Quantifying Risk in Finance With the Help of Dispersion

In the previous publication, we looked at how to calculate the mean - referred to sometimes also as an expected return of an investment. In practice though, the actual returns often differ from expectations—and that variability matters just as much as the average.

1. Dispersion as a Measure of Risk and Uncertainty

This variability is known as dispersion. It describes how spread out investment returns are around a central value, such as the mean. The greater the dispersion, the higher the uncertainty and risk. Understanding dispersion helps analysts gauge how consistent or volatile an investment’s returns may be over time. It’s a fundamental step in assessing the reliability of an investment—not just its potential reward.

1.1. Range

Simply put, the range is the difference, or the spread between the minimal possible value (for example the minimal return achievable) and the maximum possible value. It is the simplest way of measuring and defining the dispersion.

$$\text{Range}=\text{Max. value}-\text{Min.value}$$

The upside of this measure is the fact that it’s simple to compute, in fact all it takes is two values. This simplicity of it though is also it’s biggest weakness. Measuring dispersion with this method only gives us so much information, and doesn’t account for the distribution of the returns.

An another weakness of this measure is the fact that it’s very sensitive to outliers by definition, as it takes the lowest and the highest values.

1.2. Mean Absolute Deviation

To address the limitations of the range as a measure of dispersion, we can employ the Mean Absolute Deviation (MAD). Unlike the range, which only considers the two most extreme values in a dataset, MAD incorporates all data points, providing a more comprehensive picture of variability.

$$\text{MAD}=\dfrac{1}{n}\sum_{i=1}^n|X_i-\overline{X}|$$

This calculation involves taking the absolute deviation of each observation from the mean. The sum of these absolute deviations is then averaged over the number of observations $n$, which, in our case, corresponds to the number of return samples.

MAD is particularly robust in the presence of outliers, making it a reliable metric for datasets that include extreme values. While this helps prevent the cancellation of positive and negative deviations, the use of the absolute values complicates further mathematical manipulation.

Another limitation is that MAD, in its standard form, requires processing the entire dataset (the population), which can be computationally challenging when working with large datasets, such as those commonly encountered in finance.

1.3. Variance

As a solution to the limitations of MAD, we turn to the gold standard of dispersion measures in statistics and finance: the variance. In financial contexts, where datasets are often extremely large, it can be difficult—or even infeasible—to compute metrics like the mean or median for the entire population. Sample variance addresses this challenge by allowing us to make inferences based on a representative sample drawn from the population.

$$s^2=\dfrac{\sum_{i=1}^n{(X_i-\overline{X})^2}}{n-1}$$

Similar to MAD, variance measures the average deviation from the mean. However, instead of taking the absolute value of the deviations, it squares them. This not only avoids the issue of deviations canceling out, but also makes the measure more easier to work with when it comes to mathematical transformations.

A key difference lies in the denominator: instead of dividing by $n$, we divide by $n-1$. This adjustment, known as Bessel’s correction, improves the statistical properties of the estimator by correcting for bias when estimating population variance from a sample.

The main drawback of variance is interpretability. Because the deviations are squared, the resulting unit of measurement is also squared. For instance, if the original data is expressed in euros (€), variance is expressed in euros squared (€²), which lacks intuitive meaning in practical terms.

1.4. Standard Deviation

To simplify the interpretation of the variance, we can use the standard deviation. This metric is nothing else, but a square root of the variance. It allows us to return the squared units e.g the euros squared (€²) to the normal units, which are easier to do comparisons with e.g. the euros (€).

$$s=\sqrt{\dfrac{\sum_{i=1}^n{(X_i-\overline{X})^2}}{n-1}}$$

Standard deviation then indicates the amount of dispersion from the mean, and these two are very often also used together. For example we could say that on average, a portfolio manager’s return is $15\%\text{ p.a }\pm3\%$ - which stands for $\text{the mean}\pm\text{standard deviation}$.

1.5. Downside Deviation

The previously mentioned MAD, variance and standard deviation take into account how the observed value moves upside, or downside from the mean.

