Can We Measure Extreme Risk? Should we Measure it? › dev › files › Douady Slides.pdf · Copyright © 2009 Riskdata. All rights reserved. Poly-Models A Collection of Single-Factor

Copyright © 2009 Riskdata. All rights reserved. Copyright 2008 Riskdata. All rights reserved.

Can We Measure Extreme Risk?

Should we Measure it? Taleb’s Fragility and Poly-models

Raphael Douady

IAQF Seminar Series

Dec 15, 2014

Kimmel center New York

Copyright © 2009 Riskdata. All rights reserved.

Pure Performance Analysis

2

95

100

105

110

115

120

125

130

135

Sharpe = 1.3

Annualised Volatility = 2.4%

Annualised return = 6.5%

VaR 99 = 0.9% (1.3 sigma)

Peak to valley = 1.1%

Skew = +0.6

Excess Kurtosis = 0.2

WILL IT LAST? This performance series would attract any investor who is only focused

on past performances. The sequel shows he might be disappointed.

Copyright © 2009 Riskdata. All rights reserved. 3

95

100

105

110

115

120

125

130

135

Sharpe = -0.25



VaR 99 = 3.5% (3.5 sigma)


Skew = -1.0


Could such a loss be anticipated, only looking at the past fund performances?

Pure Performance Analysis


What is Value-at-Risk?

> Black Hole Horizon Effect in Sovereign Debt

• Beyond Critical Level: Deadly Debt Spiral

> Debt Burden is too Heavy

> Growth cannot Sustain the Burden

> Markets Remove Confidence

> Interest Rates Increase

> Debt Burden is Heavier

> As long as Markets have other possibilities, no country can sustain

a Deadly Debt Spiral

Interest Rate Probability Distribution

Normal

Mode Crisis Mode

rcrit VaR

4


What is a Risk Measure?

> A Risk Measure is an Ex-ante measure

• Question 1: “What is the Range of possible Future returns?”

• Answers: Expectations, Value-at-Risk, ex-ante Volatility

Today

Past Future

Observed ex-post

volatility

Future performance

expectation

VaR = Pessimistic

scenario

Optimistic

scenario

Observed ex-post

performance


Random Dynamics

6

Pure

Diffusion

Dynamics

Distribution

of random

dynamics

The Term distribution of a random dynamical system is a “cornuate ellipsoid”


Taleb’s Fragility and Antifragility

> Account for nonlinear impact of stressed variable

> Taleb’s Heuristic: include stress uncertainty

½ {Stress(σ + δσ) + Stress(σ – δσ)} >> Stress(σ)

> Even more so with threshold effects (“Fukushima effect”)

7


Error Skewness

> Model uncertainty impact is asymmetric

• An estimated 1% probability event could be 0.5% or… 10%

> The lower the probability, the higher the skew

• High quantile VaR (e.g. 99.9%) are meaningless

• Yet, a risk measure is meaningful if we consider only two probabilities:

“likely” and “unlikely”

• It corresponds to what may likely happen, using all available info at

current date and time

8

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

3 3.5 4 4.5 5

s = 1

s = 0.8

s = 1.25


Error Skewness

> Implied volatility smile in stochastic volatility models

½ {BS_Call(K,σ + δσ) + BS_Call(K,σ – δσ)} = BS_Call(K,σ’)

σ’ > σ if K is out of the money

9

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

3 3.5 4 4.5 5

Call Price Black-Scholes

σ = 1

σ = 0.8

σ = 1.25

σ average


Statistical Undecidability

> Statistical Process

• Distribution estimation from data: Maximum Likelihood

• Model comparison tests (p-value) and selection, with assumptions on

uncertainty

• Decision criterion computation: Expected utility, hedging cost, risk

measure, capital adequacy…

• Model error: inverse Hessian of MLE => Error on criterion

10

MLE MLE Uncertainty


Statistical Undecidability

> Proper process

• Assume Meta-distribution in the Space of probability measures µ

• Decision criterion F(µ) has an a priori distribution

• Conditional distribution of F(µ), given data (+ info…)

• Conditional expectation and uncertainty on F(µ) is very different from F(E(µ))

and computed error from MLE Inverse Hessian

> Statistical Decidability

• Conditional expectation of F(µ) to given data exists and “continuously” depends

on data sample

• Conditional distribution of F(µ) converges to Certainty (“Dirac mass”) when the

sample size tends to +∞

> Conjecture:

• If Meta-distribution support is not locally compact (i.e. depends on a finite

number of parameters), then the problem is generally Statistically Undecidable

11


Myopic Risk Estimation

> Probability distribution using recent past

• Asset Value = Discounted Expectation of future Cash Flows

• Derivatives depend on Volatility Estimates

• Credit spreads depend on Solvability and Asset value estimates

> Procyclicality

• Regulatory procyclicality:

Drop in asset value => Increased put price => Increased volatility => Increased

VaR => Sell-off => Drop in asset value…

• Minsky Instability Hypothesis

> Drop in asset value => Downgrading => Increased interest rates => Increased

debt burden => Solvability at stake => Downgrading => Drop in asset value…

• Procyclicality is inherent to Marked-to-Market!

