**3. The PID controller acting on the experimental portfolio**

The most important attributes of the PID controller are illustrated in this section. It is vital to understand the functioning of this engineering feedback system since it underlies and stands at the basis of the new asset allocation technique presented herein. The PID (Proportional-Integral-Derivative) controller is broadly used and implemented in several industrial production plants; "it is been successfully used for over 50 years and it is used by more than 95% of the plants processes. It is a robust and easily understood algorithm, which can provide excellent control performance in spite of the diverse dynamic characteristics of the process plant" (Gandolfi et al., 2007). In industrial environments such as chemical plants, power plants, and engineering industries, numerous processes need to be accurately controlled to conform to the required specifications of the resulting products. PID control is straightforward, easily implementable method, still currently preferred by engineers and scientists to more complex systems (Skogestad, 2010). In finance, financial market assets comprising a portfolio or a market benchmark represent the process plant, controlled by the PID controller.

The PID controller is a feedback system. It has an input and returns an output. An iterative process forms it. The inputs of the system are the set-point, or desired value, and the controlled variable that is subject to the effect of the PID controller. The PID controller, working on the input variable, returns as output the same variable operated on by the PID operators. The output variable, in turns, is fed back as an input during the following iteration. The simplest and most basic PID control is formed by the linear combination of three components: the Proportional (P), Integral (I), and Derivative (D) components. During each iteration, the current output is compared to the set-point yielding an error. The goal of the PID control is to diminish this error to the minimum (Gandolfi et al., 2007). The continuous time expression of the PID controller is given by:

$$u(t) = k\_p \left( e(t) + k\_i \lceil e(\tau) \rceil d\tau + k\_d \frac{de(t)}{dt} \right).$$

where: ( ) output Proportional Constant Integral Constant Derivative Constant ( ) error *p i d u t k k k e t* (1) In this present work, the following recurrence relation, obtained by discrete time formulation and simple-lag implementation of the integral part (Gandolfi et al., 2007) yields:

$$\begin{aligned} \mathbf{u}\_{n} &= \left(\mathbf{k}\_{\mathrm{p}} \mathbf{e}\_{\mathrm{n}} + \mathbf{k}\_{\mathrm{i}} (\mathbf{e}\_{\mathrm{n}} \mathbf{\cdot}\_{\mathrm{n} \cdot \mathrm{n}}) + \mathbf{u}\_{n \cdot 1} + \mathbf{k}\_{\mathrm{d}} (\mathbf{e}\_{\mathrm{n}} \mathbf{\cdot}\_{\mathrm{e}} \mathbf{n} \mathbf{\cdot})\right) \\ \text{where:} \\ \mathbf{u}\_{n} &= \text{output at time n} \\ \mathbf{u}\_{n \cdot 1} &= \text{output at time n} \cdot \mathbf{1} \\ \mathbf{k}\_{\mathrm{p}} &= \text{Proportional Constant} \\ \mathbf{k}\_{\mathrm{d}} &= \text{Integrality Constant} \\ \mathbf{k}\_{\mathrm{d}} &= \text{Derivative Constant} \\ \mathbf{e}\_{\mathrm{n}} &= \text{error at time n} \cdot \mathbf{1} \end{aligned} \tag{2}$$

A block diagram of the PID controller follow:

234 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

used in industrial plant production and engineering, to tactical financial portfolio asset allocation. The goal of their model was to attain long-term performance steadiness over time by controlling the risk adjusted return variable of portfolios. The main attribute to perceive was the achieved constancy and consistency of the Sharpe Ratio of the experimental portfolio (i.e. the portfolio managed by the PID methodology) in comparison to the benchmark. In the present work, the authors build up a new application based on this novel strategy. The target here is to seek a portfolio (Portfolio "A") capable of enhanced long-term risk adjusted performance and risk stability than the Buy-and-Hold portfolio (Portfolio "B").

