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- 1. 60 Years of Portfolio Optimization: Practical Challenges and Current Trends Petter Kolm Courant Institute (NYU) & the Heimdall Group, LLC Please Do Not Cite without Authors’ Permission Celebrating the 60th Anniversary of Modern Portfolio Theory Conference Honoring Nobel Laureate Harry Markowitz November 27, 2012, Fordham University, New York City60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 1
- 2. Presenter BiographyPetter Kolm is the Director of the Mathematics in Finance Masters Program and Clinical AssociateProfessor at the Courant Institute of Mathematical Sciences, New York University and the Principal ofthe Heimdall Group, LLC. Previously, Petter worked in the Quantitative Strategies Group at GoldmanSachs Asset Management where his responsibilities included researching and developing new quantitativeinvestment strategies for the groups hedge fund. Petter coauthored the books Financial Modeling of theEquity Market: From CAPM to Cointegration (Wiley, 2006), Trends in Quantitative Finance (CFAResearch Institute, 2006), Robust Portfolio Management and Optimization (Wiley, 2007), andQuantitative Equity Investing: Techniques and Strategies (Wiley, 2010). He holds a Ph.D. inmathematics from Yale, an M.Phil. in applied mathematics from Royal Institute of Technology, and anM.S. in mathematics from ETH Zurich. Petter is a member of the editorial board of the Journal ofPortfolio Management (JPM), International Journal of Portfolio Analysis and Management (IJPAM),Journal of Investment Strategies (JOIS), and the board of directors of the International Association ofFinancial Engineers (IAFE). As a consultant and expert witness, he has provided his services in areassuch as algorithmic and quantitative trading strategies, econometrics, forecasting models, portfolioconstruction methodologies incorporating transaction costs, and risk management procedures.Email: kolm@cims.nyu.eduWeb: http://www.theheimdallgroup.com http://www.cims.nyu.edu/~kolm60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 2
- 3. Happy 60th Anniversary to Harry & “Portfolio Selection”Markowitz (1952)60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 3
- 4. Outline Introduction o Markowitz’s modern portfolio theory in a nutshell Mean-variance optimization o Its tremendous impact o Some of its practical challenges Some current trends in portfolio optimization o A brief “deep dive”: Multi-period portfolio optimization with transaction costs and alpha decay (time permitting)Appendices: Mitigating estimation error in mean-variance optimization Portfolio optimization with market impact costs60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 4
- 5. Modern Portfolio Theory in a NutshellProvides a framework to Quantify risk and return Determine the trade-off between risk and returnAn investor allocates his wealth by solving the mean-variance optimization(MVO) problem max w ¢m - lw ¢Sw w s.t. w Î Cwhere w: portfolio weights m : expected returns S : covariance matrix of return l : risk aversion C: set of constraints (typically linear equality/inequality constraints)60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 5
- 6. Tremendous Impact of MVO Used extensively throughout the financial industry: o Asset and risk management, pension funds, hedge funds and prop desks o Financial planning, 401Ks The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1990 was awarded jointly to Harry Markowitz, Merton Miller and Bill Sharpe “for their pioneering work in the theory of financial economics” 18,338 articles in Google Scholar cites Markowitz’s original paper “Portfolio Selection” MVO is taught in thousands of undergraduate finance and economics, MBA and MFE programs around the world This morning, searching for “modern portfolio theory” returned o 650,000+ hits in Google o 12,500+ hits in Google Scholar o Many thousands of tweets on Twitter o 500+ YouTube videos o 200+ books on Amazon60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 6
