Have you already crossed your inflection point?

Perhaps some of you still remember the days when you sat on school benches studying mathematics to your heart's content. If this isn't your favorite subject, don't worry, I'll be very brief on the subject. My aim is not to demonstrate a theorem to you, but just to see how the concept of the inflection point can be applied to our own lives, particularly as investors and/or future annuitants.

Let's briefly recall what an inflection point is, according to Wikipedia: if, at a point on the representative curve of a continuous function, the concavity changes from "convex" to "concave" (or vice versa), we call this point the "inflection point of the curve". Since pictures are always better than words, here's what it looks like in concrete terms:

I'm almost done with the technical explanations, just a little more effort. The second derivative indicates the variation in slope :

• if it is positive on an interval, the slope increases, the curvature is upward, the function is said to be "convex" on this interval (the blue "valley" above);
• if it is negative on an interval, the slope decreases, the curvature is downward, the function is said to be "concave" on this interval (the "red" mountain above) ;
• if the second derivative cancels and changes sign, we have a inflection pointthe curvature of the curve reverses.

I think so far, despite the rather barbaric terms, you're following me. But you're probably wondering where I'm going with this. Our lives are made up of a succession of crises and rebirths, just like that of the financial markets. We think we're at the top, and then we fall. Conversely, we see no way out, and suddenly the wheel turns.

The most interesting thing about this is that during the climb, as during the fall, there's an obscure force at work here which gently but surely pushes us in the other direction. The inflection point corresponds to the exact moment when the second derivative changes sign and this force begins to unfold its effects. And yet, even as it does so, we keep going down, or up.

To take a very concrete example, let's imagine you're on the road with your car. You're driving at full speed and you let go of the gas pedal. The number of meters you cover remains significant at first, but gradually your car decelerates and you start to be overtaken by other cars. At some point, you decide to put your foot back on the gas pedal. For the first few seconds, you don't notice much, as the weight of the car and the resistance of the air prevent you from picking up speed. Gradually, you start swallowing up the kilometers again, passing the cars that had overtaken you.

The example of acceleration is a good illustration of the force that takes you to new horizons. In physics, acceleration is the second derivative of distance, and force is equal to mass times acceleration.

As an investor and/or future annuitant, you have capital that you have invested wisely, you receive dividends and/or you save a greater or lesser proportion of your income. If so, you've already passed that inflection point. The force of compound interest will gradually lead you to other, more clement climes. But this force, as in the example of the gas pedal, takes time to produce its effects. During this period, you may feel like you're stagnating, or even regressing.

Unfortunately, many investors give up too soon, disappointed by losses or results that never materialize. However, if the framework conditions have been set, the inflection point has been passed and the wind is already blowing in the sails of the boat that will take you to a new life. Sailors know all about this phenomenon. It takes a long time for the boat to get underway and change course, due to its considerable inertia. Conversely, once it's underway, it's equally difficult to stop or reverse course.

When you've put everything in place to reach your goal, when your savings rate is high and your dividends are growing steadily, even if it feels like you're stagnating or regressing, stay the course. Soon your efforts will pay off, perhaps even beyond your expectations.

So, have you already passed your inflection point?