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Models and Tools for Portfolio Planning

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### Models and Tools for Portfolio Planning

Plenary Talk APMOD 2006 June 19

Gautam Mitra

also included research results of:

John Beasley, Ken Darby-Dowman, Frank Ellison

Cormac Lucas, Diana Roman

Acknowledgement: Industry sponsors, UBS Investment Research, INSIGHT ( HBOS )

Outline

2. Modelling and solving: key issues

2.1 Modelling issues

2.2 Methods and algorithms

- Background: current status

3. Computational results: a selection

4. New developments

4.1 Shaping the distribution I 4.2 Shaping the distribution II

5. Summary

6. References

A historical perspective

- Markowitz ..mean-variance 1952,1959

- Hanoch and Levy 1969, valid efficiency criteria
- individual’s utility function

- Kallberg and Ziemba’s study.. alternative utility functions

- Sharpe ..single index market model 1963

- Arrow- Pratt.. absolute risk aversion

Mean-Variance Model

- Markowitz (1952,1959)

- alternative formulation

- QP (Arrow-Pratt absolute risk aversionparameter)

A historical perspective..cont

- Sharpe 64, Lintner 65, Mossin 66…CAPM model

- Rosenberg 1974 multifactor model

- Ross.. Arbitrage Pricing Theory(APT) multifactor equilibrium model

- Text Books: Elton & Gruber also Grinold & Kahn

- LP formulation 1980s.. computational tractability
- Konno MAD model.. also weighted goal program

- Perold 1984 survey…

Portfolio Planning

- Mean variance

Current practice and R&D focus:

- Factor model

- Rebalancing with turnover limits

- Index Tracking(+enhanced indexation) [Style input and goal oriented model]

- Cardinality of stock held: threshold constraints

- Cardinality of trades: threshold constraints

Portfolio Planning

- Tracking error as a constraint…[discuss ]

Emerging industry focus and research challenges:

- Nonlinear transaction cost /market impact[discuss ]

- Trade scheduling =algorithmic trading.. [discuss ]

- Resampled efficient frontier

- Risk attribution and risk budgeting

Modelling and solving: key issues

- Index TrackerMax
- Turnover Limit
Rebalancing model

- G.O. Side Constraints

Modelling and solving: key issues

- Integer Decision Variables & Cardinality Constraints
□

□

□

Specifies cardinality limit to no. of assets x

Modelling and solving: key issues

Groups of discrete logical variablesB = whether to buy

S = whether to sell (only if x*>0 )

K = whether to keep part or all the initial hold

(only if x* > 0)Threshold constraints for the -variables are:

minB xBB

minS xSS

minK x K

where min is the minimum weight that can be traded or held.

Cardinality constraints are:

B + S <= Tmax (maximum trades)

(B|x*=0) + K <= Hmax (maximum stocks)

And extra constraints that may be needed

B + S <= 1, (only if x* > 0)

Modelling and solving: key issues

Discrete Constraint Efficient Frontier (DCEF) (3)

Example:

- 4 stock universe, EF
- introduce cardinality constraint, i.e. build a portfolios containing 2 stocks only
Effect:

Discontinuities in the

Efficient Frontier

Modelling and solving: key issues

Discrete Constraint Efficient Frontier (DCEF) (2)

Why discontinuities?

- take investment opportunity set
- delete all inefficient portfolios (dominated points)

Modelling and solving: key issues

MV-global mean-variance portfolio

E – a portfolio on (global) efficient frontier ( = risk of benchmark)

P – on Tracking Error frontier but higher risk!

Trade scheduling

Statement of the problem:

Current holding is defined as pre trade portfolio

rebalance the portfolio ( buy and sell amounts )

=> trade orders to dealers/brokers

this leads to post trade portfolio

Trading risk is the volatility of a

long/short Portfolio = portfolio of trade orders

Trade scheduling = algorithmic trading

We want to minimise the loss of value in

getting our portfolio from what it is to what we want it to be..!

- Trading N shares of one stock
- Trading Ni shares of i=1,..m stocks
- Trading Ni shares of i=1,..m stocks
over t = 1,…T periods

Trade scheduling

Trade cost analysis:

Agency cost

Bid/ask spread

Market impact ( my trade moves the price)

Trend cost (other people’s trade moves the price…may be in my favour )

Influences: Trade in Ford impacts Trade in GM

Consider the two sets of orders

-Buy 5 million Ford// buy 5 million GM

-Buy 5 million Ford// sell 5 million GM

Modelling and solving: key issues

Non-linear transaction cost

Cost

Segment 1

Segment 3

Segment 2

Transaction amount

Modelling and solving: key issues

Actual T-costs

Buy cost

Sell cost

Buy rate

Sell rate

Buy Amount

Sell Amount

Cost

Slope r4

Slope r3

v4

Slope r2

Slope r1

v3

v2

v1

a1

a2

a3

a4

Separable programming = special ordered sets

Computational models

WE MOVE BACK TO THE MODELS WITH WHICH WE ARE CURRENTLY PREOCCUPIED..!

