Last week I traveled internationally in Business class with my wife and three year son on Turkish airlines. It was by far one of the best flight journeys for us. I typically enjoy flights anyway, but this was so much more comfortable and fun. The food was amazing, the service great, and the access to the lounges: wow I hadn’t really known about this. There was so much good food everywhere.

Before you judge me for showing off, let me tell you that I didn’t pay for this trip. I had used my Amex points to buy the tickets. Between the 100k points I got when I signed up, years of overspending on the credit card, and using it to buy several domestic flights (which gave 5x points per dollar spent), I had accumulated a little over 300k points. The flight costed me 100k points per passenger (one way). So I spent $0 overall. If I had to buy them, it would have costed me between 11k to 12k USD. I don’t fancy spending this much for a 24 hour long trip. Of course, I bought return tickets in the cattle class (because I didn’t have enough points for buying return tickets).

So why am I telling you this? Because the ROI here is insane: only a few hours of research in picking the Amex card a few years back paid off quite substantially. The asymmetry between effort and reward in this case is stunning.

In some sense, this is a like a life hack. A life hack is something where a small change produces some neat result. In another sense, this is like a leverage point. You put in a little bit of effort but can go half way around the world in top-notch comfort.

But I have a better name for the outcome I experienced: convexity.

To understand convexity, let’s understand linearity. If I am a freelancer and make $20/hr, I will make $20 if I put in one hour, and $40 if I put in two hours. The amount I make is always linear to how much time I put in.

Let’s revisit the above example of accumulating points and using them. If I had spent only half an hour learning about the basics, I would have likely not benefitted much. But after spending a bit more (let’s say a sum total of two hours), the benefit I got is lot more: it is not 4x.

In fact, if I spend some more time here, I could optimize more and unlock a lot more points. I have a friend who actually does this. He maintains a spreadsheet and spends a little time here and there, but he seems to have accumulated quite a lot of points. Moreover, he blogs about this online and that brings another source of revenue for him. Of course, if I spend all my time doing only this, the benefits will plateau.

We can therefore draw a curve depicting the reward vs. effort making some reasonable assumptions (and juxtapose linearity as well:

The precise mathematical definition of convexity is not important here. An example of convexity is the exponential function, which we are all intimately familiar with by now, thanks to Covid19.

With this image, it becomes clearer that lifehacks and leverages are not quite the same. They concern only a specific point, such that reward vs. effort ratio is bigger than 1. A straight line whose slope is greater than 1 (and passes through zero) will always have a good leverage, but it is not convex.

It is of course useful to find out things that give you a lot of leverage, but convexity is even better, and one of the best convex functions that can be attained in real life is exponential functions. Let’s look at a few examples of convexity.

#### Examples of convexities.

**Reading books**: Reading books already unlocks a lot of value. But doing it smarter is convex. 10% more time spent can produce 2x the value. I wrote about this, but not in terms of convexity, in my previous post on How to Read Books.

**Investing**: Investing (smartly) for the long-term using index funds is convex thanks to the exponential nature of compounding. Things start to move much faster after a while, which is why time is your best friend. This is why you will see examples where people who started much earlier can come out way ahead as compared to those who started later even if they put in a lot more money. Unfortunately for me, I didn’t learn about investing in my 20s (I want to write about this — in the context of privilege and knowledge passed down via generations).

**Expertise**: By being a little better, it is possible to make a lot more money. I showed the following curve in my previous Curves of Competence post.

You can see that the value produced follows a convex curve.

#### Why does it matter?

Convexity helps you figure out how to invest your time (and resources) and on what. But if you look closely, there is some stuff happening in the convex curve. The progress may look pitiful initially (see the flatter region above at the beginning), and some people may give up at that stage. Especially when you compare against something else that is linear where you see steady reward at the beginning.

My favorite example of this is when a stone-mason strikes a stone 99 times and it doesn’t budge but splits at the 100th time. Would you give up before the 100th try? And did the first 99 strikes help?

So what are some examples of convexities that you have experienced in your life?

In fact, this was not my first experience with award tickets — I was able to travel once before (albeit coach). At that time, what made me so amazed and somewhat confused was the fact that such an opportunity was out there in plain sight but I hadn’t known about it earlier. Once I came to know about it, I was able to do more research online and benefit from it. A single line from my friend “check out flyer talk” opened a world of opportunity which helped me get a free international flight (round trip no less). It was as if gold coins were on the ground, and my friend pointed me roughly in the right direction.

That left me wondering, what other kinds of such knowledge is out there with convex rewards, but that I am not aware of. And why did my friend know about it but not me. I don’t have good answers right now, but I will reflect on this when I am jetlagged and write about it later!

So yeah I am always on the lookout for convexities. Please share what you know of!