- 7. Some Quotes from Practitioners on MVOSenior portfolio manager at a large hedge fund: “We don’t use MVO, Black-Litterman or any other off-the-shelf methodology. We always combine signalsusing homegrown techniques that we develop to suit our needs. The same holds forportfolio construction: There are a lot of competing objectives in play that make ithard to cast as a classical optimization problem. That’s why we ended up with alot of techniques that look ad-hoc, but are robust and work well for us. They areprobably not optimal in a mathematical sense, but personally I consider optimalityto be ill-defined in a noisy system like the stock market. We never spend a lot oftime trying to make things optimal. Robustness is far more important to us.”Portfolio manager at a large hedge fund: “We use different portfolio allocationapproaches for different strategies. Personally, I like using the Black-Littermanmodel with MVO, while others here find it too complicated and rely on simplerule-based approaches.” 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 7
- 8. Some Practical Challenges in Using MVO (1/2)While portfolio theory has experienced tremendous success there are a number ofchallenges that need to be addressed in order to successfully use it in practice: Sensitivity to estimation errors / small changes in the inputs o Estimation errors in returns are one order of magnitude more important than the estimation errors in the covariance matrix Simple portfolio rules (e.g. equal weighting) often outperforms MVO Optimal portfolios are not necessarily well diversified and result in “corner solutions” 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 8
- 9. Challenges in Using MVO (2/2) How to incorporate t-costs o Market impact models (quadratic vs. 3/2 power) o Per-trade or per-position costs (discrete elements that make the problem non-convex and difficult to solve) o How to trade-off alpha/risk/t-costs given that they are measured in different units and apply over different horizons Alpha and risk model misalignments o Prevalent in many factor model setups Constraints o Why do people use constraints (model insurance?) o The ability to incorporate constraints makes MVO flexible for different kinds of portfolios o Do constraints eat alpha?60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 9
- 10. Some New Directions in Portfolio Optimization “Diversification” methods that are less sensitive to estimation and model errors o 1/N, minimum variance portfolios, risk parity portfolios Higher moments and other utility functions o CVaR, tail risk Multi-period optimization o Alpha sources with different decay rates (e.g. reversal vs. value) and t- costs Overlaying multiple levels of alpha (e.g. combining country views with industry or equity views in a global portfolio) Tax-efficient portfolio optimization60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 10
- 11. A Brief “Deep Dive”: Multi-Period Portfolio Optimization with Transaction Costs and Alpha Decay 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 11
- 12. Multi-Period Portfolio Optimization with Transaction Costs and Alpha Decay1 In practice multi-period models are seldom used o Difficult to estimate return/risk for multiple periods o Computationally burdensome, especially if the universe is large In general, these problems lead to complicated PDEs (Hamilton- Jacobi-Bellman (HJB) equations) o Existing models do not handle constraints o Instead, it is common to use a single-period (myopic) model and rebalance from period to period Why use multi-period models? o Alpha decay depends on strategy o Our trading today impacts asset prices in the future o Minimize our trading costs (temporary and permanent) o Portfolio in/out-flow (additional investments, redemptions, taxes, etc.)60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 12
