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Quantitative Finance
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## Mathematical definition, mapping, and detection of

(anti)fragility
a b
N. N. Taleb & R. Douady
a
NYU-Poly, 6 Metrotech Center, Brooklyn, New York, NY 11201, USA.
b
Centre dEconomie de la Sorbonne, 2 rue dUlm, Paris, France.
Published online: 04 Dec 2013.

To cite this article: N. N. Taleb & R. Douady (2013) Mathematical definition, mapping, and detection of (anti)fragility,
Quantitative Finance, 13:11, 1677-1689, DOI: 10.1080/14697688.2013.800219

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Quantitative Finance, 2013
Vol. 13, No. 11, 16771689, http://dx.doi.org/10.1080/14697688.2013.800219
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2013 iStockphoto LP

## Mathematical denition, mapping, and detection

of (anti)fragility
N. N. TALEB* and R. DOUADY
NYU-Poly, 6 Metrotech Center, Brooklyn, New York, NY 11201, USA
Centre dEconomie de la Sorbonne, 2 rue dUlm, Paris, France

## (Received 24 June 2011; accepted 22 October 2012)

1. What is fragility? harm, which is the case of all used monomodal probability
distributions. Further, what has exposure to tail events suf-
The notions of fragility and antifragility were introduced by fers from uncertainty; typically, when systemsa building,
Taleb (2012). In short, fragility is related to how a system a bridge, a nuclear plant, an airplane, or a bank balance
suffers from the variability of its environment beyond a cer- sheetare made robust to a certain level of variability and
tain preset threshold (when the threshold is K, it is called stress but may fail or collapse if this level is exceeded, then
K-fragility), while antifragility refers to when it benets they are particularly fragile to uncertainty about the distribu-
from this variabilityin a similar way to vega of an tion of the stressor, hence to model error, as this uncertainty
option or a nonlinear payoff, that is, its sensitivity to vola- increases the probability of dipping below the robustness
tility or some similar measure of scale of a distribution. level, bringing a higher probability of collapse. In the oppo-
Simply, a coffee cup on a table suffers more from large site case, the natural selection of an evolutionary process is
deviations than from the cumulative effect of some particularly antifragile, indeed a more volatile environment
shocksconditional on being unbroken, it has to suffer increases the survival rate of robust species and eliminates
more from tail events than regular ones around the center those whose superiority over other species is highly depen-
of the distribution, the at-the-money category. This is the dent on environmental parameters.
case of elements of nature that have survived: conditional on Figure 1 shows the tail vega sensitivity of an object
being in existence, then the class of events around the mean calculated discretely at two different lower absolute mean devi-
should matter considerably less than tail events, particularly ations. We use for the purpose of fragility and antifragility, in
when the probabilities decline faster than the inverse of the place of measures in L2 such as standard deviations, which
restrict the choice of probability distributions, the broader
2013 Taylor & Francis
1678 Feature

## tribution and generate a fragility-detection heuristic cover-

ing both physical fragility and model error.

preferences

## 2.1. Avoidance of the psychological

We start from the denition of fragility as tail-vega sensitiv-
ity, and end up with nonlinearity as a necessary attribute of
the source of such fragility in the inherited casea cause of
the disease rather than the disease itself. However, there is a
long literature by economists and decision scientists embed-
Figure 1. A denition of fragility as left tail-vega sensitivity; the ding risk into psychological preferenceshistorically, risk
gure shows the effect of the perturbation of the lower semi- has been described as connected to risk aversion as a result
deviation s on the tail integral n of (x X) below K, with X a of the structure of choices under uncertainty with a concav-
entering constant. Our detection of fragility does not require
ity of the muddled concept of utility of payoff (Pratt 1964,
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## the specication of f, the probability distribution.

Arrow 1965, Rothschild and Stiglitz 1970, 1971). But this
measure of absolute deviation, cut into two parts: lower and utility business led to the circularity, expressed by Machina
upper semi-deviation above the distribution center X. and Rothschild (2008), risk is what risk-averters hate.
This article aims at providing a proper mathematical de- Indeed, connecting risk to aversion to concavity of prefer-
nition of fragility, robustness, and antifragility and examines ences is a quite unhappy resultthe utility curve cannot be
how these apply to different cases where this notion is possibly monotone concave, but rather, like everything in
applicable. nature necessarily bounded on both sides, the left and the
right, convexconcave and, as Kahneman and Tversky
(1979) have debunked, both path dependent and mixed in its
1.1. Intrinsic and inherited fragility nonlinearity.
Our denition of fragility is two-fold. First, of concern
is the intrinsic fragility, the shape of the probability 2.2. Beyond Jensens inequality
distribution of a variable and its sensitivity to s, a parame-
ter controlling the left side of its own distribution. But we Furthermore, the economics and decision-theory literature
do not often directly observe the statistical distribution of reposes on the effect of Jensens inequality, an analysis
objects, and, if we did, it would be difcult to measure their which requires monotone convex or concave transforma-
tail-vega sensitivity. Nor do we need to specify such a tionsin fact limited to the expectation operator. The world
distribution: we can gauge the response of a given object to is unfortunately more complicated in its nonlinearities.
the volatility of an external stressor that affects it. For Thanks to the transfer function, which focuses on the tails,
instance, an option is usually analysed with respect to the we can accommodate situations where the source is not
scale of the distribution of the underlying security, not its merely convex, but convexconcave and any other form of
own; the fragility of a coffee cup is determined as a mixed nonlinearities common in exposures, which includes
response to a given source of randomness or stress; that of nonlinear doseresponse in biology. For instance, the appli-
a house with respect of, among other sources, the distribu- cation of the transfer function to the KahnemanTversky
tion of earthquakes. This fragility coming from the effect of value function, convex in the negative domain and concave
the underlying is called inherited fragility. The transfer in the positive one, shows that it decreases fragility in the
function, which we present next, allows us to assess the left tail (hence more robustness) and reduces the effect of
effect, increase or decrease in fragility, coming from the right tail as well (also more robustness), which allows
changes in the underlying source of stress. us to assert that we are psychologically more robust to
changes in wealth than implied from the distribution of
1.2. Transfer function such wealth, which happens to be extremely fat-tailed.
Accordingly, our approach relies on nonlinearity of expo-
A nonlinear exposure to a certain source of randomness sure as detection of the vega-sensitivity, not as a denition
maps into tail-vega sensitivity (hence fragility). We prove of fragility. And nonlinearity in a source of stress is neces-
that sarily associated with fragility. Clearly, a coffee cup, a
Inherited Fragility house or a bridge do not have psychological preferences,
, Concavity in exposure on the left side subjective utility, etc. Yet they are concave in their reaction
to harm: simply, taking z as a stress level and P(z) the
of the distribution; harm function (as a negative function), it sufces to see
that, with n > 1,
and build H, a transfer function giving an exact mapping of
tail-vega sensitivity to the second derivative of a function. P(n z)\n P(z); for all 0\n z\Z  ;
The transfer function will allow us to probe parts of the dis-
Feature 1679