The most important attributes of the PID controller are illustrated in this section. It is vital to understand the functioning of this engineering feedback system since it underlies and stands at the basis of the new asset allocation technique presented herein. The PID (Proportional-Integral-Derivative) controller is broadly used and implemented in several industrial production plants; "it is been successfully used for over 50 years and it is used by more than 95% of the plants processes. It is a robust and easily understood algorithm, which can provide excellent control performance in spite of the diverse dynamic characteristics of the process plant" (Gandolfi et al., 2007). In industrial environments such as chemical plants, power plants, and engineering industries, numerous processes need to be accurately controlled to conform to the required specifications of the resulting products. PID control is straightforward, easily implementable method, still currently preferred by engineers and scientists to more complex systems (Skogestad, 2010). In finance, financial market assets comprising a portfolio

or a market benchmark represent the process plant, controlled by the PID controller.

continuous time expression of the PID controller is given by:

where:

 

*p i d*

*u t k k k e t*

( ) output

( ) error

The PID controller is a feedback system. It has an input and returns an output. An iterative process forms it. The inputs of the system are the set-point, or desired value, and the controlled variable that is subject to the effect of the PID controller. The PID controller, working on the input variable, returns as output the same variable operated on by the PID operators. The output variable, in turns, is fed back as an input during the following iteration. The simplest and most basic PID control is formed by the linear combination of three components: the Proportional (P), Integral (I), and Derivative (D) components. During each iteration, the current output is compared to the set-point yielding an error. The goal of the PID control is to diminish this error to the minimum (Gandolfi et al., 2007). The

( ) () () ( )

*de t ut et e d kk k dt*

*pi d*

 

(1)

Proportional Constant

Integral Constant Derivative Constant

**3. The PID controller acting on the experimental portfolio** 

Fig. 1. PID control block diagram - This figure presents dynamics and processing of the error, Set-Point and controlled variable while subjected to the PID control action. Set-point = Desired value. Error = (Output – Set-Point).

## **4. Mechanisms of action of the new asset allocation technique**

This section presents an original method and system for allocating numerous assets in portfolios, via tactical asset allocation in order to achieve better return and long-term target stability (volatility control) over a desired time horizon. In particular, the present work illustrates a method and system for asset allocation of the 20 securities having each one, its own level of risk and return. The methodology consists in stabilizing the portfolio return . hence the decreasing of portfolio volatility based on the PID feedback control. By applying our strategy to a financial portfolio, financial market assets represent the process plant, controlled by PID controlling action. The assets mix of the portfolio determines the total portfolio return. The action of rebalancing the portfolio alters its return. In various aspects, this work offers methods and systems as an innovative approach to active strategy portfolio management. It is worth noting that the rebalancing of the experimental portfolio (Portfolio "A") is not dictated by a forecast analysis of the various prices of the assets belonging to the portfolio. There is no use of a vector of expected returns and there is no need of determining a variance-covariance matrix. The rebalancing is rather driven by an asset selection

An Innovative Systematic Approach to Financial Portfolio Management via PID Control 237

basis. The constraints for rebalancing are the following: every asset can take on a minimum or a maximum weight within the portfolio. The minimal weight has been defined to be equal to 1% and the maximal weight has been defined to be equal to 20% under the effect of

The set-point value of this procedure, in order for the new model to achieve its target, is set to be equal to 0.5% monthly target portfolio return. The mechanism of action of this model is similar to a dynamic Exchanged Traded Fund (ETF), replicating an index in terms of underlying assets. On the opposite, it is different in terms of relative weights and, therefore,

The algorithm and implementation of the new model is the outlined in the following steps,

(TR) 6%

iBoxx Euro Index World Wide Performance Overall 10% Market iBoxx € Financials Total Return Index 10% Market iBoxx € Non Financials Total Return Index 10%

MSCI Daily TR Gross Europe Local Currency 5% STOXX 600 Total Return Index EUR 5% STOXX Style Index TMI Growth Return Index EUR 5%

EUR 5% MSCI Daily TR Gross Total Return World USD 5%

S&P GSCI Tot Return Indx 2% S&P GSCI Energy Tot Ret 2% S&P GSCI Industrial Metals Index Total Return 2% S&P GSCI Agricultural Index Total Return CME 2% S&P GSCI Livestock Index Total Return. 2% S&P GSCI Crude Oil Total Return CME 2%

Table 2. Strategic Portfolio "B" composition: index specification. The table presents, for any macro-asset class, the specification of which particular selected indices form each macro-

Market iBoxx € Euro Sovereign Overall Total Return Index 10%

STOXX Europe Total Market Value (Net Return)

MSCI Emerging Markets Daily Gross Total Return USD 5%

MSCI Daily TR Gross North America Total Return USD 5%

Gold S&P GSCI Gold Index Total Return 5% Cash Out of the market 2%

asset class. Furthermore, the relative weights are indicated.

