Massachusetts Institute of Technology Sponsor: Electrical Engineering and Computer Science Cosponsor: Science Engineering and Business Club Professional Portfolio Selection Techniques: From Markowitz to Innovative Engineering Part 2 and in collaboration with MIT - Wed Jan 16, Thu Jan 17 2008, 04:00-6:00pm, 34-401 1
The process of portfolio construction Asset allocation: - strategic asset allocation - tactical asset allocation 1 st Day January 16 th G.A.M Model: a new tactical asset allocation technique PID feedback controller theory Applications and future research 2 nd Day January 17 th January October 16, 4, 2007 2008 2
INNOVATION 1. INTRODUCTION Use of the Feedback controller, widely applied in most industrial processes, as a technique for financial portfolio management.** AIM METHOD Tactical Portfolio Asset Allocation Technique. Rebalancing of Assets determined by the controlled value of Risk Adjusted Return subject to the action of the Controller. The innovative procedure consists in the controlling action over the uncertain behavior of the plurality of assets comprising the portfolio. The controller attempts to regulate the dynamics of the portfolio by rebalancing the weights of the different assets in such a way to force the portfolio risk adjusted return to approach the Set Point. (*), ** Patent Pending International - National 3
The Innovation 1. INTRODUCTION Comprises Seeking STABILITY CONSISTENCY of Portfolio Risk Adjusted Return over the time horizon by controlling Risk Adjusted Retun 4
2. BACKGROUND Strategic Asset Allocation = Selecting a Long Term Target Asset Allocation most common framework: mean-variance construction of Markowitz (1952) Tactical Asset Allocation = Short Term Modification of Assets around the Target systematic and methodic processes for evaluating prospective rates of return on various asset classes and establishing an asset allocation response intended to capture higher rewards 5
2. BACKGROUND Tactical Asset Allocation (TAA) asset allocation strategy that allows active departures from the Strategic asset mix based upon rigorous objective measures active management. It often involves forecasting asset returns, volatilities and correlations. The forecasted variables may be functions of fundamental variables, economic variables or even technical variables. 6
3. RISK ADJUSTED RETURN Portfolio Managers main Objective is to achieve a relevant Risk Adjusted Return. In literature and in the financial industry business, numerous kinds of return/risk ratios are commonly used. Sharpe Ratio Sortino Ratio Treynor Ratio Information Ratio... 7
4. SYSTEMS: MANUAL VS AUTOMATIC SYSTEMS Manual Control = System involving a Person Controlling a Machine. Automatic Control = System involving Machines Only. 8
4. SYSTEMS: MANUAL VS AUTOMATIC SYSTEMS (ESAMPLES) Manual Control: Driving an Automobile Automatic Control: Room Temperature Set by a Thermostat 9
4. SYSTEMS: REGULATORS VS TRACKING (SERVO) SYSTEMS Regulators: Systems designed to Hold a System Steady against Unknown Disturbances Servo: Systems designed to Track a Reference Signal 10
4. OPEN-LOOP SYSTEMS The Controller does not use a Measure of the System Output being Controlled in Computing the Control Action to Take. 11
4. FEEDBACK SYSTEMS Feedback Systems (Processes): defined by the Return to the Input of a part of the Output of a Machine, System, or Process. Controlled Output Signal is Measured and Fed Back for use in the Control Computation. 12
4.1 OPEN AND CLOSED LOOPS OPEN LOOP SYSTEM CLOSED LOOP SYSTEM System 1 affects system 2 System 2 affects system 1 13