## where Z is the level (not necessarily specied) at which

the item is broken. Such inequality leads to P(z) having a
negative second derivative at the initial value z.
So if a coffee cup is less harmed by n times a stressor of
intensity Z than once a stressor of n Z, then harm needs to
be concave to stressors up to the point of breaking; such
stricture is imposed by the structure of survival probabilities
and the distribution of harmful events, and has nothing to
do with subjective utility or some other gments. Just as
with a large stone hurting more than the equivalent weight
in pebbles, if, for a human, jumping one millimeter caused
an exact linear fraction of the damage of, say, jumping to
the ground from thirty feet, then the person would be
putation shows that he would have expired within hours
from touching objects or pacing in his living room, given Figure 2. Disproportionate effect of tail events on nonlinear
the multitude of such stressors and their total effect. The
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## exposures, illustrating the necessity of nonlinearity of the harm

fragility that comes from linearity is immediately visible, so function and showing how we can extrapolate outside the model to
we rule it out because the object would be already broken probe unseen fragility. It also shows the disproportionate effect of
and the person already dead. The relative frequency of the linear, but in reverse: for small variations, the linear sensitivity
exceeds the nonlinear, hence making operators aware of the risks.
ordinary events compared with extreme events is the deter-
minant. In the nancial markets, there are at least ten thou-
sand times more events of 0.1% deviations than events of particularly in the tails where they map to large tail
10%. There are close to 8000 micro-earthquakes daily on exposures, as revealed through perturbation analysis. More
planet earth, that is, those below 2 on the Richter scale generally, every nonlinear function will produce some kind
about 3 million a year. These are totally harmless, and, with of positive or negative exposure to volatility for some parts
3 million per year, you would need them to be so. But of the distribution (gure 2).
shocks of intensity 6 and higher on the scale make the
newspapers. Accordingly, we are necessarily immune to the
cumulative effect of small deviations, or shocks of very 3.1. Fragility and model error
small magnitude, which implies that these affect us dispro- As we saw, this denition of fragility extends to model
portionally less (that is, nonlinearly less) than larger ones. error, as some models produce negative sensitivity to uncer-
Model error is not necessarily mean preserving. s, the tainty, in addition to effects and biases under variability.
lower absolute semi-deviation, does not just express So, beyond physical fragility, the same approach measures
changes in overall dispersion in the distribution, such as, model fragility, based on the difference between a point
for instance, the scaling case, but also changes in the estimate and stochastic value (i.e. full distribution). Increas-
mean, i.e. when the upper semi-deviation from to innity ing the variability (say, variance) of the estimated value
is invariant, or even declines in a compensatory manner to (but not the mean), may lead to a one-sided effect on the
make the overall mean absolute deviation unchanged. This modeljust as an increase of volatility causes porcelain
would be the case when we shift the distribution instead of cups to break. Hence sensitivity to the volatility of such
rescaling it. Thus the same vega-sensitivity can also express value, the vega of the model with respect to such value is
sensitivity to a stressor (dose increase) in medicine or other no different from the vega of other payoffs. For instance,
elds in its effect on either tail. Thus s() will allow us to the misuse of thin-tailed distributions (say Gaussian)
express the sensitivity to the disorder cluster (Taleb 2012): appears immediately through perturbation of the standard
(i) uncertainty, (ii) variability, (iii) imperfect, incomplete deviation, no longer used as point estimate, but as a distri-
knowledge, (iv) chance, (v) chaos, (vi) volatility, (vii) bution with its own variance. For instance, it can be shown
disorder, (viii) entropy, (ix) time, (x) the unknown, how fat-tailed (e.g. power-law-tailed) probability distribu-
(xi) randomness, (xii) turmoil, (xiii) stressor, (xiv) error, tions can be expressed by a simple nested perturbation and
(xv) dispersion of outcomes. mixing of Gaussian ones. Such a representation pinpoints
the fragility of a wrong probability model and its conse-
quences in terms of underestimation of risks, stress tests
3. Detection heuristic and similar matters.
Finally, thanks to the transfer function, this paper proposes
a risk heuristic that works in detecting fragility even if we 3.2. Antifragility
use the wrong model/pricing method/probability distribu-
tion. The main idea is that a wrong ruler will not measure It is not quite the mirror image of fragility, as it implies
the height of a child; but it can certainly tell us if he is positive vega above some threshold in the positive tail of
growing. Since risks in the tails map to nonlinearities (con- the distribution and absence of fragility in the left tail,
cavity of exposure), second-order effects reveal fragility, which leads to a distribution that is skewed right.
1680 Feature

