Daniel Satchkov CFA - Quaffers.org

Reverse Stress Testing:
The Unknown Knowns

Definitions
Outline
• Stress Testing – Crash Testing Your Portfolio
• The Roots of the Present Problem
• Can Power Law Copulas help?
• Key Factor Level: RiXtrema’s Tail Copula
• Reverse Stress Testing Concepts
• RiXtrema Methodology
• The So What: How much does your big bet have
to lose to hurt?
• A Place for Reverse Stress Testing 2

Definitions
What is a function of a Paradigm?
• Tells a researcher what the world is like
• Provides foundations/allows to focus on specifics
• Postulates which entities do/do not inhabit the
researcher’s world
• Provides a theory of how entities interact (more on this
later)
• Provides “exemplars”
• E. defines acceptable questions to ask
• E. defines acceptable solutions to a problem
• E. defines acceptable methods for obtaining solutions
3

Definitions
Portfolio Stress Testing is like…
• Counting all of the possible permutations (aka MIT
Blackjack team) and getting ahead?
• Getting really creative and imagining unexpected
scenarios (aliens landing!)?
• Car Crash Testing
• Risk manager is not concerned with thinking of all of the
possible (infinite in number) scenarios
• Risk managers is like an engineer whose task is to crash
test a car
• Deal with classes of scenarios: Frontal impact, side
impact, rear impact 4

Definitions
Type of Stress Testing
• Naïve – move one parameter/factor/asset class; leave
the rest untouched (no distribution)
• Historical – use realized performance over some
historical period (no distribution)
• Factor – use joint distributions and use shock inputs to
formulate conditional distributions
• Historical + Factor – use realized historical performance
for key factors, but today’s loadings (no distribution)
• Reverse Stress Testing – specify a loss level and find the
most likely scenarios that can lead to similar levels of loss 5

Definitions
Factor Stress Testing
Given: an asset vector :
x1
x
x2
possessing multivariate distribution and zero mean with
covariance matrix:
C
C
C
• Conditional distribution of given has a mean
11
21
C
C
12
22
of:
C
12
1
*C *a
22
and variance of:
C
1
11
C12
C22
*
* C
21
6

Current state in risk modeling
Distribution trouble: A Rise in Correlations
•One often hears of “rise in correlations’ or ‘correlations going to one’
•Do we observe correlations?
•Shifting correlations mean that the model is not working and a plug is needed
•Can we use this problem to our advantage?
•“Wrong number to put into a wrong formula to get the right price”
7

Current state in risk modeling
Correlations Assymetries: Not All Going to One
Source: Chua et. al. (2009)
8

Current state in risk modeling
Beware of “Conditioning Fallacy”
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Theoretical Tail Correlations : Japan vs China
(given volatility in U.S. Equities)
0
Initial 1 Std Move 2 Std Move 3 Std Move
Source: Chua et. al. (2009)
9

Solution by RiXtrema
RiXtrema’s Tail Copula
• Risk Manager Motto: “Wrong number to put into a wrong formula to get the right
price”
• Real world is neither Gaussian nor Student
• Use devices available to make the best possible decisions under the conditions of
uncertainty
• Gaussian still represents the most tractable model
• Use Tail Implied Correlations
- Overweight tail observations (left tail is of more interest)
- Uncondition correlations to avoid ‘conditioning fallacy’
10

Solution by RiXtrema
Reverse Stress Testing – The Basics
• Specify a level of loss
• Examine the outcomes that can lead to that level of loss
• Given our same asset vector , weights :
w ( w 1
,..., )
w n
T
and portfolio return =
p
w
T *
X
Which posesses univariate normal distribution with zero mean
and variance =
w T * C *
w
11

Solution by RiXtrema
Reverse Stress Testing – Loss Distribution
• We care about all scenarios
x R
n
w T
such that (1)
• Denote H the hyperplane in
*
x
R
L
defined by the equation (1), that is:
n
H
H
( w,
L)
{ x : w * x L}
T
• The conditional distribution of X given X belongs to H is clearly
normal with mean
12

Solution by RiXtrema
Reverse ST – Initial Distribution (figure 1)
13

Solution by RiXtrema
Reverse ST: The Pivot
• The initial distribution is degenerate, because it is overparameterized (we
removed one degree of freedom when we specified a loss, but left
parametrization unchanged)
• To do that we will pivot the distribution, so that weights are moved to the vector:
we froze the initial vector to reduce the number of parameters in the
distribution, but kept the length of the limiting orthogonal vector from the
figure 1
• This is easily achieved by starting with a matrix of full rank with w as the first row
and applying the Gram-Schmidt orthogonalization procedure and denoting the
resulting matrix P , in other words, making the transform
14

Solution by RiXtrema
Reverse ST: The Pivot Continued
• The covariance matrix of the transformed vector becomes
• The transformed hyperplane is defined by the equation
• For we have
• Denote
15

