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SP XI Vienna August 28, 2007 Scenario Generation for the Asset Allocation Problem Diana Roman Gautam Mitra The asset allocation problem • An amount of money to invest Asset Allocation Problem • N stocks with known current prices S10,…,SN0 • Decision to take: how much to invest in each asset Mean-CVaR Model • Goal: to get a profit as high as possible after a certain time T Financial SG • The stock prices (returns) at time T are not known: random variables (stochastic processes) Hidden Markov Models Computational results • xi=fraction of wealth invested in asset i portfolio (x1,…,xn) • Ri=the return of asset i at time T • The portfolio return at time T: Rx=x1R1+…+xNRN (also r.v.!) How to choose between portfolios? A modelling issue! Mean-risk models for portfolio selection • Mean – risk models: maximize expected value, minimise risk Asset Allocation Problem Mean-CVaR Model • Risk: Conditional Value-at-Risk (CVaR) = the expected value of losses in a prespecified number of worst cases. Confidence level =0.01 consider the worst 1% of cases Financial SG Hidden Markov Models Computational results The optimisation problem: Max (E(Rx) ,- CVaR(Rx)) over x1,…,xn Min CVaR(Rx) over x1,…,xn S.t.: E(Rx)d ………… (1) Scenario Generation Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results • The (continuous) distribution of stock returns: approximated by a discrete multivariate distribution with a limited number of outcomes, so that (1) can be solved numerically: scenario generation. scenario set (single-period case) or a scenario tree (multi-period case). Scenario Generation • S Scenarios: Asset Allocation Problem Mean-CVaR Model Financial SG • pi=probability of scenario i occurring; • rij=the return of asset j under scenario i; • The (continuous) distribution of (R1,…,RN) is replaced with a discrete one Hidden Markov Models Computational results … asset1 asset2 asset n probability scenario 1 r11 r12 … r1N p1 scenario 2 … scenario S r21 … rS1 r22 … rS2 … … … r2N … rSN … … pS The mean-CVaR model • Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results Scenarios a LP (Rockafellar and Uryasev 2000) Min 1 S N i 1 j 1 p [ v x r ] i j ij v Subject to: N x j 1 j N x j 1 j j d 1 rij= the scenarios for assets’s returns x j 0, j We only solve an approximation of the original problem; The quality of the solution obtained is directly linked to the quality of the scenario generator (“garbage in, garbage out”). The quality of scenario generators Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results • The goal of scenario models is to get a good approximation of the “true” optimal value and of the “true” optimal solutions of the original problem (NOT necessarily a good approximation of the distributions involved, NOT good point predictions). • Difficult to test • There are several conditions required for a SG The quality of scenario generators Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results In-sample stability: different runs of a scenario generator should give about the same results. If we generate several scenario sets (or scenario trees) with the same number of scenarios and solve the approximation LP with these discretisations, we should get about the same optimal value. (not necessarily the same optimal solutions: the objective function in a SP can be “flat”, i.e. different solutions giving similar objective values) The quality of scenario generators Out-of-sample stability: Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results -Generate scenario sets of the same size -Solve the optimisation problem on each different optimal solutions -These solutions are evaluated on the “true” distributions “true” objective values -The true objective values should be similar • In practice: use a very large scenario set generated with an exogenuous SG method as the “true” distribution The quality of scenario generators -Out-of-sample stability: the important one Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results -No (simple) relation between in-sample and out-ofsample stability Hidden Markov Models • applied in various fields, e.g. speech recognition Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results • still experimental for financial scenario generation • Motivation: financial time series are not stationary; unexpected jumps, changing behaviour Hidden