## 4. Fragility and transfer theorems fragilizing errors in the application of Ricardos

comparative advantage without taking into account
The central table 1 introduces an exhaustive map of second-order effects.
possible outcomes, with four mutually exclusive categories (d) Example: hidden tail exposures to budget decits
of payoffs. Our steps in the rest of the paper are as follows. nonlinearities to unemployment.
(e) Example: hidden tail exposure from dependence on
(a) We provide a mathematical denition of fragility, a source of energy, etc. (squeezability argument).
robustness and antifragility. (f) Hidden bankruptcy risk due to leverage and debt.
(b) We present the problem of measuring tail risks
and show the presence of severe biases attending
the estimation of small probability and its nonlinearity 5. Tail vega sensitivity
(convexity) to parametric (and other) perturbations.
(c) We express the concept of model fragility in terms We construct a measure of vega in the tails of the
of left tail exposure, and show correspondence to the distribution that depends on the variations of s, the semi-
concavity of the payoff from a random variable. deviation below a certain level , chosen in the L1 norm in
(d) Finally, we present our simple heuristic to detect the order to ensure its existence under fat-tailed distributions
possibility of both fragility and model error across a with nite rst semi-moment. In fact, s would exist as a
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broad range of probabilistic estimations. measure even in the case of innite moments to the right
side of .
Conceptually, fragility resides in the fact that a smallor Let X be a random variable, the distribution of which is
at least reasonableuncertainty on the macro-parameter of one among a one-parameter family of pdf f, 2 I  R.
a distribution may have dramatic consequences on the result We consider a xed reference value and, from this refer-
of a given stress test, or on some measure that depends on ence, the left-semi-absolute deviation
the left tail of the distribution, such as an out-of-the-money
Z X
option. This hypersensitivity of what we like to call an 
out-of-the-money put price to the macro-parameter, which s (k) (X  x) fk (x) dx:
1
is some measure of the volatility of the distribution of the
underlying source of randomness. We assume that !s() is continuous, strictly increasing
Formally, fragility is dened as the sensitivity of the and spans the whole range R 0; 1), so that we may
left-tail shortfall (non-conditioned by probability) below a use the left-semi-absolute deviation s as a parameter by
certain threshold K to the overall left semi-deviation of the considering the inverse function k(s) : R ! I, dened by
distribution. s((s)) s for s 2 R .
This condition is satised, for instance, if, for any given
x < , the probability of being lower than x is a continuous
4.1. Examples and increasing function Rx of . Indeed, denoting
Fk (x) Prfk (X\ x) 1 fk (t) dt, an integration by part
(a) Example: a porcelain coffee cup subjected to random yields
daily stressors from use.
Z X
(b) Example: tail distribution in the function of the
arrival time of an aircraft. s (k) Fk (x) dx:
1
(c) Example: hidden risks of famine to a population
subjected to monocultureor, more generally,

## Table 1. Exhaustive taxonomy of all possible payoffs y = f (x).

Nonlinear payoff
function
Right tail y = f(x) derivative, Derivatives Effect of fatailedness of
Left tail (gains where x is a random equivalent f (x) compared with
Type Condition (loss domain) domain) variable (Taleb 1997) primitive x
1 Fragile Fat (regular Fat Mixed concave left, Long up-vega, More fragility
(type 1) or absorbing convex right (fence) short down-
barrier) vega
2 Fragile Fat Thin Concave Short vega More fragility
(type 2)
3 Robust Thin Thin Mixed convex left, Short up-vega, Mixed effect
concave right (digital, long down-
sigmoid) vega
4 Antifragile Thin Fat Convex Long vega More antifragility
(thicker
than left)
Feature 1681

## This is the case when is a scaling parameter, i.e.

X  + (X1 ); indeed, one has in this case F
  F
xX @Fk Xx
Fk (x) F1 X ; (x) fk (x);
k @k k
s (k) ks (1):

## It is also the case when is a shifting parameter, i.e.

X  X0  k; indeed, in this case, Fk (x) F0 (x k) and F K

## @s =@k Fk (X). FK

For K < and s 2 R , let FK
K
Z K
n(K; s ) (X  x) fk(s ) (x) dx: Figure 3. The different curves of F(K) and /0k (K) showing the
1 difference in sensitivity to changes at different levels of K.
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## Letting = (s) and integrating by part yields

In particular, (, s) = s. We assume, in a rst step, that
the function n(K; s ) is differentiable on (, ]  R . Z K Z X
The K-left-tail-vega sensitivity of X at stress level K < n(K; s (k)) (X  K)Fk (K) Fk (x) dx FkK (x) dx;
and deviation level s > 0 for the pdf f is 1 1

@n
V (X ; fk ; K; s ) (K; s ) where FkK (x) Fk ( min (x; K)) min (Fk (x); Fk (K)), so that
@s
Z K   1 RX
@fk ds
(X  x) (x) dx : @n (@FkK =@k)(x) dx
1 @k dk V (X ; fk ; K; s ) (K; s ) R1
X
:
@s (@Fk =@k)(x) dx
1

## As in many practical instances where threshold effects

are involved, it may occur that does not depend smoothly For nite differences
on s. We therefore also dene a nite difference version of Z X
the vega-sensitivity as follows: 1
V (X ; fk ; K; s ; Ds) (DFk;Ds
K
(x)) dx;
2Ds 1
 1
V (X ; fk ; K; s ; Ds) (n(K; s Ds)  n(K; s  Ds))
2Ds
Z K where s+ and s are such that s(k  
fk(s Ds) (x)  fk(s Ds) (x) s ) s Ds, s(ks )
(X  x) dx: s  Ds and DF K (x) F K (x)  F K (x).
2Ds k;Ds ks ks
1

Hence omitting the input s implicitly assumes that 6. Mathematical expression of fragility
s ! 0.
Note that n(K; s ) Efk X jX \ K Prfk (X \ K). It can In essence, fragility is the sensitivity of a given risk mea-
be decomposed into two parts: sure to an error in the estimation of the (possibly one-sided)
deviation parameter of a distribution, especially due to
n(K; s (k)) (X  K)Fk (K) Pk (K); the fact that the risk measure involves parts of the
Z K distributiontailsthat are away from the portion used for
Pk (K) (K  x) fk (x) dx; estimation. The risk measure then assumes certain extrapo-
1
lation rules that have rst-order consequences. These conse-
quences are even more amplied when the risk measure
where the rst part (X  K)Fk (K) is proportional to the applies to a variable that is derived from that used for esti-
probability of the variable being below the stress level K mation, when the relation between the two variables is
and the second part P(K) is the expectation of the amount strongly nonlinear, as is often the case.
by which X is below K (counting 0 when it is not). Making
a parallel with nancial options, while s() is a put
6.1. Denition of fragility: The intrinsic case
at-the-money, (K, s) is the sum of a put struck at K and
a digital put also struck at K with amount X  K; it can The local fragility of a random variable X depending on
equivalently be seen as a put struck at with a down- parameter , at stress level K and semi-deviation level s() with
and-in European barrier at K (gure 3). pdf f is its K-left-tailed semi-vega sensitivity V(X, f, K, s).
1682 Feature

