Professional Portfolio Selection Techniques: From ...web.mit.edu/eecsgsa/www/events/2008/ppst/session2.pdfProfessional Portfolio Selection Techniques: From Markowitz to ... Antonella

January 16, 20081

Antonella SabatiniGino GandolfiMonica Rossolini

Sponsor: EECS, Science Engineering and Business Club, Graduate Student Council 1

Massachusetts Institute of Technology

Sponsor: Electrical Engineering and Computer Science

Cosponsor: Science Engineering and Business Club Graduate Student Council

Professional Portfolio Selection Techniques: From Markowitz to

Innovative Engineering Part 2

Antonella Sabatini and Monica Rossoliniin collaboration with Gino Gandolfi

MIT - Wed Jan 16, Thu Jan 17 2008,

04:00-6:00pm, 34-401

January 16, 20082


Sponsor: EECS, Science Engineering and Business Club, Graduate Student Council

The process of portfolio constructionAsset allocation:- strategic asset allocation- tactical asset allocation

G.A.M Model: a new tactical asset allocation techniquePID feedback controller theoryApplications and future research

October 4, 20072

1st DayJanuary

16th

2nd DayJanuary

17th

January 16, 20083



1. INTRODUCTION

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.

INNOVATION Use of the Feedback controller, widely applied in most industrial processes, as a technique for financial portfolio management.**

AIM Tactical Portfolio Asset Allocation Technique.

METHOD Rebalancing of Assets determined by the controlled value of Risk Adjusted Return subject to the action of the Controller.

(*), ** Patent Pending – International -National

January 16, 20084



1. INTRODUCTIONThe Innovation

Seeking

STABILITY

CONSISTENCY

Comprises

of Portfolio Risk Adjusted Return

over the time horizon

by “controlling” Risk Adjusted Retun

January 16, 20085



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

January 16, 20086



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.

January 16, 20087



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– ...– …

January 16, 20088



4. SYSTEMS: MANUAL VS AUTOMATIC SYSTEMS

• Manual Control = System involving a Person Controlling a Machine.

• Automatic Control = System involving Machines Only.

January 16, 20089



4. SYSTEMS: MANUAL VS AUTOMATIC SYSTEMS(ESAMPLES)

• Manual Control: Driving an Automobile

• Automatic Control: Room Temperature Set by a Thermostat

January 16, 200810



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

January 16, 200811



4. OPEN-LOOP SYSTEMS

• The Controller does not use a Measure of the System Output being Controlled in Computing the Control Action to Take.

January 16, 200812



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.

January 16, 200813



4.1 OPEN AND CLOSED LOOPS

System 2 affects system 1

System 1 affects system 2

OPEN LOOP SYSTEM

CLOSED LOOP SYSTEM

January 16, 200814



4.1 CLOSED LOOP (EXAMPLE)

• Household Furnace Controlled by a Thermostat:

Room TemperatureRoom TemperatureTHERMOTHERMO--

STATSTATGas

ValveHOUSE

Desired Temperature

+

Fig. 01 – BLOCK DIAGRAM

FURNACE-

Qout

Qin

THERMOTHERMO--STATSTAT

Gas Valve

Desired Temperature FURNACE

-

Qout

Qin

HOUSE

Fig. 01 – BLOCK DIAGRAM

THERMOTHERMO--STATSTAT

Gas Valve

Desired Temperature FURNACE

-

Qout

Qin

January 16, 200815



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 OFFRoom Temperature will drop until below Reference point

•Gas Valve ON……

January 16, 200816



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.

January 16, 200817



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

January 16, 200818



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

January 16, 200819



4.3 DYNAMICS

• Feedback allows the Dynamics (Behavior) of a System to be modified:– Stability Augmentation.– Closed Loop Modifies Natural Behavior.

January 16, 200820



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

∫∞

∞−

=− )()()( tfdtf ττδτ

est

January 16, 200821



4.3 DYNAMICS – Transfer Function

• Exponential input

• Output of the form

• Where:• S can be complex

• Transfer Function = Transfer gain from U(S) to Y(S) =– Ratio of the Laplace Transform of Output to Laplace Transform of Input

etu st=)(

esHty st)()( =

ωσ jS +=

)()()( SH

SUSY

=

January 16, 200822



4.3 DYNAMICS – Laplace TransformDefinition

∫∞ −=0

)()( dtetfsF st

January 16, 200823



4.3 DYNAMICS – Laplace TransformS-Plane

( )

( )

ks

ks

ksse

kses

e

skt

kkt

k

t

t

ks

st

t

kt

kt

kt

22

22

2

2

2

)cos(

)sin(

1

1

1

1

1)(1

1)(

+⇔

+⇔

⇔−

⇔

⇔

+⇔

⇔

⇔

+

+−

−

−

δ ImpulseImpulse

)()(

1)(

)()(

)()(1)(

kSFtf

kSF

kktf

SFktf

SHtth

e

ee

kt

kS

kt

+⇔

⎟⎠⎞

⎜⎝⎛⇔

⇔−

⇔=

−

−

−

January 16, 200824



4.3 DYNAMICS – Frequency Response( )

[ ]

