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Which [type of] Mathematicsfor Financial Resilience?
Dr DJ MarsayC.Math FIMASenior Researcher, ISRS
IMA Conference on Mathematics in FinanceHeriot-Watt
Edinburgh 8th Apr ‘13
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I am a boffin, interested in the actual and potential role of mathematics in connection with crises.
This talk considers finance, linked to Cabinet Office thinking to illustrate the kind of thing that is needed.
“Anyone who believes exponential growth can go on forever in a finite world is eithera madman or an economist.”
Attrib, Kenneth Boulding, Evolutionary economist, 1973
“[T]he financial crisis has shown that … there are serious dangers if fancy mathematical tools are used by people who do not understand their limitations.”
David Spiegelhalter, Mathematics Today, Aug 2012
Finance
Mathematics is not only about:efficient computational models /
algorithmsbut alsochallenging ‘given’ theories.
Mathematics
6
opportunityrisk
happened
has not happened
Recovery!Y2k crash:usefulorganisationalchanges
2008 crash!Due to Big Bang?
End ofCold War!
Big-Bang
A subjective view
Usefuldebates, largelyforgotten
Crises
To recognize and work within the limits of our mathematics, and to expand them.
Preferably, mathematics that:1. Supports immediate, recognized
‘wants’.2. Can be used to address longer-term
issues.3. Is meaningful to clients (e.g. policy-
makers).4. Lacks misleading ‘baggage’.
But where to start?
Aspiration
The Ratio Club
8
Turing, Good, Ashby et al.Donald MacKay, for Ratio Club, 1952
Consistent with Whitehead and Russell. Even Gordon Brown knows who Turing is.
Turing considers a model of potential instability: ∂tq = D q + R(q),where q is a vector of properties, D is a diagonal diffusion matrix and is a (local) reaction.
Without diffusion one has a single epoch with regular oscillations.
Moderate diffusion leads to a variety of ‘emergent behaviours’, with greater impact near critical instabilities, where is degenerate.
One does not always get a new stable epoch: one can get ‘wandering’ behaviour near critical instabilities.
Turing’s Morphogenesis
Turing’s application was to the development of form in biology, which tends to stabilise.
The theory can also be applied to processes, such as evolution, which do not stabilise.
Crises are symptomatic of instability, which is associated with momentum.
We can model this using 2 variables, or equivalently as .
Morphogenesis with Momentum
The theory can be judged, not by its ability to predict values q(t) (which it claims to be impossible), but by the correspondence of scenarios and behaviours:
Correspondence
Morphogenesis
ConditionsDiffusionReactionInertia
Critical instabilitiesEmergent properties‘Wandering’
Behaviours?
Experience
Aim to fit the long-term ‘big picture’, not the short-run fine detail.
With the same initial conditions, one gets a variety of temporary oscillations, plus some ‘wandering’ behaviours.
Diffusion may happen to reduce or increase variability.
Transitions are most likely near the critical instabilities ( 0, i.e. near q=0,1).
time
, q-q3
q
For common stochastic systems, one can form ‘rational expectations’ by extrapolating the probable behaviour without diffusion and adding variability: q(D,t) q(D=0,t) + σ(D,t)
But uncertainty concerning instabilities is not comparable with variability.
So conventional probabilistic reasoning is not appropriate. Can you extrapolate from a caterpillar to a
butterfly? Or do you need ‘scenario analysis’?
Reasoning through instabilities
Turing developed, with Jack Good, a notion of ‘weight of evidence’, with a formula W( E | H : C ) s { s ( W( Es | H : C ) )},for a hypothesis H, a context C, total evidence, E, consisting of evidences Es from sources s, and where each s( ) is a discounting function, depending on confounding factors.
For precise hypotheses, not discounting, with a little manipulation, yields Bayes’ rule: P(H|E)=P(E|H).(P(H)/P(E)),and hence conventional probabilistic reasoning.
But are we justified in making these assumptions? Or can there be emergence?
Reasoning through instabilities
Diffusion, reaction and momentum are present Loss of confidence and panic can spread
quickly. Crises are almost defined by critical
instabilities. Actions have most impact near critical
instabilities. Experience and models from before an
instability can be very misleading (‘reversed levers’).
Outcomes seem emergent, unless bold action eliminates the instabilities.
But exogenous changes in expectation and policy can distort and complicate the picture.
Crises
16
An ‘Exploratorium’
Inspired by analysis for conflict resolution. ‘Synthetic Modelling of Uncertain
Temporal Systems’, based on Smuts’ theory of emergence.
Various UK and Allied Defence and Security presentations 1999-2010
Shows emergence, uncertainty, entropy, ‘abstract indicators’.
Informed UK crisis management.
placid random
turbulent
• internal adjustment• independent activity • closed models
• adaptation at all levels • coherent activity • open appreciation
zone 1 2 3 4 5
“swimmingwith the fishes”
Implications for collaboration
?
An emergent newdefinite context isnot inevitable …and may notbe desirable.
CollaborativeWorking document,2007
Indicators of criticality: Developments (economic, industrial, urban)
seem emergent where no one factor dominates.
Markets are more prone to crises where speculation (momentum / impulse ) is more significant than fundamentals (reaction / propagation).
Conventional methods of analysis and prediction seem unusually unreliable.
Implications: We should recognise that conventional
methods are limited to non-critical cases. It may be our assumptions that turn
emergence into disaster.
Finance and Economics
The Nexus, 2008
To US, UK and PRC, for example, the issues seemed very different.
But they were not just using different models, but were concerned with different types of uncertainty and implicit time-frames.
We need help to situate these different views.
