In the previous chapters we have seen how financial planning and investment scenarios are helpful in guiding our investment decisions. But so far, everything we learned was on a qualitative basis. In this chapter, I introduce charts and metrics we can use to quantify these aspects. This includes my preferred measures when explaining investing, but also the measures you will likely find when researching investment products and portfolios. Understanding all of them will help you find your way around, compare investments, and make good choices.

For the most part, I gloss over the mathematical detail required to calculate the metrics or construct the charts. After all, this book is about investing and not about math. If you are a curious investor, I encourage you to take the time to dive a little deeper and also understand ‘how the sausage is made.’

Ex-Post Measures

Ex-post is a Latin term for “after the event” and is a retrospective explanation of something that has already happened. When it comes to investing, ex-post measures are created from historical data. This type of measure is helpful, because it provides us with a frame of reference for how investments behaved and performed in the past. However, past performance does not necessarily indicate future results, and extrapolating the past into the future is a tricky, inaccurate business at best.

Measuring Returns

When thinking about investments, most people immediately wonder about the possible returns. I disagree with that view, and firmly believe that sound investment decisions can only be made when considering returns, risks, and the time horizon together. Nonetheless, we start by looking at typical return measures.

Start and End Balance

The easiest way to measure returns is by comparing the starting and ending balance. This method makes returns very tangible, especially when compounding over prolonged periods. However, this is also highly misleading. Because the measure does not adjust for time, it is impossible to directly compare the results to others, measured over a different period. This will almost always be the case when comparing results from dissimilar sources.

Further, this crude measure leads to surprisingly high numbers, which triggers the worst part of the human psyche: greed. Based on this measure, investors tend to overestimate their annual return, and the edge one investment might have over another. And with that, ignore all investment risks.

After dividing the ending by the starting balance, we arrive at a slight variation to this, expressing the investment profits as a percentage increase. Needless to say, this method suffers from the same limitations.

Most of the time, when either of these measures is used without also being accompanied by other, more suitable metrics, it is with the intention to deceive emotional investors.

Compound Annual Growth Rate

The Compound Annual Growth Rate (often abbreviated CAGR) is the most frequently used number to characterize investments. The number represents the average annual return from an investment, if we re-invest and compound all gains. This concept is easy to grasp, which explains its popularity.

It is a big step up from the previous measure, because we can compare CAGR values, even if they have been taken over different periods. However, especially in the shorter term, CAGR is highly misleading. The number is only an average, and we must assume our returns will be lower half of the time. And when this happens several years in a row, our investment outcome can significantly differ from what the CAGR suggests. But because we don’t know how much lower returns might turn out to be, we cannot plan for that. Therefore, CAGR is irrelevant when investing toward a goal, or for an income.

Of course, over many years the fluctuating returns will even out. But most investors will be surprised to learn that even after 25 years there is still a significant spread between the average result and less fortunate outcomes.

Only when investing excess funds does this number become relevant. Under these circumstances, we don’t need to plan for underperformance but optimize for the expected outcome – which is the average.

Annual Bars

Instead of representing investment returns with a single number describing the full range of historical data, we can also split this range into multiple periods, each represented by its own number. We often see calendar years (or quarters), but recent periods of varying lengths (e.g., 1 year, 5 years, 10 years) are also frequently used. Because humans love visualizations, these numbers are often represented as a bar chart.

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Breaking the number up into multiple smaller chunks helps investors gain a better understanding of how the returns might fluctuate. However, this method typically underrepresents these fluctuations, making investments appear less risky than they are. This is because market events seldom align with the periods shown. Events shorter than measured periods are hidden ‘inside’ the bars, and longer events (think multi-year recessions) are spread across multiple bars and easily misinterpreted.

Consequently, the annual bars are only slightly better than a single CAGR number, and most of the time, not the right tool to make investment decisions.

