Solving real-life portfolio problem using stochastic programming and - - PowerPoint PPT Presentation

Solving real-life portfolio problem using stochastic programming and Monte-Carlo techniques

Martin Branda

Charles University in Prague Faculty of Mathematics and Physics

International Conference on Computational Management Science July 28-30, 2010, Vienna

Martin Branda (Charles University) 2010 1 / 36

Contents

1 General formulation and approximation schema 2 Investment problem formulations 3 Sample approximation using Monte-Carlo techniques 4 Numerical comparison

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General formulation and approximation schema

Contents

1 General formulation and approximation schema 2 Investment problem formulations 3 Sample approximation using Monte-Carlo techniques 4 Numerical comparison

Martin Branda (Charles University) 2010 3 / 36

General formulation and approximation schema

Formulation and approximation schema

using stochastic programming

1. 2. 3. Stochastic Sample Solution programming approximation validation formulation (SA) Program with a random − → Chance constrained − → SA CCP − → Reliability factor problem (CCP) ց ↓ Penalty function − → SA PFP − → Reliability problem (PFP) Martin Branda (Charles University) 2010 4 / 36

General formulation and approximation schema

Goal

We will compare the ability of both sample approximated problems to generate a feasible solution of the original problem chance constrained problem. Similar study was performed by J. Dupaˇ cov´ a, et al. (1991) and E. ˇ Zampachov´ a (2009).

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Investment problem formulations

Contents

1 General formulation and approximation schema 2 Investment problem formulations 3 Sample approximation using Monte-Carlo techniques 4 Numerical comparison

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Investment problem formulations

Investment problem

We consider 13 most liquid assets which are traded on the main market (SPAD) on Prague Stock Exchange. Weekly returns from the period 6th February 2009 to 10th February 2010 are used to estimate the parameters of distributions. Suppose that the small investor trades assets on the ”mini-SPAD”

• market. This market enables to trade ”mini-lots” (standardized

number of assets) with favoured transaction costs.

Martin Branda (Charles University) 2010 7 / 36

Investment problem formulations

Loss random variable

The random loss function Z(x, y, R) = −

n

• i=1

(Ri − ci)Qixi +

n

• i=1

fiyi, Qi the quotation of the ”mini-lot” of security i, fi the fixed transaction costs, ci the proportional transaction costs, Ri the random return of the security i, truncated normal distribution and multivariate skewed t-distribution (A. Azzalini et al (2003)) were used to model the random returns, xi the number of ”mini-lots”, yi binary variables which indicate, whether the security i is bought

• r not.

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Investment problem formulations

Set of feasible solutions

The set of feasible solutions contains a budget constraint and the restrictions on the minimal and the maximal number of ”mini-lots” which can be bought, i.e. X = {(x, y) ∈ Nn × {0, 1}n Bl ≤ n

i=1(1 + ci)Qixi + n i=1 fiyi ≤ Bu,

liyi ≤ xi ≤ uiyi, i = 1, . . . , n}, where Bl and Bu are the lower and the upper bound on the capital available for the portfolio investment, li > 0 and ui > 0 are the lower and the upper number of units for each security i.

Martin Branda (Charles University) 2010 9 / 36

Investment problem formulations

Portfolio problem with a random factor

” min

(x,y)∈X ” − n

• i=1

(Ri − ci)Qixi +

n

• i=1

fiyi

• r equivalently

” min

(r,x,y)∈R×X ”r

−

n

• i=1

(Ri − ci)Qixi +

n

• i=1

fiyi ≤ r But the random returns Ri are random ...

Martin Branda (Charles University) 2010 10 / 36

Investment problem formulations

Formulation and approximation schema

using stochastic programming

1. 2. 3. Stochastic Sample Solution programming approximation validation formulation (SA) Program with a random − → Chance constrained − → SA CCP − → Reliability factor problem (CCP) ց ↓ Penalty function − → SA PFP − → Reliability problem (PFP) Martin Branda (Charles University) 2010 11 / 36

Investment problem formulations

Chance constrained problem

Value at Risk problem

The chance constrained portfolio problem can be formulated as follows min

(r,x,y)∈R×X r

P

• −

n

• i=1

(Ri − ci)Qixi +

n

• i=1

fiyi ≤ r

• ≥ 1 − ε,

for some ε ∈ (0, 1).

Martin Branda (Charles University) 2010 12 / 36

Investment problem formulations

Solving chance constrained problems

In general, the feasible region is not convex even if the functions are convex, it is even not easy to check feasibility because it leads to computations of multivariate integrals. Hence, we will try to reformulate the chance constrained problem using a penalty function and incorporate it into the objective function...