QP for Portfolio Planning

- Problem Statement: KKT optimality conditions

- Solution of continuous convex QP

- LP extended to QP: Wolfe’s Algorithm

- Beale, Dantzig, Cottle and Van-de-Panne’s pivotal method

- Hohenbalken’s Method: Successive LPs

- Goldfarb’s dual approach

- Interior point method (IPM)

most general statement of the quadratic program :

is a statement of the general, QP problem. For a given variable index j, when absent specifies default value , when absent specifies a default value , that is, no finite upper bound.

The Algorithm

- Sparse-Simplex
And

Symmetry/skew symmetry in the generalised version

Optimum via complementary bases

Complementary bases:

Better C.B. via neighbouring bases (1 pair basic, 1 pair non-basic)

Algorithm sequence

- Primal Version
- b values are feasible
- c infeasible

- Dual version
- c values feasible
- b infeasible

- 1. If the tableau in Standard form choose incoming as the dual complement of the infeasible primal.
- 2. If the tableau is nearly complementary nonStandard form
- take as the incoming variable the primal
- 3. Follow normal LP procedure to update column
- 4. Select outgoing variable by ratio test and include the complement in the test.
- Terminates**** Either =>Primal and Dual feasibility [optimal ]
- or =>unbounded ray [dual]… primal infeasible

Family of models

- Efficient Frontier:
Succession of λ-values in monotonic sequence

Succession of models --- each very similar to its predecessor

- Models laid out on B&B Tree
Continuous ( 0 ≤ ≤ 1 ) Discrete ( =0 or 1) by succession of FIXES□ Succession of fixes

Succession of models – each very similar to its predecessor

Family of models…in sequence

A sequence of models P0, P1,… Pk

Let optimum bases for these problems be denoted

Let number of iterations be denoted

Let number of basis changes from neighbour be denoted

Typically is of the order of 1 percent or less

Warm Start

- Basis Save (compact form)
- Restart by factorising the basis

- Basis Factors Save
- Restart and continue (in situ)

- Solution Save
- Restart with “CRASH” and “PUSH”

- External Save
- Supply “SAVE” material to start run

Branch fix & Relax

In the search as takes the value zero, also becomes zero, the values of and all become fixed, and constraints (1) and (2) and (3) now become redundant; these may now be relaxed.

[Optimum]Basis Prediction

- Take Primal non-zeros into the basis
- Add dual complements in the opposite status
- When updating the model
- Fix a variable
- Primal variable leaves basis
- Dual complement enters basis

- Free a constraint
- Primal constraint (logical) enters basis
- Dual complement leaves basis

- Fix a variable

Computational result: a selection

The Results:The ModelsSolver PerformanceEfficient FrontierBranch, Fix and RelaxComputational result: a selection

The ModelsComputational result: a selection

Relaxed QP Performance

Computational result: a selection

Integer Stage Performance

Computational result: a selection

Continuous Efficient Frontier (Cfu9583)

Computational result: a selection

Discrete EF (Cfu508)

Computational result: a selection

Branch, Fix and Relax: Cfu9583 illustration

Computational result: a selection

Comparisons with CPLEX

Three principles of choice(under uncertainty)

- Expected utility maximisation

- Mean-risk model

- Stochastic dominance

The first approach:

Portfolio construction based on stochastic dominance and target return distributions

First and Second Order Stochastic Dominance

Definition: Rx is preferred to Ry with respect to FSD (Notation: Rx>FSD Ry) if and only if Fx(r)Fy(r), for every r, with at least one strict inequality.

Intuitively, this describes the preference for larger outcomes.

Definition: Rx is preferred to Ry with respect to SSD (Notation: Rx>SSD Ry) if and only if F(2)x(r)F(2)y(r), for every r, with at least one strict inequality,

Where: .

First and Second Order Stochastic Dominance- discrete random variables

Consider now the special case where the two random variables Rx and Ry are discrete with T equally probable outcomes. In order to make the comparison, we rank the outcomes of Rx and Ry in ascending order to obtain two T-dimensional vectors: (1,…, T) and (1,…, T) such that 1…T and 1…T.

Rx>FSD Ry if and only if ii for every i{1,…,T} with at least one strict inequality.

Rx>SSD Ry if and only if for every i{1,…,T}

with at least one strict inequality.

Motivation and contribution

The attempts to combine the practicality of mean-risk models with the theoretical solid basis of stochastic dominance mean-risk models consistent with SD.

- The proposed model that finds a portfolio which is:
- non-dominated with respect to SSD ( optimal for every rational and risk averse investor).
- has a return distribution close to a reference (target, goal, aspiration) distribution which is introduced as input data.

The entire distribution of portfolio return is taken into account, not just summary statistics.

This is achieved using a computationally tractable procedure which uses linear inequalities and LP [Goal] models.

