20120717 lazzari wong - pca interest rate models

Shaun Lazzari

Celine Wong

Presented to SIAS, 17 July 2012

Dimension reduction techniques and

forecasting interest rates

Agenda

1) Interest rate modelling

2) Dimension reduction techniques

3) A few methodological considerations

4) Dimension reduction in a capital model

5) Other applications

Closing remarks

Discussion

1

1) Interest rate modelling Why do insurers and pension funds need to model interest rates?

2

Rising rates

• Reduce MV of fixed assets already in portfolio

• Encourage early contract terminations

Inte

rest ra

te level

Falling rates

• Reduce returns on newly purchased assets

Negative impacts

1) Interest rate modelling Features of interest rates

Each term point on the curve represents a

risk, having its own history and properties

3

Sample equity data Sample interest rate data

The yield curve is a curve!

A single point risk

2,858

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

0 5 10 15

0 5 10 15

Term

0 5 10 15

Term

0 5 10 15

Term

1) Interest rate modelling Features of interest rates

4

Level

0 5 10 15

PC1

PC2

PC3

Level

Slope

Curvature

Slope Curvature

These moves can be represented by simple shapes…

Agenda






Closing remarks

Discussion

5

2) Dimension reduction techniques Purpose – 2D example

6

1

:

:

:

:

:

40

X1 X2 Two-dimensional

data set:

X1

X2

1

:

:

:

:

:

40

X1’ One-dimensional

data set:

X1

X2

Identify intuitive

meanings

Reduce the size

of the dataset

Information lost

2) Dimension reduction techniques Principal components analysis – best at explaining variance

7

X1 X2 1

:

:

:

:

:

:

:

40

X1’ 1

:

:

:

:

:

:

:

40

Calculate

covariance

matrix

Covariance matrix:

Var(X1) Cov(X1, X2)

Cov(X2,X1) Var(X2)

Two-dimensional

data set:

One-dimensional

data set:

X1

X2

Derive

components

X1

X2 X1’ X2’

Dimension

reduction

X1

X2 X1’

The term points on the curve are highly correlated to each other

8

Sample interest rate data:

2) Dimension reduction techniques How can PCA be used to model interest rates?

0%

1%

2%

3%

4%

5%

6%

0 5 10 15

Historical interest

rate data


9

Step 1 - Deriving the principal components:

1 2 3 4

…

5 6 T

T-dimensional data set:

• One-year changes

• Demeaned

50

yrs …

1 2 3 4 5 6 T


10

Historical interest

rate data

1 2 3

3-dimensional data set:

Multipliers

Covariance

matrix

3 Components

chosen

…

T

T

T = no. of maturities

Dimension

reduction

T components

derived

Calculate

covariance

matrix

Derive

components


1 2 3 4

…

5 6 T



• Demeaned

50

yrs


11

3 Components

chosen

1 2 3


Multipliers

M1, M2, M3

M1 M2 M3

Principal

Components

PC1, PC2, PC3

Fit

distributions

to multipliers

Step 2 – Simulating interest rates


12

3 Components

chosen

1 2 3


Multipliers

M1, M2, M3

M1 M2 M3

Principal

Components

PC1, PC2, PC3

)(

)(3

)(2

)(1

)(3

)(2

)(1

tYC

iM

iM

iM

tP

tP

tP

For simulation i, at each maturity term t = 1 to T:

Fit

distributions

to multipliers

Fixed across all

simulations i but vary

across maturity terms t

The ith set of simulated multipliers

based on fitted distributions;

Fixed across maturity terms t


2) Dimension reduction techniques There is a spectrum of methods

13

Aspects of

components

Explaining the

data

Interpretable

shapes

Principal

components

explains data

variance (but only

considers data

variance)

may resemble

level, slope,

curvature.

may not be smooth

Independent

components

Identify “drivers of

data”

non-intuitive

shapes

Polynomial

components

components not

directly derived

from data

Components have

simple form,

smooth


14

Aspects of

components

Explaining the

data

Interpretable

shapes

Principal

components

explains data

variance (but only

considers data

variance)

may resemble

level, slope,

curvature.

