Convexity and Concavity
For a convex payoff over a specified range, equal upward and downward changes can yield more gain than loss; a concave payoff has the reverse pattern. Taleb connects these responses to antifragility and fragility. The variable, range and direction of benefit matter when applying the idea.
THE IDEA
When to Smile and When to Frown
A nonlinear response can be convex, concave or mixed, with different shapes over different ranges.
Figures 10 and 11 show the following simplifications of nonlinearity: the convex and the concave resemble a smile and a frown, respectively.
FIGURE 10. The two types of nonlinearities, the convex (left) and the concave (right). The convex graph is smile-shaped; the concave graph is frown-shaped.
FIGURE 11. Smile! A better way to understand convexity and concavity. The convex graph looks like a smile; the concave graph looks like a frown. The convex (left) is antifragile, the concave (right) is fragile (has negative convexity effects).
I use the term “convexity effect” for both, in order to simplify the vocabulary, saying “positive convexity effects” and “negative convexity effects.”
Why does asymmetry map to convexity or concavity? Simply, if for a given variation you have more upside than downside and you draw the curve, it will be convex; the opposite for the concave. Figure 12 shows the asymmetry reexpressed in terms of nonlinearities. It also shows the magical effect of mathematics that allowed us to treat steak tartare, entrepreneurship, and financial risk in the same breath: the convex graph turns into concave when one simply puts a minus sign in front of it. For instance, Fat Tony had the exact opposite payoff than, say, a bank or financial institution in a certain transaction: he made a buck whenever they lost one, and vice versa. The profits and losses are mirror images of each other at the end of the day, except that one is the minus sign times the other.
Figure 12 also shows why the convex likes volatility. If you earn more than you lose from fluctuations, you want a lot of fluctuations.
FIGURE 12. Pain More than Gain, or Gain More than Pain. Assume you start from the “You Are Here” spot. In the first case, should the variable x increase, i.e., move to the right on the horizontal axis, the gains (vertical axis) are larger than the losses encountered by moving left, i.e., an equivalent decrease in the variable x. The graph illustrates how positive asymmetry (first graph) turns into convex, smile-shaped curving and negative asymmetry (second graph) turns into concave, frown-shaped curving. To repeat, for a set deviation in a variable, in equivalent amounts in both directions, the convex gains more than it loses, and the reverse for the concave.
When to use it
Plotting a venture’s gains and losses across plausible outcomes before deciding whether variability would help.
Comparing a lender’s credit payoff with an option holder’s payoff as the underlying asset value changes.
Examining a commission schedule: a flat percentage is linear, while tiers, caps or bonuses can change the shape of the payoff.
Examining whether a research program limits the cost of failed trials while retaining gains from successful ones.
FROM THEORY TO EVERYDAY LIFE
In practice
Entrepreneur and Lender Payoffs
In a simplified model, an entrepreneur with limited downside and a claim on large gains may have a convex payoff over a range of outcomes. A lender’s payoff as the borrower’s assets increase is capped at repayment, while losses arise below that level. The comparison concerns payoff curves, not a claim that every entrepreneur benefits from every kind of volatility.
Fat Tony vs. Financial Institution
As described in the text, Fat Tony’s transactions are mirror images of those of a bank. When the bank loses money, Fat Tony gains, illustrating how the sign reversal of a convex payoff turns it concave (or vice versa).
A MOMENT TO REFLECT
Put it to work.
Think of a decision you face where the payoff is asymmetric. Is it convex (more upside potential than downside) or concave (more downside than upside)? How does this affect your willingness to take risks?
What variable would you put on the horizontal axis to test whether a startup’s or bank’s payoff is convex? Over what range would the model apply?
Consider an investment you made. Was the payoff curve convex or concave? How did volatility impact the outcome?
Often confused with
References
- Antifragile: Things That Gain from Disorder — Nassim Nicholas Taleb (2012)
CONTINUE THE CONVERSATION
Source texts
- Antifragile (Nassim Nicholas Taleb)