The nite-difference fragility of X at stress level K and semi- One may use nite differences to compare the fragility of
deviation level s (k)  Ds with pdf f is its K-left-tailed nite- two random variables: V(Y, g, L, u, u) > V(X, f, K, s,
difference semi-vega sensitivity V(X, f, K, s, s). s). In this case, nite variations must be comparable in
In this denition, the fragility relies in the unstated size, namely Du=u Ds=s.
assumptions made when extrapolating the distribution of X Let us assume, to start with, that u is differentiable,
from areas used to estimate the semi-absolute deviation strictly increasing and scaled so that Y u() . We
s(), around , to areas around K on which the risk mea- also assume that, for any given x < , @Fk =@k(x) [ 0. In
sure depends. this case, as observed above, k ! s k is also increasing.
Let us denote Gk (y) Prgk (Y \y). We have
6.2. Denition of fragility: The inherited case
Gk (u(x)) Pr (Y \ u(x)) Pr (X \ x) Fk (x):
gk fk
We consider here the particular case where a random
variable Y = u(X) depends on another source of risk X, itself
subject to a parameter . Let us keep the above notation for
X, while we denote by g the pdf of Y, Y = u() and u() Hence, if (L, u) denotes the equivalence of (K, s)
the left-semi-deviation of Y. Given a strike level L = u(K), with variable (Y, g) instead of (X, f), then we have
let us dene, as in the case of X, Z Z
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X X
Z K  du
f(L; u (k)) GLk (y) dy FkK (x) (x) dx:
f(L; u (k)) (XY  y)gk (y) dy: 1 1 dx
1

The inherited fragility of Y with respect to X at stress Because u is increasing and min(u(x),u(K)) = u(min(x, K)),
level L = u(K) and left-semi-deviation level s() of X is the in particular
partial derivative Z X
  du
@f u (k) f(X; u (k)) Fk (x) (x) dx:
VX (Y ; gk ; L; s (k)) (L; u (k)) 1 dx
@s
Z K   1
@gk ds
(XY  y) (y) dy : The L-left-tail-vega sensitivity of Y is therefore
1 @k dk
RX
 (@FkK =@k)(x)(du=dx)(x) dx
Note that the stress level and the pdf are dened for the V (Y ; gk ; L; u (k)) R1
X
:
variable Y, but the parameter which is used for differentia- 1
(@Fk =@k)(x)(du=dx)(x) dx
tion is the left-semi-absolute deviation of X, s(). Indeed,
in this process, one rst measures the distribution of X and
its left-semi-absolute deviation, then the function u is For nite variations
applied, using some mathematical model of Y with respect Z X
to X and the risk measure is estimated. If an error is made  1 du
V (Y ; gk ; L; u (k); Du) DFk;Du
K
(x) (x) dx;
when measuring s(), its impact on the risk measure of Y 2Du 1 dx
is amplied by the ratio given by the inherited fragility.
Once again, one may use nite differences and dene the
nite-difference inherited fragility of Y with respect to X, by where k   
u and ku are such that u(ku ) u Du, u(ku )
replacing, in the above equation, differentiation by nite 
u  Du and Fk;Du (x) Fk (x)  Fku (x).
K K K

## differences between values + and , where s (k )

u
Next, theorem 1 proves how a concave transformation
s Ds and s (k ) s  Ds. u(x) of a random variable x produces fragility.

6.3. Implications of a nonlinear change of variable on Theorem 1 (Fragility Transfer Theorem): Let, with the
the intrinsic fragility above notation, u : R ! R be a twice differentiable function
such that u() = and, for any x < , (du=dx)(x) [ 0. The
We study here the case of a random variable Y = u(X), the
random variable Y = u(X) is more fragile at level L = u(K)
pdf g of which also depends on parameter , related to a
and pdf g than X at level K and pdf f if, and only if, one has
variable X by the nonlinear function u. We are now inter-
ested in comparing their intrinsic fragilities. We shall say, Z X
d2 u
for instance, that Y is more fragile at the stress level L and HkK (x) (x) dx \ 0;
left-semi-deviation level u() than the random variable X, 1 dx2
at stress level K and left-semi-deviation level s() if the
L-left-tailed semi-vega sensitivity of Y is higher than the where
K-left-tailed semi-vega sensitivity of X:  
@PkK @PkK @Pk @Pk
HkK (x) (x) (X)  (x) (X);
V (Y ; gk ; L; u )[V (X ; fk ; K; s ): @k @k @k @k
Feature 1683
Rx
and where Pk (x) 1 Fk (t) dt is the price R x of the put
option on X with strike x and PkK (x) 1 FkK (t) dt is
H
that of a put option with strike x and European down-
and-in barrier at K.
H can be seen as a transfer function, expressed as the dif-
K
ference between two ratios. For a given level x of the random
variable on the left-hand side of , the second one is the ratio
of the vega of a put struck at x normalized by that of a put Figure 4. The transfer function H for different portions of the
at-the-money (i.e. struck at ), while the rst one is the distribution: its sign ips in the region slightly below .
same ratio, but where puts struck at x and are European
down-and-in options with triggering barrier at the level K. As this is a strict inequality, it extends to an interval on the
right-hand side of K, say (1; K 0 ] with K \K 0 \X. But,
Proof: Let on the other hand,
Z X Z X Z
@Fk @FkK @Pk @PK X
@Fk @Fk
I Xk (x) dx; IXKk (x) dx; (X)  k (X) (x) dx  (X  K) (K):
1 @k 1 @k @k @k @k @k
Z X Z X K
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@Fk du @FkK du
IYk (x) (x) dx; IYLk (x) (x) dx:
1 @k dx 1 @k dx
For K negative enough, (@Fk =@Fk )(K) is smaller than its
  average value over the interval [K, ], hence
One has V (X ; fk ; K; s (k)) IXKk =IXk and V (Y ; gk ; L; u
(k)) IYLk =IYk , hence @Pk @PK
(X)  k (X) [ 0:
@k @k
IYLk IXKk
V (Y ; gk ; L; u (k))  V (X ; fk ; K; s (k)) 
IYk IXk
! We have proven the following theorem.
IXKk IYLk IYk
 :
IYk IXKk IXk Theorem 2 (Fragility Exacerbation Theorem): With the
above notation, there exists a threshold < such that,
Therefore, because the four integrals are positive, if K 6 , then HkK (x)[0 for x 2 (  1, ] with K <
V (Y ; gk ; L; u (k))  V (X ; fk ; K; s (k)) has the same sign as < . As a consequence, if the change of variable u is con-
IYLk =IXKk  IYk =IXk . On the other hand, we have IXk cave on (1, ] and linear on [, ], then Y is more
(@Pk =@k)(X), IXKk (@PkK =@k)(X) and fragile at L = u(K) than X at K.
One can prove that, for a monomodal distribution, <
Z X < (see discussion below). So whatever the stress level
@Fk du
IYk (x) (x) dx K below the threshold , it sufces that the change of var-
1 @k dx
Z X iable u be concave on the interval (1, ] and linear on
@Pk du @Pk d2 u [, ] for Y to become more fragile at L than X at K. In
(X) (X)  (x) 2 (x) dx;
@k dx 1 @k dx practice, as long as the change of variable is concave
Z X
@Fk du
K around the stress level K and has limited convexity/concav-
IYLk (x) (x) dx ity away from K, the fragility of Y is greater than that of X.
1 @k dx
Z X Figure 4 shows the shape of HkK (x) in the case of a
@PkK
du @PkK d2 u
(X) (X)  (x) 2 (x) dx: Gaussian distribution where is a simple scaling parameter
@k dx 1 @k dx ( is the standard deviation ) and X 0. We represented
An elementary calculation yields K 2k while in this Gaussian case, Hk 1:585k.

