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22nd European Conference on Operational Research
Prague, July 8-11, 2007
Financial Optimisation I, Monday 9th July, 8:00-9.30am
Mathematical Programming Models for
Asset and Liability Management
Katharina Schwaiger, Cormac Lucas and Gautam Mitra,
CARISMA, Brunel University West London
Outline
• Problem Formulation
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
• Scenario Models for Assets and Liabilities
• Mathematical Programming Models and Results:
–
–
–
–
Linear Programming Model
Stochastic Programming Model
Chance-Constrained Programming Model
Integrated Chance-Constrained Programming
Model
• Discussion and Future Work
Problem Formulation
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
• Pension funds wish to make integrated financial
decisions to match and outperform liabilities
• Last decade experienced low yields and a fall in
the equity market
• Risk-Return approach does not fully take into
account regulations (UK case)
use of Asset Liability Management Models
Pension Fund Cash Flows
Sponsoring
Company
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Figure 1: Pension Fund Cash Flows
Discussion
• Investment: portfolio of fixed income and cash
Mathematical Models
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
• Different ALM models:
– Ex ante decision by Linear Programming (LP)
– Ex ante decision by Stochastic Programming (SP)
– Ex ante decision by Chance-Constrained
Programming
• All models are multi-objective: (i) minimise
deviations (PV01 or NPV) between assets and
liabilities and (ii) reduce initial cash required
Asset/Liability under uncertainty
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
• Future asset returns and liabilities are random
• Generated scenarios reflect uncertainty
• Discount factor (interest rates) for bonds and
liabilities is random
• Pension fund population (affected by mortality)
and defined benefit payments (affected by final
salaries) are random
Scenario Generation
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
• LIBOR scenarios are generated using the Cox,
Ingersoll, and Ross Model (1985)
• Salary curves are a function of productivity (P),
merit and inflation (I) rates
sx  s y
merit ( x)
[(1  I )(1  P)]( x  y )
merit ( y)
• Inflation rate scenarios are generated using
ARIMA models
Linear Programming Model
• Deterministic with decision variables being:
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
–
–
–
–
–
–
–
Amount of bonds sold
Amount of bonds bought
Amount of bonds held
PV01 over and under deviations
Initial cash injected
Amount lent
Amount borrowed
• Multi-Objective:
– Minimise total PV01 deviations between assets and
liabilities
– Minimise initial injected cash
Linear Programming Model
• Subject to:
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
–
–
–
–
–
Cash-flow accounting equation
Inventory balance
Cash-flow matching equation
PV01 matching
Holding limits
Linear Programming Model
PV01 Deviation-Budget Trade Off
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
Stochastic Programming Model
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
• Two-stage stochastic programming model with
amount of bonds held z b , sold y b and bought x b
and the initial cash C being first stage decision
variables
• Amount borrowed brt s, lent letsand deviation of
asset and liability present values ( LPVt s, BPV t b ,s) are
the non-implementable stochastic decision
variables
• Multi-objective:
– Minimise total present value deviations between
assets and liabilities
– Minimise initial cash required
SP Model Constraints
• Cash-Flow Accounting Equation:
B
 (1   ) P x
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
b 1
b
B
b
 C   (1   ) Pb y b
b 1
• Inventory Balance Equation:
zb  Ob  xb  yb
b
• Present Value Matching of Assets and Liabilities:
B
B
b 1
b 1
b,s
b,s
s
BPV
z

BPV
x

(
1

r
)
le
 t b  t b
t
t 1

devots  devuts  LPVt s  (1  rt )brt s1
SP Constraints cont.
• Matching Equations:
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
B
b
b
s
s
s
(
c
z

c
x
)

br

L

le
 1 b 1 b 1 1 1
s
b 1
B
b
b
s
s
s
s
s
(
c
z

c
x
)

br

(
1

r
)
br

L

le

(
1

r
)
le
 t b t b t
t
t 1
t
t
t
t 1
b 1
B
s, t  2..T  1
b
b
s
s
s
s
(
c
z

c
x
)

(
1

r
)
br

L

le

(
1

r
)
le
 Tb T b
T
T 1
T
T
T
T 1
b 1
s
Discussion
• Non-Anticipativity:
br[1, s ]  br[1, s1]
le[1, s ]  le[1, s1]
s, s1
Stochastic Programming Model
Deviation-Budget Trade-off
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
Chance-Constrained Programming
Model
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
• Introduce a reliability level  t, where 0   t  1, which is
the probability of satisfying a constraint and  t is the
level of meeting the liabilities, i.e. it should be greater
than 1 in our case
• Scenarios are equally weighted, hence  s   1  ...   S  
• The corresponding chance constraints are:
N
s
t 1
 t L
S
s
t 1
  t br  A le
s
t 1
s
t 1
s
t 1
s, t  1..T  1
   ts1  1   t
t
 ts  0,1
s, t
s 1
CCP Model
Cash versus beta
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
CCP Model
SP versus CCP frontier
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
Integrated Chance Constraints
Outline
Problem
Formulation
• Introduced by Klein Haneveld [1986]
• Not only the probability of underfunding is important,
but also the amount of underfunding (conceptually close
to conditional surplus-at-risk CSaR) is important
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
At s1 le t s1  t Lst1   t brt s1  shortagets  0
S
  shortagets  Lˆt
s 1
Where  is the shortfall parameter
s, t
t
Discussion and Future Work
Generated Model Statistics:
Outline
Problem
Formulation
Obj. Function
Scenario Models
LP
SP
CCP
1 linear
22 nonzeros
1 linear
13500 nonzeros
1 linear
6751 nonzeros
Linear
Programming
CPU Time
Stochastic
Programming
(Using CPLEX10.1 0.0625
on a P4 3.0 GHZ
machine)
28.7656
1022.23
No. of
Constraints
633
All linear
108681 nonzeros
66306
All linear
2538913 nonzeros
53750
All linear
1058606 nonzeros
No. of
Variables
1243
all linear
34128
all linear
20627
6750 binary
13877 linear
ChanceConstrained
Programming
Discussion
Discussion and Future Work
• Ex post Simulations:
Outline
Problem
Formulation
Scenario Models
Linear
Programming
Stochastic
Programming
ChanceConstrained
Programming
Discussion
– Stress testing
– In Sample testing
– Backtesting
References
• J.C. Cox, J.E. Ingersoll Jr, and S.A. Ross. A Theory of the
Term Structure of Interest Rates, Econometrica, 1985.
• R. Fourer, D.M. Gay and B.W. Kernighan. AMPL: A
Modeling Language for Mathematical Programming.
Thomson/Brooks/Cole, 2003.
• W.K.K. Haneveld. Duality in stochastic linear and dynamic
programming. Volume 274 of Lecture Notes in Economics
and Mathematical Systems. Springer Verlag, Berlin, 1986.
• W.K.K. Haneveld and M.H. van der Vlerk. An ALM Model
for Pension Funds using Integrated Chance Constraints.
University of Gröningen, 2005.
• K. Schwaiger, C. Lucas and G. Mitra. Models and Solution
Methods for Liability Determined Investment. Working
paper, CARISMA Brunel University, 2007.
• H.E. Winklevoss. Pension Mathematics with Numerical
Illustrations. University of Pennsylvania Press, 1993.