- 13. Related Work Merton (1969; (1990) o The investment-consumption problem Kritzman, Myrgren et al. (2007) o Portfolio transitions Engle and Ferstenberg (2007) o “Execution risk”, the interplay between transaction costs and portfolio risk Grinold (2006) Garleanu and Pedersen (2009) o Interplay between alpha-decay and transaction costs (only temporary one-period impact)60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 13
- 14. Myopic Case (1/2) ì ï ü ïx ¢a - g x ¢Sx - 1 x ¢Lx ï max í t t ï tý xt ï ï î 2 t t 2 t ï ï þFOC: at - gSx t - Lx t + Lx t -1 = 0Therefore, x t = (gS + L)-1 at + (gS + L)-1 Lx t -1 = alpha portfolio + drag from current holdings New holdings » weighted average of all past alphas Higher t-costs & less frequent trading → greater drag60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 14
- 15. Myopic Case (2/2) x t = (gS + L)-1 at + (gS + L)-1 Lx t -1 Does not incorporate dynamics of alpha o Different strategies operate on different time scales Crude way to model transaction costs o We want both temporary and permanent impact costs May want to include other forecasts (e.g. volume, volatility) (not discussed in this talk)60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 15
- 16. Let’s Look at This Intuitively . . . Q t60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 16
- 17. Stochastic Optimal ControlAn optimal control problem consists of the following components: Process dynamics Returns and alpha decay for all N assets Permanent and temporary market impact for all assets Observable quantities Cost functionNote: In general, optimal control problems lead to complicated PDEs (Hamilton- Jacobi-Bellman equations) Large dimension (number of assets)60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 17
- 18. Process Dynamics (1/2): Alpha Returns rt +1 = mt + at + etr+1 where rt +1 º pt +1 - pt Î N , mt are the fair security returns, Et (etr+1 ) = 0 , and Vart (etr+1 ) = S Alpha driven by K mean-reverting factors ( K N )2 at = Bft + eta Dft +1 = -Dft + etf+1where ft Î K (factors), B Î N ´K (factor loadings) D Î K´K , positive definite matrix of mean-reversion coefficients Et (eta ) = Et (etf+1 ) = 0 , Vart (etf+1 ) = Sf , and Vart (eta ) = Sa error terms etr+1, eta , etf mutually independent60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 18
- 19. Short Excursion: Idealized Market Impact Modelprice t 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 19
- 20. Process Dynamics (2/2): Market Impact Costs Temporary and permanent impact (Almgren and Chriss (2000) dynamics)3 at = Bft + eta + PDx t + H Dx t - H Dx t -1 permanent temporary a = Bft + e + PDx t + H (Dx t - Dx t -1 ) t If temporary impacts persist for several periods, we can model this by introducing a new state variable4 ht = Ght -1 + (I - G )H Dx t -1 + eth Combining this with the alpha process, we have at = Bft + eta + (P + H )Dx t - ht 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 20
- 21. Process Dynamics as a State Space ModelConsider the “augmented state” st = ( ft , x t , ht )¢We write the state dynamics as æI - D 0 0 ö æ0 ö æe f ö ç ÷ ÷ ç ç ÷ ÷ ç t÷ ç ÷ ç ç 0 st = ç ÷ ÷ s + çI I 0 ÷ t -1 ç ÷ Dx + ç 0 ÷ ÷ ÷ t ç ÷ ç ÷ ç ÷ ç ÷ ÷ ç ç 0 ç 0 G÷ ÷ ÷ ç ÷ ç(I - G )H ÷ ç ÷ çe ÷ ç h÷ ç ÷ è ø è ø è tø ˆ ˆ = Ast -1 + B Dx t + et æ 0 0ö çS f ç ÷ ÷ ÷where Vart -1(et ) = ç 0 ç ç 0 0÷ ÷ ç ÷ ÷ ç ç0 ÷ 0 Sh ÷ è ø 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 21
- 22. Objective Function (1/2)Multi-period optimization problem: éìT -1 ï ü ï ù max E ê êíå (1 - r)t (x ¢a - l x ¢Sx - 1 Dx ¢LDx )ý + (1 - r)T (x ¢ a - l x ¢ Sx )ú ï ïDx1 ,Dx 2 ,... êëï t =1 ï î t t 2 t t 2 t t ï ï þ T T 2 T T úúûwhere r Î (0,1) is a discount factor, l is a risk aversion coefficient, S is thecovariance matrix of returns, and x 0 are the initial portfolio holdingsRemark: In this model, the trade size D x t is our control variable Model contains Grinold (2006), Engle and Ferstenberg (2007), Garleanu and Pedersen (2009) as special cases60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 22