 K 1 Z X
IYLk IYk @Pk @PkK d2 u
  (X) (x) 2 dx
IXKk IXk @k 1 @k dx 7. Discussion
 1 Z X
@Pk @Pk d2 u
(X) (x) 2 dx 7.1. Monomodal case
@k 1 @k dx
Z X We say that the family of distributions ( f) is left-monomodal
du
2
 HkK (x) 2 dx: if there exists lk < such that @fk =@k P 0 on
1 dx
 (-1, ] and @fk =@k 0 on [lk , ]. In this case, @Pk =@k is
Let us now examine the properties of the function HkK (x). a convex function on the left half-line (-1, ], then concave
For x K; we have (@PkK =@k)(x) (@Pk =@k)(x)[0 (the after the inexion point . For K 6 , the function @PkK =@k
positivity is a consequence of that of @Fk =@k), therefore coincides with @Pk =@k on (-1, K], then is a linear exten-
HkK (x) has the same sign as (@Pk =@k)(X)  (@PkK =@k)(X). sion, following the tangent to the graph of @Pk =@k in K (see
graph below). The value of (@PkK =@k)(X) corresponds to the
1684 Feature

## G K When we apply a nonlinear transformation u, the action

of the parameter is no longer a scaling: when small nega-
H <0 tive values of X are multiplied by a scalar , so are large
G
negative values of X. The scaling applies to small nega-
tive values of the transformed variable Y with a coefcient
(du=dx)(0), but large negative values are subject to a differ-
H >0 ent coefcient (du=dx)(K), which can potentially be very
P / different and much larger, in particular.

P K/
7.3. Fragility drift
K
Fragility is dened as the sensitivity, i.e. the rst partial
Figure 5. The distribution G and the various derivatives of the derivative, of the tail estimate with respect to the left
unconditional shortfalls. semi-deviation s. Let us now dene the fragility drift:

## intersection point of this tangent with the vertical axis. It

increases with K, from 0 when K ! -1 to a value above @2n
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VK0 (X ; fk ; K; s ) (K; s ):
(@Pk =@k)(X) when K = . The threshold corresponds to @K@s
the unique value of K such that (@PkK =@k)(X) (@Pk =
@k)(X). When K < , then
  K In practice, fragility always occurs as the result of fragil-
@Pk @Pk @PK @Pk
Gk (x) (x) (X) and GKk (x) k (x) (X) ity drift, indeed, by denition, we know that (, s) = s,
@k @k @k @k hence V(X, f, , s) = 1. The fragility drift measures the
speed at which fragility departs from its original value 1
when K departs from the center .
are functions such that Gk (X) GKk (X) 1 and which are
proportional for x 6 K, the latter being linear on [K, ].
7.4. Second-order fragility
On the other hand, if K < , then (@PkK =@k)(X)\
(@Pk =@k)(X) and Gk (K)\GKk (K), which implies that The second-order fragility is the second-order derivative of
Gk (x)\GKk (x) for x 6 K. An elementary convexity analysis the tail estimate with respect to the semi-absolute devia-
shows that, in this case, the equation Gk (x) GKk (x) has a tion s:
unique solution with < < . The transfer function @2n
HkK (x) is positive for x < , in particular when x 6 and Vs0 (X ; fk ; K; s ) (K; s ):
(@s )2
negative for < x < (gure 5).

## As we shall see below, the second-order fragility drives

7.2. Scaling parameter the bias in the estimation of stress tests when the value of
We assume here that is a scaling parameter, i.e. X = + s is subject to uncertainty, through the Jensen inequality.
(X1 ). In this case, as we saw above, we have
8. Denitions of robustness and antifragility
   
1 xX xX
fk (x) f1 X ; Fk (x) F1 X ; Antifragility is not the simple opposite of fragility, as we
k k k
  saw in table 1. Measuring antifragility, on the one hand,
xX
Pk (x) kP1 X ; s (k) ks 1: consists of the ipside of fragility on the right-hand side,
k but on the other hand requires a control on the robustness
Hence, of the probability distribution on the left-hand side. From
that aspect, unlike fragility, antifragility cannot be summa-
 
KX rized in one single gure but necessitates at least two.
n(K; s (k)) (X  K)F1 X When a random variable depends on another source of
k
  randomness, Y = u(X), we shall study the antifragility of
KX
kP1 X ; Y with respect to that of X and to the properties of the
k
function u.
@n 1 @n
(K; s )  (K; k)
@s  s (1) @k 8.1. Denition of robustness
1
 (Pk (K) (X  K)Fk (K) Let (X) be a one-parameter family of random variables
s (k)
with pdf f. Robustness is an upper control on the fragility
(X  K)2 fk (K)): of X, which resides on the left-hand side of the distribution.
Feature 1685