• Instability is necessary for economic efficiency

12


From a systemic stand point, liquidity risk is the accumulation of

“hidden market risks” which leads to a global liquidity crisis

Liquidity Risk

Price

# of bidders

DEMAND

SUPPLY

Unexpected

behavior of an

asset price

Change in asset

liquidity

13


Market Discipline

14


MEASURED RISK

HIDDEN RISK

Optimization

Maximizes the

Ratio of Hidden/

Measured Risk

MEASURED

RISK

HIDDEN RISK

Real Risk

15

Optimizers Failed, However Advanced...

Expected

Return

Risk

MAX

MIN

Optimizer

Optimizers, however sophisticated, simply maximize expected return whilst minimizing measured risk. They

therefore by design maximize the proportion of the risk they can’t measure – the hidden risk – leading

automatically to portfolios which eventually deliver very nasty surprises….


Capital Allocation Medium Performance Medium Risk Control

Risk Allocation Weak Performance Risk Control OK

Markowitz Performance OK Bad Risk Control

FOFiX Performance OK Good Risk Control

16

Pay for “Hedging Costs”, Relaxing

“Business as Usual” Risk Constraints

___

___

___

___

Track Record of Quant Driven Portfolios With a 3% Tail Risk Budget

100

105

110

115

120

125

130

135 Capital Allocation

Risk allocation

FOFIX

Markovitz

Cash (Risk free rate)

.

An additional risk

premium can be

extracted by

relaxing “normal”

risk constraints.

An extreme risk

budgeted

portfolio (FOFIX)

over-performs

Markovitz in

business normal

conditions

despite exhibiting

a lower shape

ratio, simply

because they

have a higher

volatility.

While in troubled

times, Markovitz

collapses while

FOFIX stands up

well.


What are you looking for?

Did you lose

your key there?

No, on the

other side, but

here

I have light!

17

Copyright © 2009 Riskdata. All rights reserved. 18

Credit driven fund:

• Long AAA bonds, Short T-bonds, duration 10Y

95

100

105

110

115

120

125

130

135

Sharpe = -0.25



VaR 99 = 3.5% (3.5 sigma)


Skew = -1.0


Could such a loss be anticipated, only looking at the past fund performances?

Yes with nonlinear factor analysis

Pure Return-Based Analysis


Factor Analysis

19

Credit driven fund vs. AAA spread over T-Bonds:

• This fund was just surfing the good wave during the analysis period

The fund returns mostly depend on the AAA credit spread, in a nonlinear (optional) way.

The grey curve is obtained by cumulating this nonlinear function of the credit spread changes over the years.

This leads us to the way extreme risk can be anticipated, through the concept of STRESS VAR.


Factor Analysis

20

> Credit driven fund vs. AAA spread over T-Bonds:

• The driving factor experienced in the past many jumps

comparable to the crisis

One can see that the loss experienced in 2007 had several similar precedents. The loss of the

fund is in line with its Stress VaR, which itself is derived from “extrapolated” losses of the fund,

prior to its actual track record.


The Data Wall

> 10,000+ Hedge Funds

• A few years of history => 10’s of returns

• Unreliable, incomplete, delayed, fast changing positions

• Large variety of strategies and trading universe

> 10,000’s Market Factors

• All asset classes

• Long term history, including many crises, cycles

• Hedge Funds often uncorrelated to Markets

• Correlation only appear during Crises

> Too many Models, too few Information

> IMPOSSIBLE TO SELECT AND CALIBRATE A MODEL


Poly-Models

A Collection of Single-Factor Models

> Step 1: Identify a LARGE Set of Factors

• LONG HISTORY (20 Yrs incl. crises)

• As many factors as potential Risk Sources Several 100’s

> Step 2: Scan Factors One at a Time

• Select only factors with a strong statistical relationship with the fund Score

• Focus on EXTREME MOVES Nonlinear Models

> Step 3: Stress Selected Factors

• Information Ratio = Impact of Factor / Uncertainty

• Merge single-factor models to maximize Information Ratio

22

Poly-models are aimed at breaking the “data wall”. The major innovation is in the way the distribution of future

returns is estimated; using a very long history of markets in order to include past crises, a large number of factors

in order to account for all possible risk sources and a collection of nonlinear models in order to account for

extreme risks and, in particular, the impact of liquidity gaps. The short fund history is optimally used.


