Convexity and Concavity in Investing, Life and Decision Making

Risk is one of those things that took me a hell of a long time to understand.

It’s not intuitive, mostly non linear, has exponential properties and it just hurts our monkey brains to comprehend. 🐒

I’ve spend the majority of my adult life overly focused on rewards and payouts neglecting risk and ruin.

This has caused me to make endless dumb decisions, bad investments and poor life choices.

Call this behaviour whatever you want (greed, ambition or arrogance) but the end result is the same.

We are all trying to maximize the good things and avoid the bad things in life.

The problem is we don’t have a full picture of what is going on in the world around us. It’s a pretty messy place and we deal with massive information asymmetries on a daily basis.

So how can we get even with probability while optimizing for luck and mitigating risk?

One answer is understanding the difference between convexity and concavity.

At first glance this may seems overly simplistic but I think it provides a balanced framework for educated decision making. This is also vitally important when making highly leveraged decisions.

Let’s look at the differences between the two and get into some examples.

Convex and Concave: A Simple Definition

The basic difference between something convex and something concave is pretty simple.

The words are more often used in science and mathematics than investment strategies, where they describe the shape of an object:

  • Convex is used to describe a shape that curves inward, like a sideview mirror of a car. 
  • Concave is the opposite: a shape that curves outward. An American football is a good example of this.

For you visual people out there:

Both convex and concave shapes are commonly used in engineering, and most often in the design of lenses. Both types of shape will bend light in a different way, and allow it to be focused on the sensor of a camera or other apparatus.

There is an easy way to remember the difference. A concave shape curves inward like the mouth of a cave, and a convex shape is the opposite.

Convexity and Concavity in Investing

The difference between the two shapes might seem like an abstract discussion, but it has important consequences when it comes to life and investing decisions. In particular, the difference between these two types of shape is very important when it comes to analyzing risk.

The most famous application of these concepts to investment risks has been outlined in the book Antifragile: Things That Gain From Disorder, by Nassim Nicholas Taleb. This book is complex, long, and often infuriating, but Taleb’s basic ideas are easy to understand. (see my Fooled by Randomness Notes)

He argues that the return or loss on investments can be seen as varying according to a central variable. An example of this might be a simple stock investment, where an investor will make a gain from an increase in the price of a stock.

How much they gain or lose as the price varies can be plotted on a simple graph, and this graph will show either a convex or a cancave shape.

A graph with a convex shape will look like this:

As you can see, the shape of the curve on this graph curves inward, making it convex.

This means that the investment here will benefit from uncertainty. If the price of the stock goes up (here called “variable X”), the investor stands to gain significantly. If, on the other hand, the price goes down, the loss made will not be as great.

This is why hedging your bets is almost always a good move.

Taleb’s general point is that our world is characterized by uncertainty, and that investment strategies should be designed with this in mind. 

Any investment is, of course, based on a prediction that the price of a particular stock or commodity will increase: otherwise it would not be made.

However, Taleb argues, investors should also realize that even the best predictions are never perfect. They should therefore prepare themselves for volatility by making their investments convex.

To see why, it’s instructive to consider the opposite kind of strategy: one that would appear concave when plotted on the same graph above.

The loss / gain profile of such a strategy, given the same market volatility, would be opposite to that of a convex risk profile. If the price of a stock goes up, an investor will make gains, but these will be smaller than the losses incurred by a similar-sized reduction in the same variable.

Convexity and Antifragility

Though Taleb’s ideas are built on the concepts of convexity and concavity, he doesn’t use these terms after giving a basic definition of the idea. Instead, he uses the term “antifragility”, from which he also draws the title of his book.

“Antifragility” is an extension of the idea of concavity. Taleb’s central argument is that most banks and large financial institutions have taken for granted that market fluctations will hurt them, and so continue to implement concave strategies. 