may not be smooth

Independent

components


data”

non-intuitive

shapes

Polynomial

components

components not

directly derived

from data

Components have

simple form,

smooth

Explaining the data

Interpretable shapes


15

Aspects of

components

Explaining the

data

Interpretable

shapes

Principal

components

explains data

variance (but only

considers data

variance)

may resemble

level, slope,

curvature.

may not be smooth

Independent

components


data”

non-intuitive

shapes

Polynomial

components

components not

directly derived

from data

Components have

simple form,

smooth

Explaining the data



16

Aspects of

components

Explaining the

data

Interpretable

shapes

Principal

components

explains data

variance (but only

considers data

variance)

may resemble

level, slope,

curvature.

may not be smooth

Independent

components


data”

non-intuitive

shapes

Polynomial

components

components not

directly derived

from data

Components have

simple form,

smooth

Explaining the data



17

Aspects of

components

Explaining the

data

Interpretable

shapes

Principal

components

explains data

variance (but only

considers data

variance)

may resemble

level, slope,

curvature.

may not be smooth

Independent

components


data”

non-intuitive

shapes

Polynomial

components

components not

directly derived

from data

Components have

simple form,

smooth

Explaining the data


2) Dimension reduction techniques Independent component analysis – best at identifying drivers

Principal components Independent components

Uncorrelated

components

Independent

components

18

Unless multivariate

normal


Unless multivariate

normal


Uncorrelated

components

Independent

components

19


Unless multivariate

normal

• Best at explaining variance in data


Uncorrelated

components

Independent

components

20


Unless multivariate

normal


• Components always orthogonal


Uncorrelated

components

Independent

components

21


Unless multivariate

normal

• Best at identifying structure


Uncorrelated

components

Independent

components

22




Unless multivariate

normal

• Best at identifying structure

• Components not constrained to be

orthogonal


Uncorrelated

components

Independent

components

23



0 5 10 15

Term

PC1

PC2

PC3 0 5 10 15

Term

IC1

IC2

IC3


24

• Components with interpretable shapes –

level, slope and curvature • Components with more economic

meaning – attributed to underlying risk

drivers of yield curve movements

Unless multivariate

normal


Uncorrelated

components

Independent

components

2) Dimension reduction techniques Polynomial components – Components with imposed structure

25

Principal components may:

• not always be intuitive

• not always be smooth

…but explain most variance of the data

Polynomial components are:

• easier to interpret

• smooth by construction

…but explain less variance of the data

0 5 10 15

Term

PC1

PC2

PC3

0 5 10 15

Term

Comp1

Comp2

Comp3

Principal components Polynomial components

Agenda






Closing remarks

Discussion

26

27

3) A few methodological considerations Techniques very general, so lots!

Covariance

matrix

2) Dimension reduction techniquesHow can PCA be used to model interest rates?

9

Historical interest

rate data

3 Components

chosen

…

1 2 3

T

T


1 2 3 4

…

5 6 T

Dimension

reduction

T components

derived



Multipliers


• Demeaned

Calculate

covariance

matrix

Derive

components


50

yrs


11

3 Components

chosen

1 2 3


Multipliers

M1, M2, M3

M1 M2 M3

Principal

Components

PC1, PC2, PC3

)(

)(3

)(2

)(1

)(3

)(2

)(1

tYC

iM

iM

iM

tP

tP

tP


Fit

distributions

to multipliers

Fixed across all







28


Covariance

matrix


9

Historical interest

rate data

3 Components

chosen

…

1 2 3

T

T


1 2 3 4

…

5 6 T

Dimension

reduction

T components

derived



Multipliers


• Demeaned

Calculate

covariance

matrix

Derive

components


50

yrs

• Data history

• Data transformation

• Missing data

• Pre-processing of the data

(e.g. smoothing, interpolation, extrapolation)

• Frequency and timing of the data

• Maturity terms to use

- implicitly weighting the terms on the curve

• Spots or forwards

• Compounding type

29


Covariance

matrix


9

Historical interest

rate data

3 Components

chosen

…

1 2 3

T

T


1 2 3 4

…

5 6 T

Dimension

reduction

T components

derived



Multipliers


• Demeaned

Calculate

covariance

matrix

Derive

components


50

yrs

• Based on covariance or correlation

matrix?

