Scenario Generation for the Asset Allocation Problem Diana Roman Gautam Mitra EURO XXII Prague July 9, 2007

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG The asset allocation problem x i =fraction of wealth invested in asset i  portfolio (x 1,…,x n ) R i =the return of asset i at time T The portfolio return at time T: R x =x 1 R 1 +…+x N R N (also r.v.!) How to choose between portfolios? A modelling issue! An amount of money to invest N stocks with known current prices S 1 0,…,S N 0 Decision to take: how much to invest in each asset Goal: to get a profit as high as possible after a certain time T The stock prices (returns) at time T are not known: random variables (stochastic processes)

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Mean-risk models for portfolio selection Mean – risk models: maximize expected value, minimise risk Risk: Conditional Value-at-Risk (CVaR) = the expected value of losses in a prespecified number of worst cases. The optimisation problem: Min CVaR(R x ) over x 1,…,x n S.t.: E(R x )  d ………… Max (E(R x ),- CVaR(R x )) over x 1,…,x n (1) Confidence level  =0.01  consider the worst 1% of cases

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Scenario Generation 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 ).

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Scenario Generation S Scenarios: p i =probability of scenario i occurring; r ij =the return of asset j under scenario i; The (continuous) distribution of (R 1,…,R N ) is replaced with a discrete one

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG The mean-CVaR model Scenarios  a LP (Rockafellar and Uryasev 2000) Min Subject to: r ij = the scenarios for assets’s returns 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”).

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG The quality of scenario generators 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

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG 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

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Out-of-sample stability: -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 The quality of scenario generators In practice: use a very large scenario set generated with an exogenuous SG method as the “true” distribution

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG -Out-of-sample stability: the important one -No (simple) relation between in-sample and out-of- sample stability The quality of scenario generators

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Hidden Markov Models applied in various fields, e.g. speech recognition still experimental for financial scenario generation Motivation: financial time series are not stationary; unexpected jumps, changing behaviour

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Hidden Markov Models Real world processes produce observable outputs – a sequence of historical prices, returns… A set of N distinct states: S 1,…,S N 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.

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Assumptions: First order Markov chain: at any time point, system’s state depends only on the previous state and not the whole history: P(q t =S i | q t-1 =S j, q t-2 =S k,….)= P(q t =S i | q t-1 =S j ) with q t =system’s state at time t Time independence: a ij =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 Hidden Markov Models Computational results Financial SG The output distributions: mixtures of normal distributions Mixtures of normal density functions can approximate any finite continuous density function. Hidden Markov Models M mixtures: =the normal density function with mean vector  j and covariance matrix  j

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Hidden Markov Models c 11,…,c 1M  11,…,  1M  11,…,  1M 11 22 33 a 11 a 12 a 21 a 13 a 31 N=3 M mixtures a 32 a 23  c 31,…,c 3M  31,…,  3M  31,…,  3M c 21,…,c 2M  21,…,  2M  21,…,  2M

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG The parameters of a HMM: Number of states N Number of mixtures M Initial probabilities of states:  1,…,  N Transition probabilities: A=(a ij ), i,j=1…N For each state i, parameters of the output distributions: Hidden Markov Models  Mixture coefficients c i1,…,c iM  Mean vectors  i1,...,  iM  Covariance matrices  i1,…,  i1.

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Historical data: O=(O 1,…,O T )=(r tj, t=1…T,j=1…N) is used to “train” the HMM. Meaning: Find the parameters =( , A, C, ,  ) s.t. P(O| ) maximised 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 Hidden Markov Models

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Training HMM’s 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 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

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Training HMMs: initial parameters How choose 0 ? Not important for  i and a ij (could be 1/N or random) Very important for C,  and  – but no”best” way to estimate them k-means clustering algorithm: separate historical data into M clusters starting parameters: Based on the mean vectors and covariance matrices of the clusters

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Training HMM: parameters re- estimation Forward probabilities: time t, state i Need to calculate additional quantities: Backward probabilities: time t, state i In calculus: the multi-variate normal density: Calculated recursively after time Use Baum-Welch algorithm (EM):

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Additional quantities: Probability of the historical observation to be generated by the current model: Training HMM: parameters re- estimation

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Training HMM: parameters re- estimation

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG HMM: estimation of the current state The state of the system at the current time? Via Viterbi algorithm Given an observation sequence O=(O 1,…,O T ) and a model, find an “optimal” state sequence Q=(q 1,…,q T ) i.e., that best “explains” the observations: maximises P(Q|O, )

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG HMM for scenario generation  A scenario: a path of returns for times T+1,…,T+TP  Estimate the current state (time T); say, q t =S i { Transit to a next state S j according to transition probabilities a ij Generate a return conform to the distribution of state j } …. t=1t=2t=3t=Tt=T+1t=T+2t=T+TP Historical data: estimation of Estimation of the system’s state at time T Generation of scenarios

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG HMM – implementation issues Number of states? Still a very much unsolved problem. 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

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Computational results Historical Dataset: 5 stocks from FTSE monthly returns: Jan 1993-Dec 2003  Generate scenario returns for 1 month ahead  500, 700, 1000, 2000, 3000 scenarios

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Computational results For each scenario size: -Run 30 times  generate 30 different discretisations for the assets’ returns (R 1,…,R N ) -Solve mean-CVaR model with these discretisations  get 30 solutions x 1,…,x 30 -Similar solutions as scenario size increases: (x 2 =x 3 =0, x 5 >=50%) -Evaluate these solutions on the “true” distribution?

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Computational results Out-of-sample stability: -The “true” distribution: generated with Geometric Brownian motion, scenarios -Each of the 30 solutions was evaluated on this distribution  30 “true” objective values (=portfolio CVaRs)

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Computational results Geometric Brownian motion (GBM) -The standard in finance for modelling stock prices -Stock prices are approximated by continuous time stochastic processes (accepted by practitioners…) S 0 : the current price  : the expected rate of return  : the standard deviation of rate of return {W t }: Wiener process - the “noise” in the asset’s price.

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG Computational results Statistics for the series of “true” objective functions 500 scen700 scen1000 scen2000 scen3000 scen Mean St Deviation Range Minimum Maximum Quality of solutions improve with larger scenario sets (as expected!) Reasonably small spread; pretty similar objective values

Asset Allocation Problem Mean-CVaR Model Hidden Markov Models Computational results Financial SG 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 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.