Asset class Index Weight

Monetary Deutsche Borse EUROGOV Germany Money Market

the PID control action.

using the expression:

Bonds

Equity

Commodities

the model is a dynamic strategy.

technique consisting in the stabilization of return by means of the PID feedback control modeling procedure. The new model simply tends to follow and not predict the financial market oscillations and market variability, adjusting to such variations and oscillations. It takes into consideration past and current portfolio dynamics. It tunes to financial market fluctuations by performing smoothing and anticipatory actions in the attempt to hold as close as possible to the target, hence minimizing the error generated by the difference between the set-value and current portfolio return. The controlled process plant, namely the return variable, does not need to be modeled or defined by a mathematical closed form equation; assumptions, linearization, and simplification procedures on the dynamics of the plant are not required. The PID control modifies the portfolio asset weights, according to the PID algorithm. The methodology starts by presenting two initially identical portfolios: the benchmark, namely Portfolio "B", and the experimental portfolio, or Portfolio "A". The procedure uses a 12-year monthly frequency time-series per each of the securities of the Portfolio "B", covering the period February 1999 - February 2011. Portfolio "A" assets are rebalanced at the end of each month, according to the PID procedure. At the end of the observation period, namely in February 2011, the two portfolios, the Benchmarked Portfolio "B" and the experimental portfolio, Portfolio "A", are observed and compared, targeting to verify the efficiency of the new model compared to benchmarking. In this work, the comparison is carried out without taking into consideration tax and transaction costs. Portfolio "B" , namely the benchmark is composed by 20 assets chosen in such a way to form a well diversified portfolio. In particular, the following assets have been considered: a monetary index, 4 fixed-income (or bonds) indices, 7 stocks (equity) indices, 6 commodities indices, gold and a risk-free asset class denominated "cash". The inclusion and use of a riskfree asset in the experimental portfolio is been indicated by the consideration that the new model permits partial disinvestment of the risky portfolio by partially reallocating risky assets in risk-free assets (Qian, 2003). The following table illustrates how the strategic asset allocation of the well-diversified portfolio has been defined. The right-end-side column indicates the respective weights of each asset class:


Table 1. Strategic composition for macro-asset class of Portfolio "B". The table illustrates Portfolio "B" composition, namely the benchmark composition. It specifies the various macro-asset classes and their relative assigned weights.

After having presented which the strategic macro-asset classes are, for the benchmark portfolio, the following table is presented. It exhibits for each asset class, which are the selected indices in order to form the well diversified portfolio with its respective assigned weights. Firstly, Portfolio "A" has the identical composition as that of Portfolio "B". Next, Portfolio "A" asset weights are varied following the PID signals. The rebalancing occurs on a monthly

technique consisting in the stabilization of return by means of the PID feedback control modeling procedure. The new model simply tends to follow and not predict the financial market oscillations and market variability, adjusting to such variations and oscillations. It takes into consideration past and current portfolio dynamics. It tunes to financial market fluctuations by performing smoothing and anticipatory actions in the attempt to hold as close as possible to the target, hence minimizing the error generated by the difference between the set-value and current portfolio return. The controlled process plant, namely the return variable, does not need to be modeled or defined by a mathematical closed form equation; assumptions, linearization, and simplification procedures on the dynamics of the plant are not required. The PID control modifies the portfolio asset weights, according to the PID algorithm. The methodology starts by presenting two initially identical portfolios: the benchmark, namely Portfolio "B", and the experimental portfolio, or Portfolio "A". The procedure uses a 12-year monthly frequency time-series per each of the securities of the Portfolio "B", covering the period February 1999 - February 2011. Portfolio "A" assets are rebalanced at the end of each month, according to the PID procedure. At the end of the observation period, namely in February 2011, the two portfolios, the Benchmarked Portfolio "B" and the experimental portfolio, Portfolio "A", are observed and compared, targeting to verify the efficiency of the new model compared to benchmarking. In this work, the comparison is carried out without taking into consideration tax and transaction costs. Portfolio "B" , namely the benchmark is composed by 20 assets chosen in such a way to form a well diversified portfolio. In particular, the following assets have been considered: a monetary index, 4 fixed-income (or bonds) indices, 7 stocks (equity) indices, 6 commodities indices, gold and a risk-free asset class denominated "cash". The inclusion and use of a riskfree asset in the experimental portfolio is been indicated by the consideration that the new model permits partial disinvestment of the risky portfolio by partially reallocating risky assets in risk-free assets (Qian, 2003). The following table illustrates how the strategic asset allocation of the well-diversified portfolio has been defined. The right-end-side column