Equity Chart

The human brain is incredibly good at recognizing patterns and trends. Therefore, representing the account value as a graph, the so-called equity curve, is an appealing and intuitively understood visualization. Technically, such a chart contains all the available information about the investment, solving the issues we saw with CAGR or annual returns. Furthermore, we can use this type of chart to compare investments to a benchmark, which adds context and aids interpretation.

Unfortunately, not all these charts are the same, and the vertical axis is of primary interest here. This axis can be scaled linearly, or logarithmically. Because investments compound over time, the account value grows exponentially (the shape of a hockey stick). And while this looks impressive, this type of chart tends to cloak drawdowns and hides most of the detail from less-recent history. Further, linearly scaled charts make it almost impossible to compare between periods or investments in a meaningful manner.

Equity curves should always be scaled logarithmically to compensate for their exponential growth. With this scaling, constant returns result in straight lines, making it easy to spot any irregularities. In addition, when two investments grow at the same rate, their curves are parallel, allowing for easy comparisons between periods or investments.

When it comes to visualizing investment risks, equity curves provide only limited information. It is easy to spot periods of prolonged losses, and how long it took to recover from these. But measuring the depth of such drawdowns from the chart is cumbersome at best.

Rolling Returns

As an alternative to charting the account value, we can also chart the rolling returns over a fixed lookback period. Instead of an ascending chart, we now have a (mostly) horizontal chart, which saves space and allows a higher zoom level. Especially when an investment is plotted against a benchmark, these charts provide valuable detail and insight. Specifically, we can distinguish periods of positive or negative performance, and identify periods where the investment outperforms or underperforms its benchmark.

But there are also issues with these charts. When charting returns over shorter periods (e.g., 3 months), the charts tend to become noisy and hard to read. Consequently, rolling return charts work best to plot returns over 1-year or longer periods. However, of course, charts using these longer periods hide any events shorter than that lookback period.

While returns are positive, these charts are easy and intuitive to interpret. But, when returns turn negative, investors naturally like to understand how much in total they lost, and how long it took to recover from these losses. Unfortunately, rolling return charts provide no information about these crucial aspects.

Tracking Chart

Oftentimes, investors are interested in an investment’s performance compared to a benchmark. The best way to show this is with a tracking chart, constructed by dividing the investment account values by the benchmark. When the investment outperforms its benchmark, the chart slopes upwards.

This chart provides excellent insight into the nature of the investment and how it compares to its benchmark. Some investments outperform their benchmark continually, and we see their tracking chart climbing smoothly. Other investments only shine during specific periods, e.g., during recessions, and their tracking charts show some bumps along the way.

Of course, this chart also has disadvantages. Most importantly, tracking charts only show relative performance, and carry no information about absolute performance. If the investment and its benchmark decline in tandem, this is completely invisible. Unfortunately, such behavior is more of the default than the exception, as we routinely pick benchmarks with characteristics similar to the investment.

Summary

As we can see, all ex-post measures of returns suffer from severe limitations. Most of them hide or underrepresent the investment risks, and require the combination with other measures to paint a more conclusive picture of an investment’s nature.

Measure	Interpretation	Pro	Contra
Compound Annual Growth Rate	Higher numbers indicate higher returns	* Easy to understand and use * Relevant when investing excess funds	* Misleading in the shorter term * Irrelevant when investing toward a goal or for income
Annual Returns	Taller bars indicate higher returns	* Easy to understand and compare * Provides limited insight into return fluctuations	* Underrepresents the range of investment outcomes
Equity Chart	Steeper slopes indicate higher returns	* Intuitively understood * Hints at fluctuating investment returns	* Provides only little insight into investment risks * Difficult to compare
Rolling Return Chart	Higher curve indicates higher returns	* Intuitively understood * Shows absolute and relative performance * Shows correlation	* Provides only little insight into investment risks * Requires lengthy periods to reduce noise * Hides short-term fluctuations
Tracking Chart	Positive slope indicates outperformance	* Intuitively understood * Shows relative performance	* No information about absolute performance * Hides investment risks

Measuring Risk

Investors are typically much more concerned with their investments’ returns than the risks involved. And while I admire this optimism, it is essential to consider investment risks and how they might affect the investment outcome.