Solution by RiXtrema
Reverse ST – Distribution Turn
16

Solution by RiXtrema
Reverse ST: The Pivot
• The initial hyperplane H goes to another hyperplane:
T
HY y : wY
*
• The new distribution has a covariance matrix:
• Now this distribution (by analogy with original conditional distribution) is normal
with
mean:
y L
a
Y
H
d
11
L
* w
D
2
Y
* D
d
D
I1
11
I1
D
d
1I
II
covariance matrix:
1
Y
DH
DII
* DI1
* D1
I
d11
17

Solution by RiXtrema
Reverse ST: Getting Some Variety
• We have generated a conditional hyperplane in a mathematically
tractable form
• Now the question arises: what scenarios do we want to see?
• The criteria:
a. They are likely
b. They are at least somewhat different (otherwise we
could just do with one scenario to describe all)
c. They are not missing any danger scenarios
We look for M+1 (one being the origin) points
• Points lie inside a given radius (to satisfy point (a), since likelihood
drops when getting further away from the origin)
• Points are equidistant from one another (satisfying (b) and (d)) 18

Solution by RiXtrema
Reverse ST: Getting Some Variety Continued
• The problem is states as follows: denote the origin, and find
points
so that
• subject to
• where stands for Euclidean distance
19

Solution by RiXtrema
Reverse ST: How Do We Get Variety?
• Two possible solutions for M=3
20

Solution by RiXtrema
Reverse ST: Now What?
• We have our conditional distribution with the center a(H) – our key
danger scenario
• We have M+1 points that cover the space surrounding the a(H) to
give us some flavor of different possibilities that could hurt us
• Now, we need to situate our M+1 points in such a way that they
represent the most likely scenarios among those located at equal
distances (remember, there is an infinity of possible M+1 sets, we
have to pick the one that contains the more likely scenarios)
• Principal component analysis will help us to place our points on the
axes of importance
21

Solution by RiXtrema
Reverse ST: Using PCA to Place Points
• Denote V the (n-1)x(n-1) matrix, whose columns are
Y
eigenvectors of D H
, sorted in descending order of
eigenvalues and apply the transform
z
j
V
T
j
* x , j
0,1,...,
• Given the desired level of relative likelihood we
calculate the radius
M
• and finalize the transform
22

Solution by RiXtrema
Reverse ST: Using PCA to Place Points
• Denoting the conditional density in we calculate the
relative likelihood of scenarios by
• and sort the scenarios in descending order of their relative
likelihood (see next figure)
• Then just make the backward pivot to the initial placement
by
23

Solution by RiXtrema
Reverse ST: Relative likelihood
Most likely scenario
(interior ellipse)
Less likely scenarios
(exterior ellipses)
24

Solution by RiXtrema
Reverse Stress Testing
Loss Level = 15
Name aH x1 x2 x3 x4 x5
Greece Bond(10 GOV TR) -46 -3 -100 -22 -64 -44
Portugal Bond(10 GOV TR) -25 8 -60 -16 -39 -17
Taiwan Equity(TAIEX) -36 -40 -32 -28 -13 -70
Singapore Equity(Straits Times) -42 -43 -32 -39 -21 -74
Hong Kong Equity(Hang Seng) -43 -48 -34 -40 -24 -71
South Africa Bond(EMBI Global) -40 -60 -40 -25 -47 -28
Ireland Bond(10 GOV TR) -12 1 -26 -10 -14 -11
US Equity(S&P 500) -28 -23 -18 -29 -30 -42
Indonesia Bond(EMBI Global) -38 -43 -29 -35 -31 -50
AUD -16 -15 -13 -16 -9 -29
CMBX AAA -5 -6 -6 -12 -3 0
Italy Bond(10 GOV TR) -12 -7 -19 -10 -12 -11
Spain Bond(10 GOV TR) -10 -5 -17 -8 -11 -9
US Bond(10 GOV TR) 5 5 3 4 4 6
Relative likelihood 1.00 0.29 0.26 0.15 0.10 0.10
25

Solution by RiXtrema
Conclusions
• Any stress testing technique must be mindful of the crippling shifts
in correlations that render the results dangerous to use
• Once the tail copula is calibrated, we can then go one step further
and look for sets of likely scenarios that will produce certain user
specified losses without specifying the shocks
• After loss level is specified we use Gram-Schmidt orthogonalization
procedure to specify a tractable distribution, find the center of the
conditional loss distribution and find points around in a likelihood
region to show possible danger scenarios
• Distinguish between factors that produce high impact in all
scenarios, versus ‘sleeper’ factors that are dangerous in specific
circumstances
26

Solution by RiXtrema
Practical Importance Summary
• Check intuitions by seeing how much segments of
the portfolios have to lose before you see a large
P&L impact
• Understand alternative vulnerabilities
• Create more economic hedges
27