We say that f is b-robust beyond stress level K < if Note that robustness is in effect impervious to changes of
V(X, f, K, s()) b for any K K. In other words, the probability distributions. Also note that this measure of
robustness of f on the half-line (, K] is R(1;K (Xk ; fk ; robustness ignores rst-order variations since owing to their
K; s (k)) maxK 0 K V (Xk ; fk ; K 0 ; s (k)), so that b-robust- higher frequency, these are detected (and remedied) very
ness simply means R(1;K (Xk ; fk ; K; s (k)) b. early on.
We also dene b-robustness over a given interval
[K1, K2] by the same inequality being valid for any K 2 8.2. Examples of robustness (barbells)
[K1, K2]. In this case we use RK1 ;K2  (Xk ; fk ; K; s (k))
maxK1 K 0 K2 V (Xk ; fk ; K 0 ; s (k)). (a) Trial and error with bounded error and open payoff.
Note that the lower R, the tighter the control and the (b) For a barbell portfolio with allocation to numeraire
more robust the distribution f. securities up to 80% of the portfolio, no perturbation
Once again, the denition of b-robustness can be trans- below K set at 0.8 of valuation will represent any
posed, using nite differences V(X, f, K, s(), s). difference in results, i.e. b = 0. The same for an
In practical situations, setting a material upper bound b insured house (assuming the risk of the insurance
to the fragility is particularly important: one needs to be company is not a source of variation), no perturba-
able to come with actual estimates of the impact of the tion for the value below K, equal to minus the insur-
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error on the estimate of the left-semi-deviation. However, ance deductible, will result in signicant changes.
when dealing with certain classes of models, such as Gauss- (c) A bet of amount B (limited liability) is robust,
ian, exponential of stable distributions, we may be lead to as it does not have any sensitivity to perturbations
consider asymptotic denitions of robustness, related to cer- below 0.
tain classes.
For instance, for a given decay exponent a > 0, assuming
that f(x) = O(eax) when x , the a-exponential asymp- 8.3. Denition of antifragility
totic robustness of X below the level K is
The second condition of antifragility regards the right-hand
 a(XK 0 ) 0  side of the distribution. Let us dene the right-semi-deviation
Rexp (Xk ; fk ; K; s (k); a) max (e V (Xk ; fk ; K ; s (k))):
0 K K of X:

0 0 Z
If one of the two quantities ea(XK ) fk (K 0 ) or ea(XK ) V (Xk ; 1

fk ; K 0 ; s (k)) is not bounded from above when K , then s (k) (x  X) fk (x) dx;
X
Rexp = + and X is considered as not being a-exponentially
robust. Similarly, for a given power > 0, and assuming that
f(x) = O(x) when x , the -power asymptotic
robustness of X below the level K is and, for H > L > ,
Z H
Rpow (Xk ; fk ; K; s (k); a) max ((X  K 0 )a2 V (Xk ; fk ; K 0 ; s (k))): n (L; H; s (k)) (x  X) fk (x) dx;
0
K K L

@n (L; H; s )
W (X ; fk ; L; H; s )
@s
If one of the two quantities (X  K 0 )a fk (K 0 ) or (X  K 0 )a2 Z H 
@fk
V (Xk ; fk ; K 0 ; s (k)) is not bounded from above when (x  X) (x) dx
@k
K , then Rpow = + and X is considered as not being L
Z 1 1
-power robust. Note that the exponent 2 is used with the @fk
 (x  X) (x) dx :
fragility, for homogeneity reasons, e.g. in the case of stable X @k
distributions, when a random variable Y = u(X) depends on
another source of risk X.
When Y = u(X) is a variable depending on a source of
Denition 2a, Left-robustness (monomodal distribution):
noise X, we dene
A payoff y = u(x) is said to be (a,b)-robust below L = u(K)
for a source of randomness X with pdf f assumed monomo- WX (Y ; gk ; u(L); u(H); s )
dal if, letting g be the pdf of Y = u(X), one has, for any K Z u(H) 
@gk
K and L = u(K) (y  u(X)) (y) dy
u(L) @k
Z 1 1
VX (Y ; gk ; L0 ; s (k)) aV (X ; fk ; K 0 ; s (k)) b: (4) @fk
(x  X) (x) dx :
X @k
The quantity b is of order deemed of negligible utility
(subjectively), that is, does not exceed some tolerance level
in relation to the context, while a is a scaling parameter Denition 2b, Antifragility (monomodal distribution): A
between variables X and Y. payoff y = u(x) is locally antifragile over the range [L, H]
if
1686 Feature

(1) it is b-robust below for some b > 0, higher-order partial derivatives, where n is large
(2) WX (Y ; gk ; u(L); u(H); s (k)) P aW (X ; fk ; L; H; enough so that the interval [s ns] covers the
s (k)); where a u (k)=s (k). range of possible values of s. Indeed, in this case,
the nite difference estimate of fragility uses evalua-
The scaling constant a provides homogeneity in the case tions of at points spanning this interval.
where the relation between X and Y is linear. In particular,
nonlinearity in the relation between X and Y impacts robust-
ness. The second condition can be replaced by nite differ- 9.3. Unconditionality of the shortfall measure
ences u and s, as long as u/u = s/s. Many, whenR K presentingRshortfall,
K
deal with the conditional
shortfall 1 x f (x) dx= 1 f (x) dx; while such a measure
might be useful in some circumstances, its sensitivity is not
9. Remarks
indicative of fragility in the sense used in this discussion.
RK
9.1. Fragility is K-specic The unconditional tail expectation n 1 x f (x) dx is
more indicative of exposure to fragility. It is also preferred
We are only concerned with adverse events below a certain to the raw probability of falling below K, which is
pre-specied level, typically the breaking point. Exposures RK
1 f (x) dx, as the latter does not include the conse-
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## A can be more fragile than exposure B for K = 0, and RK