Volatility

The most popular measure of risk is historical volatility, calculated as the standard deviation of investment returns. Usually, this number is annualized, so that it stands for a percentage change in the asset price, that, with a probability of 68%, won’t be exceeded over the span of 1 year.

It is important to notice that volatility is a non-directional measure and describes the unsteadiness of returns, regardless of their direction. With that, sudden rallies can have the same effect on volatility as quick selloffs have, which is why volatility does not necessarily equate to the risk of incurring investment losses.

Curious investors might wonder about the difference between volatility and the standard deviation of returns. Mathematically speaking, there is none. But when investors use the term volatility, they typically refer to a short-term measure. In contrast to that, they prefer the term standard deviation of returns when describing longer periods of time.

Unfortunately, for many investments the annualized volatility is about twice the annual return. Consequently, 1-year returns may very well be negative, even in the presence of a positive long-term trend. Luckily, this situation improves with longer investment periods. This is because returns scale proportionally to time, while volatility only scales with the square-root of time. Therefore, the ratio between returns and volatility improves if the investment horizon is sufficiently long.

The calculation of volatility assumes that the (logarithmic) returns are normally distributed. However, this is an oversimplification: Typical investment assets have excess kurtosis, or fatter tails than the normal distribution. Therefore, volatility underestimates the probability of extreme outcomes, e.g., black swan events.

Maximum Drawdown

The Maximum Drawdown is an often-used metric to describe investment risk. It represents the largest loss, measured from an earlier peak, an investment suffered over the observation period. The maximum drawdown adds tremendous value, because of its intuitive interpretation. Only when investors feel comfortable losing as much as the maximum drawdown suggests is an investment suitable for their risk appetite.

However, the maximum drawdown is also misleading because no statistical properties are associated with measuring a maximum. Therefore, investors don’t know the likelihood of their investment losing more than the stated maximum. This lack of definition also makes comparing two assets based on the maximum drawdown tricky. Unless the assets are closely related, we don’t know if two maximum drawdowns have equal severity in terms of mathematical probability.

We can improve this situation by making the observation period sufficiently long. But even with exceedingly lengthy periods, there are still no guarantees. Thus, we can be certain that at some point in the future, drawdowns will exceed the presumed maximum drawdown.

Maximum Time to Recover

Closely related to drawdowns are the Maximum Time to Recover or Maximum Flat Days. This metric describes the maximum time it has taken to recover from a drawdown. Because of its intuitive interpretation, this metric also has tremendous value, especially when paired with the maximum drawdown.

Even though this measure is remarkably simple, it does something particularly useful: it creates a relationship between losses and returns. For an investment to make sense, the investment period should exceed the maximum time to recover losses.

My criticism of this metric is like the maximum drawdown: there are no statistical properties associated with this metric, and investors never know how likely it might be to exceed this value. Therefore, investors should take this number with a grain of salt – at some point in the future, losses might take longer to recover.

Ulcer Index

The Ulcer Index is another measure of investment risk, calculated as the root-mean-square of drawdowns. It not only measures the depth of all drawdowns, large or small, but also considers their duration. These features make the metric an excellent way to characterize the downside risk of an investment.

Interpreting the Ulcer Index, investors should assume to frequently experience drawdowns of that magnitude. Further, during bear markets or recessions, investments might easily reach four or five times as deep drawdowns.

While, in my opinion, the Ulcer Index is a much better metric to measure historical investment risk than volatility, it is, on the other hand, mathematically much less relevant. The only measure I am aware of that is derived from the Ulcer Index is the Ulcer Performance Index, or Martin Ratio.