[ ]

)(,)()cos(

)()(2

)(

)()(

)()(2

)(

)()(

)(

)()(

)(

2)cos()(

)()(

)(

ωϕωϕω

ωω

ωω

ωω

ω

ω

ω

ω

ϕωϕω

ωϕ

ωω

ω

ω

ω

ω

ωω

jHjHMtAM

MMAty

MjH

jHjHAty

jHty

tu

jHty

tu

js

AtAtu

eee

eee

ee

e

ee

tjtj

j

tjtj

tj

tj

tj

tj

tjtj

==

+=

+=

=

−+=

−=

=

=

=

=

+==

+−+

−

−

−

−

January 16, 200825



4.3 DYNAMICS – Frequency Response Bode Plot

)cos()(

tan

1

1)(

1)(

:1kfor

1

22

ϕω

ωϕ

ω

ωω

+=

⎟⎠⎞

⎜⎝⎛

−=

+=

+=

+=

=

−

tAMtyk

kM

kjjH

kssH

January 16, 200826



4.4 BLOCK DIAGRAM

January 16, 200827



4.4 BLOCK DIAGRAM

Gc Gp

R e U Y+

-

Fig. 01Transfer Function = Linear Mapping of the Laplace Transform of the Input, R, to the Output YGG

GGpc

pc

RY

+=1

Where Y = Process Output; R = Set-Point; Gp = Process Gain; GcController Gain

)()()( SH

SUSY

=

January 16, 200828



4.5 STABILITY – Poles & Zeros

∞=→=

=

)(0)()()()(

sHsasasbsH Such S-values

Poles of H(s) Transfer Function

Denominator factors

0)(0)()()()(

=→=

=

sHsbsasbsH Such S-values

Zeros of H(s) Transfer Function

Numerator Factors

January 16, 200829




00)(1)(

1)(

<→>

=

+=

−

sktth

kssH

e kt

00)(1)(

1)(

>→<

=

+=

−

sktth

kssH

e kt

k1

=τ

Exponential decay Exponential decay StabilityStability

Exponential growth Exponential growth InstabilityInstability

τ = Time Constant= Time Constant

January 16, 200830




23

11

2312)( 2 +

++

−=

++

+=

sssssH

s

January 16, 200831




EXPLORING THE S-PLANE.....

January 16, 200832




January 16, 200833



4.5 Complex Poles

ωωω

ζ 22

2

2)(

nn

n

SsH

s ++=

ζ

ωn

Damping RatioDamping Ratio

Natural Natural FrequencyFrequency

ζθ sin 1−=

ωσ djs ±−=

ωζσ n=

ζωω2

1−= nd

January 16, 200834



4.5 Impulse ResponseFor Low For Low Damping Damping

OscillatorOscillatory y

ResponseResponse

For High For High Damping Damping

(near 1) (near 1) No No

OscillationsOscillations

σσ < 0 < 0 UnstableUnstable

σσ > 0 > 0 StableStable

σσ = 0 = 0 n.a.n.a.

January 16, 200835



4.5 Step Response (Unit Step Response) Time Domain Specifications

• RISE TIME – Time necessary to Approach Set Point (tr)

• SETTLING TIME – Time necessary for Transient to Decay (ts)

• OVERSHOOT – % of Overshoot value to Steady State Value (M%)

• PEAK TIME – Time to reach highest point (tp)

January 16, 200836




• RISE TIME – Time necessary to Approach Set Point (tr)• SETTLING TIME – Time necessary for Transient to Decay (ts)• OVERSHOOT – % of Overshoot value to Steady State Value (M%)• PEAK TIME – Time to reach highest point (tp)

January 16, 200837




• RISE TIME – Time necessary to Approach Set Point (tr)

ωnrt

8.1≅

For 5.0=ζ

Rise Time

January 16, 200838



• PEAK TIME – Time to reach highest point (tp)

ωπ

dpt ≅

For 5.0=ζ

Peak Time


ζωω2

1−=nd

January 16, 200839



• OVERSHOOT – % of Overshoot value to Steady State Value (M%)

eM pζπζ

21−−=

For 5.0=ζ

Overshoot


January 16, 200840



• SETTLING TIME – Time necessary for Transient to Decay (ts)

σζ ω6.46.4

==n

st

For 5.0=ζ

Settling Time


ωζσn

=

January 16, 200841



• Specify tr, Mp and ts:

t

Mt

s

p

rn

6.4

)(

8.1

≥

≥

≥

σ

ζζ

ω

4.5 Step Response (Unit Step Response) Time Domain Specifications Design

January 16, 200842



• Specify tr, Mp and ts:

sec5.16.4

6.0)(

sec/0.38.1

sec3

%10

sec6.0

≥⇒≥

≥⇒≥

≥⇒≥

≤

=

≤

σσ

ζζζ

ωω

t

Mt

tMt

s

p

nr

n

s

p

r

rad


ωnζsin 1−

σ

January 16, 200843



• 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


January 16, 200844



• For a 2°-order system with no zeros:

• Zero in LHP Increase Overshoot• Zero in RHP Decrease Overshoot• Pole in LHP Increase Rise Time the denominator

pure integration


σ

ζω

6.4

5.0%,16

8.1

≅

=≅

≅

tM

t

s

p

nr

January 16, 200845



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

January 16, 200846



• GAM Model Transient Response to a step function representing the SP value = Desired value of the Returns.

