Nested Stochastics in Life Insuranceb9f3124e-8a75...This analysis requires a deep understanding of...

Making you safer.

Nested Stochastics in Life Insurance

Annual Meeting of the Swiss Association of Actuaries, AFIR Section

Martigny, August 31st, 2012

Dr. Urs Burri

Leiter Aktuariat & Steuerung

Basler Leben AG

Basler Versicherung AG

What is “nested stochastics” and why do you need “nested stochastics”

The crucial steps to make nested stochastic simulations feasible

Accurate fast models increase the transparency of the risk exposure

Why “nested stochastics” is the most efficient method for solvency and ALM

Embedding all solvency calculation techniques in a unifying framework

www.baloise.com

Core of all modern valuation frameworks: MCEV engine

2

Fund Mgt decisions

Bonus philosophy

PH-behavior

Regulatory constr.

Assets

Liabilities

Market consistent valuation

e.g. MCEV

IFRS/Local GAAP/Business plan

Fair value balance sheets for:

SST and Solvency II

Dynamic model

Various interactions/Loops

Stochastic calculation

Complex

Properties: Accurate calculation of all embedded

options and guarantees

Slow

Economic scenarios

Financial data / Assets

Policy data / Liabilities

Biometric parameters

…..

August 31st, 2012 Aktuariat & Steuerung/Nested_stochastics_in_life_insurance.ppt

www.baloise.com

Core of all modern valuation frameworks: The Baloise case

3

Fund Mgt decisions

Bonus philosophy

PH-behavior

Regulatory constr.

Assets

Liabilities

Prophet Asset Liability

Strategy Library (ALS)1

Market consistent valuation

e.g. MCEV

IFRS/Local GAAP/Business plan

Fair value balance sheets for:

SST and Solvency II

Realistic management decision rules with

good backtesting results

ALS with monthly exact "external

liabilities1"

e.g. individual life: 48 dynamic segments

Properties:

Very slow: MCEV calculation (5'000

simulations)

- individual life on 70 CPUs: 8 h

- group life on 70 CPUs: 20 minutes

Economic scenarios

Financial data / Assets

Policy data / Liabilities

Biometric parameters

…..

1 SUNGARD iWorks Prophet


www.baloise.com

Projection of market consistent balance sheets

4

t = 0 t = 1

… Real world

scenarios S

S1

S2

SN

0

2000

4000

6000

8000

10000

12000

Nu

mb

er

of

sce

nar

ios

D_RBC1

VaR 1%

E.S. 1%

Reasonably accurate calculation of

expected shortfall (1%) requires about

100'000 ("outer") real world scenarios.

Runtime of calculation becomes prohibitive

ind. life: 100'000 x 500 x 8h/5000 = 80'000 h ≈ 475 weeks

group life: 100'000 x 500 x 1/3 h/5000 = 3'333 h ≈ 20 weeks

Goal: Accurate calculation of the distribution of the RBC at t = 1

Embedded options and guarantees Stochastic calculation

Motivation: Assessment of risk exposure requires distribution of risk bearing capital (RBC)

……

t = 2 … t=40

……

risk free interest FX EUR-CHF credit spread

t = 1

…

…

…

F A

L

F A

L

F A

L

500 ("inner")

valuation

scenarios

10-20 different market risk factors


www.baloise.com

Common approach: Use of replicating portfolio (RP)

5

Requirements:

) RP(L) ue(PresentVal risk

ue(L)PresentVal risk factorfactor

ue(RP(L))PresentVal ue(L)PresentVal

Challenges:

Hard to find good RP2

Criteria for goodness-of-fit, metric2

Overfitting: Subspace of null vectors is huge2

Quality of roll forward of RP(L), quality of RP(L) in tail

Impossible to replicate cash flows of path dependent options (e.g. rolling means, legal quote,

bonus philosophy) with path independent candidate assets

F A

L

Shareholder

Cash Flows

Liability

Cash Flows

Strategy:

Ignore complexity of model (e.g. management rules)

Find replicating portfolio for liability CFs:

Candidate assets: ZCB, puts, calls, swaptions, …

assets candidate RP(L)

Predictive power questionable?

Model validation?

Our conclusion: Establishment of reliability of RP would be a formidable task Look for an alternative

Candidate assets analytically tractable Solvency capital calculation straightforward

2 Presentation J. Crugnola/A. Meister: Stochastic Uncertainties in Market Consistent Valuation


www.baloise.com

Crucial steps to make nested stochastic simulation feasible

6

Try to solve a harder problem!

Find a RP fast tool that replicates all cash flows in every projection year.

Solve the right problem! Do not ignore the dynamics of the “fund”.

