risk management in public institutions is different...

RisKontroller Global LLCShaping the future™

FormerlyRisKontrol Group GmbH

Risk management in public institutions is

different; everyone is different in its own way.

The implications for managingThe implications for managing strategic ALM in turbulent timesAnd what to do about it

Prof. Dr. Jerome L Kreuser - CEO and Founderhttp://RisKontroller.com

[email protected]

Contents and Direction What does it mean “everyone is different” What does it imply about the mathematical approach Compare some approaches Importance of trees and solutions Define stochastic processes and trees keeping in mind Define stochastic processes and trees keeping in mind

generating “real” problems Shaping multiple distributions.p g p Examples

Always keeping in mind tractable real problems(You can’t always just scale up.)

2Kreuser – ETHZ Risk Center – 11 March 2014

Thi f k i i t d i h j t I h d d t

Some HistoryThis framework originated in a research project I headed at the World Bank for sovereign risk management combining macro and micro economics and finance in a multi-period dynamic stochastic programming frameworkdynamic stochastic programming framework.

Clients/discussions/seminars: UNCTAD UNCTAD Central Bank of Colombia Brazilian hedge fund Central Bank of Jordan South American Central Banks - FLAR Reserve Bank of India Norway SWF Reinsurance companies Studienzentrum Gerzensee – SNB training central banks Many international presentations and workshops Disc ssions ith MAS and GIC

3

Discussions with MAS and GIC

Kreuser – ETHZ Risk Center – 11 March 2014

Example: central banks reserves management

Obj i d i hObjectives concerned with

More than just Safety Liquidity and Returns More than just Safety, Liquidity, and Returns Ratio of reserves to short-term-debt Preserve real purchasing power Minimize cost of reserves Controlling for contingent liabilities Si f Size of reserves Strength of balance sheet for stability of external value of

domestic currency Assets to exceed scheduled amortization of external debt over

next 12 months(Greenspan-Guidotti Rule) May also manage debt

4

May also manage debt


Sovereign wealth funds and pension funds

SWF objectives concerned with Usually long-term horizon Preserving real purchasing power Preserving real purchasing power Country welfare Inflation C diti i

Defined benefit pension funds Maximize returns

Commodities prices Alternative assets Spending rule

Meet liabilities Stability of contributions Limiting downside funding ratio

Counteract fiscal deficits Mitigate macroeconomic volatility Hedge commodity depletion

g g Limiting downside of contributions Account for changing pensioners LDI

Commodity price stabilization Limit downside tracking error

LDI


GIC’s mission is to preserve and enhance the international purchasing power of the reserves placed

Their missions GIC s mission is to preserve and enhance the international purchasing power of the reserves placed under our management by the Singapore Government. The aim is to achieve good long-term returns above global inflation over the investment time horizon of 20 years.

Teacher retirement system of Texas: The investment portfolio includes all assets invested by TRS to provide retirement, and related death, health, and disability benefits under the pension plan as well as health p , , , y p pbenefits or other services …. The pension trust portfolio is …to achieve the following objectives:Control risk …and by establishing long-term risk and return expectations; and…, achieve a long-term rate of return that:

Exceeds the assumed actuarial rate of return adopted by the Board;Exceeds the long term rate of inflation by an annualized 5%; and

The Government Pension Fund Global (The Fund) shall be managed subject to the constraints

Exceeds the long-term rate of inflation by an annualized 5%; andExceeds the return of a composite benchmark of the respective long-term normal asset mix weighting of the major asset classes.

set out below and in accordance with NBIM’s strategy plan. The objective shall be achieved in a controlled manner. The Fund should be invested to improve the risk-return relationship and exposed to different systematic risk factors.

“SWFs and other long-term funds come in many flavors There is no single optimal governance frameworkSWFs and other long-term funds come in many flavors. There is no single optimal governance framework or performance benchmark to adopt. In constructing the appropriate benchmarks for your fund, you must approach the exercise both broadly, from a political economy and government strategy point of view, and more narrowly, from a financial standpoint.”

6

(2011-11-15). Sovereign Wealth Funds and Long-Term Investing (p. 93). Columbia University Press.