Beta

The measure of Beta stems from the Capital Asset Pricing Model (abbreviated CAPM) and describes how much an asset moves in unison with a benchmark. This is in contrast to Alpha, which describes the fraction of the asset returns that is idiosyncratic and ndependent from the benchmark. Beta values larger than one describe investments that amplify the benchmark’s returns, while values lower than one indicate that an investment dampens the benchmark’s returns.

With these properties, beta is a double-edged sword. When markets go up, we may prefer a high beta as that lets us maximize our participation in those gains. But when markets decline, we prefer a lower beta as that reduces our losses. Because markets are often erratic, beta is considered a measure of risk, even though this might seem counterintuitive.

In general, investors should seek investments with a low beta but a high alpha. Such investments will do well, regardless of the market’s direction.

Drawdown Chart

So far, we have only looked at risk metrics. But, of course, there are also ways to visualize risk. One of the most-often used charts is the Drawdown or Underwater Chart. This chart shows the loss since the most-recent highest high, making losses and the time to recover very tangible. Furthermore, the chart makes spotting an investment’s correlation with its benchmark easy.

However, the chart paints an incomplete picture. While it provides good guidance on everyday drawdowns, the charts typically do not cover enough history to allow for a conclusive evaluation of recession performance. Instead, the same criticism I had for maximum drawdown and days to recover applies here. Investors need to apply some common sense and be prepared for the future to exceed the limits suggested by the chart.

Summary

Like return metrics, all risk measures come with significant limitations. Specifically, most of them underestimate tail risks and rely on anecdotal evidence to set risk limits. Consequently, no single measure is sufficient to describe investment risk and guide investment decisions holistically.

Measure	Interpretation	Pro	Contra
Volatility	Lower value indicates more docile investment	* Basis for many mathematical methods	* Non-directional measure * Unintuitive interpretation * Underestimates tail risks
Maximum Drawdown	Lower value indicates safer investment	* Intuitive interpretation * Measures downside risk	* Limited reliability when comparing assets
Maximum Time to Recover	Lower value indicates faster recovery	* Intuitive interpretation * Connects losses with returns	* Limitations when comparing assets
Ulcer Index	Lower value indicates less risky investment	* Measures downside risk	* Less versatile usage * Unintuitive interpretation
Beta	Lower value indicates less dependency on the market	* Effective way of determining market risk	* Not necessarily a risk measure
Drawdown Chart	Shallower graph indicates lower risk	* Intuitive interpretation	* May underrepresent risk

Risk-Adjusted Returns

Investors often look for simple answers when comparing investment alternatives. One solution to address this need is the use of risk-adjusted metrics. These metrics express the possible investment returns relative to the risk taken. In simple terms, risk-adjusted metrics measure an investment’s quality or the money-manager’s skill.

Sharpe Ratio

The Sharpe Ratio is the most-popular risk-adjusted return metric. It is based on an investment’s excess return over the risk-free (think Treasury Bill) investment, divided by volatility.

Of course, my criticism of this measure matches its ingredients. Because it is based on volatility, it underestimates tail risks. Further, volatility is not necessarily a measure of risk, as it is non-directional and treats upside and downside variations the same. As a result, the Sharpe Ratio might not always achieve its objective of measuring an investment’s quality.

Martin Ratio

The Martin Ratio or Ulcer Performance Index is another often-used risk-adjusted return metric. It is calculated by dividing an investment’s excess return by its Ulcer Index.

This metric is an improvement over the Sharpe Ratio, as the Ulcer Index is a much better representation of investment risk. Consequently, the Martin Ratio is the better and more relevant metric.

Summary

To summarize, the Sharpe and Martin Ratio are valuable metrics for comparing the quality of investments. And because these measures are invariant to leverage, they allow fair comparisons between investments even if they have vastly different characteristics.

However, investors would be ill-advised to base their investment decisions purely on these risk-adjusted metrics. To reach our financial goals, we must meet some minimum returns. And while we could leverage up a high-quality investment until we meet the desired returns, retail investors do not have access to the same level of leverage as institutional investors do. Furthermore, applying leverage also incurs borrowing costs.

