How Jensen’s Inequality Can Help You Avoid Hidden Investment Risks

During their recent episode, Taylor, Carlisle, and Brendan Hughes discussed How Jensen’s Inequality Can Help You Avoid Hidden Investment Risks. Here’s an excerpt from the episode:

Tobias: JT, top of the hour. Do you want to do– give the people what they– [unintelligible 00:30:45]?

Jake: There are dozens. All right, this week, we’re going to be talking about this mathematical concept called Jensen’s inequality. Any familiarity with that, to start?

Brendan: No.

Tobias: Yeah, I know the name, but I can’t for the life of me think what it is.

Jake: That’s a different Jensen, no. It’s this fundamental concept in math, and particularly in convex analysis and probability theory. And convexity, just to make it more simple, is like the shape of the graph, if it curves up in a nonlinear way, it keeps getting steeper. And Jensen provides this insight into how convex or concave functions behave when applied to the expected value of random variables. It’s like, here’s the curve. Let’s try to guess what we’re going to get from that statistically. And it has broad applications across economics, statistics, finance, machine learning, physics. And it really explains how nonlinear functions interact with averages and expectations and gives us some useful tools, especially around risk. So, who is this mysterious Jensen? His full name, and this is quite a mouthful, is Johan Ludwig William Valdemar Jensen, born 1859, died–

Tobias: Is it really Voldemort?

Jake: It’s not Voldemort, but it’s V-A-L-D-E-M-A-R. Died 1925. He was a Danish mathematician. And despite his significant contributions, he’s somewhat of a lesser-known figure. And really unlike a lot of prominent mathematicians, he didn’t spend his life in academia. His primary occupation was an engineer at the Copenhagen Telephone Company. So, in a lot of ways, he was kind of Claude Shannon, before Claude Shannon at Bell Labs.

And so, he worked in telephony and communications in the early 20th century. And you really had to have a deep understanding of electrical engineering, signal processing, a lot of these places where calculus and numbers theory actually played a crucial role. He was this engineer by day and mathematician by night, and kind of makes him an intriguing figure in history. And that’s kind of been ignored largely so. And there’s really not very much recorded about his personal life and hobbies and relationships. So, I don’t have anything there.

But let’s get into some practical examples that kind of illustrate what’s going on with Jensen’s inequality. And imagine that you’ve got two routes that you can go for a road trip, and one is this scenic mountain pass. It’s got steep curves, unpredictable weather, or there’s a really long, flat, boring highway that gets you there. And both routes take the same time to get from point A to point B. Here’s the catch though. One of them has these wild up and downs, the other one smooth. And what Jensen’s inequality says is that when you’re dealing with this curvy path, like in math terms, like a convex function, the average outcome from all the twists and turns is always more extreme than just taking the average point directly.

So, if you’re driving through these curvy mountains, your fuel efficiency, your stress levels are probably going to be all over the place, and you’re perhaps better off choosing the highway if you want this predictable ride. And when things are curvy or convex, the averages of the wild ride will be much more extreme than you would otherwise guess. There’s a mathematical proof of that I won’t get into, because then we’d really be down in the weeds.

Let’s take another example that might help, let’s say food portions. Say you’re at a restaurant and you have two options for the amount of food that you’ll get. One is a fixed 500 calories, let’s call it every time you order. Or you can order this variable portion where sometimes you’re going to get 300, sometimes you’re going to get 700, the average is still 500, okay?

So, according to Jensen’s inequality, even though the average is 500, your satisfaction is going to be lower with the variable one, because some days you’re left feeling hungry with the 300 calories. Other days, you’re too full. You don’t even want to eat it all with the 700. So, the overall experience is worse than just getting the 500 consistency. I’ll leave out kind of hormetic stressor and all of that stuff out of the out of the equation at this point.

But let’s take a last example, study time. You’re preparing for a big exam. There’s two study plans that you can choose from. Plan A is you study consistently for 2 hours every single day. Plan B is you study for 6 hours on some days, goof around the other days. Which one do you think is probably going to get you to a better outcome? It’s probably the more consistent one, more will stick. That irregular study pattern in plan B is going to be more stressful, less retention of information, even if the total amount of study time was the same between the two.

Now, let’s see if we can get back to finance and stick the landing. The world of investing is an absolute playground for Jensen’s inequality. It’s all nonlinearity, reinforcing feedback loops, convexity, it’s everywhere. And let’s say, typically, sometimes you can choose between investing in let’s say a stock that has a lot of potential gain, but also major downside. Well, if you use Jensen’s inequality, when you average out all these ups and downs, the overall return is likely to be lower than if you’d chosen a safer, more predictable route. Because this convexity doesn’t play nice with averages. Anyhow, we’ve covered this kind of before when we’ve talked about ergodicity and variance strain and the difference between different types of averages if you guys remember that.

Basically, if you use convex functions at all, like risk measurements, and you’re assessing these potential losses from different outcomes, Jensen’s inequality tells you it’s generally riskier to average the outcomes first and then apply a risk function rather than calculating the risk of each outcome and then averaging them afterwards. So, the order of operation matters. Basically, anytime you’re dealing with anything curvy, we have to be especially careful when taking averages. Otherwise, you’re inviting errors structurally, mathematically. And the more unpredictable the situation, the more extreme the outcomes, the fatter the tails, good or bad, the more that Jensen’s inequality is going to influence your final result.

Tobias: That’s interesting, JT. In terms of just– one more time, how we apply that. If you have fat tails, what’s the approach to that?

Jake: Taking the average of any kind of thing with fat tails is going to–

Tobias: What is that average?

Jake: It’ll systematically underrepresent the actual risk that you’re taking.

Tobias: That’s interesting.

Jake: Right. And I think this is probably where—like, LTCM probably ran into this.

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