4.1 CLOSED LOOP (EXAMPLE) Household Furnace Controlled by a Thermostat: Qout Desired Temperature THERMO- STAT Gas Valve FURNACE Qin + - HOUSE Room Temperature Fig. 01 BLOCK DIAGRAM 14
4.1 CLOSED LOOP (EXAMPLE) Household Furnace Controlled by a Thermostat: Plot of Room Temperature and Furnace Action Initially Room Temperature << Reference (or SET POINT) Temperature. Thermostat ON Gas Valve ON Heat Qin supplied to House at rate > Qout (Heat loss) Room temperature will rise until > Reference Point Gas Valve OFF Room Temperature will drop until below Reference point Gas Valve ON 15
4.1 CLOSED LOOP (Components) ACTUATOR = Gas Furnace PROCESS = House OUTPUT = Room Temperature Disturbances = Flow of Heat from the house via wall conduction, etc. PLANT = Combination of Process and Actuator CONTROLLER = components which compute desired controlled signal SENSOR = Thermostat COMPARATOR = Computes the difference between reference signal and sensor output. 16
4.2 FEEDBACK SYSTEM PARAMETERS Set-Point = Target Value that an Automatic Control System will aim to Reach. Output = Current Output of the System. Error = Difference between Set Point and Current Output of the System. Block Diagram of Plant = Mathematical Relations in Graph Form 17
4.3 DYNAMICS Dynamic Model = Mathematical Description via equation of motion of the system Three domains within which to study dynamic response S-plane Frequency Response State Space 18
4.3 DYNAMICS Feedback allows the Dynamics (Behavior) of a System to be modified: Stability Augmentation. Closed Loop Modifies Natural Behavior. 19
4.3 DYNAMICS - Superposition PRINCIPLE OF SUPERPOSITION if input is a sum of signals Response = Sum of Individual Responses to respective Signals It works for Linear Time-Invariant Systems Used to solve Systems by System responses to a set of elementary signals Decomposing given signal into sum of elementary responses Solve subsystems General response = sum of single subsystem solutions Elementary signals Impulse = Intense Force for Short Time Exponential e st f ( τ ) δ ( t τ ) dτ = f ( t) 20
4.3 DYNAMICS Transfer Function Exponential input u( t)= e st Output of the form y( t) = H( s) e st Where: S can be complex S = σ + jω Transfer Function = Transfer gain from U(S) to Y(S) = Y ( S) = H ( S) U ( S) Ratio of the Laplace Transform of Output to Laplace Transform of Input 21
4.3 DYNAMICS Laplace Transform Definition F ( s) = f ( t) dt e st 0 22
23 4.3 DYNAMICS Laplace Transform S-Plane ( ) ( ) k s k s k s s e k s e s e s kt k kt k t t k s s t t kt kt kt 2 2 2 2 2 2 2 ) cos( ) sin( 1 1 1 1 1 ) 1( 1 ) ( + + + + + δ Impulse Impulse ) ( ) ( 1 ) ( ) ( ) ( ) ( ) 1( ) ( k S F t f k S F k kt f S F k t f S H t t h e e e kt ks kt + =
24 4.3 DYNAMICS Frequency Response ( ) [ ] [ ] ) ( ), ( ) cos( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 ) cos( ) ( ) ( ) ( ) ( ω ϕ ω ϕ ω ω ω ω ω ω ω ω ω ω ω ϕ ω ϕ ω ω ϕ ω ω ω ω ω ω ω ω j H j H M t AM M M A t y M j H j H j H A t y j H t y t u j H t y t u j s A t A t u e e e e e e e e e e e t j t j j t j t j t j t j t j t j t j t j = = + = + = = + = = = = = = + = = + +
4.3 DYNAMICS Frequency Response for k= 1: 1 H( s) = s+ k 1 H( jω) = jω+ k 1 M = 2 ω + k 1 ω ϕ= tan k y( t) = AM cos( ωt+ ϕ) 2 Bode Plot 25
4.4 BLOCK DIAGRAM 26
4.4 BLOCK DIAGRAM + R e U Y - G c G p Fig. 01 Y R GcG p = 1+ GcG p Transfer Function = Linear Mapping of the Laplace Transform of the Input, R, to the Output Y Where Y = Process Output; R = Set-Point; G p = Process Gain; G c Controller Gain Y ( S) U ( S) = H ( S) 27
4.5 STABILITY Poles & Zeros b( s) H ( s) = a( s) a( s) = 0 H ( s) b( s) H ( s) = a( s) b( s) = 0 H ( s) = = 0 Such S-values Poles of H(s) Transfer Function Denominator factors Such S-values Zeros of H(s) Transfer Function Numerator Factors 28