In these strategies, a reduction in a particular variable – say the stock price of a large bank – will incure losses for investors. That’s to be expected, of course. But Taleb points out that in concave strategies, these losses do not vary linearly against variation in the given variable. In other words, if a stock price drops 5%, it may lead to losses of 10% for investors. 

This second point is really important, due to leveraged concave bets the risks are compounded by many orders of magnitude. The risk of ruin is essentially amplified when you use leverage.

This, argues Taleb, is because a small number of large banks are responsible for most of the world economy, and they themselves are fragile. This means that the world economy, as a whole, is also fragile, and does not respond well to uncertainty. 

The best approach for investors, then, is the opposite: to make investments that are convex, and therefore offer increasing returns (or at least decreasing losses) in times of market instability. To be “antifragile”, in Taleb’s terminology.

Real-Life Examples of Convexity and Concavity

If you are not running an investment bank, however, it can be difficult to see how the concepts of convexity and concavity apply to you. So let’s look at two everyday examples of them: one drawn from everyday life, and one that relates to investing. 

First, let’s explain how we can apply the concepts to an everyday situation. If you have just one job, then your income will likely vary according to how many hours you work: in other words, the type of graphs we saw above would be a straight line. So let’s consider someone with two jobs, someone who – for instance – is an aspiring professional Baskball player, but also gives Basketball lessons to children.

This person would be, in Taleb’s terms, “antifragile”. That’s because their overall income will be a curve when plotted against their uncertain future. Kids will always need Basketball lessons, so the income from this job will steadily decrease if our teacher works fewer hours. But let’s consider how their other job will effect their income. If they land a contract with a major team, their income will increase dramatically. If they don’t ever make it as a professional player, they have lost nothing. The curve of their income will be convex, with the potential losses from uncertainty much smaller than the potential gains.

Another example of a similar idea, this time drawn from investing, is that of “Trend-Following Strategies”. These are investment strategies that are conciously designed to take advantage of market fluctuations. Instead of seeking a large, repeated dividend from investments, these investors are content to see low annual gains in pursuit of one-off payday.

The most extreme examples of these strategies take the concept of convexity even further. Some are designed so that even drops in an underlying variable will result in profit. During times of market stability, these strategies will make small annual losses, but any volatility will result in profits. 

Should I be Convex or Concave?

It will come as no surprise, if you’ve read this far, that there is a simple answer to this question: you should be convex. 

That might sound obvious, but it is counter-intuitive to most people. Most of us like to believe that we live in a predictable, stable world, whether it comes to our everyday lives or our investment decisions. Accordingly, we naturally think that it is better to take a reliable monthly paycheck over a one-off payment in the event of market fluctuations.

But if the past decade – one that contained the 2008 crash and now the Covid-19 pandemic – has taught us anything, it’s that we do not live in a stable world. About the only certainty we can have, in other words, is ongoing uncertainty.

This is the realization that convex investment and life decisions are based on, and why you should try to become convex in everything you do.

How To Become Convex

The best way of becoming convex will depend on the field in which you work, but will always rest on one key principle: diversity.

As we saw with the example of the Basketball coach above, is to diversify your income streams. No strategy – whether it be for life or investment – can be convex if it responds in a straightforward way to just one variable. In life, this means ensuring that you have side-projects that might, one day, pay off big time. 

In designing an investment strategy, achieving convexity means something a little more complex. Instead of going for just a diversity of investments, you should also think about how they effect each other. Ideally, you need to design a strategy in which falls in price of one investment lead to larger gains for your other investments.

Summing Up

In short, we should all recognize that we live in an uncertain, risky world, and design their strategies to reflect this. Whether you call this strategy “convex”, “antifragile”, or something else, the basic principle is the same: market volatility should increase your profits more than it reduces them.

By becoming convex, the impact of risk on your investments will be hugely reduced. In fact, it might be that you come to embrace risk as an inherent part of your strategy, rather than living in constant fear of market fluctuations.