In a first step it might help to integrate replicating portfolio techniques into the

dynamic model3

Then think about the run-time-inefficiencies in your models, look at a TEV-

calculation for example…

Analyse your MCEV model in detail

In a further step replace in your “vision“ the word “proxy” by

“perfect analytical / algebraic replication”

3 Case study with SUNGARD

This analysis requires a deep understanding of your MCEV model,

on the actuarial as well as on the soft- and hardware level.


www.baloise.com

Our approach: Construction of a fast model

7

Fund

Mgt Decisions

PH-Behavior

Accounting

Assets

driven by

economic

scenarios

mapped to

ALS classes

grouped

Liabilities

full-fledged

model

Market consistent Balance Sheet

RBC

BEL

MV(A)

speed up

Fast model:

In Prophet

MCEV model:

Prophet ALS

Fund

Mgt Decisions

PH-Behavior

Accounting

Assets

driven by

economic

scenarios

mapped to

ALS classes

grouped

Liabilities Find good basis

functions that

communicate

efficiently with

fund further grouping

yearly cash flows

SH Cash Flows

RBC

Generic framework: accommodates

wide range of applications

Dedicated framework: trimmed to

speed up specific applications

0 10 20 30 40

SH

Ca

sh

Flo

w

Year

MCEV model Fast model

0 10 20 30 40

SH

Ca

sh

Flo

w

Year


0 10 20 30 40SH

Ca

sh

Flo

w

Year


0 10 20 30 40SH

Ca

sh

Flo

w

Year


Full complexity of

path-dependent

embedded options is

retained

"Stochastic commutation

functions"


Aim: Replication of cash flows in every scenario in every year.

www.baloise.com

Quality test of fast model: Group life

8

Comparing sample valuations at t = 1 using the "MCEV model" and the "fast model".

Sampling over the full range of the distribution of RBC at t = 1:

The quality is uniform over the whole range of scenarios from worst to best.

0.0

01

%0

.00

2%

0.0

03

%0

.00

4%

0.0

05

%0

.00

6%

0.0

07

%0

.00

8%

0.0

09

%0

.01

0%

0.0

20

%0

.03

0%

0.0

40

%0

.05

0%

0.0

60

%0

.07

0%

0.0

80

%0

.09

0%

0.1

00

%0

.20

0%

0.3

00

%0

.40

0%

0.5

00

%0

.60

0%

0.7

00

%0

.80

0%

0.9

00

%1

.00

0%

1.1

00

%1

.20

0%

1.3

00

%1

.40

0%

1.5

00

%1

.60

0%

1.7

00

%1

.80

0%

1.9

00

%2

.00

0%

3.0

00

%4

.00

0%

5.0

00

%6

.00

0%

7.0

00

%8

.00

0%

9.0

00

%1

0.0

00

%2

0.0

00

%3

0.0

00

%4

0.0

00

%5

0.0

00

%6

0.0

00

%7

0.0

00

%8

0.0

00

%9

0.0

00

%1

00

.00

0%

D_R

BC

1-

RB

C0

Quantile

Delta of Risk free discount of RBC at t = 1 and RBC at t = 0

MCEV model

Fast model

Run-Times per sample valuation (500 Simulations, 20 CPUs):

• "MCEV model": ca. 7 min

• "Fast model": ca. 0.2 s

Speed-up by a factor of more than 1'000


www.baloise.com

Advantages of nested stochastics techniques:

Modeling Aspects I

9

Conceptually very clean method

Sound economic interpretation of the results

50 Mio. simulations in 12 h instead of 475 weeks (for individual life)

Calculation of distribution of RBCt=1 in 12 h VaRα, ESα straightforward

Validation "fast" versus "MCEV model" in 2-3 days

Calibration, i.e. transformation of "MCEV model" "fast model" with

fast robust algorithm, no "art" or "trial and error"


www.baloise.com


Modeling Aspects II

10

Dynamic model calculation without grouping of model points seems to be

accessible. Main problem: memory (de)allocation

"Fast model" completely equivalent to "MCEV model" (e.g. update of

management decision report with over 160 sensitivities)

The group life model allows for an analytical and economically

interpretable basis of "candidate liabilities" (summarized in 3 pages of

concise formulae)

Whether 50 Mio. simulations are sufficient or not is an inherent problem of

all stochastic techniques. An acceleration of O(1) is always possible by

brute force hardware improvements.


www.baloise.com


ALM aspects

11

The "fast models" are complete ALM-tools

If you change tactic or strategic asset allocation, there is no need

for a recalibration

ALM and SST calculations are "consistent"

Good platform for developing optimal hedging strategies

Efficient "risk dimension reduction methods" can be implemented


Focus on the tail

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Solvency aspects

12

Easy to verify other proxy methods4

Confidence intervals for expected shortfall easily accessible:

Apply standard statistical methods to the tail of the pertinent distribution,

which is conveniently described by generalized Pareto-distribution5

Other model enhancements:

Replace by

= 1, 2, ..., 5 years:

ORSA-like

solvency reporting

= 1, 2, ..., 12 months:

monthly SST

…

1 year 1 year


4 Ambrus et al., Interest Rate Risk: Dimension Reduction in the Swiss Solvency Test 5 Fuhrer, Schmid, On the accuracy of solvency calculations, in preparation

…

www.baloise.com 13

Embedding all Solvency calculation methods in one

unifying framework

• Initial data (Policy and Asset Data):

• Scenario data:

• Contribution of one simulation path to the RBC after one year:

• Tensor Product Decomposition6:

(exact equality may require "infinite sums")

In principle, all "proxy techniques" used for Solvency calculations can be written in this way, e.g.:

SST Standard model ("Delta-Gamma") and other "formula fitting" methods:

RP: For fixed , the are the candidate assets, the are the coefficients

determined by regression:

Our nested stochastics approach: and are both determined algebraically using algorithms.