F l t t t i i

The criteria for our framework Focus on long-term strategic issues. Implement multiple institutional objectives. Implement shaping of distributions. Future rebalancing => multi-stage Mix factors in macroeconomics and finance. Integrate assets liabilities derivatives alternatives cash flows etc Integrate assets, liabilities, derivatives, alternatives, cash flows, etc. Estimate changing stochastics, correlations, etc. Extremely flexible and open models Hedge against bubbles, crashes, regime changes, etc. Dynamic uncertainty structure (=> trees) generated using history,

theories, implied prices, expert views, behaviors, etc., p p , p , , Easily translate “executive speak” into “risk speak”.

Need a tree based uncertainty structure with dynamic rebalancing

7

y y g


Basic model

( ) ( )min

,PE f x ω( ) ( ),P fx X P∈

1.P is distribution of random ω

2.X is nonempty closed set of decisionsp y

3.P does not depend on x

4. f is random objective of loss or cost

5.From now on we drop expectation E from notation.

8

p p


Multi-objective problemu t object e p ob e

Suppose now we have K functions pp

An efficient solution of the multi-objective x̂problem is one such that no x exists with

ˆ( ) ( )f x f xω ω≤ ( , ) ( , ) ˆand ( , ) ( , )

i i

i i

f x f xf x f x iω ω

ω ω≤

≠ ∀

Generally, this is not what we want because not all objectives are created equal rather we want

9

objectives are created equal, rather we want …


To consider two basic options

( )1

min,f x

xω

P bl I

( , ) 2,3,..., k k

xf x k K x Xω ε≤ = ∈

Problem IStochastic constrained problem

( , ) 1, 2,..., k kf x k K x Xω ε≤ = ∈

pSCP Or

min( , ) k kp f x x X

xω ∈

Problem IIStochastic shortfall kx shortfall objectiveSSO


Some observations

1.The Pareto optimal case is generally not used

2.Useful if f is convex

3. f may be linear or nonlineary

4.We compare Problems I (SCP)and II (SCO)

5 I th f th d ti f P bl I5. In the case of the second option for Problem I:

min0 0

,( , ) 1, 2,...,

k k kk

k k k

p t tx tf x t k K x Xω ε

≥

− ≤ = ∈


Example: Stochastic shortfall objective Problem II

( ), ,MAX t t t e t eW penalty SHFδ π − Γ

( )

tt T e X∈ ∈

, ,

, 0, ,

t e t t e

t e t

SHF target TWSHF t e e X

≥ −≥ ∀ ∋ ∈

Ziemba


Compare methodsRisk constraints (SCP) Shortfall in obj (SSO)

Feasibility Can be initially infeasible No infeasible

Executive Confidence level and limits Difficulty in picking andExecutiveunderstanding

Confidence level and limits are clear

Difficulty in picking and interpreting weights

Shape distribution By confidence level or CVaRlimit value

By changing weight ornonlinear functionlimit value nonlinear function

Linearization Multiple constraints with different confidence levels

Piecewise estimation of nonlinear function

Nonlinear – increase marginal penalty with increasing loss

Loss functions can be nonlinear

Combinations of nonlinear functions

Linear Works well with linear loss Can be unstable if allLinear Works well with linear loss => bigger trees

Can be unstable if all functions are linear

From now on I will focus on SCP

13

From now on I will focus on SCP


What characterizes a good result?

Good solution today Well shaped densities Well shaped densities

Its all in the shape!Its all in the shape!

Decisions are made basedU th h

“The problem with this approach [like M-V] is that the return distribution

Upon the shape.

The problem with this approach [like M V] is that the return distribution implied by a particular portfolio at one point in time may be quite different at another point in time – and investors want the distribution, not the portfolio.” Kritzman (Windham Capital Management, LLC and MIT, Sloan School),

14

( p g , , ),2013 on Risk Disparity


How to grow a great treeM th dMethods Conditional sampling and importance sampling Sampling from specified marginals and correlations Moment matching (often unfairly disparaged)g ( y p g ) Path-based methods and bundling Optimal discretization of the entire tree

Create from stochastic processes

Let the processes do the talking

Capture the important stuff

Get good pdfs

15

Get good pdfs


The dreaded “curse of dimensionality”The “curse of dimensionality”: It is usually assumed that only a tree of a few factors can be built. The reality is that using:factors can be built. The reality is that using: scenario reduction, large-scale solvers, ffi i t t h i t t t defficient techniques to generate trees, and large workstations a large number of factors can be included.g

Example: A tree of 27 factors over three periods requires a binary tree of 812scenarios.