Motivation and contribution

Close to a reference distribution means “close or better”:

3 cases:

1. The reference distribution is SSD efficient portfolio whose return distribution is the reference distribution.

2. The reference distribution is not SSD efficient portfolio whose return distribution is SSD efficient - better than reference distribution (outperforms the goal!)

3. The reference distribution is not attainable (too high outcomes) portfolio whose return distribution is SSD efficient and comes close to the reference distribution.

SSD efficiency and multiple criteria optimisation

The portfolio selection problem: case of T equally probable scenarios.

portfolio x return (f1(x),…,fT(x)):

random variable seen as a T-dimensional vector, where

(the portfolio return in scenario i=1,…,T)

SSD efficiency and multiple criteria optimisation

How do we express that z1=(f1(x1),…fT(x1)) dominates z2=(f1(x2),…fT(x2)) with respect to SSD relation?

The sum of the worst k outcomes of z1 is greater than the the sum of the worst k outcomes of z2, k=1,…,T.

Equivalently: the CVaR of z1 at confidence level k/T is smaller than the the CVaR of z2, at confidence level k/T, k=1,…,T.

Example:

Order the outcomes

(1,4,3,2) and (3,5,0,2)

(1,2,3,4) and (0,2,3,5)

Cumulate the outcomes

Pareto comparison

(1,3,6,10) and (0,2,5,10)

(1,4,3,2) dominates (3,5,0,2) with respect to SSD

SSD efficiency and multiple criteria optimisation

The sum of the worst k outcomes of a vector z=(z1,…,zT) is the optimal value of a LP problem:

Max

Such that:

for i=1,…T

for i=1,…T

(Uryasev and Rockafellar 2000, Ogryczak 2002)

tk is the k-th worst outcome of the vector z

SSD efficiency and multiple criteria optimisation

The set of SSD non-dominated solutions is the set of Pareto non-dominated solutions of the multi-objective LP:

(1)

Such that:

for i, k=1,…T

for i, k=1,…T

for j=1,…n.

is the portfolio return in scenario i.

where:

The k-th objective function is the sum of the worst k outcomes of the portfolio return

6. Summary

- We have presented a number of research problems from industry and academic perspectives
- The modelling aspects focus on industry’s requirements
- The success in computing solutions to real world practical problems has opened up new opportunities of real time trade scheduling applications
- We have also given examples which vindicate our choice of SSX algorithm which processes a family
of problems. The role of ‘warm start’ is high lighted

- A new approach to portfolio planning in which the distribution is shaped is proposed

6. References (CARISMA) / Mitra et al

6.1 Computational Aspects of Alternative Portfolio Selection Models in the Presence of Discrete Asset Choice Constraints, Quantitative Finance Vol. 1. (2001) 1-13, N Jobst, M Horniman, C Lucas and G Mitra. (J).

6.2 A Review of Portfolio Planning: Models and Systems, (2003) an invited chapter, pp1 – 39 in:Advances in Portfolio Construction and Implementation, S E Satchell, A E Scowcroft (Eds.), Butterworth & Heinemann, Oxford. G.Mitra

6.3 Portfolio Optimisation Models and Properties of Return Distributions, Mathematical Programming Journal, 2006, D Roman, G.Mitra, KH Darby-Dowman

6.4 Quadratic Programming for Portfolio Planning: Insights into Algorithmic and Computational Issues, CARISMA Research Report 2006, G.Mitra, E F D Ellison, A Scowcroft, to appear in Journal of Asset Management.

6.5 Mean-Risk Models Using Two Risk Measures, A Multiobjective Approach, Quantitative Finance, Special Issue of Risk Decisions (Submitted 2006), D Roman, G.Mitra, K H Darby-Dowman

CARISMA/Others

6.6 Heuristics for cardinality constrained portfolio optimisation , J Beasley, T.-J. Chang, N. Meade and Y.M. Sharaiha) Computers & Operations Research, vol.27, 2000, pp1271-1302

6.7 Is enhanced indexation a viable momentum strategy? J Beasley and N. Meade CARISMA Technical Report 2005

Tracking Error

6.8 Jorion, Philippe, (2003), “Portfolio optimization with tracking-error constraints”, Financial Analysts Journal, Sept-Oct, AIMR, pp 70 – 82.

Non linear Transaction cost

6.9 Konno, Hiroshi, Akishino, Keiserke, Yamomoto, Rei., (2005), “Optimization of long-short portfolio under non-convex transaction costs”, in Special issue, Optimization and Risk Modelling, Guest Editor: Gautam Mitra, Computational Optimization and Application, Vol 32, Numbers 1/2, pp 112 – 132.

Trade Scheduling = Algorithmic Trading

6.10 An Algorithmic Approach to Optimal Trade Scheduling,

Dan di Bartolomeo, Northfield Systems, New Directions in

Financial Modelling,CARISMA meeting, 22-25 May 2006, London

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