Poly-Models

> Multi-factor Model

Fund = 1 Fact1 +…+ n Factn +

• Coefficient i are fixed

• Factor set {Fact1,…,Factn} is frozen

> Poly-model: Collection of models:

• Linear: Fund = i Facti + i i = 1…n

• Nonlinear + lags:

Fund = i(Facti) + i(Facti(t-1)) + i Fund(t-1) + i i = 1…n

• Score each model by relevance in Extreme scenarios


Poly-Models

> Relation with Multi-factor Models: the Linear case

• Fund = i Facti + i i = 1…n

• Fund = 1 Fact1 +…+ n Factn +

• <Fund, Facti> = i Var(Facti) = j <Facti, Factj>

• (1,…, n) = Cov(Fact)-1 (1,…, n)

• The uncertainty on I’s depends on factors colinearity

• Badly conditioned covariance matrix Low Information Ratio

> Nonlinear Modelling

• Hermitte Polynomials Hk: i(Facti) = ik Hk(Facti) + i

• Nonlinear Multi-factor model by inverting Cov(Hk(Facti))

• Improve Information Ratio with LOESS Regression


Poly-Models

> Model Selection

• For each subset of indices I = (i1,…,iq), merge models as above

• Compute the Information Ratio = Merged Impact / Uncertainty

• Find the subset I with the highest Information Ratio

> Stepwise Regression

• Find the factor Facti1 with highest Information Ratio

• Take this factor as given. Find the second factor Fact i2 such as, jointly

with Facti1, the Information Ratio is maximum

• Repeat until the Information Ratio cannot be increased

• Try to remove factors while increasing the Information Ratio

• Stop when it is not possible to add, nor remove factors


Poly-models

> Handle 100’s of Market Factors

> Model Rare Events (“Black Swans”)

> More accurate when needed than when not needed!

• Tail concentration effect

> Suited for Risk Measurement and Stress Scenarios

• Prediction from individual factors can be merged

• Risk measure = STRESS VaR (worst case) includes Hidden Risks

> Can be aggregated for a portfolio

• Risk Contributions involve Extreme Correlations

• Superior Allocation and Optimization


Wrong Answer: Multi-linear Factor Analysis

y = 0.3684x2 + 0.4048x - 0.0038

R2 = 0.2925

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Funds

FoF

HFRI

Diag

Poly. (Funds)

Fung-Hsieh 7 Factors Model: Misses large negative events

HFR funds returns in Sep 08, Risk analysis as of Aug 08

Actual Performance

Model P

redic

tion


Nonlinear Single Factor analysis

HFR funds Sep 08, Risk analysis as of Aug 08

174 Riskdata factors, adaptive p-value computation, “most relevant” factor

y = 0.8846x - 0.011

R2 = 0.8121

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Pred Sep 08

Diag

Linear (Pred Sep 08)


Can We Anticipate the Impact of Time Bombs?

Actual vs. predicted performances of hedge funds during the 2008 crisis

14% 16% 5%

7%

36%

1%

7% 5%

8%

-35%

-30%

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

-40% -30% -20% -10% 0% 10% 20% 30% 40% Predictions

Ac

tua

l P

erf

orm

an

ce

s

Error of Type 1: Predicted Winner, Actual Loser

Error of Type 2: Predicted Loser, Actual Winner

29

This graph compares the actual performance of hedge funds during Sep-Oct 08 vs. what could have been

predicted by FOFiX’s nonlinear factor analysis (using fund data until Mar 08 only).

Assuming an investor anticipated the market crisis, the set of funds that appeared to be actual losers and

winners was quite predictable.

In the following slides we will see the techniques put in place to reach such a result.


Stress VaR

Combine STRESS TESTS and Value-At-Risk

> Step 1: Identify a LARGE Set of Factors

• 99% confidence interval of each factor based on LONG HISTORY (20 Yrs)

> Step 2: Scan Factors One at a Time

• Select only factors with a strong statistical relationship with the fund

• Focus on EXTREME MOVES

> Step 3: Stress Each Selected Factor

• Measure the impact on the fund

• Use NONLINEAR model

30

99% STRESS VaR = WORST STRESS TEST + SPECIFIC Stress VaR is a risk measure that combines stress tests and value-at-risk. It relies on “poly-models” for the estimation

of the distribution of future returns is estimated. It uses market history that includes past crises, sufficiently many

factors in order to account for all possible risk sources. Nonlinear models capture extreme risks and, in particular, the

impact of liquidity gaps. The Stress VaR unveils hidden risks by identifying drivers of returns


Capital Allocation Medium Performance Medium Risk Control

Risk Allocation Weak Performance Risk Control OK

Markowitz Performance OK Bad Risk Control

FOFiX Performance OK Good Risk Control

31

Pay for “Hedging Costs”, Relaxing

“Business as Usual” Risk Constraints

___

___

___

___

Track Record of Quant Driven Portfolios With a 3% Tail Risk Budget

100

105

110

115

120

125

130

135 Capital Allocation

Risk allocation

FOFIX

Markovitz

Cash (Risk free rate)

.

An additional risk

premium can be

extracted by

relaxing “normal”

risk constraints.

An extreme risk

budgeted

portfolio (FOFIX)

over-performs

Markovitz in

business normal

conditions

despite exhibiting

a lower shape

ratio, simply

because they

have a higher

volatility.

While in troubled

times, Markovitz

collapses while

FOFIX stands up

well.

Documents

Can We Measure Extreme Risk? Should we Measure it? › dev › files › Douady Slides.pdf · Copyright © 2009 Riskdata. All rights reserved. Poly-Models A Collection of Single-Factor