efficient portfolio optimizations in s-plus / nu-optfrey/instruction/priorsemesters... · portfolio...
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2Efficient Portfolio OptimizationsEfficient Portfolio Optimizations
in Sin S--Plus / NUOPTPlus / NUOPTStatistical Modeling and Computation in FinanceStatistical Modeling and Computation in FinanceMarriott Marquis, New York Marriott Marquis, New York -- 28 September 200028 September 2000
Renaissance Technologies Corp.Renaissance Technologies Corp.Dr. Robert J. Frey, Managing DirectorDr. Robert J. Frey, Managing Director
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2•• Began April 1999Began April 1999
•• Over $500 millionOver $500 million
•• U.S. equity market neutral U.S. equity market neutral across multiple factorsacross multiple factors
•• “Low” turnover and modest “Low” turnover and modest leverageleverage
•• Closed June 2000
EquimetricsEquimetrics
Closed June 2000
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2•• Annualized return 28.7%Annualized return 28.7%
•• Twice the S&P 500 return with Twice the S&P 500 return with only twoonly two--thirds the riskthirds the risk
•• Strict, modelStrict, model--driven strategydriven strategy
•• Portfolio optimization is used Portfolio optimization is used extensivelyextensively
•• Typically holds over 1,000 Typically holds over 1,000 positions from a universe of positions from a universe of 1,500 stocks1,500 stocks
•• Portfolio is rebalanced daily
Equimetrics’Equimetrics’PerformancePerformance
&&CharacterCharacter
Portfolio is rebalanced daily
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2•• Portfolio efficiency is a key Portfolio efficiency is a key
element in investment element in investment managementmanagement
•• Conceptually difficult and Conceptually difficult and computationally expensivecomputationally expensive
•• Black box or mechanical Black box or mechanical approaches are dangerous approaches are dangerous
•• R&D requires an efficient R&D requires an efficient solution approach
PortfolioPortfolioOptimizationOptimization
solution approach
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2•• Up to one thousand stocks as Up to one thousand stocks as
potential investmentspotential investments
•• Minimize variance of returnMinimize variance of return
•• Achieve at least a target Achieve at least a target expected returnexpected return
•• Invest all available capitalInvest all available capital
•• No short positionsNo short positions
•• No more than 2% in any one No more than 2% in any one position
Case StudyCase Study
position
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02.001
tosubject
minimize
min
21
≤≤=
≥
xx1xµ
Qxx
T
T
T
rMathematicalMathematicalFormulationFormulation
II
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2
02.00
1
tosubject
minimize
min
,21
≤≤
=
≥
∑
∑
∑∑
i
ii
iii
jii j
ji
x
x
rx
xx
µ
σ
MathematicalMathematicalFormulationFormulation
I’I’
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2SIMPLE I…
model.00 <- function(Q, r, minR, minP) {Stock <- Set()i <- Element(set = Stock)j <- Element(set = Stock)pQ <- Parameter(Q, index = dprod(Stock, Stock))pR <- Parameter(list(1:ncol(Q), r), index = Stock)vX <- Variable(index = Stock)oF <- Objective(minimize)oF ~ 0.5 * Sum(vX[i] * Sum(pQ[i, j] * vX[j], j), i)Sum(pR[i] * vX[i], i) >= minRSum(vX[i], i) == 1vX[i] >= 0vX[i] <= minP }
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2•• MultiMulti--factor model: factor model:
rr = = BfBf++ εε
•• WOLG assume ten factors, allWOLG assume ten factors, allorthonormalorthonormal
•• Covariance matrix can be Covariance matrix can be restated asrestated as
Q = BBQ = BBTT + D
Risk ModelRisk Model
+ D
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2•• Factor loading for a portfolio Factor loading for a portfolio
with positions with positions xxy = y = BBTTxx
•• Systematic variance from Systematic variance from factorsfactors
yyTTy y = = xxTTBBBBTTxx
•• Unsystematic variance Unsystematic variance (residual risk)(residual risk)
xxTTDx
Factor Factor LoadingLoading
Dx
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0yxBx
x1xµ
xy
DI
xy
=−
≤≤=
≥
T
T
T
T
r
02.001
tosubject
minimize
min
21
MathematicalMathematicalFormulationFormulation
IIII
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jyxbx
x
rx
xy
ji
iji
i
ii
iii
iii
jj
∀=−
≤≤
=
≥
+
∑
∑
∑
∑∑
,002.00
1
tosubject
minimize
,
min
22221
µ
ν
MathematicalMathematicalFormulationFormulation
II’II’
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2SIMPLE II…model.01 <- function(B, v, r, minR, minP) {
Stock <- Set()Factor <- Set()i <- Element(set = Stock)k <- Element(set = Factor)pB <- Parameter(B, index = dprod(Stock, Factor))pV <- Parameter(list(1:length(v), v), index = Stock)pR <- Parameter(list(1:length(r), r), index = Stock)vX <- Variable(index = Stock)vY <- Variable(index = Factor)oF <- Objective(minimize)oF ~ 0.5 * Sum((pV[i] * vX[i])^2, i) + Sum(vY[k]^2, k)Sum(pR[i] * vX[i], i) >= minRSum(vX[i], i) == 1.0vX[i] >= 0.0vX[i] <= minPSum(pB[i,k] * vX[i] - vY[k], i) == 0.0 }
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2I versus II…
oF ~ 0.5 * Sum(vX[i] * Sum(pQ[i, j] * vX[j], j), i)
oF ~ 0.5 * Sum((pV[i] * vX[i])^2, i) + Sum(vY[k]^2, k)Sum(pB[i,k] * vX[i] - vY[k], i) == 0.0
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SampleSampleRun TimesRun Times
n I II
100
250
500
1,000
Pentium III – 667 MHz. – 512 Mb.
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SampleSampleRun TimesRun Times
n I II
100 0.62 sec.
250 8.68
500 57.97
1,000 445.76
Pentium III – 667 MHz. – 512 Mb.
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SampleSampleRun TimesRun Times
n I II
100 0.62 sec. 0.08 sec.
250 8.68 0.16
500 57.97 0.24
1,000 445.76 0.47
Pentium III – 667 MHz. – 512 Mb.
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SampleSampleRun TimesRun Times
n I II
100 0.62 sec. 0.08 sec.
250 8.68 0.16
500 57.97 0.24
1,000 445.76 0.47
2,500 1.43
5,000 3.29
7,500 5.82
10,000 8.14
Pentium III – 667 MHz. – 512 Mb.
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log(Size)
log(
Run
time)
5.0 5.5 6.0 6.5
02
46
Optimization Formulation I
Slope = 2.9
~ cubic
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log(Size)
log(
Run
time)
5 6 7 8 9
-2-1
01
2
Optimization Formulation II
Slope = 1.1
~ linear
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2•• Efficiency matters because Efficiency matters because
optimization is both pervasive optimization is both pervasive and expensiveand expensive
•• Exploit the special structure of Exploit the special structure of a problem whenever possiblea problem whenever possible
•• Test alternatives to see what Test alternatives to see what tradetrade--offs make senseoffs make sense
•• Build tools to create a Build tools to create a productive environment
ConclusionsConclusions
productive environment
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Thank You!Thank You!
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