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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