much less fragile if K is, say, four mean deviations below quences. For instance, two such measures 1 f (x) dx and
RK
0. We may need to use nite s to avoid situations as we 1 g(x) dx may be equal over broad values of K, but the
will see of vega-neutrality coupled with short left tail. RK
expectation 1 x f (x) dx can be much more consequential
RK
than 1 x g(x) dx as the cost of the break can be more
9.2. Effect of using the wrong distribution f severe and we are interested in its vega equivalent.
Comparing V(X, f, K, s, s) and the alternative distribution
V(X, f , K, s, s), where f is the true distribution, the
measure of fragility provides an effective indication of the 10. Applications to model error
sensitivity of a given outcome, such as a risk measure, to
model error, provided no paradoxical effects perturb the In the cases where Y depends on X, among other variables,
situation. Such paradoxical effects are, for instance, a often x is treated as non-stochastic, and the underestimation
change in the direction in which certain distribution percen- of the volatility of x maps immediately into the underesti-
tiles react to model parameters, such as s. It is indeed pos- mation of the left tail of Y under two conditions:
sible that nonlinearity appears between the core part of the
distribution and the tails such that when s increases, the (a) X is stochastic and its stochastic character is ignored
left tail starts fattening, giving a large measured fragility, (as if it had zero variance or mean deviation);
then steps back, implying that the real fragility is lower (b) Y is concave with respect to X in the negative part
than the measured one. The opposite may also happen, of the distribution, below X.
implying a dangerous under-estimate of the fragility. These
nonlinear effects can stay under control provided one makes
some regularity assumptions on the actual distribution, as 10.1. Convexity bias or Jensens inequality effect
well as on the measured one. For instance, paradoxical
effects are typically avoided under at least one of the fol- Further, missing the stochasticity under the two conditions
lowing three hypotheses. (a) and (b), in the event of the concavity applying above X
leads to negative convexity bias from the lowering effect on
(a) The class of distributions in which both f and f are the expectation of the dependent variable Y.
picked are all monomodal, with monotonous depen-
dence of percentiles with respect to one another.
(b) The difference between percentiles of f and f has 10.2. Case 1: Application to decits
constant sign (i.e. f is either always wider or
always narrower than f at any given percentile). Example: A government estimates unemployment for the
(c) For any strike level K (in the range that matters), the next three years as averaging 9%; it uses its econometric
fragility measure V monotonously depends on s on models to issue a forecast balance B of 200 billion decit
the whole range where the true value s can be in the local currency. But it misses (like almost everything
expected. This is particularly the case when partial in economics) that unemployment is a stochastic variable.
derivatives kV/sk all have the same sign at mea- Employment over 3-year periods has uctuated by 1%, on
sured s up to some order n, at which the partial average. We can calculate the effect of the error with the
derivative has that same constant sign over the following:
whole range on which the true value s can be
expected. This condition can be replaced by an unemployment at 8%, balance B(8%) = 75 bn
assumption on nite differences approximating the (improvement of 125 bn);
unemployment at 9%, balance B(9%) = 200 bn;
Feature 1687

## 10.5. Missing effects

The study of model error is not to question whether a
model is precise or not, whether or not it tracks reality, it is
to ascertain the rst- and second-order effect from missing
the variable, ensuring that the errors from the model do not
have missing higher-order terms that cause severe
unexpected (and unseen) biases in one direction because of
convexity or concavity, in other words, whether or not the
model error causes a change in z.

## 10.6. Model bias, second-order effects, and fragility

Figure 6. Convexity effects allow the detection of both model bias
and fragility. Illustration of the example; histogram from Monte Having the right model (which is a very generous assump-
Carlo simulation of government decit as a left-tailed random tion), but being uncertain about the parameters will invari-
variable simply as a result of randomizing unemployment of ably lead to an increase in model error in the presence of
which it is a convex function. The method of point estimate would convexity and nonlinearities. As a generalization of the def-
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## assume a Dirac stick at 200, thus underestimating both its

expected decit (312) and the skewness (i.e. fragility). icit/employment example used in the previous section, say
we are using a simple function:

f (xja); (5)
unemployment at 10%, balance B(10%) = 550 bn
where a is assumed to be the average expected rate, where
(worsening of 350 bn).
we take u as the distribution of a over its domain }a
The convexity bias from underestimation of the decit is Z
by 112.5 bn, since B(8%) B(10%)=2 312:5. a a u(a) da: (6)
Further look at the probability distribution caused by the }a

## missed variable (assuming, for simplicity, the decit is

Gaussian with a mean deviation of 1%) (gure 6). The mere fact that a is uncertain (since it is estimated)
might lead to a bias if we perturb from the outside (of the
integral), i.e. stochasticize the parameter deemed xed.
10.3. Adding model error and metadistributions Accordingly, the convexity bias is easily measured as the
difference between (a) f integrated across values of potential
Model error should be integrated in the distribution as a
and (b) f estimated for a single value of deemed to be
stochasticization of parameters. f and g should subsume the
its average. The convexity bias xA becomes
distribution of all possible factors affecting the nal out- Z Z
come (including the metadistribution of each). The so-called xA
f (xja) u(a) da dx
perturbation is not necessarily a change in the parameter }x }a
Z  Z 
so much as it is a means to verify whether f and g capture 
the full shape of the nal probability distribution.  f x  a u(a) da dx: (7)
}x }a
Any situation with a bounded payoff function that organi-
cally truncates the left tail at K will be impervious to all
xB , the missed fragility, is assessed by comparing the two
perturbations affecting the probability distribution below K.
integrals below K, in order to capture the effect on the left
For K = 0, the measure equates to mean negative semi-devi-
tail:
ation (more potent than negative semi-variance or negative
semi-standard deviation often used in nancial analyses). Z K Z
xB (K)
f (xja)u(a) da dx
1 }a
Z Z 
10.4. Model error and semi-bias as nonlinearity from K 
missed stochasticity of variables  f (x  a u(a) da) dx; (8)
1 }a
Model error often comes from missing the existence of a
random variable that is signicant in determining the which can be approximated by an interpolated estimate
outcome (say option pricing without credit risk). We cannot obtained with two values of a separated from a mid-point
detect it using the heuristic presented in this paper but as by Da, the mean deviation of a, and estimating
mentioned earlier the error goes in the opposite direction as Z K
the model tends to be richer, not poorer, from overtting. 1
xB (K)
( f (x j
a Da) f (x j
a  Da)) dx
But we can detect the model error from missing the sto- 1 2
chasticity of a variable or underestimating its stochastic Z K
character (say option pricing with non-stochastic interest  f (x j
a) dx: (9)
1
rates or ignoring that the volatility can vary).
1688 Feature