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Investment problem formulations

Penalty function portfolio problem

Corresponding penalty function problem using the simple penalty function [·]+ is min

(r,x,y)∈R×X r + N · E

• −

n

• i=1

(Ri − ci)Qixi +

n

• i=1

fiyi − r + . Setting N = 1/(1 − ε) we obtain the CVaR problem, c.f. R.T. Rockafellar,

• S. Uryasev (2002).

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Investment problem formulations

Asymptotic equivalence

• f chance constrained and penalty function problems

Under the following assumptions, the asymptotic equivalence of the problems can be shown: Continuity of constraints and probabilistic functions. Compactness of the fixed set of feasible solutions. Existence of a permanently feasible solution. See Y.M. Ermoliev, et al (2000) - one joint chance constraint and particular penalty function (sum of positive parts), M.B., J. Dupaˇ cov´ a (2008) - one joint chance constraint and arbitrary penalty function, M.B. (2010A) - several joint chance constraints and arbitrary penalty functions.

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Investment problem formulations

Deterministic vs. stochastic penalty method

Deterministic - penalizes the infeasibility with respect to the decision vector, cf. M.S. Bazara, et al. (2006). Stochastic - penalizes violations of the constraints jointly with respect to the decision vector and to the random parameter...

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Sample approximation using Monte-Carlo techniques

Contents

1 General formulation and approximation schema 2 Investment problem formulations 3 Sample approximation using Monte-Carlo techniques 4 Numerical comparison

Martin Branda (Charles University) 2010 17 / 36

Sample approximation using Monte-Carlo techniques

Formulation and approximation schema

using stochastic programming

1. 2. 3. Stochastic Sample Solution programming approximation validation formulation (SA) Program with a random − → Chance constrained − → SA CCP − → Reliability factor problem (CCP) ց ↓ Penalty function − → SA PFP − → Reliability problem (PFP) Martin Branda (Charles University) 2010 18 / 36

Sample approximation using Monte-Carlo techniques

Sample approximated chance constrained problem

(SA CCP)

The problem can be formulated as a large mixed-integer linear program using additional binary variables zs, s = 1, . . . , S ˆ ϕCPP,S

γ

= min

(r,x,y,z)∈R×X×{0,1}S r

−

n

• i=1

(Rs

i − ci)Qixi + n

• i=1

fiyi − M(1 − zs) ≤ r, 1 S

S

• s=1

zs ≥ 1 − γ, for some level γ ∈ (0, 1) (γ = ε in general).

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Sample approximation using Monte-Carlo techniques

Sample approximated penalty function problem

(SA PFP)

The variables which are necessary to replace the positive parts are nonnegative continuous and the resulting problem is ˆ ϕS

N

= min

(r,x,y,v)∈R×X×RS

+

r + N S ·

S

• s=1

vs vs ≥ −

n

• i=1

(Rs

i − ci)Qixi + n

• i=1

fiyi − r. for some penalty parameter N > 0.

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Sample approximation using Monte-Carlo techniques

Estimated sample sizes

Estimates for the sample sizes which are necessary to generate a lower bound for the optimal value of the original chance constrained problem, cf. S. Ahmed, A. Shapiro (2008): S ≥ 2ε (γ − ε)2 ln 1 δ , a feasible solution of the original chance constrained problem, slight modification of J. Luedtke, S. Ahmed (2008): S ≥ 2 (ε − γ)2

• ln 1

δ + 13 ln 116 + 13 ln 2 + ln

• 2

(ε − γ)

• + ln

2D τ , which is based on the decomposition of the set of feasible solutions into the integer and real bounded part, where |X| ≤ 11613 · 213, and we set τ = 10−6, D = 2 · 106 (difference between the worst loss and the best profit for the distribution with bounded support).

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Sample approximation using Monte-Carlo techniques

Sample sizes - lower bound

Table: Sample sizes - lower bound

ε γ δ S 0.1 0.2 0.01 93 0.05 0.1 0.01 185 0.01 0.02 0.01 9211 0.1 0.2 0.001 139 0.05 0.1 0.001 277 0.01 0.02 0.001 13816

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Sample approximation using Monte-Carlo techniques

Sample sizes - feasibility

Bounded losses

Table: Sample sizes - feasibility

ε γ δ S 0.1 0.05 0.01 86496 0.05 0.025 0.01 348199 0.01 0.005 0.01 901792970 0.1 0.05 0.001 88338 0.05 0.025 0.001 355567 0.01 0.005 0.001 920213650

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Numerical comparison

Contents

1 General formulation and approximation schema 2 Investment problem formulations 3 Sample approximation using Monte-Carlo techniques 4 Numerical comparison

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Numerical comparison

Monte-Carlo simulation study

Truncated normal distribution and multivariate skewed t-distribution (A. Azzalini et al (2003)) were simulated using R 2.10.1. 100 samples for each sample size S = 100, 250, 500, 750, 1000. Decreasing levels ε = 0.1, 0.05, 0.01. Increasing penalty parameters N = 1, 10, 100, 1000. Resulting mixed-integer linear programming problems were solved using GAMS 23.2 and IBM ILOG CPlex 12.1. Reliability (feasibility) of the optimal solutions of the SA problems was verified on an independent sample of 10 000 realizations.