Consequently, these metrics must be combined with measures of return to provide a more conclusive characterization of an investment.

Measure	Interpretation	Pro	Contra
Sharpe Ratio	Higher value indicates higher quality investment	* Most-commonly used	* Underestimates tail risk * Volatility not representative of investment risk
Martin Ratio	Higher value indicates better investment	* Uses good representation of investment risk	* Less frequently used

Ex-Ante Measures

Ex-Ante means “before the event” and describes a prediction of the future investment outcome. Of course, these predictions are based on past data, but they do not directly extrapolate ex-post measures into the future. Instead, additional steps are taken to make these predictions more relevant and easier to understand.

Range of Investment Outcomes

The ex-ante measures used for this book are based on Monte Carlo Simulations. In these simulations, random samples are drawn from the investment’s historical distribution of returns to create a swarm of several thousand hypothetical investment outcomes. All these outcomes share the exact same statistical properties as the original historical data, and no assumptions about the shape of the distribution are made.

Based on this swarm of outcomes, we can establish a range by picking the values at defined percentiles. For my simulations, I use the 5th and the 95th percentile, resulting in 90% of the outcomes landing between the upper and lower limits.

It is important to note that these percentiles are by no means the worst possible outcomes. Quite to the contrary: at the 5th percentile, one in twenty paths will go below the lower limit. Consequently, one might wonder if more extreme percentiles, e.g., at 1% and 99%, might lead to better predictions. However, unfortunately, this is not the case. Investing requires a leap of faith, and when using overly pessimistic limits, the only rational decision is to invest in Treasury Bills. This directly contradicts our life experience: investing in risky assets typically leads to better outcomes than a T-Bill.

We have seen these cones of investment outcomes in the chapter about investment scenarios. They are a helpful and intuitive way to compare assets with each other. However, modelling the investment outcome in absolute (dollar) terms is inconvenient when further calculations are required.

Range of Investment Returns

Most often, investors are more interested in the range of investment returns than the investment outcome in absolute terms. Luckily, this is easy to achieve. By dividing the investment outcome by time, we can convert it to a range of annualized returns.

As with the investment outcome, the horizontal axis records time. On the vertical axis, we have the upper and lower bounds for annualized returns. The charts show how volatility dominates the outcome over short periods, leading to a wide range of returns, including negative ones. But with time, the cone becomes narrower, and the trend takes over.

This visualization achieves something none of the other charts and metrics did: to establish a relationship between the three most important investment variables, return, risk, and time. Most importantly, these charts provide an estimate for investment returns under pessimistic conditions and as a function of time. This estimate is a powerful tool to identify the best possible investment and risk level for a given investment scenario and time horizon.

Most investors will be surprised by the relevance that time has on their investments. For example, at the 5th percentile, it takes years for most investments to break even, and even after 25 years, there is still a wide gap between the pessimistic and the optimistic outcome. We will see many examples of this in the section about investment assets.

Less obvious is that ranking returns at the 5th percentile leads to different results than ranking average returns: Under pessimistic assumptions, more aggressive investments oftentimes have lower returns than safer alternatives. Because the width of the return range scales with volatility, this intuitively makes sense.

The finding has important implications. When investing toward a goal or for income, we need to optimize for the pessimistic investment outcome, and the ex-ante range of investment returns is the right tool for the job. With this tool in our arsenal, the perspective on ‘beating the market’ changes. We no longer attempt to beat a benchmark’s average return but beat the benchmark at the level of pessimism that matches our investment scenario.

Sadly, we do not see these charts too often; in fact, I believe that my TuringTrader software is the first software to routinely create such charts. Regardless, they are so helpful and intuitive that it has become my second nature to explain investing with their help.

Recession Outcomes

In the discussion about ex-post metrics, we have seen how the maximum drawdown and the maximum time to recover are intuitive, but misleading measures. However, we can improve them with the help of Monte Carlo simulations.

