### Convexity Made Simple

Convexity is a concept that all traders and investors must understand. While it can be complex, it isn’t difficult to grasp; at its core, convexity is simply an expression of non-linearity. What makes it so important is that it applies *outside* of finance, to life itself – as a philosophy of making small sacrifices for large potential returns.

### Convexity In Simple Terms: Think Of A Moving Car

What is convexity?

Let’s begin by establishing the technical definition of convexity. Mathematically, convexity is the second derivative of a curve, that is, the rate of change *of* the rate of change.

Putting this in a real life context, think about a car moving at a velocity of 60 km/h. Its velocity is how fast it is moving, which is the *rate of change* *of its position*. Now, the car starts to accelerate, which means that its velocity is changing. Therefore, acceleration is the *rate of change* of velocity, or the *rate of change* of the *rate of change of its position. *

Consider this graphically. If the car accelerates at a constant rate, the graph is an upward sloping line, and if the car accelerates at an increasing rate, the graph is an upward sloping *curve*. Finally, if the car stays at the same speed of 60 km/h, it has an acceleration of 0, and the graph is just a flat line.

If we were to calculate the second derivatives at these 3 different parts of the graph, we would get different results. When acceleration is constant and the graph is an upward sloping *line*, the second derivative is greater than 0, but it is the same value as long as the line slopes at the *same angle*.

When acceleration is increasing and the graph is an upward sloping *curve,* the value of the second derivative changes; the steeper the curve, the higher its value, and vice versa. Lastly, when acceleration is 0 and the graph is a flat line, the second derivative has a value of 0.

Now, focus on the steepest part of the curve, a.k.a the most convex part of the curve. You can observe that a *small change* in time corresponds with a *large change* in the car’s position. This *asymmetric relationship* is the power of convexity. * *

For bonds, this translates into a small change in interest rates leading to a large change in the bond’s price, but only at the convex parts of the bond’s curve.

From the diagram above, you can see that Bond 1 has a more convex curve than Bond 2 (Bond 2’s curve is noticeably flatter). For a 1% change in interest rates (yield), the price of Bond 1 changes by a considerably larger amount than Bond 2, which is clearly illustrated by the difference in size between the purple and orange rectangles.

Moreover, how much a bond’s price changes when interest rates change depends on which part of the curve the change occurs in. When the change occurs in areas where the curve is less convex, price doesn’t change as much. Conversely, when the change occurs in areas where the curve is *more* convex, the bond’s price moves a lot more.

It really is the same phenomena as with the accelerating car, just with different parameters.

*To be continued…*

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### Convexity In Simple Terms: Think Of Asymmetric Butterflies

Therefore, technically defined, a bond’s convexity is the rate of change of its *duration*. If we were to unpack that russian doll of a statement, we would get: Convexity is the rate of change *of* the rate of change of its price with respect to a change in interest rates.

Which is more than a mouthful, and if that statement isn’t helpful to you, then perhaps this one will be: Convexity is how quickly a bond’s sensitivity to interest rates changes.

Understanding this will help you to see what is actually going on in a bond’s price when interest rates change. If we are in the convex part of the curve, we know that duration will change very quickly when interest rates change (this is what having high convexity means). This means that as interest rates change, the bond’s price will become *even more *sensitive to movements in interest rates, the result of which is *larger* changes in the bond’s price!

Hence, convexity is what is responsible for large asymmetric price moves when markets and risk appetites shift, which makes it *extremely important* for traders/investors to understand.

At this point, it is important to draw the distinction between the asymmetric nature of convexity and nonlinear dynamics in complex systems. While convexity, when described as a small change in one variable leading to a large change in another, can sound very similar to the butterfly effect, they are not exactly the same.

The difference between both really is one of scale. Change, in the context of convexity and complex systems, can lead to nonlinear and asymmetric outcomes in both, *but*, convexity is often contained to individual entities, not entire systems. Bond prices are a good example of this, a small change in interest rates that leads to a large change in a bond’s price only affects the bond’s price. On the other hand, small changes in one variable of a complex system can cause feedback loops across many different, but interrelated parts in the same system.