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Numerical comparison 1. 2. 3. Stochastic Sample Solution programming approximation validation formulation (SA) Program with a random − → Chance constrained − → SA CCP − → Reliability factor problem (CCP) ց ↓ Penalty function − → SA PFP − → Reliability problem (PFP) Martin Branda (Charles University) 2010 26 / 36

Numerical comparison

Reliabilities

Reliability S γ min max mean st.dev 100 0.1 0.8844 0.9967 0.9592 0.0255 100 0.05 0.9054 0.9869 0.9516 0.0189 100 0.01 0.8939 0.9941 0.9456 0.0250 250 0.1 0.9546 0.9968 0.9824 0.0098 250 0.05 0.9545 0.9950 0.9820 0.0086 250 0.01 0.9555 0.9950 0.9807 0.0115 500 0.1 0.9744 0.9982 0.9903 0.0043 500 0.05 0.9744 0.9982 0.9903 0.0043 500 0.01 0.9726 0.9982 0.9906 0.0043 750 0.1 0.9849 0.9994 0.9952 0.0033 750 0.05 0.9849 0.9994 0.9952 0.0033 750 0.01 0.9866 0.9994 0.9953 0.0032 1000 0.1 0.9870 1.0000 0.9966 0.0025 1000 0.05 0.9870 1.0000 0.9966 0.0025 1000 0.01 0.9870 1.0000 0.9966 0.0025

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Numerical comparison

S N min max mean st.dev 100 1 0.7622 0.9480 0.8770 0.0303 100 10 0.8967 0.9976 0.9581 0.0220 100 100 0.8967 0.9976 0.9581 0.0219 100 1000 0.8967 0.9976 0.9581 0.0218 250 1 0.8330 0.9290 0.8888 0.0199 250 10 0.9495 0.9950 0.9788 0.0101 250 100 0.9571 0.9973 0.9841 0.0089 250 1000 0.9571 0.9973 0.9840 0.0089 500 1 0.8716 0.9270 0.9016 0.0134 500 10 0.9723 0.9955 0.9871 0.0044 500 100 0.9813 0.9996 0.9935 0.0033 500 1000 0.9813 0.9995 0.9934 0.0033 750 1 0.8697 0.9330 0.8990 0.0108 750 10 0.9785 0.9950 0.9878 0.0036 750 100 0.9890 0.9995 0.9957 0.0026 750 1000 0.9890 0.9993 0.9956 0.0026 1000 1 0.8739 0.9253 0.8976 0.0097 1000 10 0.9753 0.9964 0.9886 0.0038 1000 100 0.9900 0.9999 0.9966 0.0023 1000 1000 0.9900 0.9999 0.9966 0.0023

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Numerical comparison

Optimal values

ˆ ϕCPP,S

γ

S γ min max mean st.dev 100 0.1 29739.36 66854.82 41784.66 7525.69 100 0.05 29739.36 66854.82 41821.60 7465.46 100 0.01 29680.35 69513.05 42312.34 7612.11 250 0.1 37609.63 121252.72 52429.77 9887.54 250 0.05 37609.63 121252.72 52431.23 9884.16 250 0.01 38260.62 121972.21 52626.23 9909.60 500 0.1 45085.97 125638.34 67824.32 15849.91 500 0.05 45085.97 125638.34 67824.32 15849.91 500 0.01 45085.97 125638.34 67942.02 15757.14 750 0.1 48562.73 160984.79 74655.08 19435.11 750 0.05 48562.73 160984.79 74652.82 19436.71 750 0.01 48562.73 155469.46 74679.40 19187.28 1000 0.1 59129.41 187831.95 93390.26 28293.28 1000 0.05 59129.41 187831.95 93414.25 28269.13 1000 0.01 59129.41 187831.95 93384.85 28264.63