- 23. Objective Function (2/2)We observe that (at time t) l 1 l 1x t¢at - x t¢Sx t - Dx t¢LDx t = x t¢(Bft + eta + (P + H )Dx t - ht ) - x t¢Sx t - Dx t¢LDx t 2 2 2 2 æs ö¢ æ R S öæs ö ÷ ÷ç t ÷ =ç t ÷ ç ç çDx ÷ ç÷ çS ¢ Q ÷çDx ÷ + O(et ) ÷ç ÷ a ç t÷ ç è øè ÷è t ø ÷ç ø ÷ ÷ º c(st , Dx t ) + O(eta )where æ 1 ö ç 0 ç B¢ 0 ÷ ÷ ÷ æ 0 ö ç ç 2 ÷ ç ÷ ÷ ç1 ÷ ç ç1 ÷ ç l 1 ÷ ÷ ç ÷ 1 R = ç B - S - I ÷, ç ÷ S = ç (P + H )÷, and Q = - L ç2 ÷ ÷ ç2 ç 2 2 ÷ ÷ ç ç ÷ ÷ 2 ç 1 ÷ ÷ ç 0 ÷ ç 0 - I 0 ÷ ç è ÷ ÷ ø ç ç ÷ ÷ ç è 2 ø60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 23
- 24. Multi-Period Portfolio Optimization with Alpha Decay and Market Impact as aStochastic LQ Regulator ProblemProcess dynamics: ˆ ˆ st = Ast -1 + B Dx t + etObjective function: é T -1 ù max E ê å (1 -r)t c(s , Dx ) + (1 - r)T C (s )ú Dx1 ,Dx 2 ,...,DxT -1 ê t =1 t t T ú ë ûwhere æ ÷¢ æ ö öæ ÷ ÷ç ö çs ÷ ç R S ÷çs ÷ ç c(s, Dx ) = ç ÷ ç ç C (s ) = s ¢Rs ÷ ÷ and çDx ÷ çS ¢ Q ÷çDx ÷ è ÷èø ÷ç ÷ øè øand R = R¢ ³ 0, S ³ 0,Q = Q ¢ > 060 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 24
- 25. Main Results (1/2) The optimal trade is linear in the state variable, i.e. Dx t = Ltst ˆ ˆ ˆ ˆ where Lt = -(Q + (1 - r)B ¢Kt +1B)-1(S + (1 - r)B ¢Kt +1A), t < T , where K t satisfies the Riccati equation ˆ K t = R + (1 - r)A¢ K t +1Aˆ ˆ ˆ ˆ ˆ ˆ ˆ -(S ¢ + (1 - r)A¢ K t +1B )(Q + (1 - r)B ¢K t +1B )-1(S + (1 - r)B ¢K t +1A) Can show that optimal portfolio is weighted average of o Current holdings o Alpha portfolio o Expected optimal portfolio based on future periods alpha and expected temporary and permanents market impacts60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 25
- 26. Main Results (2/2) Computationally, we can efficiently solve this model for a large universe Any positive expected value strategy is tradable at some rate of trading (fixed costs aside) A case against “lazy portfolios”: Rebalancing less frequently does not reduce market impact costs – it increases them 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 26
- 27. The Constrained Multi-Period Portfolio Optimization ProblemMulti-period portfolio optimization with transaction costs: ˆ ˆ st = Ast -1 + B Dx t + et é T -1 ù max E ê å (1 -r)t c(s , Dx ) + (1 - r)T C (s )ú Dx1 ,Dx 2 ,... ê t =1 t t T T ú ë ûWe introduce linear inequality constraints of the form GS st + G Dx Dx t £ g, t = 1, 2, 3,...where Gs Î L´(K +2N ) , GDx Î L´N , and g Î LRemarks: Does not have a closed form solution; no Riccati equation (cf. QP) The deterministic counterpart (i.e. et º 0 ) can be shown to have a piecewise linear optimal control defined on a polyhedral partitioning of the state space o Can be difficult to calculate the partitioning60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 27
- 28. Summary Traditional view is suboptimal: Instead, portfolio construction, risk analysis, and execution should all be integrated Tractable framework for dynamic portfolio analysis: o Alpha decay o Market impact costs o Realistic portfolio constraints Other applications of this framework: o High-frequency automatic market making o Index tracking and enhanced indexing with transaction costs o Risk management: Portfolio and liquidity risk “Prediction”: In the next years, we will see more multi-period portfolio optimization used in practice60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 28
- 29. Appendix: Mitigating Estimation Error in MVO 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 29
- 30. Mitigating Estimation Error in MVO Constrain portfolio weights (“the practitioner’s solution”) o No short-selling constraints (see for example, Frost and Savarino (1988), Chopra (1991), Gupta and Eichhorn (1998), Grauer and Shen (2000), Jagannathan and Ma (2003)) o “Diversification indicators” (Bouchard, Potters et al. (1997)) Improve estimation o Bayesian techniques: James-Stein estimation (Jobson and Korkie (1981), Ledoit and Wolf (2003)) Black-Litterman model (Black and Litterman (1990)) o Robust statistics (Trojani and Vanini (2002), DeMiguel and Nogales (2006)) Incorporate estimation error in portfolio allocation o Adjustment of risk aversion factor (Horst, Roon et al. (2000))60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 30
