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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

Preview

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

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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

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Clear epochs,with ‘crisis’ in

late 80s.

Consistent with Turing et al.

PerestroikaGlasnost