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Dynamic portfolio optimization with stochastic programming

TIØ4317, H2009

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Dynamic Trading Strategies

A sequence of buy and sell decisions, including short term borrowing and lending

Rebalancing portfolio weights at discrete times

• Simple decisions rules for portfolio rebalancing• Stochastic dedication• Stochastic linear programming

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Modeling portfolio decisions in discrete time• Portfolio decisions can be made at a finite number of

points in time called trading dates.– No decisions are assumed to be taken between one trading date

and the other

• Prices over time are modeled following the structure of a binary lattice.

• Let us denote– s as the index of a possible state

– as the set of the state indices at time tt

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A binary lattice

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Linear scenario structures

• Scenario– A particular scenario is denoted by

set of the stages in t+1 that can be reached by a state in t

set of the stages in t that can reach a particular stage in t+1

1 20, 1,..., ... Ts s sT l

ss

s s

tst n l

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Linear scenario structure

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Non-anticipativity

• Trading strategies can not depend on what happens in the future• Two scenarios with the same history up to time require the

same strategy to be implemented up to that time (non-anticipativity)

• To model non-anticipativity we can recombine the scenarios on an event tree

• With a binary lattice every node has two predecessors– We do not know the node we come from

• With an event tree we have just an predecessor for each node– We always know the history of the process

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Event Tree

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Some formal definitions

• Event Tree: directed graph• is the set of the nodes• is the set of the possible links• is the set of possible states at time t

,

t

T

tt

0

1

1tv

ttv

t ss ,

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Event tree properties

• 1

• 2

Every state has a unique predecessor

000 s

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Scenarios: formal definition

such that for each

last trading date for the scenario strategy l

v tts

0 10 1, ,..., l

l

vv vs s s

11 ,v t v t

t ts s E

Tt ll ;,...,1

l

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Decision rules for Dynamic portfolio strategies• Buy-and-hold• Constant mix• Constant proportion • Option based portfolio insurance

– Simple decision rules for rebalancing of portfolios

– No optimality

– Easy to specify and compute

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Buy-and-hold

• Specify the proportion of the initial wealth invested in the risk free and in the risky asset at time 0.– The portfolio is held until maturity under all the scenarios

• Let us define the growth of the risky asset value

with fixed since time 0• This portfolio has a minimum value of

stI 1

st

spt IxVxVV 10000 1

0x00xV

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Constant mix Strategy

• Specifies that the proportion of value of the risk free and the risky asset wrt the portfolio value remains constant for all scenarios/trading times

• Values of risky and risk free assets in the portfolio at time t, scenario s

st

stp

sIt

stp

sft

IxVV

xVV

101

01

1

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Constant mix Strategy (Cont’d…)

1 0 1 0 1 0

1 0 1 0 1 0

1

1 1

s s s sft p t p t t

s s s sIt p t p t t

V V x V x I x

V V x V x I x

• Constant mix strategy rebalancing condition

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Constant proportion strategy

• A fixed proportion of the portfolio is invested in the risky asset

• This proportion stays fixed all along the lifetime of the investment by rebalancing

• This strategy provides for a floor g, below which the asset value is not allowed to fall.

gVV spt

sIt

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Option-based portfolio insurance

• A mix of risk free and risky asset such that the payoff scheme matches the one of a portfolio composed of risk free assets and call options

• The risk free assets are kept equal to the floor of the portfolio and any excess value is invested in call options.

• If the portfolio value drops down its minimum value is given by the value of the risk free investment.

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Stochastic dedication

• The model optimizes short term borrowing and lending decisions as new information arrives

• It does not account for portfolio rebalancing• Portfolio decisions are optimized at time 0

– Together with the borrowing-lending decisions using the surplus-shortage between assets and liabilities.

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Necessary conditions for immunization: definitions• Discount factor

• Present value of asset i in scenario l

• Present value of liabilities in scenario l

0 1

1

tln

ft

l

rd

T

t

lnti

lt

li FdP

10

1

Tl l lL t t

t

P dL

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Necessary conditions for immunization• Necessary condition for scenario immunization

• This condition can be very expensive or even impossible to satisfy for all scenarios

lPxP lL

n

ii

li

10

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Relaxation

• What the model seeks to find is a trade-off between reward, when the asset portfolio outperforms the liabilities against the risk when the portfolio underperforms.

• Present value of the asset-liabilities portfolio in scenario l

lL

n

ii

li

l PxPPxV 1

00;

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Relaxation (cont’d…)

• Define as the initial budget and as the maximum risk accepted

0 01

0 01

;

max 0;

max 0;

l l l

nl l l

i i Li

nl l l

L i ii

V x P y y

y P x P

y P P x

0v

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Trade-off formulation

0 0 0

0 0

max

. .

, 0

l l

l

li i

l l

l

l l l li i L

l l

py

sc P x v

py

y y P x P l

y y

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about the immunization

• To satisfy the immunization condition we have to include borrowing and lending decisions

• The price of a portfolio of assets will be covered by liabilities and loans

00

00

1

00000

vLvvxFn

iii

0v st

st vv ,

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Cashflow matching

• The stochastic cashflow matching equation, encompassing borrowing and lending decisions is given by

CF+interests+borrowed funds = liabilities+lended money+ debts

s

tstf

st

st

st

st

stf

n

ii

sit vrvLvvrxF 1111

101 11

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Stochastic dedication

0

0 0 00 0 0 0 0 0

1

1 0 1 1 1 11

min

. .

