very hard. If you can avoid doing that, it’s a win. If portfolio returns are very volatile, you could end up starting your investment journey at the worst possible time and instead of the expected 7-10% annual nominal returns (as has been the case with equities over the long term) you can be looking at 50% drawdown and 10 to 20 years of unrealized losses ¹ or barely outperforming holding cash. Portfolio B makes the timing of your investment less important. Presumably, you can start investing at any time and still get a positive return.

Why is this relevant? Long periods of positive equity returns have often been followed by equally long periods of negative returns. For example while from 1948-1966 annualized excess returns of equity vs cash were 13.5%, from 1966-1982 those same returns were -3%. While from 1982-2000 annualized excess returns were 12.7%, from 2000-2013 excess returns were just 1.2%. From 2013 until now (2024) annualized nominal rate of return for S&P500 is about 13.05% again. Who’s to say that then next 15 years will be like the last? And who’s to say when exactly the long term equity cycle switches from being overall positive to overall negative for investors?

Second, most people don’t know how they’ll react to a 50% loss even if it’s unrealized. It’s easy to say that equity investors always come out winners in the long run ². However, it’s difficult to predict what your individual response will be. To quote Mike Tyson here “Everyone has a plan: until they get punched in the face”. Will you be tempted to sell it all at the worst possible moment? Keep in mind that most people are psychologically more affected by losses than by wins, even though on paper we should be indifferent ³.

In other words, the inherent uncertainty of outcome and our psychological response to it makes the path your portfolio takes extremely important.

Now let’s look at the commonly used metrics that allow us to understand overall portfolio performance and its volatility going from the basics to more complex measures.

Absolute Return

This is quite simply a ratio of your initial investment value to its final value expressed as a percentage. It measures the total appreciation or deprecation of your portfolio during a specified time period.

$$A = {Ve + D - Vs \over Vs} * 100$$

Where:

• $A$ is the absolute return,
• $Ve$ is the final investment value,
• $Vs$ is the initial investment value,
• $D$ is the dividends or distributions received during the specified time period.

Alpha or Relative Return

Alpha (commonly shown as $\alpha$) is the relative return of the portfolio vs a benchmark. It is usually represented as a single number, like 2 or -4. However, the number actually indicates the percentage above or below a benchmark index that the portfolio achieved given a specified time period.

An alpha of 1.0 means the investment outperformed its benchmark index by 1%. An alpha of -1.0 means the investment underperformed its benchmark index by 1%. If the alpha is zero, its return matches the benchmark.

$$\alpha = {Rp - Rb}$$

Where:

• $Rp$ is the portfolio return,
• $Rb$ is the benchmark return.

Why is relative return or alpha relevant? At the end of the day, all the decisions taken while managing a portfolio are made to increase absolute return and decrease volatility. If that return cannot beat the benchmark which usually involves no decision making it's important to understand why.

Arithmetic Mean

This is the sum of a series of numbers, (daily, weekly, monthly or yearly portfolio returns in our case), divided by the number of items in that series. The formula for the arithmetic mean is simple and is very commonly used to find an average for a data set.

$${\displaystyle \mu={\frac {1}{n}}\sum {i=1}^{n}a_{i}={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}}$$

Where:

• $\mu$ is the arithmetic mean
• $a_i$ is the individual portfolio return value in the data series,
• $n$ is the total number of portfolio return values.

It’s a good way to ballpark portfolio performance over multiple time periods. It is also used in combination with other metrics, such as standard deviation and skew discussed below to understand portfolio volatility.

The downside of arithmetic mean is that it can be skewed by extreme values and can therefore mask the volatility that may be present in the portfolio.

Geometric Mean

The geometric mean for a series of numbers is calculated by taking the product of these numbers and raising it to the inverse of the length of the series. It is best used to calculate the average of a series of data where each item has some relationship to the others. That’s because the formula takes into account serial correlation. This is useful when comparing portfolio returns or any financial series that involve compounding. Compounding affects the return for each succeeding period measured.

Additionally, it reduces the distortion caused by volatility and extreme values (effectively smoothing out impact of large variations) on a data series. Therefore, it provides a more reliable measure when returns fluctuate significantly.

$$G=\left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}$$

Where:

• $G$ is the geometric mean,
• $xi$ is the individual portfolio return value in the data series,
• $n$ is the total number of portfolio return values.

Geometric mean is always lower than arithmetic mean unless all the returns in the series are identical. For volatile returns it can be significantly lower.

note

You can find the approximate geometric mean value given the arithmetic mean and the standard deviation.

$$G \approx \mu - {\sigma^2 \over 2}$$

Where:

• $G$ is the geometric mean,
• $\mu$ is the arithmetic mean,
• $\sigma$ is the standard deviation.

Maximum Drawdown

This number captures your worst observed unrealized loss as a percentage. It measures the difference from peak to trough of your portfolio over a specific time period. This is something that can help you compare different investments while you are evaluating your options or can help you set some rules for what to do in worst case scenario. It can also highlight what are some of the outliers in your portfolio returns are if combining it with arithmetic and geometric mean.

$$MDD = {Vt - Vp \over Vp} * 100$$

Where:

• $MDD$ is the maximum drawdown,
• $Vt$ is the trough value of the investment,
• $Vp$ is the peak value of the investment.

Sharpe Ratio

The Sharpe ratio is a widely used metric for measuring the risk-adjusted return of an investment or portfolio. It helps investors compare different portfolios by considering both their returns and volatility. The formula for calculating the Sharpe ratio is:

$$Sh = {Rp - Rf \over \sigma p}$$

Where:

• $Sh$ is the Sharpe ratio,
• $Rp$ is the average (arithmetic mean) return of the portfolio,
• $Rf$ is the risk-free rate of return (such as the yield on government bonds),
• $\sigma p$ is the standard deviation of the portfolio's returns.