However, the asymmetric change in the bond’s price *might *go on to cause a cascading feedback loop through the rest of the financial system, simply because of how interconnected components of the financial system are. The Great Reflation of 2021 is a good example of this as the selloff in mainly the 10 and 30 year part of the UST curve went on to spark rallies/sell-offs in other asset classes.

Ultimately, convexity really is an expression of nonlinearity, which in turn can be observed on the micro, individual level, as well as the macro, system-wide one.

*To be continued…*

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### Convexity In Simple Terms: How It Applies To Life

Convexity is much more than a mathematical/financial concept, it is also a way of living, a philosophy, if you will. Understanding what it is and how it applies is not just crucial for traders, but for everyday life as well, where small changes that are *made* or *not made* can lead to much larger consequences.

Interestingly, the concept of convexity isn’t foreign to most of us – we just do not see it in terms of asymmetric outcomes. Consider these aphorisms that are often delivered as grandmotherly wisdom; “Short term pain, long term gain”, and “Penny wise pound foolish”.

Both of these sayings reference gains that outweigh costs. In the former, one is encouraged to put up with short term discomfort in order to reap longer term rewards. For example, to endure the short term unpleasantness of exercise in order to reap the benefits of a stronger and more healthy body (and mind) in the future. In other words, small changes that lead to a big effect.

The latter is slightly different, in that it infers, rather than directly references a time frame for costs and payoffs. The saying is often directed at those who look to save small amounts of money on some purchases, then going on to splurge on others. A good example of this is someone who always looks to save a few dollars (or even cents) on the food they buy, but often splurges on luxury items.

There is no way that the tiny amounts saved from scrimping on food add up to the price of even one such luxury item; which is the point of the aphorism. Saving small amounts will not help one’s *future* financial security if one spends irresponsibly. If a person really wants to secure their financial future via saving, then saving on big ticket items matters more than scrimping on small ones.

But how does this relate to convexity? By telling us to focus on the gains that matter (*not* splurging large amounts unnecessarily), and not the small savings that don’t.

Another way to think about convexity in life is in terms of *paying* or *not paying* a small fee. Someone who *pays* the small fee is **long** convexity. Others who choose to *not pay* the small fee are **short** convexity.

Take flossing one’s teeth for example. The payment here is the couple of minutes a night it takes to perform the teeth cleaning exercise. People who floss are paying this couple of minutes, and people who *do not* floss are receiving this couple of minutes (they get 2 more minutes to do something else).

But, what is the payment for? Prevention of cavities, and other tooth/gum diseases, of course. The cost of two minutes + dental floss every day is ultimately very cheap compared to the much more expensive cost of filling cavities, not to mention much less physically painful.

If there were to be a pithy slogan for flossing that also demonstrates the power of convexity, it would be something along the lines of, 2 minutes a day keeps the drilling at bay! If that sounds familiar, it’s because it is closely related to everyone’s favorite adage, “An apple a day keeps the doctor away!” Notice the convexity lurking behind the words here as well – the small cost of an apple to avoid the much larger cost of ill health.

The point here isn’t about flossing or one’s health, although both are important, but that a small payment is regularly made in order to avoid a much more expensive one later. Again, small changes that contribute to future asymmetric outcomes.

How are *you* positioned against convexity?

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### Focus On Convexity: Trading On The Short Side

The financial version of apples and doctors (or flossing and dentists) is the darkly humorous “Picking pennies in front of a steamroller.”

But how does the concept of convexity apply in trading?

Just like in life, convexity is about the relationship between small payments and asymmetric outcomes.

However, in finance, being short convexity can actually earn one an income. This applies specifically to options trading, where traders who sell options receive the option premium upfront.

By selling far out of the money contracts, that is options that have a very high probability of expiring out of the money and thus worthless, a trader can earn a consistent, albeit relatively small income.

This sounds like a dream to most people, especially inexperienced or new traders/investors.

However, a strategy that is overly focused on selling options is a very dangerous strategy – akin to not flossing one’s teeth.

Why?

Simply because financial history has shown that rare events, which lead to massive disruption in markets, occur a lot more frequently than current financial statistics account for.

This means that by consistently selling far out of the money options, a trader is also consistently exposing themselves to such massive disruptions.