Exponential decay Stability τ = 1 k 4.5 STABILITY Poles & Zeros H ( s) h( t) k > = 0 = e kt 1 s + k s 1( t) < 0 Exponential growth Instability τ = Time Constant H ( s) h( t) k < 0 = = s 1 + e kt s 1( t) > k 0 29
4.5 STABILITY Poles & Zeros 2s + 1 H ( s) = 2 + 3s + s 2 = 1 s + 1 + s 3 + 2 30
4.5 STABILITY Poles & Zeros EXPLORING THE S-PLANE... 31
4.5 STABILITY Poles & Zeros 32
4.5 Complex Poles H ( s) = 2 ω n ω n s + 2ζ S + 2 2 ω n ζ Damping Ratio s = σ ± j ω d ω n Natural Frequency σ = ζ ω n ω d = n ω θ = sin 1 ζ 1 ζ 2 33
For Low Damping Oscillator y Response For High Damping (near 1) No Oscillations 4.5 Impulse Response σ < 0 Unstable σ > 0 Stable σ = 0 n.a. 34
4.5 Step Response (Unit Step Response) RISE TIME Time necessary to Approach Set Point (t r ) SETTLING TIME Time necessary for Transient to Decay (t s ) OVERSHOOT % of Overshoot value to Steady State Value (M % ) PEAK TIME Time to reach highest point (t p ) Time Domain Specifications 35
4.5 Step Response (Unit Step Response) Time Domain Specifications RISE TIME Time necessary to Approach Set Point (t r ) SETTLING TIME Time necessary for Transient to Decay (t s ) OVERSHOOT % of Overshoot value to Steady State Value (M % ) PEAK TIME Time to reach highest point (t p ) 36
4.5 Step Response (Unit Step Response) Time Domain Specifications RISE TIME Time necessary to Approach Set Point (t r ) Rise Time t r 1.8 ω n For ζ = 0. 5 37
4.5 Step Response (Unit Step Response) Time Domain Specifications PEAK TIME Time to reach highest point (t p ) Peak Time t p π ω d For ζ = 0. 5 ω d = n ω 1 ζ 2 38
4.5 Step Response (Unit Step Response) Time Domain Specifications OVERSHOOT % of Overshoot value to Steady State Value (M % ) Overshoot M p = e πζ 1 2 ζ For ζ = 0. 5 39
4.5 Step Response (Unit Step Response) Time Domain Specifications SETTLING TIME Time necessary for Transient to Decay (t s ) Settling Time t s = 4.6 ζ ω n = 4.6 σ σ = ζ ω n For ζ = 0. 5 40
4.5 Step Response (Unit Step Response) Time Domain Specifications Design Specify t r, M p and t s : ω n ζ ζ ( σ t 1.8 t 4.6 s r M p ) 41
4.5 Step Response (Unit Step Response) Time Domain Specifications Design Specify t r, M p and t s : sin 1 ζ ω n σ t r M t s ω n 0.6sec 3sec ζ ζ ( σ p = 10% 1.8 ω n ) ζ 3.0rad 0.6 4.6 σ 1.5sec t t r M s p / sec 42
4.5 Step Response (Unit Step Response) Time Domain Specifications Design Adding a Zero Adding a Derivative Effect Increase Overshoot Decrease Rise Time Adding a Pole s-term in the denominator pure integration finite value stability Integral of Impulse Finite Value Integral of Step Function Ramp Function Infinite Value 43
4.5 Step Response (Unit Step Response) Time Domain Specifications Design For a 2 -order system with no zeros: t t r M s 1.8 ω n p 16%, ζ = 0.5 4.6 σ Zero in LHP Increase Overshoot Zero in RHP Decrease Overshoot Pole in LHP Increase Rise Time the denominator pure integration 44
4.6 Model From Experimental Data Transient Response input an impulse or a step function to the system Frequency Response Data exciting the system with sinusoidal input at various frequencies Random Noise Data 45
4.6 Model From Experimental Data GAM Model Transient Response to a step function representing the SP value = Desired value of the Returns. 46
5.1 FEEDBACK CONTROLLER Several parameters characterize the process. The difference ("error ) signal is used to adjust input to the process in order to bring the process' measured value back to its desired Set-Point. In Feedback Control the error is less sensitive to variations in the plant gain than errors in open loop control Feedback Controller can adjust process outputs based on History of Error Signal; Rate of Change of Error Signal; More Accurate Control; More Stable Control; Controller can be easily adjusted ("tuned") to the desired application. 47
