Distributions, Distributions, Distributions

Hi, and welcome to the first post in our series about the quantitative methods we use at Element 14 Capital. We love our investments in AI and with the models becoming better and better, it has never been easier to use refined statistical methods and modeling to aid our decision making.

However, whether something is easy depends not only on your level of understanding, but also on experience — and on catching many nuances and small details. Our blog series will feature key investment takeaways, short and to-the-point explanations of the mathematical models, and some of the experiences and observations we made. We hope you find these posts useful, and we would love to hear your feedback, as well as suggestions for topics to cover next.

Oh, and feel free to share your own observations too!

Probability distribution. Simply integrate to get probability!

In statistics, pretty much everything revolves around the probability distribution, or, in the continuous case, the probability density function. Visually, the function itself shows well how the values are distributed. Then, let's take a look at Figure 1. The plot shows the Gaussian distribution, also known as the bell curve, which is one of the most fundamental probability distributions and something many of you are probably already familiar with. Most of the time, when someone mentions standard deviation or percentiles, they are referring to this type of distribution. The interpretation, and the reason why it is useful, is pretty straightforward. It has the highest value at its center, i.e. the mean, and the farther we move away from the mean, the lower the curve becomes. This, of course, corresponds to those outcomes being less likely.

Figure 1Illustrative

a

b

Show the underlying values

Range	Probability
Below the mean (0)	50.0%
Above the mean (0)	50.0%
Below the threshold (1.5)	93.3%
Above the threshold (1.5)	6.7%

Also, in Figure 1a, we see that the vertical line divides the function in half. The area under the curve for values ranging from 0 to infinity is the same as the area for values ranging from negative infinity to 0, and each is equal to ½. Figure 1b shows an example where we determine the probability of the value being above the threshold of 1.5. Also, a key feature of all probability distributions is that they must be normalized so that the total area under the curve is equal to 1, because 1 is the maximum probability, i.e. 100%.

Now, I want you all to also remember that this is a continuous function. What does that mean? It means that it can take any value in a range, no matter how finely you zoom in between two numbers. The opposite would be a discrete function, which can be thought of as a set of separate points or bars rather than one continuous range of values. It is essentially the difference between a ramp and a staircase: the staircase only lets you stand on fixed steps, while the ramp lets you stand anywhere along it. Now, why is it important whether the function is continuous or discrete?

It all comes down to the fact that, for a continuous variable, probabilities are calculated using integrals. As you may remember from calculus classes, a key feature of integrals is that they can be evaluated over ranges, such as from 0 to infinity. Because a continuous variable can land anywhere in a range rather than on separate values, the probability must therefore be determined over a range rather than for one exact number. To determine the probability, we calculate the area under the distribution across that interval.

Venture returns are commonly described by a power law

When you do a quick search on how to model venture capital returns, the answer you'll get is a power-law distribution. This is intuitive because it captures the 80/20 rule which states that roughly 20% of the inputs generate roughly 80% of the outputs, which is the most important logic governing venture capital. This has a consequence in producing a long tail in which very large outcomes are possible albeit rare, just like the unicorns. A distribution that follows this behavior is the Lomax distribution, which is essentially a Pareto distribution, but one that starts at the value of zero rather than at the value of one. Let's look at it in Figure 2. Indeed, the density is highest near 0×, which corresponds to the likelihood of a failed investment being high.

Figure 2Model output

a

b

Tail index α — lower values put more weight in the extreme tail

2.50

Median

1.43×

Mean

2.98×

Below the mean

72.1%

Above 5×

15.3%

Showing our published parameters.

Reset

Show the underlying values

Statistic	Value
Median	1.43×
Mean	2.98×
Share below the mean	72.1%
Share above 5×	15.3%

Let's also take a look at something interesting. Figure 2a shows the median while Figure 2b shows the mean. We can see that the median is 1.43× while the mean is 2.98×. How do we interpret this?

Mean and median mean more than two ways of averaging numbers

When we are given a set of numbers, the mean is calculated by adding all the values and dividing by how many values were in the set. The median is found by arranging the numbers from smallest to largest and selecting the number in the middle. In probability, however, these two values carry a more important meaning.

The median is the point where the probability from the leftmost value to the median totals 50%, and the probability from the median to infinity also totals 50%. Essentially, the median tells us what point splits the probability in half. The mean, on the other hand, is the probability-weighted average of all possible outcomes, which actually translates to, well, the average return. The difference between the mean and the median very much reflects the investment world. Most investments will fall short of the mean MOIC because the mean return is driven by the precious outliers. We may draw 0.05×, 0.5×, and 0.8×, but one time we hit 20× and that one outcome greatly changes the average.

This also shows how brittle the nature of venture investing can be, because so much depends on an improbable outlier. A big fund can allow a few not-great investment batches because it has more opportunities for a rare winner to appear. But small funds such as ours don't have this liberty. That uncertainty is precisely something we model to aid in our decision making at Element 14, and we will cover it in subsequent posts.

Is pure power law actually a good descriptor?

Some of you reading this might have already noticed that something is not quite right with the Lomax plots. Yes, they do capture the 80/20 rule, but the way they handle losses doesn't match reality. The Lomax spreads failing investments smoothly across small positive returns like 0.2× or 0.6×, whereas if a company is struggling, a complete shutdown is often more likely than a tiny positive exit, especially in the Silicon Valley environment. The "go big or go home" mindset is very much real.

Single probability-distribution functions, such as the Gaussian and Lomax, have neat formulas for both the distributions themselves and their respective integrals — which is convenient, but we should model returns in a more nuanced way, such as the one shown in Figure 3. This is our "mix" distribution, which is a mix of a continuous function with a discrete probability for failed, 0× investments.

Figure 3Model output

Show the underlying values

Component	Value
Probability at exactly 0×	67%
Probability of a positive return	33%
Mean	2.98×
Median	0× (more than half of investments fail)
Smallest positive return	2.5×
Cap on the positive component	100×

Instead of using one smooth Lomax curve for the entire range of possible returns, we split the distribution into two parts. In our current model, roughly two-thirds of the probability is placed at exactly 0×, representing complete or near-complete failure, while the remaining third is described by a continuous distribution of positive returns. Now, we are aware that a failed investment does not always result in a complete zero, but this is a practical approximation of the "go big or go home" behavior, especially in the kind of pre-seed investing we do.