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Numerical comparison

S N min max mean st.dev 100 1 24011.20 45692.02 33403.52 4311.27 100 10 30739.36 67854.82 42830.41 7489.58 100 100 30739.36 67854.82 42902.79 7484.36 100 1000 30739.36 67854.82 42903.93 7474.20 250 1 29745.59 83386.57 37382.48 6017.16 250 10 37848.38 118742.60 52156.49 9360.82 250 100 39630.90 122252.72 53493.47 9862.21 250 1000 39630.90 122252.72 53458.34 9898.87 500 1 32345.07 71348.70 43537.39 8424.45 500 10 45481.55 110479.11 63886.92 13472.75 500 100 46085.97 126638.34 68995.38 15851.31 500 1000 46085.97 126638.34 68914.67 15748.83 750 1 33415.20 94959.23 44922.49 9914.34 750 10 47249.47 150732.32 68251.45 17167.97 750 100 49562.73 157103.91 75669.31 19379.62 750 1000 49562.73 157103.91 75541.31 19234.11 1000 1 34061.30 98653.13 51840.01 12169.11 1000 10 51622.86 162568.42 82550.78 23493.47 1000 100 59121.39 182075.76 94331.08 27977.78 1000 1000 59121.39 182561.86 94357.45 28209.17

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Numerical comparison

Penalty term S N min max mean st.dev 100 1 2622.61 15554.88 7731.06 2530.95 100 10 0.00 309.34 3.09 30.93 100 100 0.00 0.00 0.00 0.00 100 1000 0.00 0.00 0.00 0.00 250 1 5787.60 20394.96 9673.10 2115.72 250 10 162.04 12197.59 2570.37 2255.62 250 100 0.00 0.00 0.00 0.00 250 1000 0.00 0.00 0.00 0.00 500 1 6682.76 27887.84 11865.88 3520.77 500 10 886.09 12248.70 5109.98 2719.34 500 100 0.00 0.00 0.00 0.00 500 1000 0.00 0.00 0.00 0.00 750 1 7304.04 43597.69 13191.21 5100.68 750 10 1995.04 18457.88 7328.08 3405.62 750 100 0.00 0.00 0.00 0.00 750 1000 0.00 0.00 0.00 0.00 1000 1 8886.75 42469.68 16647.91 7013.17 1000 10 2795.02 29325.36 9591.71 5108.62 1000 100 0.00 0.00 0.00 0.00 1000 1000 0.00 0.00 0.00 0.00

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Numerical comparison

Conclusion

The penalty term decreases with increasing penalty parameter N and reduces violations of the stochastic constraint. The reliability of the obtained solutions increases with increasing levels γ and penalty parameters N for each sample size S. Both problems are also able to generate comparable solutions for the same sample sizes.

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Numerical comparison

References

• S. Ahmed, A. Shapiro (2008). Solving chance-constrained stochastic

programs via sampling and integer programming. In Tutorials in Operations Research, Z.-L. Chen and S. Raghavan (eds.), INFORMS.

• E. Angelelli, R. Mansini, M.G. Speranza (2008). A comparison of MAD

and CVaR models with real features. Journal of Banking and Finance 32, 1188-1197.

• A. Azzalini, A. Capitanio (2003). Distributions generated by

perturbation of symmetry with emphasis on a multivariate skew t-distribution. Journal of the Royal Statistical Society: Series B 65, 367-389. M.S. Bazara, H.D. Sherali, C.M. Shetty (2006). Nonlinear programming: theory and algorithms. Third Edition, John Wiley & Sons, New Jersey.

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Numerical comparison

References

M.B., J. Dupaˇ cov´ a (2008). Approximations and contamination bounds for probabilistic programs. SPEPS 2008-13. M.B. (2010A). Reformulation of general chance constrained problems using the penalty functions. SPEPS 2010-2. M.B. (2010B). Solving real-life portfolio problem using stochastic programming and Monte-Carlo techniques. Accepted to Proceeding of MME 2010.

• J. Dupaˇ

cov´ a, A. Gaivoronski, Z. Kos, T. Szantai (1991). Stochastic programming in water management: A case study and a comparison

• f solution techniques. European Journal of Operational Research 52,

28-44.

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Numerical comparison

References

Y.M. Ermoliev, T.Y. Ermolieva, G.J. Macdonald, and V.I. Norkin (2000). Stochastic optimization of insurance portfolios for managing exposure to catastrophic risks. Annals of Operations Research 99, pp. 207-225.

• J. Luedtke, S. Ahmed (2008). A sample approximation approach for
• ptimization with probabilistic constraints. SIAM Journal on

Optimization, vol.19, pp.674-699. R.T. Rockafellar, S. Uryasev (2002). Conditional Value-at-Risk for General Loss Distributions. Journal of Banking and Finance, 26, 1443-1471.

• E. ˇ

Zampachov´ a (2009). Approximations in stochastic optimization and their applications. Ph.D. thesis, Brno University of Technology.

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Numerical comparison

Thank you for your attention.

e-mail: branda@karlin.mff.cuni.cz

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