Especially in finance though, the target value might not be the mean, and we might not want to measure both the upside and the downside. In fact, investors are generally significantly more concerned about the downside risk, and not so much of the upside risk, which the downside deviation can help us measure. This is where the Downside Deviation, also called standard target semi-deviation comes into play.

For example when investing, we might be concerned about the size of the returns below our target rate of return. To calculate the standard target semi-deviation, we first need to specify a target, let’s call it $T$.

$$S_{\,target}=\sqrt{\dfrac{\sum^{n}_{\text{for all }X_i<T}{(X_i-T)^2}}{n-1}}$$

Once we have the target defined, we need to calculate the deviations of the population from the target. Once we have this, we only sum the square of the deviations which are lower than our target, this means we do a sum for all of the $X_i$ where $X_i<T$. After that, similarly to variation, we divide it by $n-1$ and finally to conclude we take the square root.

1.6. Coefficient of Variation

One might wonder how to compare the dispersion among different assets or benchmarks in a meaningful way. So far, most measures of dispersion have been expressed in absolute terms. However, this makes it difficult to compare assets like bonds and stocks, since their underlying characteristics and absolute price levels differ.

To address this, we can use the coefficient of variation (CV). The CV standardizes dispersion by expressing the standard deviation as a proportion of the mean. This allows us to compare variability across different assets by showing how much an asset's returns deviate from the mean relative to the size of the mean itself.

$$\text{CV}=\dfrac{\text{standard deviation}}{\overline{X}}$$

We can use coefficient of variation either as a percentage (150%) or also as a multiple (1.5) - both of these are widely used and acceptable. CV is a scale free measure.

When it comes to using it in Finance, it’s also very useful as we already mentioned to compare different assets. There is some corner cases that work in the maths, but won’t make sense in finance realias. One of them is the case when the mean $\overline{X}$ is negative, this causes the coefficient to be negative, which makes it not relevant.

2. Real Life Example of Measuring Dispersion in Finance

We can take up on the example from before and have a look at how the returns of bonds and equities compare. Let’s assume the following table contains the yearly returns for a stock (A) portfolio and a bond (B) portfolio respectively.

If we calculate the mean returns for both portfolios over the past ten years, we find that they each have the same average annual return of 6.2%. But does that mean they are equivalent? Not quite—we can't stop our analysis there.

As investors concerned with more than just returns, we need to go a step further and assess the volatility of each portfolio. This is done by calculating the standard deviation, which measures how much returns deviate from the mean. According to the table, the standard deviation is 12.30% for the stock portfolio and 3.08% for the bond portfolio.

While this gives us valuable insight into the individual risk profiles of each investment, it's still not enough to make a comparison. To properly compare them relative to their returns, we need to take one more step: calculate the coefficient of variation.

Now we get to the key insight. The stock portfolio has a coefficient of variation of 1.98, meaning its returns deviate by 198% relative to the mean return. In contrast, the bond portfolio has a coefficient of 0.50, indicating a 50% deviation from its mean return.

In simpler terms, although both portfolios have the same average return of 6.2% per year, the bond portfolio achieves this return with significantly less relative volatility. This makes it the more stable investment when comparing risk-adjusted performance.

3. Conclusion

While average return is often the first metric investors consider, it tells only a part of the story. As we've explored in this article, dispersion—or the variability around that average—is just as important when evaluating an investment's risk and reliability.

We began with simple measures like range, moved to more comprehensive ones like mean absolute deviation and variance, and progressed to refined tools such as standard deviation, downside deviation, and the coefficient of variation (CV). Each method offers a different lens through which to view risk, and each has its strengths and limitations depending on the context.

The real-world example comparing a stock and bond portfolio makes the takeaway clear: returns alone do not define the quality of an investment. Even when two assets have identical average returns, their risk profiles can differ dramatically.

Ultimately, understanding and applying these measures of dispersion equips investors with a more complete picture—allowing for smarter, risk-adjusted decisions in an uncertain financial world.