- 31. o Resampled efficiency (Michaud (1998), Jorion (1992), Scherer (2002), Markowitz and Usmen (2003))Robust optimization (El Ghaoui and Lebret (1977), Ben-Tal and Nemirovski(1998; (1999))60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 31
- 32. Appendix: Portfolio Optimization with Market Impact Costs 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 32
- 33. Portfolio Optimization with Market Impact CostsTransaction costs extensions of the mean-variance framework are typically of theform max w ¢m - lw ¢Sw - hTC (x ) w s.t. w ¢e + TC (x ) £ w preve ¢ w ÎCwhere w: portfolio weights; m : vector of expected returns S : covariance matrix of returns; l : risk aversion coefficient x = w - w prev : trade; C : other constraints e = (1,1,...,1)¢ h : “aversion to t-costs” Transaction costs for trade x : { TC i (x i ) = max ai ⋅ | x i |, bi ⋅ | x i |2 +ci ⋅ | x i |1+b }60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 33
- 34. Practical Considerations Typically, TC (x ) = MIC (x )¢ ⋅ x where MIC is the market impact cost function Transaction costs models often involve nonlinear functions Software for general nonlinear optimization problems available, but computational time required for solving such problems is often too long for realistic investment management applications (large universes with thousands of assets) Efficient and reliable software is available for linear (LP), quadratic (QP), and second-order cone programs (SOCP)→ Can approximate more complex nonlinear optimization problem by simplerproblems that can be solved quickly (piecewise linear approximations)60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 34
- 35. Separable TCsTransaction cost penalty function is a separable function dependent only on theportion to be traded x = w - w 0 , where w 0 is the original portfolio and w is thenew portfolio after rebalancing. We express this as N TC (x ) = åTC i (x i ) i =1where TC i is the transaction cost function for security i and x i is the portion ofsecurity i to be traded60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 35
- 36. Linear-Quadratic TCsParameterize the transaction cost function TC i as a quadratic function of theform TC i (x i ) = ai ⋅ c{x ¹0} + bi x i + gi x i 2 i = fixed cost + proportional cost + quadratic (market impact) cost c{x ¹0} is the indicator function: equal to one when x i ¹ 0 and zero otherwise5 i ai , bi , and gi may be different for each asset60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 36
- 37. Ignoring fixed costs, we obtain the problem ( max w ¢m - lw ¢Sw - lTC b ¢ x + x ¢Gx w )subject to the usual constraints, where b ¢ = (b1,..., bN ) and ég 0 0 ùú ê i ê0 g2 úú ê G = êê úú ê 0 úú ê ê0 0 gN úú êë û This is QP problem that can be solved with exactly the same software that we use for solving the classical mean-variance optimization problem But remember: Typical transaction cost functions are of the form { TC i (x i ) = max ai ⋅ | x i |, bi ⋅ | x i |2 +ci ⋅ | x i |1+b } So can we do better? How does it differ?60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 37
- 38. Using the Impact Model Separate temporary (non-linear) and permanent impact (linear) as in Almgren, Thum et al. (2005): o Perform non-linear regression to find temporary impact parameters (g, h, b ) : nt I tperm = g ⋅ st ⋅ + etperm Vt b nt I ttemp = h ⋅ st ⋅ sign(nt ) ⋅ + ettemp Vt ⋅ T o Expected t-cost (in dollars): æ ö ç 1 I perm + I temp ÷ TC (S t ⋅ nt ) = St ⋅ nt ⋅ ç t ÷ ç2 t ÷ ÷ è ø 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 38
- 39. Market Impact (June – Sept, 2007)6 b = 0.63 0.054 [0.60 0.038] g = 2.31 0.073 [0.31 0.041] h = 0.22 0.028 [0.14 0.006] Order imbalance Daily volume60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 39
- 40. Estimated Market Impact Cost Function (with Standard Errors) 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 40
- 41. Piecewise Linear Approximation (2 Pieces) (a) Type of fit; (b) No. of segments (c) Placement of breakpoints60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 41