Can’t evaluate over the lifetime of the How to go about it?

16

universe How to go about it?


( )d t

The stochastic process

,( ) ( , ) ( , ) ( , ) ( )( )i

i i j j jji

ds t s t dt b s t s t d ts t

μ σ ω= +

0 2 2, ,

1 1

1( ) exp ( ) 1, 2,...,2

L L

i i i i j j i j j jj j

s t s b t b t i Lμ σ σ ω= =

= − + ∀ =

{ } ( )1 20 0 01 2( ) , , , Lt t t

nE s t s e s e s eμ μ μ=

( ) { } { } 2, ,

1cov ( ), ( ) ( ) ( ) exp( ) 1

L

l k l k l j k j jj

s t s t E s t E s t t b b σ=

= −

( ) { }( )2 2 2,

1var ( ) ( ) exp 1

L

l l i j jj

s t E s t t b σ=

= −

Kreuser and Wets

17

and Wets


Estimating processes

Assume are constant over intervals, ,bμ σ

( ) ( )( ) (0)exp Let (0)i i i i iE s t s t sμ θ= =

( ),( ) exp 0,1,...,i l i l i ls t t l kθ μ η= + ∀ =

( )ln 0,1,...,i li l i l

s t t l k iη μ

= − = ∀ , ln 0,1,..., i l i li

t l k iη μθ

∀

Kreuser and Wets

18

and Wets


Estimating the stochastic process

Let and solve lt lh=

g

2min ( )ln

ki l

is t lh iμ

− ∀

0 ln

,

With

ili i i

lh iμθ μ θ=

∀

( ) ( )With

exp , i i iKh f K K kθ μ μ= >

Correlations – estimated similarlyDifferent lookback length? Brownian Bridge and time weighted Kreuser

and Wets

19

Different factors? Scaling removed and Wets


Estimating short term fundamentals using theoriesTransient estimation

Estimating short-term fundamentals using theories, expert views, implied prices, behaviors, etc.

160

180

ex

Transient Process Estimation

120

140

ic T

otal

Ret

urn

Inde

Anchor it in the future

80

100

STO

XX A

sia

Paci

fi

Actual ValuesExpected Value LineFuture Expectations

Jan07 Apr07 Jul07 Oct07 Jan08 Apr08 Jul08 Oct08 Jan09 Apr09 Jul09 Oct0940

60

Ti

DJ

± 1 SD

20

Time


Long-term estimation

fundamental (long-term) value, bubbles, theories, etc.

Stationary Estimation

180

200

ex

Stationary Estimation

Actual ValuesExpected Value LineFuture Expectations

140

160

Tota

l Ret

urn

Inde

p

± 1 SD

100

120

TOX

X A

sia

Paci

fic

60

80

DJ S

T

21

n98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04 Jan05 Jan06 Jan07 Jan08 Jan09 Jan10 Jan11 Jan12 Jan1360

Time


Combining short and long


Build tree from processes: issues to consider

Stability←Size ←

Capture important “stuff” ←

ArbitrageBias

Design good branching strategy (time and Bias

Quality Control

gy (state) ←

Keep tree small enough


Matching moments Spreadfunction

( ) 2 2, , 1 , 2 ,

,, ,

, , k n k n k k k l

k n k k ln k n k l

MINf u d d

u dα α

ρ ≠

− + + function

( )

( )

,

2 2 2

exp

exp 2 exp 1

n k n kn

L

u t

u t t b d

ρ μ

ρ μ σ

=

= +

( )

( )( )

, , ,1

2

exp 2 exp 1

exp( ) 1

n k n k k l l k kn l

L

n k n l n k l l j k j j k l

u t t b d

u u t t b b d

ρ μ σ

ρ μ μ σ

=

− = − +

− + = − +

( )( ), , , , ,1

p( )

1

n k n l n k l l j k j j k ln j

n

ρ μ μ

ρ

=

=

( )3 2k kn

+ +≥

0 1 n

nρ≤ ≤ , 0

k n

N L N

u

R u Rρ ×

>

∈Ρ ⊂ ∈Μ ⊂

( )2 1n

k≥

+

For k=27 n≈15For k 50 n 26

24

For k=50 n≈26


Readable by computer and

What is a model?computer and human - OPEN

Easy and efficient toEasy and efficient to create/modify and separate from tree

Optimal objectivesInequalities

LinearNonlinearIntegerComplementarityComplementarityEquilibriumRobustSpecial structures