5.1 FEEDBACK CONTROLLER The ideal version of the Feedback Controller is given by the formula: 1 de( t) u( t) = k e( t) + + e( τ ) dτ p T d dt i where u = Control Signal; e = Control Error; R = Reference Value, or Set-Point. T Proportional Term P Control Signal = Integral Term I Derivative Term D 48
5.2 FEEDBACK COTROLLER Proportional Term, P Adjusts Output in Direct Proportion to Controller Input (Error, e). Parameter gain, K p. Effect: lifts gain with no change in phase. Proportional - handles the immediate error, the error is multiplied by a constant K p (for "proportional"), and added to the controlled quantity. 49
5.3 FEEDBACK CONTROLLER Integral Term, I The Integral action causes the Output to Ramp. Used to eliminate Steady State Error. Effect: lifts gain at low frequency. Gives Zero Steady State Error. Infinite Gain + Phase Lag. Integral - To learn from the past, the error is integrated (added up) over a period of time, and then multiplied by a constant K i and added to the controlled quantity. Eventually, a well-tuned Feedback Controller loop's process output will settle down at the Set-Point. 50
Derivative Term, D 5.4 FEEDBACK CONTROLLER The derivative action, characterized by parameter K d, anticipates where the process is going by considering the derivative of the controller input (error, e). Gives High Gain at Low Frequency + Phase Lead at High Frequency Derivative - To handle the future. The 1st derivative over time is calculated, and multiplied by constant K d, and added to the controlled quantity. The derivative term controls the response to a change in the system. The larger the derivative term, the more the controller responds to changes in the process's output. A Controller loop is also called a "predictive controller." The D term is reduced when trying to dampen a controller's response to short term changes. 51
6. METHOD - the G.A.M. model Novel approach to Portfolio Tactical Asset Allocation. Recalling TAA Constant Proportion, Core Satellite and Active Strategies. Portfolio Assets Rebalancing is dictated by an Asset Selection Technique Consisting in the Optimization of Risk Adjusted Return by means of the G.A.M. model. 52
6.1 METHOD - the G.A.M. model Tests performed using the following data: Time horizon: 11 years Frequency: Monthly Number of Assets: 9 Period: January 1996 October 2006 9-asset Monthly Data Portfolio [10 years]; Comparison between the G.A.M. Portfolio and the Buy-and-Hold Portfolio. 53
6. METHOD - the G.A.M. model Risk Adjusted Return is not Optimized via Rebalancing of Asset Weights following a Forecasting Methodology of the Expected Return Vector. Investors seek Consistent and Stable Portfolio Performance over Time. Risk Adjusted Return is induced towards Stability Risk Adjusted Return is Controlled. 54
6. METHOD - the G.A.M. model For a portfolio to be tactically managed over a time horizon by means of the G.A.M. model: Given an initial asset allocation mix (Initial Portfolio), the assets are rebalanced at a predetermined frequency (monthly, or bimonthly, or quarterly); the rebalancing process is determined by choosing that particular mix of assets such at, at each iteration (monthly, or bimonthly, or quarterly), the current risk adjusted return approaches the current controlled system output. 55
