reverse optimization of growth optimal portfolio selection
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REVERSE OPTIMIZATION OF
GROWTH OPTIMAL PORTFOLIO SELECTION
Márk HorváthBudapest University of Technology and Economics, Department of Stochastics,
HungaryMHORVATH@MATH.BME.HU
András Urbán
Budapest University of Technology and Economics, Department of Finance,Hungary
URBAN@FINANCE.BME.HU
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
MT
iT
MT
iT
MMT
iiT
MT
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=
T
t
T
tT
RE
RT
G
)(
)()(
)(
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
R
E
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
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