25

Special structures


( ) 0 and for somee e e tZ X ALPHA Z X t≥ Λ ≥ ∀ ∈

Multiple objectives and shaping via CVaR constraints

( ) , 0, , and for some

and

e e e tZ X ALPHA Z e X t≥ Λ − ≥ ∀ ∈

is a convex loss function

( )( ), 1 t

t e e

e X

Z clevel ALPHAρπ ρ∈

≤ − −( )eXΛ ,t eπ is a convex loss function

is the probability of event e at time tALPHA is the determined VaR valueρ is the confidence value

( )XΛ π,t eπ

ρ is the confidence value is the level for confidence valueclevelρ

Rockafellar and Uryasev

The directional derivative of the second constraint provides its marginal value.

CVaR constraints are generally cheap => you can have many => you can shape distributions with them

26

The directional derivative of the second constraint provides its marginal value.


Density crusher for shaping

( )1

1MAX t

t tt t t t

t tr

W pr q pE δ − − Θ ( )

1

2t

t trt T r

q p∈ −

for 0λ λ

≤

Push probability mass into [p,q]

( ) ( )2

for 01

for 0 1 2e

ee

λ λλ

λ λ λ

≤ −Θ = − ≤ ≤

[p q]

0 5

1

1.5

2

2.5x 10-3 Density Function Estimator for New Process

1 for 12

ee eλ λ− + ≥ 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

x 104

-0.5

0

0.5

Wealth in basket terms

2

2.5x 10

-3 Density Function Estimator for New Process

1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9-0.5

0

0.5

1

1.5

Kreuser

27

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

x 104 Wealth in basket termsand Wets


Estimating pdfs by maximum likelihood estimation

1

min ln ( )

th t ( ) 1

t

qk e

kke X

qk

u

d

ϕ ξ

ξ ξ

=∈

−

Estimation works well even for small number of e

d t1

so that ( ) 1

( ) 0, ,

kk

kq

kk

u d

u

ϕ ξ ξ

ϕ ξ ξ

=

=

≥ ∀ ∈

as opposed to other methods.

Also non-1

1 , 1,

kq

kk k

ku A u kϕ

=

=

∈ ∈ = negativeityconstraints are cheap

Dong and W t

28

Wets


Sample estimation and modifications INITIAL AFTER SHAPING


Different regimes via mixture models

800

1000

1200

h

Scenarios from Stochastic Solution

Generate crash trees Q

See, for example, Tascaand Battiston, 2014, LSE

Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan160

200

400

600

Periods

Tota

l Exp

ecte

d W

ealthGenerate crash trees

or regime changes

Graft or join trees at root

Q

Graft or join trees at root

P“Base” tree

New distribution

P

We also have New distribution

(1 )D P Qλ λ λ= − +multi-normalshere


Bubbles, crashes, regime changes

1. Estimate long-term price Using Sornette Indicators™ etc.

2. Estimate future crash distributions

3. Estimate distribution of crash size over future times

4. Generate regime change tree and recoveries

crash size over future times

5 G ft t “ l” t

31

5. Graft to “normal” tree


Framework successfully projects dip and mean-reversion on 5 Sep 2008!Solution saves US$ 20 billion

In 6 months

800

900Scenarios from Stochastic Solution

Model run 5 Sept 2008

100

200

300

400

500

600

700

FTS

E A

W A

ME

RIC

AS

- P

RICE

INDE

X

Compare estimation onMarch 2012

Jan08 Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18 Jan19 Jan20 Jan21 Jan22 Jan23 Jan24 Jan25 Jan260

Periods

Actual = 3,496 billion NOK

Model expectation = 3,476


RBI Model

RBI ModelRun 2009Success still in 2013

Branches areBranches are NOT equal probability


RBI objectives Lower limit on the size of reserves – US$ 200 billion, adjusted

for nominal GDP growth(%) Lower limit on the ratio of NFA to (NFA+NDA) Lower limit on the ratio of NFA to (NFA+NDA) Upper limit on the (%) fall in value of reserves in any period in

US dollars. Mark-to-market value of reserves not to lag behind the

expected value, as measured in the composite currency F i h ld d h i i f Foreign currency assets should exceed the amortization of

external debt over the next 12 months Ratio of short-term external debt to reserves should not

exceed a pre-set level Limit the liquidity at risk

All expressed as CVaR constraints at every

b l i i t 21

34

rebalancing point = 21


Estimated (from Jan 2009)versus actual


Systemic RisksPotential threats to financial stability for U.S. financial systemPotential threats to financial stability for U.S. financial system

OFR Testimony on “Monitoring Systemic Risk”

Disruptions in wholesale funding markets, such as repurchase agreements, or repo.