No fitting of coefficients is involved as in other approaches.

p

p Rmmmm ),,,( 21

ss R ),,,( 21

),(1 mRBC

)()()()(),()( 1ps

j

j

jps RFRFmBcmRBCRRF

...2

1

,

)2()1(

01 lk

kllk

l

ll BrrBrRBCRBC

)()( jj Ac )(mBB jj

j

j

j BAmRBC )(),(1

)(jc )(mB j

Space of functions of the scenario data

Space of functions of the initial data

First derivatives of the RBC w. r. t. the risk factors ("Delta")

Second derivatives of the RBC w. r. t. the risk factors ("Gamma")


6 Stone-Weierstrass theorem

m

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Conclusions

14

Full nested stochastics is challenging but feasible

Prerequisites: Understand your model, your software and your hardware

Once the right problems are identified and solved, the model structure is

preserved in full integrity: Powerful and transparent method for

ALM/solvency considerations

Access to the distribution of RBC opens the door to a plethora of new

applications

Nested stochastics models allow one to address the relevant task: Providing concise information for the management to drive decisions


www.baloise.com

References

15

Case Study SUNGARD,

http://www.sungard.com/~/media/financialsystems/casestudies/iworks_casestudy_baloise.ashx

http://www.prophet-web.com/assets/Datasheets/BaloiseCaseStudy.pdf

J. Crugnola-Humbert, A. Meister, Stochastic Uncertainties in Market Consistent

Valuation, SAV General Assembly, August 28th 2009, Luzern

M. Ambrus, J. Crugnola-Humbert, M. Schmid: "Interest Rate Risk: Dimension

Reduction in the Swiss Solvency Test", EAJ 1 (2) (2011), 159-172

http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s13385-011-0041-1

http://dx.doi.org/10.2139/ssrn.1935378

M.H. Stone, "Applications of the Theory of Boolean Rings to General

Topology", TAMS, 41 (3), (1937), 375


www.baloise.com

Back-up

16 August 31st, 2012 Aktuariat & Steuerung/Nested_stochastics_in_life_insurance.ppt

www.baloise.com

QIS 4 Standardmodell

QIS 5 internal model

SST

internal model

IAS 19

Evolution of valuation frameworks & software @Baloise.Switzerland

17

PVFP Embedded value TEV

publ.

MCEV

publ.

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Local GAAP

LAT Accounting IAS /

US GAAP

IFRS 4

Phase I

..IFRS 4

Phase II

SST

Fieldtest Swiss solvency test SST

Standard Model

Solvency II

Prophet

international Software

Test

Asset/

Global

RP

Tool ALS

ext. liab.

PC Hardware Server 1

20 CPUs

Server 2

24 CPUs

Server 3

32 CPUs

deterministic Mathematics stochastic

calc.

market

cons. val. RP nested

stochastic

Grid

Test

Life

DFA

MCEV

proj.


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Confidence intervals for expected shortfall:

August 31st, 2012 18

Theorem of extreme value theory: For a large class of distributions F of the variable X, the distribution of the excess losses Fu

over the threshold u,

converges to a generalized Pareto distribution7 Gξ,β with parameters and .

The pertinent tail of the distribution of -ΔRBC can accurately be described by a generalized

Pareto distribution, given the estimate with Nu the number of data points

with values above threshold u in a sample of size N. VaRα and ESα have the simple form

Standard statistical methods allow for a straightforward determination of confidence intervals8 of

VaRα and ESα for α close to 1 (in our case α ≥ 0.92 roughly).

]P[)( uXyuXyFu

7 for details, see

Balkema, A., and de Haan, L. (1974), Annals of Probability,

2, 792–804.

Pickands, J. (1975), Annals of Statistics, 3, 119–131.

Aktuariat & Steuerung/Nested_stochastics_in_life_insurance.ppt

NNNuF u /)()(

8 for an overview of the method, see e.g.

McNeil A., Frey R., Embrechts P., Quantitative Risk Management: Concepts,

Techniques and Tools, (2005), Princeton University Press.

. 1

VaRES , 11VaR

u

N

Nu

u

)(u

Making you safer.

Contact:

Basler Versicherungen

Dr. Urs Burri

Aeschengraben 21

CH-4002 Basel

[email protected]

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