Typically, when such rare events occur, losses incurred by people who were selling options tend to *far *outweigh whatever gains that they had made.

A good example of this would be the large number of traders who were selling US equity index options going into March 2020 (specifically puts), just before global lockdowns. These people were “short vol”, which means they were short volatility, and is another way of describing selling options.

Since an option’s price increases when volatility increases, selling an option is taking the bet that volatility will *not* increase by much in the future; hence the trader is in effect *shorting volatility*.

As we know, global markets grew extremely volatile extremely quickly, wiping out anyone who was short vol. In other words, they were *flattened by the proverbial steamroller* because they were busy picking up pennies by selling puts.

It is important to note that selling options is not the only short convexity strategy used in trading the financial markets.

Arbitrage strategies (e.g. merger-arb, index-arb) tend to be short convexity as well, along with any other strategy that, in general, exposes a trader to potential losses that are much larger than potential gains.

All these strategies run the (high) risk of being flattened by the steamroller, which is of course, a representation of the second part of the convexity deal – asymmetric outcomes.

Unfortunately, people tend to forget about this once they see the possibility of earning consistent returns. This inability to respect, and/or recognize the nature of convexity in markets could be due to a combination of psychological and statistical blindness.

Psychological because the vast majority of humans seem to be wired in such a way as to emotionally prefer consistent returns.

They simply make us feel better, and in seeking the emotional security of feeling better, we expose ourselves to negative asymmetric outcomes.

Statistically, the same inability to recognize that rare events really are *not* that rare in financial markets contributes to our failure to factor convexity into our thinking. This is primarily due to the ubiquitous misperception that markets (and life) follow a normal distribution, a.k.a, the bell curve.

Unfortunately for those who learn, believe, and eventually come to model their perspectives on it, markets (and life) are *not* normally distributed.

Instead, they follow Pareto, or more generally, Power Law distributions, where rare events are much less rare than in bell curves.

Given that this is the reality in which we live, would it not be wiser to adapt to convexity, rather than be crushed by it?

*To be continued…*

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### Focus On Convexity: Trading On The Long Side

If a trader being short convexity runs the risk of getting flattened by the proverbial steamroller, what about traders who are long convexity?

How do they fit into the penny-picking-steamrolling aphorism?

First of all, we need to understand what being long convexity in trading actually means.

Long convexity traders are those who make money from the low probability events that inevitably occur in markets.

A good example of this would be the *opposite* of the option seller described in Part 1, that is traders who *purchase* out of the money options (often far out of the money). They do so in anticipation of markets making a big directional move, netting them a handsome profit in the process.

In other words, they are *long* *volatility *(long vol), and hence also long convexity. Metaphorically, these traders are the ones trying to jump aboard the steamroller as it flattens everything in its path.

Being flattened by a steamroller brings to mind financial disasters such as the Great Financial Crisis in 2008, or the massive global selloff in markets during March/April of Pandemic 2020.

However, it is important to note that the low probability events that long convexity traders profit from need not be so dramatic.

For instance, CTA’s (trend followers) are generally long convexity traders, and the low probability events that they trade are markets entering/exiting trends.

While this may sound strange, markets actually spend most of their time trading sideways (70% of the time, according to market lore). Hence, a market starting to trend is actually a low probability event, and if traded properly, a great source of profits over time.

The problem with being long convexity in financial markets is that one might go bankrupt before the “long term gain” comes along. This is often referred to in financial circles, again in a darkly humorous way, as “death by a thousand cuts”.

But what exactly are these cuts, and how do they come about?

The exact nature of how the cuts are made differ by strategy.

For CTAs, they come about from being stopped out multiple times in choppy, sideways markets that refuse to trend for a sustained period of time.

For traders who go long convexity via purchasing options, the cuts come from their options expiring worthless as they patiently wait for volatility to explode.

Subsequently, “Death” comes about when long convexity traders lose all their money before the low probability event comes about. Since their trading accounts deplete slowly over time, with small loss after small loss, rather than all at once as happens with traders caught being short convexity, the process is referred to as a “thousand cuts”.

Of course, this begs the question: If being short convexity leads to being flattened by a steamroller, and being long convexity to a slow death by bleeding, aren’t both strategies equally unviable?

*To be concluded…*

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