Levels
Epoch horizons can be based on Turing: Short: Until the next instability. Medium: Until the next instability whose
potential is not adequately understood. Long: Seeking to work through instabilities
with uncertain potential behaviours. But further aids seem needed.
Locating the key finance issues?
probabilistic
ambiguous
uncertain
Low data volume,human pace
High data volumefor time available
Efficiency / overloadproblem
Effectiveness /Uncertainty
problem
Most Applicationsof Mathematics
Different Mathematics?
Adapted from IMA ‘Mathematics in Defence 2009’
factual
Current capability
No requirement
Dynamics of Financial Issues
Risks become
apparent.
Typical crisis kinematics
New contextemerges,.
Time
Energy
Extrapolation ‘works’
A
B
CA’
Extrapolation appears to work, but hidden risks are building.
Incoherence
Extrapolationbecoming effective.
Where will any new context come from?Will it be favourable?
Adapted from IMA ‘Mathematics in Defence 2009’
Are Bubbles a good metaphor?Baggage?
+ What are the limitations of conventional methods?
What happened?
2004
Q2 2004
Q3 2004
Q4 2004
2005
2005
Q3 2005
Q4 2005
2006
Q2 2006
Q3 2006
Q4 2006
2007
Q2 2007
Q3 2007
Q4 2007
2008
Q2 2008
Q3 2008
Q4 2008
2009
Q2 2009
Q3 2009
Q4 2009
2010
Q2 2010
Q3 2010
Q4 2010
2011
Q2 2011
Q3 2011
Q4 2011
2012
Q2 2012
90
95
100
105
110
115
120
125
130
Index (2004=100)
15%
Source: Thomson Reuters
Simple linear trend used for pre-crisis trends
But could change ofgovernment been a factor? Or EU? …
No ‘negative bubble effect’.
Fits Turing et al.
Outline
Evolutionary fitness:The concept of Resilience to
Crises
Evolutionary patterns:Learning the lessons of history?
Shock, surprise & irresilience:Managing neutered risk is not
enough
Evolving strategies:Real options for gaining traction
in time
A Cab.Off. Perspective
Clear links to mathematical issues?
IMPACT ASSESSMENTVULNERABILITY THREAT
POLICY
ENDS
MEANS
Value Loss
Cost & Criticality
Probable & Expected
Weakness
What & Which
Exploitation
Who & How
Exposure
Where & When
Capability
What & Which
Opportunity
Where & When
Intent
Who & How
InformationRequirements
InformationRequirements
COUNTER-MEASUREASSESSMENT
Cost Benefit
Contingencies
Conventional
risk analysis
Atomisedcontrol freakery
left, right & centre
A C.O. Perspective, 1999 --
Mathematically,This seems to assume a single epoch.
LearningMyopia
Exploitationsqueezes outExploration
EXPLORE
Diversity Select & Grow Variable Durations
EXPLOIT
Fixed timescales Niche specialists Reward narrow
Rational behaviour
Inte
racti
ve
Kn
ow
led
ge
Value?
Blend of Investments(BoI)
&(RoI)
Return on Investments
Yield
Thwartingpeacetime
transformation
A Cab.Off. Perspective, 1999 --
Epochs?
Resilience Issues
Crises have their own imperatives.
Our mathematics and scientific approach should not get in the way, and ideally should help.
As a minimum, we need to appreciate our limitations.
Resilience Issues?
If Whitehead, Turing et. al. are right, resilience also relies on: Identifying critical instabilities. Appreciating which data and judgements to
trust or discount, when. Recognition that the ‘mechanism’ can change.
And, ideally, on reasoning through instabilities: Reformed concepts of uncertainty, value etc.
Appreciating the nature of crises and of uncertainty as well as normal domain expertise. E.g., by recognizing or creating an extra
dimension.
Financial ‘prediction’
Critical instabilities matter in the medium-term
There may be no ‘regression to the mean’.
Source: Bank of England Inflation Reports, 3/13.
Volatility, ‘tail-risk’ and fan-charts are only meaningful in the short-term.
Source: Bank of England Inflation Reports, 2/07.Rescaled to match 3/13.
Towards Sustainable Recovery?
Estimation of ‘the multiplier’ (growth/cuts) Discount evidence before 2009 (different
epoch). Look at which other country’s data? (‘Frame’)
Is there potential to transform the economy? Could Brown’s Green agenda have
transformed? Is there scope for initiatives around inequality?
(E.g. German-style boards.) What scope for regulation?
E.g. should the government promote understanding of speculation and crises? Would this be enough?
Conclusions
Mainstream financial and economic mathematics have myopically focussed on ‘exploit’. To inform policy, they need to address ‘explore’ as well.
The mathematical framework of Turing et al provides a good ‘seed’, with worked examples.
‘Understanding limitations’ would be a good start.
Developing a reformed concept of ‘value’ might be a good aspiration.
But the issues are broader than just finance.
Dr DJ Marsay C. Math FIMASenior Researcher, ISRSdjmarsay@ucl.ac.uk, djmarsay.wordpress.com, LinkedIn.
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Comments & Questions?
Which type of Mathematicsfor Financial Resilience?
IMA Conference on Mathematics in Finance
Heriot-WattEdinburgh 8th Apr ’13
Graphic from Peter Allen, Cranfield
Taking Whitehead and Russell seriously. Normal approaches are only valid in stable ‘epochs’.This limitation is no longer so appreciated.
A general ontological approach (1979)
Empirical Support: Conflict Deaths 1945 - 2005
36
Clear epochs,with ‘crisis’ in
late 80s.
Consistent with Turing et al.
PerestroikaGlasnost
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