A Map and Simple Heuristic to Detect Fragility, Antifragility, and Model Error

The main results are 1) definition of fragility, antifragility and model error (and biases) from missed nonlinearities and 2) detection of these using a single “fast-and-frugal”, model-free, probability free heuristic. We provide an expression of fragility and antifragility as negative or positive sensitivity to convexity effects, i.e., dispersion and volatility (a variant of negative or positive “vega”) beyond Jensen’s Inequality, across domains, and show similarities to model errors coming from missing hidden convexities -model errors treated as left or right skewed random variables. Broadening and formalizing the methods of Dynamic Hedging, Taleb (1997), we present the effect of nonlinear transformation (convex, concave, mixed) of a random variable with applications ranging from exposure to error, tail events, the fragility of porcelain cups, deficits and large firms and the antifragility of trial-and-error and evolution. The heuristic lends itself to immediate implementation, and uncovers hidden risks related to company size, forecasting problems, and bank tail exposures (it explains the forecasting biases). While simple, it vastly outperforms stress testing and other such methods such as Value-at-Risk. Introduction: Main practical result of this paper: a risk heuristic that "works" in detecting fragility even if we use the wrong model/pricing method/probability distribution. The main idea is that a wrong ruler will not measure the height of a child; but it can certainly tell you if he is growing. Since (as we will see) risks in the tails map to nonlinearities (concavity of exposure), second order effects reveal fragility, particularly in the tails (revealed through perturbation)where they map to large tail exposures. Further, the misspecification in using thin-tailed distributions (say the Gaussian) shows immediately through perturbations of standard deviation when it appears to be unstable. Further here are results that shows how fat-tailed (powerlaw tail) probability distribution can be expressed by simple perturbation and mixing of the Gaussian. Why the same heuristic (detection of convexity effects) can measure both fragility and model error: Where F is a valuation “model”, Model (or Valuation) Error = E1 + E2 +E3 where we asume that the three types of errors are orthogonal hence additive. E1= linear error, the “slope”, an error about the first derivative of the model with respect to a variable (equivalent of the delta for an option), say a = F@x+DxD-F@xD Dx . The model identifies the parameter a, but has a wrong value for such parameter in, say, a regression. One can safely believe that modelers cannot easily make such error (the results of the mistracking will be immediately visible). E2= missing a stochastic variable determining F. We unfortunately do not deal with that in this paper, but have evidence (Makridakis et al, 1982; Makridakis and Hibon, 2000) that, if anything, models by overly insample fitting, include too many variables, not too few. E3 (Procrustean Bed)= missing convexity effects, the “hidden gamma”, that is, a) missing the stochastic character of a variable deemed deterministic (and fixed) and b) F is convex or concave with respect of such variable. The resulting bias causes misestimation of F , with undervaluation or overvaluation that maps to the nonlinearity. Such error being rare (and compounded by those rare large deviations), it is likely to be missed. Example of E3. A government estimates unemployment for the next three years as averaging 9%; it uses its econometric models to issue a forecast balance B of 200 billion deficit in the local currency. But it misses (like almost everything in economics) that unemployment is a stochastic variable. Employment over 3 years periods has fluctuated by 1% on average. We can calculate the effect of the error with the following: Ë Unemployment at 8% , Balance B(8%) = -75 bn (improvement of 125bn) Ë Unemployment at 9%, Balance B(9%)= -200 bn Ë Unemployment at 10%, Balance B(10%)= --550 bn (worsening of 350bn) So E3 is the convexity bias from underestimation of the deficit is by -112.5bn, since B H8 %L+B H10 %L 2 = -312.5 Further look at the probability distribution caused by the missed variable (assuming to simplify deficit is Gaussian with a Mean Deviation of 1% ) Figure 1CONVEXITY EFFECTS ALLOW THE DETECTION OF BOTH MODEL BIAS AND FRAGILITY. Illustration of the example; histogram from Monte Carlo simulation of government deficit as a left-tailed random variable simply as a result of randomizing unemployment of which it is a convex function. The method of point estimate would assume a Dirac stick at -200, thus underestimating both the expected deficit (-312) and the skewness (i.e., fragility) of it. Most significant (and preventable) model errors, as we will see, arise from E3. Now this paper will focus on a heuristic that can both detect Fragility and E3since our definition of fragility is grounded in nonlinearities. Further, the “fat tailedeness” of probability distributions is a straight application of E3, the missing of a convexity effect. Nonlinearity and Fragility: Simply, for the fragile, shocks bring higher and higher harm as their intensity increases (up to the point of breaking). Another example. For a collision, forty miles per hour causes more than four times the harm of ten miles per hour. Jumping from a level 30 feet high is more harmful than jumping 10 times from 3 feet. Every payoff one can think of in nature is nonlinear, hence subjected to some tail payoff, and some asymmetry in its distribution. And every model has some kind of Procrustean bed-style sucker problem coming with it, some error from missing the stochasticity of some variable and the nonlinear character of the payoff. The object here is to detect fragility (and, by the same process, to detect its opposite, antifragility, ability to gain from disorder). The same method that detects fragility can detect convexity biases, or model error stemming from missing the stochasticity of a variable, as well as sensitivity to the use of the wrong probability distribution. Our steps are as follows: a. We define fragility, robustness and antifragility. b. We presents the problem of measuring tail risks and show the presence of severe biases attending the estimation of small probability and its nonlinearity (convexity) to parametric (and other) perturbations . c. We express the concept of model fragility in terms of left tail exposure, and show correspondence to the concavity of the payoff from a random variable. d. Finally, we present our simple heuristic to detect the possibility of both fragility and model error across a broad range of probabilistic estimations. The central Table 1 introduces the exhaustive map of possible outcomes, with 4 exhaustive mutually exclusive categories of payoffs. The end product is f(x), which can be reduced to a scalar, and is the central variable of concern. We consider both the probability distribution of f(x) the payoff function, a "derivative" function of x, x being a "primitive" random variable, and the functional properties (concave, convex, linear). We present a series of arguments that can be proved (owing to the format of the discussion, some idiot-savant “quants” might not recognize the proof, so try a bit harder to adapt to the language). Note about the lack of symmetry between fragility and antifragility. By shrinking the left tail (in the presence of unbounded positive payoffs) you cause antifragility; but by increasing the right tail you don’t reduce fragility. Definition and Map of Fragility, Robustness, and Antifragility Table 1Introduces the Exhaustive Taxonomy of all Possible Payoffs y=f(x) 2 Heuristic.nb