We can probe xB by point estimates of f at a level of X 6 K declare the risk as grossly mis-measured (no need
for further risk assessment).
1
x0B (X ) ( f (X j
a Da) f (X j
a  Da))  f (X j
a); (10) Step 2 (second order): For all parameters p com-
2 pute the ratio of rst- to second-order effects at
the initial range p = estimated mean deviation.
so that
Z K
l0
xB (K) x0B (x) dx; H(Dp)
;
1 f ( p)
where
which leads us to the fragility heuristic. In particular, if we
    
assume that x0B (X ) has a constant sign for X K, then 0 1 1 1
B(K) has the same sign. l (Dp)
f p Dp f p  Dp :
2 2 2

## 11. The fragility/model error detection heuristic

Step 3: Note parameters for which H is
signicantly > or <1.
(detecting A and B when cogent)
Step 4: Keep widening p to verify the stability
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## of the second-order effects.

Example 1 (detecting tail risk not shown by stress test,
B): The famous rm Dexia went into nancial distress a
few days after passing a stress test with ying colors. If a 11.2. The heuristic applied to a stress test
bank issues a so-called stress test (something that has not In place of the standard, one-point estimate stress test S1,
proven very satisfactory), off a parameter (say stock market) we issue a triple, S1, S2, S3, where S2 and S3 are S1 
at 15%. We ask them to re-compute at 10% and 20%. p. Acceleration of losses is indicative of fragility.
Should the exposure show negative asymmetry (worse at
20% than it improves at 10%), we deem that their risk
increases in the tails. There are certainly hidden tail expo- Remarks
sures and a denite higher probability of blowup in addition
to exposure to model error. Note that it is somewhat more (a) Simple heuristics have a robustness (in spite of a
effective to use our measure of shortfall in the denition possible bias) compared with optimized and cali-
given above, but the method here is effective enough to brated measures. Ironically, it is from the multiplica-
show hidden risks, particularly at wider increases (try 25% tion of convexity biases and the potential errors
and 30% and see if the exposure shows an increase). Most from missing them that calibrated models that work
effective would be to use power-law distributions and per- in-sample underperform heuristics out-of-sample
turb the tail exponent to see symmetry. (Gigerenzer and Brighton 2009).
(b) Heuristics allow the detection of the effect of the
Example 2 (detecting tail risk in over-optimized system, use of the wrong probability distribution without
B): Raise airport trafc 10%, lower 10%, take average changing the probability distribution (just from the
expected traveling time from each, and check the asymmetry dependence on parameters).
for nonlinearity. If asymmetry is signicant, then declare the (c) The heuristic improves and detects aws in all other
system as over-optimized (both A and B as thus shown). commonly used measures of risk, such as CVaR,
The same procedure uncovers both fragility and the conse- and expected shortfall; stress-testing and similar
quences of model error (potential harm from having the methods have been proven to be completely ineffec-
wrong probability distribution, a thin-tailed rather than a fat- tive (Taleb 2009).
tailed one). For traders (and see the discussions of Gigerenzer (d) The heuristic does not require parameterization
and Brighton (2009) and Gigerenzer and Goldstein (1996)), beyond the choice of p in units of mean absolute
simple heuristics tools detecting the magnitude of second- deviations.
order effects can be more effective than more complicated
and harder to calibrate methods, particularly under multi- 12. Further applications
Derman and Wilmott (2009) and Haug and Taleb (2011). In parallel works, applying the simple heuristic allows us
to detect the following hidden short options problems by
merely perturbing a certain parameter p.
11.1. The heuristic applied to a model
(a) Size and negative stochastic dis-economies of scale.
Step 1 (rst order): Take a valuation. Measure (i) Size and squeezability (nonlinearities of squeezes
the sensitivity to all parameters p determining V
in costs per unit).
over nite ranges p. If materially signicant,
check if stochasticity of parameter is taken into
(b) Specialization (Ricardo) and variants of globalization.
account by risk assessment. If not, then stop and
Feature 1689

(i) Missing stochasticity of variables (price of wine). Gigerenzer, G. and Brighton, H., Homo heuristicus: Why biased
(ii) Specialization vs broad distributed exposures in minds make better inferences. Top. Cogn. Sci., 2009, 1(1), 107
143.
natural systems.
Gigerenzer, G. and Goldstein, D.G., Reasoning the fast and frugal
way: Models of bounded rationality. Psychol. Rev., 1996, 103,
(c) Portfolio optimization (Markowitz). 650669.
(d) Debt and leverage. Haug, E. and Taleb, N.N., Option traders use (very) sophisticated
(e) Budget decits: convexity effects explain why uncer- heuristics, never the BlackScholesMerton formula. J. Econ.
Behav. Organiz., 2011, 77(2), 97106.
tainty lengthens, and does not shorten expected decits.
Jensen, J.L.W.V., Sur les fonctions convexes et les ingalits entre
(f) Iatrogenics (medical) or how some treatments are les valeurs moyennes. Acta Math., 1906, 30, 175193.
concave to benets, convex to errors. Kahneman, D. and Tversky, A., Prospect theory: An analysis of
(g) Disturbing natural systems. decision under risk. Econometrica, 1979, 46(2), 171185.
(h) Collateralized debt and derivative contracts: collat- Machina, M. and Rothschild, M., Risk. In The New Palgrave Dic-
tionary of Economics, 2nd ed., edited by S.N. Durlauf and L.E.
eral or exchange margin shift the strike of the
Blume, 2008 (Macmillan: London).
option describing the exposure, making out-of-the- Makridakis, S., Andersen, A., Carbone, R., Fildes, R., Hibon,
money, hence more fragile. M., Lewandowski, R., Newton, J., Parzen, R. and Winkler,
R., The accuracy of extrapolation (time series) methods:
Results of a forecasting competition. J. Forecast., 1982, 1,
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