4.6 Model From Experimental Data

January 16, 200847



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.

January 16, 200848




⎟⎟⎠

⎞⎜⎜⎝

⎛++= ∫ dt

tdedetetu TTk di

p

)()(1)()( ττ

∑

The ideal version of the Feedback Controller is given by the formula:

where u = Control Signal;e = Control Error;R = Reference Value, or Set-Point.

Control Signal =

Proportional Term P

Integral Term I

Derivative Term D

January 16, 200849



5.2 FEEDBACK COTROLLER

• Adjusts Output in Direct Proportion to Controller Input (Error, e).

• Parameter gain, Kp.• Effect: lifts gain with no change in phase.• Proportional - handles the immediate error, the error is

multiplied by a constant Kp (for "proportional"), and added to the controlled quantity.

Proportional Term, P

January 16, 200850




• 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 Kiand added to the controlled quantity. Eventually, a well-tuned Feedback Controller loop's process output will settle down at the Set-Point.

Integral Term, I

January 16, 200851




• The derivative action, characterized by parameter Kd, 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 Kd, 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.

Derivative Term, D

January 16, 200852



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.

January 16, 200853



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 20069-asset Monthly Data Portfolio [10 years];

• Comparison between the G.A.M.® Portfolio and the Buy-and-Hold Portfolio.

January 16, 200854




• 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.

January 16, 200855




• 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.

January 16, 200856



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.

January 16, 200857




PID [Continuous]

PID [Discrete]

PID [Simple Lag Implementation]

January 16, 200858



6.2.3 METHOD - the G.A.M. model1. 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.

January 16, 200859




January 16, 200860



7. EMPIRICAL RESULTS

1.833.3713.15%4.00%27.01%16.47%2006-0.400.8213.37%5.80%-2.36%7.75%20051.46-0.1211.89%4.49%20.32%2.46%2004

-0.132.6416.06%5.61%0.87%17.80%20031.47-1.4817.54%8.05%28.79%-8.88%20020.94-1.5112.55%8.53%14.78%-9.91%20010.80-0.4617.32%7.64%16.79%-0.51%20002.011.6527.86%8.27%58.99%16.69%1999

-0.54-0.4223.42%10.47%-9.71%-1.42%19980.46-0.0412.04%8.25%8.59%2.66%19971.371.304.77%4.84%9.52%9.30%1996

GAM Portfolio Sharpe Ratio

B&H Portfolio Sharpe Ratio

GAM Portfolio

σ

B&H Portfolio

σ

GAM Portfolio Returns

B&H Portfolio Returns

January 16, 200861



7. EMPIRICAL RESULTS

-2.00

-1.00

0.00

1.00

2.00

3.00

4.001,

996

1,99

7

1,99

8

1,99

9

2,00

0

2,00

1

2,00

2

2,00

3

2,00

4

2,00

5

2,00

6

B&H Portfolio Sharpe RatioGAM Portfolio Sharpe RatioPoly. (B&H Portfolio Sharpe Ratio)Poly. (GAM Portfolio Sharpe Ratio)

January 16, 200862



8. CONCLUSIONS AND FUTURE WORK

INNOVATION •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.

FUTURE WORK•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

ELECTRICAL ENGINERING FINANCE

++ ==ENHANCING

FINANCIAL MARKET ANALYSIS

January 16, 200863



CONTACT INFORMATION

• Ing. Antonella Sabatini – [email protected], [email protected]

• Prof. Gino Gandolfi – [email protected]

• Dott.ssa Monica Rossolini – [email protected]

January 16, 200864



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

January 16, 200865



Appendix DCf/(1+%)Cf/(1+%)

^Year^Year

January 16, 200866



Appendix A Simple Lag Derivation

January 16, 200867



Appendix A Simple Lag Derivation

January 16, 200868



Appendix ZZiegler-Nichols Tuning for PID Controller

PkPkkk

ud

ui

up

125.0

5.0

6.0

≅

≅

≅

Pu=Period of oscillation

ku=Proportional gain at the edge of oscillatory behavior

January 16, 200869



• Dynamic compensation can be based on Bode Plots

• Bode Plots can be determined experimentally

Appendix FFrequency Response and Bode Plots

Documents

Professional Portfolio Selection Techniques: From ...web.mit.edu/eecsgsa/www/events/2008/ppst/session2.pdfProfessional Portfolio Selection Techniques: From Markowitz to ... Antonella