Exposure to a sudden unanticipated rise in interest ratesExposure to a sudden, unanticipated rise in interest rates. Exposure to shocks from greater risk-takingExposure to a sudden shock to market liquidity. p q yExcessive credit risk-taking and lax underwriting standards. Operational risk from automated trading systems, such as

high-frequency trading.Pressure on emerging-market currency and asset markets.


Testing Model Risk and Stability

Stability via sensitivity to tree perturbations

Convergence of tree as branches get large

Measure marginal impact of scenarios and risk

constraints

Stability can be analyzed via point-to-setStability can be analyzed via point to set

mapping and Variational Inequalities

Analyze from point of view of finite mixtureAnalyze from point of view of finite mixture

models


SummaryWe have framework for: long-term strategic issues. Integrate ALM etc.

flexible and open models Mix factors Able to hedge bubbles, crashes, Integrate ALM etc.

changing stochastics multiple institutional objectives. h i f di t ib ti

Able to hedge bubbles, crashes, Dynamic rebalancing “executive speak” to “risk speak”. M d l d t iti it shaping of distributions. Model and component sensitivity

Why aren’t comparable frameworks ubiquitous?

“For too many banks, it still makes more sense to risk failing conventionally than try to succeed unconventionally.” , Mohamed El-Erian 15/2/14 FT

“A sound banker is not one who foresees danger and avoids it, but one who, when he is ruined, is ruined in a conventional way along with his fellows, so that no one can really blame him.”, John Maynard Keynes 1931


Some referencesKreuser, Jérôme (2012) “Correcting the Financial Crisis Failures of Asset–Liability Management (ALM) Risk Management”,

in Bob Swarup (ed). Asset Liability Management for Financial Institutions: Balancing Financial Stability with Strategic Objectives. London, UK: Bloomsbury Information Ltd, May.

Kreuser J L (ed ) 2012 Risk Management for Sovereign Institutions: Innovations in strategic risk management for volatileKreuser, J.L. (ed.), 2012. Risk Management for Sovereign Institutions: Innovations in strategic risk management for volatile times, The Marketing & Management Collection, Henry Stewart Talks Ltd, London (online at http://hstalks.com/main/browse_talks.php?father_id=670&c=250 )

Claessens, Stijn and Jerome Kreuser (2010). "Strategic Investment and Risk Management for Sovereign Wealth Funds”, BIS/ECB/WB Meeting November 2008 in Central Bank Reserves and Sovereign wealth Management edited by ArjanBIS/ECB/WB Meeting November 2008, in Central Bank Reserves and Sovereign wealth Management, edited by ArjanB. Berkelaar, Joachim Coche, and Ken Nyholm, 2010, Palgrave MacMillan, Hampshire, England.

Claessens, Stijn and Jerome Kreuser (2007) “Strategic foreign reserves risk management: Analytical framework”, in Financial Modeling, Annals of Operations Research Volume 152: 79-113, Springer, Netherlands.

Claessens Stijn and Jerome Kreuser (2004) “A Framework for Strategic Foreign Reserves Risk Management ” in RiskClaessens, Stijn and Jerome Kreuser (2004). A Framework for Strategic Foreign Reserves Risk Management, in Risk Management for Central Bank Foreign Reserves, European Central Bank Publication, Frankfurt am Main, May.

Claessens, Stijn, Jerome Kreuser, Lester Seigel, and Roger J-B Wets (1998). "A Tool for Strategic Asset Liability Management," World Bank Working Paper, Research Project Ref. No. 681-23, World Bank, Washington, DC, March 12 1998March 12, 1998.

Bhattacharya, Himadri, Jerome Kreuser, and Sivaprakasam Sivakumar. “A sovereign asset–liability framework with multiple risk factors for external reserves management—Reserve Bank of India.” in Portfolio and Risk Management for Central Banks and Sovereign Wealth Funds, Edited by Joachim Coche, Ken Nyholm and Gabriel Petre, Palgrave Macmillian, New York 2011

39

New York, 2011.


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