- 42. Piecewise Linear Approximation: Simple Model T-Cost ($) 1% 2% 3% Trade size Volume ìs x , ï 0 £ x £ 1% ⋅ Vol ï1 ïTC (x ) = ïs1 (1% ⋅ Vol) + s 2 (x - 1% ⋅ Vol), í 1% ⋅ Vol £ x £ 2% ⋅ Vol ï ïs 1% ⋅ Vol + s (1% ⋅ Vol)+s (x - 2% ⋅ Vol), ï 1( ï ) 2 2% ⋅ Vol £ x £ 3% ⋅ Vol î 3where s1 < s2 < s 3 are the (known) slopes of the three linear segments60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 42
- 43. A “QP Friendly” Formulation N ( TC (x ) = å s1,i ⋅ y1,i + s2,i ⋅ y2,i + s 3,i ⋅ y 3,i , i =1 ) s1 < s2 < s 3where7 0 £ y1,i £ 0.15 ⋅ Voli 0 £ y 2,i £ 0.25 ⋅ Voli 0 £ y 3,i £ 0.15 ⋅ Voliand wi - w prev ,i = y1,i + y 2,i + y 3,i Converges to the nonlinear impact function by increasing the number of line segments The optimization may become “unstable” if too many segments are used (see, Rote (1992) and Ceria, Takriti et al. (2008))60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 43
- 44. Remark: Using Conic (SOCP) Solvers8We can incorporate 3/2-power transaction cost functions (coming from squareroot market impact cost functions) directly into a conic solver. This fact is basedon the observation that the following two optimization problems are equivalent: N1. min x1 ,...,x N åa x 3/2 i i i =1 N2. min x1 ,...,x N åa y i i y1 ,...,yN i =1 s.t. x 13/2 £ y1,..., x N £ yN 3/2Now we note that the constraints x i3/2 £ yi are equivalent with the rotatedquadratic cone inequalities x i2 £ yi ⋅ z i z i2 £ x i60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 44
- 45. Summary Impact functions are very noisy: We only need to approximate them to the order of the estimation error when used in portfolio optimization o A few linear/quadratic segments are enough o Allows us to solve very large problems and If we have access to a conic (SOCP) solver we can incorporate 3/2-power market impact functions directly60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 45
- 46. ReferencesAlmgren, R. and N. Chriss (2000). "Optimal Execution of Portfolio Transactions." Journal of Risk 3(2): 5-39.Almgren, R., C. Thum, E. Hauptmann and H. Li (2005). "Direct Estimation of Equity Market Impact." Risk 18: 57–62.Almgren, R., C. Thum, E. Hauptmann and H. Li (2005). "Equity Market Impact." Risk 18(7): 57-62.Ben-Tal, A. and A. S. Nemirovski (1998). "Robust Convex Optimization." Mathematics of Operations Research 23(4): 769- 805.Ben-Tal, A. and A. S. Nemirovski (1999). "Robust Solutions to Uncertain Linear Programs." Operations Research Letters 25(1): 1-13.Black, F. and R. Litterman (1990). "Asset Equilibrium: Combining Investor Views with Market Equilibrium." Journal of Fixed Income.Bouchard, J.-P., M. Potters and J.-P. Aguilar (1997). Missing Information and Asset Allocation. Science & Finance, Capital Fund Management.Ceria, Takriti, Tierens and Sofianos (2008). Incorporating the Goldman Sachs Shortfall Model into Portfolio Optimization. Axiomas Breakfast Research Seminar Series. New York.Chopra, V. (1991). "Mean-Variance Revisited: Near-Optimal Portfolios and Sensitivity to Input Variations." Russell Research Commentary 2: 1-15.DeMiguel, V. and F. Nogales (2006). Portfolio Selection with Robust Estimates of Risk, Citeseer.El Ghaoui, L. and H. Lebret (1977). "Robust Solutions to Least-Squares Problems with Uncertain Data." SIAM Journal on Matrix Analysis and Applications 18(4): 1035-1064.Engle, R. F. and R. Ferstenberg (2007). "Execution Risk " Journal of Portfolio Management 33(2): 34-44.Frost, P. and J. Savarino (1988). "For Better Performance: Constrain Portfolio Weights." Journal of portfolio management 15(1): 29-34.Garleanu, N. B. and L. H. Pedersen (2009). "Dynamic Trading with Predictable Returns and Transaction Costs." SSRN eLibrary.60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 46