We start by creating a swarm of drawdown outcomes. These outcomes include all simulated drawdowns, regardless of how small or short-lived they were. This swarm is like the investment outcomes, but with two notable differences. First, we only consider drawdowns, so all outcomes start in the red on day one. And further, we peg the results at zero once they recovered their prior loss.

To get a curve of a representative recession drawdown, we narrow down this swarm in two steps. First, we filter the swarm for 5% of the worst drawdowns, measured by their Ulcer Index. Then, we calculate the account value at the 5th percentile for these.

The result shows how a hypothetical recession scenario might play out. This chart is different from the ex-post underwater chart in numerous ways. For once, there is, of course, no historical precedent for this drawdown. However, investors should assume that this drawdown resembles the way a major recession (think 2008) unfolds. Furthermore, this hypothetical drawdown has clearly defined statistical properties. Consequently, we can use these to compare different assets and asset classes fairly.

When interpreting these charts, investors should keep in mind that the drawdown shown in the chart is by no means the worst case, or even the worst event recorded over the simulation. However, we can be sure that an investment with a shallower drawdown chart and faster recovery is indeed less risky than an asset with deeper drawdowns and longer recovery.

It is worth noting that these charts are much more than a fancy replacement for the maximum drawdown and days to recover metrics. Their shape also reveals a lot about the nature of the investment risk, and how drawdowns might play out. For example, looking at the chart above, we notice that the S&P can reach a 20% drawdown within just a few days, while it might take up to a year to reach a 35% drawdown.

Curious investors might wonder how they are supposed to use this information. The rational answer is to ignore it. The reason for this likely surprising answer is that the range of outcomes already contains everything there is to know for making a sound investment decision. Nonetheless, it is crucial to also consider the investor’s emotions, most importantly the fear of loss. We can define an investor’s personal risk tolerance as the loss at which the investor is tempted to second-guess the investment, regardless of the rationale behind it. If the projected recession outcome comes close to that individual threshold, the prudent investor should opt for a safer alternative.

Like the range of investment returns, these charts are not commonly used. Instead, I believe to have invented this method of estimating and visualizing investment risks. Again, I find these charts so useful that I will illustrate investment risks throughout this book with their help.

Measure	Interpretation	Pro	Contra
Range of Investment Outcomes	Range of account values over time	* Defined statistical properties	* Unintuitive representation
Range of Investment Returns	Range of investment returns over time	* Defined statistical properties * Intuitive representation * Establishes relationship between returns, volatility, and investment period * Highly relevant when investing toward a goal or for income	* Not a commonly used chart
Recession Chart	Development of deep drawdowns over time	* Defined statistical properties * Intuitive representation * Allows comparison between assets * Relevant for all investment scenarios	* Not a commonly used chart

Summary

The ex-ante charts discussed here are two tremendously helpful tools for making sound investment decisions: The range of investment returns applies while investing toward a goal or for income. And the recession chart helps investors select a comfortable risk level that is relevant for all investment scenarios.

Conclusion

In this chapter, we reviewed a list of ex-post and ex-ante measures. The important takeaway is that ex-post measures might be intuitive but are often misleading. Investors should not assume values taken from past readings to be indicative of future outcomes. This is especially true for risk metrics.

In contrast, ex-ante measures are specifically designed to make forecasts for the future. These forecasts come in the form of expected ranges, plus the statistical probability of staying within these bounds. Our ex-ante measures are based on Monte Carlo simulations and establish a clear relationship between returns, volatility, and investment period.

This relationship is of utmost relevance for investors, as it creates a direct path from financial planning, via investment scenarios, to portfolio management. We can eliminate any guess work, and simply identify the most suitable investment, given the investor’s time horizon and investment scenario. With this mechanism in place, we can decouple the personal risk-tolerance from the investment outcome. Instead of making unintuitive tradeoffs between investment returns and the associated risks, risk tolerance becomes simply a personal preference.

As a result, we have an investment process that follows analytical steps, and, equally important, helps integrate the investors’ emotions.