Covariance

matrix


9

Historical interest

rate data

3 Components

chosen

…

1 2 3

T

T


1 2 3 4

…

5 6 T

Dimension

reduction

T components

derived



Multipliers


• Demeaned

Calculate

covariance

matrix

Derive

components


50

yrs

• Determining the intrinsic dimension of data

• Smoothing methods

• How shapes of components change over time

30


0%

10%

20%

30%

40%

50%

60%

70%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Component

% data variance explained by each component

Consider removing these

components

31



11

3 Components

chosen

1 2 3


Multipliers

M1, M2, M3

M1 M2 M3

Principal

Components

PC1, PC2, PC3

)(

)(3

)(2

)(1

)(3

)(2

)(1

tYC

iM

iM

iM

tP

tP

tP


Fit

distributions

to multipliers

Fixed across all







• Consider different distributions to capture

asymmetry and fat tails

Agenda






Closing remarks

Discussion

32

4) Dimension reduction in a capital model Purpose of the yield curve model

• Yield curve model just one part of a broader model

• Model should be appropriate for its end purpose, e.g:

• For us, “end” is capital requirements 33

Navigation

Fighting western

imperialism

4) Dimension reduction in a capital model Workings of a capital model

• Want to identify extreme downside events and their impact

on business

• Feedback loops improve appropriateness of models

34

Risk modelling

Business modelling

Aggregation Reporting

4) Dimension reduction in a capital model Number of available stress tests is limited

35

Stress tests

“Heavy” business model

Capital requirement

Lots of risk scenarios

• αr12 + βr2+

γr3 + …

Proxy business model

Own funds distribution

£x

4) Dimension reduction in a capital model Component combinations as stress tests

• Realistic and relevant

• Automates process

• Compatibility with stochastic scenarios

36

Same information

0 5 10 15

Shift

Term

Component Magnitude

1 x

2 y

3 z

4) Dimension reduction in a capital model What do we want our stress tests/components to achieve?

• Realism

• Interpretability

• Diversity, but especially adversity

37

4) Dimension reduction in a capital model Business model should impact the risk model

• e.g. a portfolio approx. hedged to level yield curve

movements:

• Standard (PCA) components not good. Can we do better? 38

0%

2%

4%

0 5 10 15

Stress 1

Base

0%

2%

4%

0 5 10 15

Stress 2

Base

%Δ own funds: 0.3% %Δ own funds: 6.5%

4) Dimension reduction in a capital model How to incorporate business information?

• Hard since this is what we’re building the model to find!

• Make use of existing information

• Or do some preliminary stress tests:

39

0 5 10 15 Term

exp

osu

re

4) Dimension reduction in a capital model Weighting by prior exposure estimates

40

Yield curve data

Weights

Weighted data

Yield curve components

Dimension

reduction

0 5 10 15

0 5 10 15

0 5 10 15

4) Dimension reduction in a capital model Weighting method gives more useful components

• Interpretability of components may be diminished

• Partial weighting gives best of both worlds

41

# Components % SCR (no

weighting)

% SCR

(weighting)

1 22.1% 82.2%

2 22.9% 96.7%

3 98.4% 99.6%

Agenda






Closing remarks

Discussion

42

5) Other applications Beyond the yield curve

• Other risks

– Implied volatility surfaces

– Asset portfolios

– Mortality curves

– Arbitrary groupings

• Internal data

43

60%

80%

90%

95%

100%

105%

110%

120%

150%

200%

0.00%

20.00%

40.00%

0.5 2

4

6

8

10 Moneyness

Term

Closing remarks

• Spectrum of techniques

• Many implementation considerations

• Purpose of model should influence the calibration

44

Thanks Peter…

45

Closing remarks

• Spectrum of techniques

• Many implementation considerations

• Purpose of model should influence the calibration

• Questions and comments?

• Our contact details:

– [email protected]

– [email protected] 46

20120717 lazzari wong - pca interest rate models

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