- 47. Grauer, R. and F. Shen (2000). "Do Constraints Improve Portfolio Performance." Journal of Banking and Finance 24(8): 1253-1274.Grinold, R. (2006). "A Dynamic Model of Portfolio Management." Journal of Investment Management(2): 5-22.Gupta, F. and D. Eichhorn (1998). "Mean-Variance Optimization for Practitioners of Asset Allocation." Handbook of Portfolio Management, Frank J. Fabozzi Associates, New Hope, Pennsylvania.Horst, J., F. Roon and B. Werker (2000). Incorporating Estimation Risk in Optimal Portfolios, Working Paper Tilburg University 2000.Jagannathan, R. and T. Ma (2003). "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps." Journal of Finance: 1651-1683.Jobson, J. and B. Korkie (1981). "Putting Markowitz Theory to Work." Journal of portfolio management 7(4): 70–74.Jorion, P. (1992). "Portfolio Optimization in Practice." Financial Analysts Journal: 68-74.Kritzman, M., S. Myrgren and S. Page (2007). "Optimal Execution for Portfolio Transitions." Journal of portfolio management 33(3): 33.Ledoit, O. and M. Wolf (2003). "Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection." Journal of Empirical Finance 10(5): 603-621.Markowitz, H. and N. Usmen (2003). "Resampled Frontiers Versus Diffuse Bayes." Journal of Investment Management 1(4): 9-25.Markowitz, H. M. (1952). "Portfolio Selection." Journal of Finance 7(1): 77-91.Merton, R. C. (1969). "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case." The Review of Economics and Statistics 51(3): 247-257.Merton, R. C. (1990). Continuous-Time Finance, Blackwell Publishers.Michaud, R. (1998). "Efficient Asset Allocation: A Practical Guide to Stock Portfolio Optimization and Asset Allocation." Harvard Business School Press, Boston.Rote, G. (1992). "The Convergence Rate of the SandwichAlgorithm for Approximating Convex Functions." Computing 48(3): 337-361.60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 47
- 48. Scherer, B. (2002). "Portfolio Resampling: Review and Critique." Financial Analysts Journal: 98-109.Trojani, F. and P. Vanini (2002). "A Note on Robustness in Mertons Model of Intertemporal Consumption and Portfolio Choice." Journal of Economic Dynamics and Control 26(3): 423-435.Endnotes 1 For further details, see Kolm, “Dynamic Portfolio Analysis with Transaction Costs, Alpha Decay, and Constraints”, inpreparation.2 In the first equation, ft Î K represents the factors, B Î N´K the factor loadings, and etf the idiosyncratic components.This specification generalizes a standard static factor model, making it time-dependent. The second equation specifies thetemporal behavior of the factors. Here D Î K´K is a positive definite matrix of mean-reversion coefficients. Intuitively,the greater the elements of this matrix the faster the factors mean revert to zero.3 Note that the term H (Dwt - Dwt -1 ) reverses the effect of a trade from one period to the next, making the impact ofH Dwt effective for one period only (single period impact). Therefore, we refer to H Dwt as the temporary component andPDwt as the permanent component.4 Here the matrix G Î N´N (with G < 1 ) determines how fast the temporary impact decays. We assume Et (eth ) = 0 , andVart (eth ) = Sh .5 Fixed costs are hard to deal with as they lead to combinatorial optimization problems.60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 48
- 49. 6 The parameter estimates from Almgren et al. (2005) are reported in brackets. Note that the model used in thispresentation is a slight simplification compared to their model. In particular, we absorb the cross-sectional liquidity d æQ ö ç ÷correction term ç t ÷ into çV ÷ g. ç ÷ è tø7 Note that because of the increasing slopes of the linear segments and the goal of minimizing that term in the objectivefunction, the optimizer will never set the decision variable corresponding to the second segment, y 2,i , to a number greaterthan 0 unless the decision variable corresponding to the first segment, y1,i , is at its upper bound. Similarly, the optimizerwould never set y 3,i , to a number greater than 0 unless both y1,i , and y 2,i , are at their upper bounds. So, this set ofconstraints allows us to compute the total traded amount of asset i as y1,i + y2,i + y 3,i .8 I thank Reha Tütüncü for pointing this out to me. 60 YEARS OF PORTFOLIO OPTIMIZATION: PRACTICAL CHALLENGES AND CURRENT TRENDS, VER. 11/27/2012. © P. KOLM 49

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