Markov Models Real world processes produce observable Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results outputs – a sequence of historical prices, returns… • • • • A set of N distinct states: S1,…,SN System changes state at equally spaced discrete times: t=1,2,… Each state produces outputs according to its “output distribution” (different states ->different parameters) The “true” state of the system at a certain time point is “hidden”: only observe the output. Hidden Markov Models Assumptions: Asset Allocation Problem • Mean-CVaR Model Financial SG Hidden Markov Models • Computational results • First order Markov chain: at any time point, system’s state depends only on the previous state and not the whole history: P(qt=Si | qt-1=Sj, qt-2=Sk,….)= P(qt=Si | qt-1=Sj) with qt=system’s state at time t Time independence: aij=probability of changing from state i to state j: the same at any time t. Output-independence assumption: the output generated at a time t depends solely on the system’s state at time t (not on the previous outputs) Hidden Markov Models Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results The output distributions: mixtures of normal distributions M mixtures: M f ( x) c j f j ( x; j , j ) j 1 f j ( x; j , j ) M c j 1 j 1 =the normal density function with mean vector j and covariance matrix j Mixtures of normal density functions can approximate any finite continuous density function. Hidden Markov Models a11 Asset Allocation Problem 1 Mean-CVaR Model a12 Financial SG Hidden Markov Models Computational results 2 2 c21,…,c2M 21 ,…,2M 21,…,2M 1 a21 M c11,…,c1M 11 ,…,1M c1 j N ( x; 1 j , 1 j ) j 1 11,…,1M a31 a32 a23 a13 3 3 N=3 M mixtures c31,…,c3M 31 ,…,3M 31,…,3M Hidden Markov Models The parameters of a HMM: Asset Allocation Problem Mean-CVaR Model • Number of states N • Number of mixtures M • Initial probabilities of states: 1,…, N • Transition probabilities: A=(aij), i,j=1…N Financial SG Hidden Markov Models Computational results • For each state i, parameters of the output distributions: Mixture coefficients ci1,…,ciM Mean vectors i1,..., iM Covariance matrices i1,…, i1. Training Hidden Markov Models Historical data: O=(O1,…,OT)=(rtj, t=1…T,j=1…N) is used to “train” the HMM. Asset Allocation Problem Mean-CVaR Model Meaning: Find the parameters Financial SG =(, A, C, , ) s.t. P(O| ) maximised Hidden Markov Models Computational results •Cannot be solved analytically and no best way to find •Iterative procedures (e.g. EM, Baum-Welch) can be used to find a local maximum. •Parameters N and M are supposed to be known! Training HMM’s • Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models • Start with some initial parameters 0 ; compute P(O| 0) Re-estimate parameters 1 ; compute P(O| 1) P(O| 0) • Obtain sequence 0, 1, 2… with P(O| i) P(O| i-1) • (P(O| i))i converges towards a local maximum • Limited knowledge about the convergence speed Computational results • • • Observed sharp increase in the first few iterations, then relatively little improvement Practically: stop when P(O| i)- P(O| i-1) is small enough Use final i for generation of scenarios Training HMMs: initial parameters Asset Allocation Problem Mean-CVaR Model • How choose 0? • Not important for i and aij (could be 1/N or random) • Very important for C, and – but no”best” way to estimate them Financial SG Hidden Markov Models • Computational results • k-means clustering algorithm: separate historical data into M clusters starting parameters: Based on the mean vectors and covariance matrices of the clusters Training HMM: parameters estimation re- Use Baum-Welch algorithm (EM): Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results Need to calculate additional quantities: Forward probabilities: time t, state i t (i) P(O1...Ot , qt Si | ) Backward probabilities: time t, state i Calculated recursively after time t (i) P(Ot 1...OT | qt Si , ) In calculus: the multi-variate normal density: 1 1 1 T f ij ( x) exp ( x ) ( x ) ij ij ij N /2 1/ 2 (2 ) | det ij | 2 Training HMM: parameters estimation Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results re- Additional quantities: t (i) t (i) t (i) P(qt Si | O, ) P(O | ) t (i, j ) P(qt Si , qt 1 S j | O, ) Probability of the historical observation to be generated by the current model: M M i 1 i 1 P(O | ) T (i ) t (i ) t (i ) Training HMM: parameters estimation re- T 1 Asset Allocation Problem *i 1 (i) aij* Mean-CVaR Model Computational results t 1 T 1 t t 1 Financial SG Hidden Markov Models (i, j ) T i* t (i)Ot t 1 T t 1 t (i ) t (i ) T *i T ( i )( O )( O ) t t i t i t 1 T t 1 t (i ) HMM: estimation of the current state •The state of the system at the current time? Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results •Via Viterbi algorithm •Given an observation sequence O=(O1,…,OT) and a model , find an “optimal” state sequence Q=(q1,…,qT) •i.e., that best “explains” the observations: maximises P(Q|O, ) HMM for scenario generation Historical data: estimation of …. Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results t=1 t=2 t=3 Generation of scenarios …. t=T t=T+1 t=T+2 t=T+TP Estimation of the system’s state at time T A scenario: a path of returns for times T+1,…,T+TP Estimate the current state (time T); say, qt=Si { • Transit to a next state Sj according to transition probabilities aij • Generate a return conform to the distribution of state j } HMM – implementation issues • Number of states? Still a very much unsolved problem. Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results • The observation distributions for each state? • The initial estimates of the model’s parameters • Computational issues: lots!! • For large number of assets, large covariance matrices (at every step of re-estimation: determinants, inverses); • The quantities calculated recursively get smaller and smaller • Or the opposite: get larger and larger Computational results Historical Dataset: Asset Allocation Problem • 5 stocks from FTSE 100 Mean-CVaR Model • 132 monthly returns: Jan 1993-Dec 2003 Generate scenario returns for 1 month ahead 500, 700, 1000, 2000, 3000 scenarios Financial SG Hidden Markov Models Computational results Computational results Asset Allocation Problem For each scenario size: - Run 30 times generate 30 different discretisations for the assets’ returns (R1,…,RN) - Solve mean-CVaR model with these discretisations get 30 solutions x1,…,x30 - Similar solutions as scenario size increases: (x2=x3=0, x5>=50%) - Evaluate these solutions on the “true” distribution? Mean-CVaR Model Financial SG Hidden Markov Models Computational results Computational results Out-of-sample stability: Asset Allocation Problem Mean-CVaR Model Financial SG Hidden Markov Models Computational results - The “true” distribution: generated with Geometric Brownian motion, 30.000 scenarios - Each of the 30 solutions was evaluated on this distribution 30 “true” objective values (=portfolio CVaRs) Computational results Geometric Brownian motion (GBM) Asset Allocation Problem Mean-CVaR Model - The standard in finance for modelling stock prices - Stock prices are approximated by continuous time stochastic processes (accepted by practitioners…) Financial SG S (t ) S0 exp{( Hidden Markov Models Computational results • • • • 2 2 )t Wt } S0: the current price : the expected rate of return : the standard deviation of rate of return {Wt}: Wiener process - the “noise” in the asset’s price. Computational results Statistics for the series of “true” objective functions 500 scen 700 scen 1000 scen 2000 scen 3000 scen Asset Allocation Problem Mean 0.0035 0.0033 0.0031 0.0029 0.0026 Mean-CVaR Model St Deviation 0.0012 0.0012 0.001 0.0006 0.0003 Financial SG Range 0.0051 0.0042 0.0048 0.0022 0.0011 Minimum 0.0024 0.0023 0.0023 0.0023 0.0023 Maximum 0.0074 0.0065 0.0071 0.0045 0.0034 Hidden Markov Models Computational results • Quality of solutions improve with larger scenario sets (as expected!) • Reasonably small spread; pretty similar objective values Conclusions and final remarks • For the mean-CVaR model: SG that can capture extreme price movements • Stability is a necessary condition for a “good” SG • HMM is a discrete-time model; experimental for financial SG • Motivated by non-stationarity of financial time series Financial SG • Hidden Markov Models Two stochastic processes: one of them describes the “state of the system” • Implementation problems, especially when the number of assets is large • An initial “good” estimate for HMM parameters is essential • The number of states: supposed to be known in advance • Good results regarding out-of-sample stability • The “true” distribution when testing out-of-sample stability: with GBM - standard in finance. Asset Allocation Problem Mean-CVaR Model Computational results