6.2.1 METHOD - the G.A.M. model 1. Choose risk adjusted return parameter (Set-Point); 2. Set risk adjusted return value; 3. Set Controller parameters; 4. Choose Initial Portfolio (IP); i.e 1. All equivalents weights among the plurality of all the assets of the portfolio; or 2. Initial Portfolio could be dictated by Markovitz Asset Allocation. 56
6.2.2 METHOD - the G.A.M. model PID [Continuous] PID [Discrete] PID [Simple Lag Implementation] 57
6.2.3 METHOD - the G.A.M. model 1. Calculate first risk adjusted return value for the Initial Portfolio. 2. Controller determines the controlled value of the risk adjusted return for the portfolio. Rebalancing of the Portfolio is necessary in order for the Portfolio Risk Adjusted Return to comply with the Controller. 3. New data acquisition from financial markets is performed and the corresponding Risk Adjusted Return is calculated based on the current financial market data. 4. Tasks 2, 3 and 4 are iteratively repeated at a predetermined frequency until the chosen time horizon has been reached. 5. The purpose of performing these iterations is to Stabilize Portfolio Risk Adjusted Return via the combined contributions of the Controller and actual financial markets. Portfolio Asset Rebalancing and variation of Asset Mix is a result of both the Controller Effect and the Financial Market Dynamics. 58
6.2.4 METHOD - the G.A.M. model 59
7. EMPIRICAL RESULTS B&H Portfolio Returns GAM Portfolio Returns B&H Portfolio σ GAM Portfolio σ B&H Portfolio Sharpe Ratio GAM Portfolio Sharpe Ratio 1996 9.30% 9.52% 4.84% 4.77% 1.30 1.37 1997 2.66% 8.59% 8.25% 12.04% -0.04 0.46 1998-1.42% -9.71% 10.47% 23.42% -0.42-0.54 1999 16.69% 58.99% 8.27% 27.86% 1.65 2.01 2000-0.51% 16.79% 7.64% 17.32% -0.46 0.80 2001-9.91% 14.78% 8.53% 12.55% -1.51 0.94 2002-8.88% 28.79% 8.05% 17.54% -1.48 1.47 2003 17.80% 0.87% 5.61% 16.06% 2.64-0.13 2004 2.46% 20.32% 4.49% 11.89% -0.12 1.46 2005 7.75% -2.36% 5.80% 13.37% 0.82-0.40 2006 16.47% 27.01% 4.00% 13.15% 3.37 1.83 60
7. EMPIRICAL RESULTS B&H Portfolio Sharpe Ratio GAM Portfolio Sharpe Ratio Poly. (B&H Portfolio Sharpe Ratio) Poly. (GAM Portfolio Sharpe Ratio) 4.00 3.00 2.00 1.00 0.00-1.00-2.00 1,996 1,997 1,998 1,999 2,000 2,001 2,002 2,003 2,004 2,005 2,006 61
8. CONCLUSIONS AND FUTURE WORK INNOVATION FUTURE WORK ELECTRICAL ENGINERING + The innovation consists in using a Controller to control (to minimize the error e) portfolio risk adjusted return. Controlling = to have risk adjusted return approach and hold a steady state value as close as possible to the desired risk adjusted return. Controller needs to minimize steady state error the difference between Set-Point and the desired risk adjusted return over the time horizon. Adopting many more asset classes. Making Assets Time Series vary in frequency and length. Using other risk adjusted returns or other indices (I.e. Sortino, Information Ratio).. Take into account transaction and management fees. Use parameters setting and constraints. Use a index based portfolio as the Buy-and-Hold Portfolio ENHANCING FINANCE = FINANCIAL MARKET ANALYSIS 62
CONTACT INFORMATION Ing. as@alum.mit.edu, asabatin@mit.edu Prof. gino.gandolfi@unipr.it Dott.ssa monica.rossolini@gmail.com 63
APPENDIX βeta Ra = RFR + β (Rm- RFR) Where Ra = Return of an asset A RFR = Risk Free Rate Rm = Expected Market Return The measure of an asset's risk in relation to the market 64
Appendix D Cf/(1+%) ^Year 65
Appendix A Simple Lag Derivation 66
Appendix A Simple Lag Derivation 67
Appendix Z Ziegler-Nichols Tuning for PID Controller k p 0.6 k u k i 0.5 P u k d 0.125 P u P u =Period of oscillation k u =Proportional gain at the edge of oscillatory behavior 68
Appendix F Frequency Response and Bode Plots Dynamic compensation can be based on Bode Plots Bode Plots can be determined experimentally 69