A higher Sharpe Ratio indicates better risk-adjusted performance. The ratio quantifies how much excess return the portfolio generates per unit of risk taken.

Sortino Ratio

The Sortino ratio is an extension of the Sharpe ratio that focuses on the downside risk of an investment or portfolio. Unlike the Sharpe ratio, which considers all volatility (both upside and downside), the Sortino ratio only takes into account the downside volatility, which is essential for risk-averse investors. The formula for the Sortino ratio is:

$$So = {Rp - Rf \over σd}$$

Where:

• $So$ is the Sortino ratio,
• $Rp$ is the average (arithmetic mean) return of the portfolio,
• $Rf$ is the risk-free rate of return,
• $\sigma d$ is the downside standard deviation, which measures the volatility of negative returns.

The Sortino Ratio penalizes portfolios for experiencing large negative swings, making it more relevant for investors who prioritize capital preservation.

Standard Deviation and Skew

Standard deviation (commonly shown as $\sigma$) is a measure of how dispersed some data is around its average. For asset returns it is a measure of how risky the asset is. Individual equities typically have standard deviation of around 30% a year. Bonds usually less.

Standard deviation is calculated by taking the square root of the variance, which itself is the average of the squared differences of the mean.

$$\sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \mu)^2}$$

Where:

• $\sigma$ is the standard deviation of the data set,
• $N$ is the number of values in the data set,
• $x_i$ is each individual value,
• $\mu$ is the arithmetic mean of the values.

When analysts want to understand the risks associated with an investment they look first and foremost at its standard deviation. A quality portfolio (such as Portfolio B discussed in the Introduction) will display a low standard deviation. A portfolio with higher standard deviation will be inherently riskier.

One drawback of relying on standard deviation is that it assumes a normal (bell shaped) distribution of data values. This means the equation indicates that the same probability exists for achieving values above the mean or below the mean. Many portfolios do not display this tendency. Nevertheless, using a normal distribution is often preferrable to more esoteric options because the assumptions are clear and easily understood.

Related to standard deviation is skew. Skew is a statistical measure that indicates the asymmetry of a probability distribution. In the context of portfolio management, skew helps investors understand the likelihood and magnitude of extreme returns. Positive skew indicates that the distribution has a longer tail on the right side, suggesting a higher probability of positive outliers. Negative skew implies a higher probability of negative outliers. Understanding the skew of a portfolio is vital because investors typically aim to achieve positive returns while minimizing the risk of significant losses. Skewed distributions can have a profound impact on the risk-reward profile of an investment.

$$S = \frac{\sum_{i=1}^{n}{(x_{i} - \mu)^{3}}/n} {\sigma^{3}}$$

Where:

• $S$ is the skew,
• $n$ is the number of data points,
• $x_i$ is each individual value of the data set,
• $\mu$ is the arithmetic mean of the data set,
• $\sigma$ is the standard deviation of the data set.

And last but not least portfolio metric we'll discuss here is Beta.

Beta

Beta ($\beta$) is used in finance to denote the volatility of a security or portfolio compared to the market, usually the S&P 500 (SPY ETF for example).

The baseline number for beta is one, which indicates that the security's price moves exactly as the market moves. A beta of less than 1 means that the security is less volatile than the market, while a beta greater than 1 indicates that its price is more volatile than the market. If a stock's beta is 1.5, it is considered to be 50% more volatile than the overall market. Like alpha, beta is a historical number.

Acceptable betas vary across companies and sectors. Many utility stocks have a beta of less than 1, while many high-tech Nasdaq-listed stocks have a beta of greater than 1. To investors, this signals that tech stocks offer the possibility of higher returns but generally pose more risks, while utility stocks are steady earners.

To calculate Beta we can use the following formula:

$$\beta = {CV(R_e, R_m) \over V(R_m)}$$

Where:

• $\beta$ is the Beta coefficient,
• $CV$ is the co-variance of asset’s return $R_e$ with market return $R_m$,
• $V$ is the variance of market return $R_m$.

In this formula, covariance is used to measure the correlation in price moves between portfolio and general market. A positive covariance means they tend to move in lockstep, while a negative covariance means they move in opposite directions.

To calculate co-variance we can use the following:

$$CV = \sum {(R_e - \mu(R_e)) \times (R_m - \mu(R_m)) \over N - 1}$$

Where:

• $CV$ is the covariance of asset's return $R_e$ with market return $R_m$,
• $\mu(R_e)$ is the avarage of asset's return,
• $\mu(R_m)$ is the average of market return,
• $N$ is the number of items in the data set.

And to calculate variance, the following:

$$V = \sum {(R_e - \mu(R_e))^2 \over N }$$

Where:

• $V$ is the variance of returns
• $R_e$ is the individual return (such as daily)
• $\mu(R_e)$ is the mean return
• $N$ is the total number of returns

Conclusion

The metrics explored above should be enough to give anybody, from beginner to professional, a good understanding of portfolio performance. There's plenty of other things that can be explored but like most things in life they follow the Pareto principle "roughly 80% of consequences come from 20% of causes (the 'vital few')". In other words, 20% of available portfolio metrics (e.g. the ones discussed above) will provide 80% of valuable information. The more esoteric things become the more they become subject to declining marginal utility.

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Footnotes

1. Hard to believe in this day and edge. After the introduction the Fed + Federal Government Put since 2008 financial crisis long bear markets seem impossible but who knows? ↩

2. This has been true historically in US public markets if your time horizon is 30-50 years. ↩

3. Thinking, Fast and Slow by Daniel Kahneman. ↩