reverse optimization of growth optimal portfolio selection
DESCRIPTION
Reverse Optimization of Growth Optimal Portfolio SelectionTRANSCRIPT
REVERSE OPTIMIZATION OF
GROWTH OPTIMAL PORTFOLIO SELECTION
Márk HorváthBudapest University of Technology and Economics, Department of Stochastics,
András Urbán
Budapest University of Technology and Economics, Department of Finance,Hungary
GOALS OF THE PAPER Compare four asset pricing methods:
Historical average of returnsCapital Asset Pricing Model (CAPM)Carhart Four-Factor ModelGrowth Optimal Pricing Model (GOPM)
Basis of comparision:Squared error
T
t
d
ii
it
it
RVar
RE
1= 1)(
2)()(1,
)(
)(E
REVERSE OPTIMIZATION
investor
CovariancesExpected returns
Portfolio optimization
Market portfolio
Expected returns
Forward optimization
Reverse optimization
METHODOLOGY
time
RTETexpected
returnrealizedreturn
past future
2)()(1, )( i
tit RE
HISTORICAL AVERAGE
time
past future
1
1=
)()(T
t
iT
iT RE
CAPITAL ASSET PRICING MODEL
time
past future
))((= )()()()()( fT
MT
iT
fT
iT rRrE E
)(
),(= )(
)()()(
MT
MT
iTi
TRVar
RRCov
CAPITAL ASSET PRICING MODELDETAILS
time
))((= )()()()()( fT
MT
iT
fT
iT rRrE E
)( )(MTRE
1
1=
)(T
t
MtR
Market return is memoryless
CAPITAL ASSET PRICING MODELDETAILS
Market volatility has memoryExponential weigthed moving covariance
)(~1),(~
),(~1),(~
)(1
2)()(
1)()(
)(1
)(1
)()(1
)()(1
)()(
iT
iiT
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iT
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iT
MMT
iiT
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iT
RraVRRRRraV
RRvoCRRRRRRvoC
)(
),(= )(
)()()(
MT
MT
iTi
TRVar
RRCov
CARHART FOUR-FACTOR MODEL
)()()())((= )()()()()()()()(T
iT
iT
ifT
MT
ifT
iT MOMmomHMLhSMBsrRrE EEEE
))((= )()()()()( fT
MT
ifT
iT rRrE E
GROWTH OPTIMAL PRICINGMaximizing wealth level:
011 )1(...)1()1(= WRRRW TTT
TTGtR
T
tt
T
tT eWeWRWW 0
)1ln(
1=0
1=0 ==)1(=
)1ln(
)1ln(1
=1=
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tT
RE
RT
G
)(
)()(
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11
)11
,(1
=)(1
MT
MT
iT
iT
R
RRCov
R
E
E
TASK
1
1=
)()(T
t
iT
iT RE
))((= )()()()()( fT
MT
iT
fT
iT rRrE E
)(
)()(
)(
11
)11
,(1
=)(1
MT
MT
iT
iT
R
RRCov
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E
)()()())((= )()()()()()()()(T
iT
iT
ifT
MT
ifT
iT MOMmomHMLhSMBsrRrE EEEE
DATA
•Standard and Poor’s 500 constituents – weekly returns
Assets•Janu
ary 1970– December 2008
Interval
•US Treasury Bill – 1 month
Risk-free rate
•NYSE, AMEX and NASDAQ capitalization weighted average, including dividends
Market index
STATISTICAL TESTS
0},)(
)(
)(
)({:
1= 1)(
2)()(2,
)(
2)()(1,
0
T
t
d
ii
it
it
i
it
it
RVar
RE
RVar
REH
EE
Method SNSE diff.
Student’s p-value
Four-Factor – Average 311 2.15%
CAPM - Four-Factor 104 39.05%
GOPM - CAPM 51 3.33%
STATISTICAL TESTSDETAILS