1 1

,

, , 0

n

i iin

s s s s s s s st i i f t t t t t f t t

i

v

sc F x v v L v

F x r v v L v r v

t s

x v v

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A primer in Stochastic Programming• We are interested in finding a “solution” to the problem

• Idea of the solution: the best value which satisfies the constraints most of the time (a reliable optimal solution)

0"min" ,

. . , 0i

f x

sc f x

x X

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A primer in Stochastic Programming• To find a solution we make suitable transformations

of the functions in the system– EXAMPLE

An optimization problem can be modified with a penalty function for the constraints. The idea with SP is the same: modify the functions in a reasonable way to find a solution

,, xgxf ii

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A primer in Stochastic Programming• Then solve the problem

0min ,

. . , 0i

E g x

sc E g x

x X

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A primer in Stochastic Programming• A typical SP problem

min

.

0

xf x

sc Ax b

T x h

x

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A primer in Stochastic Programming

A possible modification:• Define a variable y and the function

• Now define

And drop all the stochastic constraints in the original problem

, min , | , 0Q x q y W y h T x y

0 , ,g x f x Q x

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A primer in Stochastic Programming• We obtain the so called recourse formulation

min ,

.

0

f x Q x

sc Ax b

x

E

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A primer in Stochastic Programming• If we consider the discrete case we can link to a

scenario every possible realization of the random variable

• In this case we can write the large scale deterministic equivalent

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A primer in Stochastic Programming• Large scale deterministic equivalent

,

1

min ,

.

,

0 0,

k

N

l l lx yk

l l l

l

f x pq y

sc Ax b

T x Wy h l

x y l

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A primer in Stochastic Programming• If it could be possible to forecast the future we could have a

different first stage decision for each scenario

,

1

min ,

.

,

0 0,

l l

N

l l l lx yk

l

l l l l

l l

f x pq y

sc Ax b

T x Wy h l

x y l

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A primer in Stochastic Programming• Since we can not forecast the future we need to enforce

nonanticipativity, setting• With nonanticipativity constraints we require that the

decisions taken in different scenarios that at a given stage “look the same” have to coincide

• This is what is automatically done in the recourse formulation, setting a unique first stage decision

lxxl

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A primer in Stochastic Programming• Multistage recourse problems• Recourse on the recourse on the recourse on the

recourse…• It is an extension of the two stage model and it is

formulated with a nested structure

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A primer in Stochastic Programming• Multistage recourse formulation

0 1 1 1

0 0 0

0 0 1 1 1 1

1 1 1

0

min min , ... min ,

.

0, 0 . .

T T T

T T T T T T T

t t

f y q y q y

sc W y h

T y Wy h

T y W y h

y y t as

E E

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A rigorous framework for optimization under uncertainty

Stochastic programming

• Is the mathematical programming tool that facilitates the optimization of dynamic strategies on event trees

• Models are optimal and satisfy non-anticipativity • Portfolio rebalancing is allowed as new information

becomes available.

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Stochastic Programming for dynamic strategies• At each trading date the manager assesses the

market conditions• The manager also assesses the potential changes in

conditions of market parameters• The new information is incorporated in a sequence of

transactions

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Model formulation

• The model encompasses two types of constraints: • Inventory balance constraints

• Cashflow balance constraints

• The model encompasses two levels of constraints:– First stage constraints

– Time-staged constraints

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First stage constraints

• Inventory balance constraintFace value of assets in the portfolio equal to what we had in the portfolio plus what we have bought minus what we have sold

• Cashflow balance equationInflows from sales of securities plus borrowed money equal the amount invested for purchasing new securities plus the amount invested in the riskless asset plus the payment of liabilities

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First stage constraints (Cont’d…)

• Inventory balance

• Cash flow balance

00

000

00 iiii yxbz

0 0 0 0 0 0 00 0 0 0 0 0 0 0

1 1

n nb ai i i i

i i

P y v v P x v L

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Time staged constraints

• At each time period we have a set of constraints for each scenario

• Decisions are conditioned by the state of the system at time t as well as the decisions taken in t-1 at the predecessor state

1ts

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Time staged constraints (Cont’d…)• Inventory balance constraints

One constraint for each security and for each scenario

• Cashflow balance constraints

One constraint for each scenario

1 1s s s s sti t i t i ti tiz z x y

1 1 1 11 1

1 11

1

1

n ns s bs s s s st i t i ti ti t t t t

i in

as s s s s sti ti t t t t t

i

F z P y r v v

P x v r v L

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End of horizon constraint

• We evaluate the terminal wealth as sum of the market value of the portfolio of assets and the money the investor has lended.

n

i

sTi

bsTi

sT

sT zPvW

1

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Objective function

• The objective function is expressed in form of expected utility of terminal wealth

maxT

s sT

s

p W U

48Stochastic program for dynamic strategies

0 0 00 0 0 0

0 0 0 0 0 0 00 0 0 0 0 0 0 0

1 1

1 1

1 1 1 11 1

11

max

. .

, ,

1

1

T

s sT

s

i i i i

n nb ai i i i

i i

s s s s sti t i t i ti ti

n ns s bs s s s st i t i ti ti t t t t

i in

as s sti ti t t t

i

p W

sc z b x y i

P y v v P x v L

z z x y t s i

F z P y r v v

P x v r

U

1

1

,s s st t

ns s bs sT T Ti Ti

i

v L t s

W v P z

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Use of stochastic programming for dynamic portfolio management

• The advantage of using a stochastic programming framework is that a set of restriction such as transaction costs, multiple state variables, market incompleteness, taxes and trading limits can be handled simultaneously within the framework

• The drawback is that the computational effort explodes as the number of scenarios and decision stages increases.