> Asset class Weight Monetary 6% Bonds 40% Equity 35% Commodities 12% Gold 5% Cash 2%

Table 1. Strategic composition for macro-asset class of Portfolio "B". The table illustrates Portfolio "B" composition, namely the benchmark composition. It specifies the various

After having presented which the strategic macro-asset classes are, for the benchmark portfolio, the following table is presented. It exhibits for each asset class, which are the selected indices in order to form the well diversified portfolio with its respective assigned weights. Firstly, Portfolio "A" has the identical composition as that of Portfolio "B". Next, Portfolio "A" asset weights are varied following the PID signals. The rebalancing occurs on a monthly

indicates the respective weights of each asset class:

macro-asset classes and their relative assigned weights.

basis. The constraints for rebalancing are the following: every asset can take on a minimum or a maximum weight within the portfolio. The minimal weight has been defined to be equal to 1% and the maximal weight has been defined to be equal to 20% under the effect of the PID control action.

The set-point value of this procedure, in order for the new model to achieve its target, is set to be equal to 0.5% monthly target portfolio return. The mechanism of action of this model is similar to a dynamic Exchanged Traded Fund (ETF), replicating an index in terms of underlying assets. On the opposite, it is different in terms of relative weights and, therefore, the model is a dynamic strategy.

The algorithm and implementation of the new model is the outlined in the following steps, using the expression:


Table 2. Strategic Portfolio "B" composition: index specification. The table presents, for any macro-asset class, the specification of which particular selected indices form each macroasset class. Furthermore, the relative weights are indicated.

An Innovative Systematic Approach to Financial Portfolio Management via PID Control 239

Table 3 illustrates information about return and volatility. Portfolio "A" has an annualized return of 7,25% compared to 5,14% of Portfolio "B". The cumulative return in the observation period (1999-2011) is 86,96% for Portfolio "A" and 61,66% for the benchmark. In terms of portfolio risk, the experimental portfolio realizes an annualized volatility of 7,93%, indicatively in line and consistent with 7,01% recorded by Portfolio "B". Portfolio "A", with only a slightly higher volatility, is able to obtain more satisfying results both in annualized

Annualized Return 7,25% 5,14% Cumulative Return 86,96% 61,66% Annualized Volatility 7,93% 7,01%

Table 3. Return and Volatility data. This table presents the comparison of annualized return, cumulative return and annualized volatility of Portfolio "A" and Portfolio "B". Period of

After having analyzed the data in the observation period, it is considered interesting to

Table 4 demonstrates monthly data; scrupulously, it is evident that the mean monthly return of the Portfolio "A"(0,60%) is superior to the Portfolio "B" mean monthly return (0,43%). The set-point or target value for the model was 0,5% monthly; thus, the experimental portfolio reaches the ideal target. The mean monthly volatility for Portfolio "A" is 2,29%,

Mean Monthly Return 0,60% 0,43% Mean Monthly Volatility 2,29% 2,02%

Table 4. Monthly Return and Volatility information. This table presents the comparison of average monthly returns and average monthly standard deviations of Portfolio "A" and

Table 5 shows, in the first and second column respectively, Portfolio "A" returns and Portfolio "B" returns for each year of the observation period. It is important to specify that each year is considered by counting from February (t-1) to February (t). This allows the yearly periods to be defined by 12 periods of 12 month each one, considering that the given time series starts in February. This table demonstrates that the new model performance is, in most cases, equivalent or better than the benchmark portfolio performance for each analyzed year, except for three years 2004-2005, 2006-2007 and 2009-2010, where Portfolio "A" underperforms Portfolio "B". The third and fourth columns of Table 5 display the annual volatility for the two portfolios. We can see that in many years, the new model presents higher volatility than Portfolio "B", but it is necessary to remember what

whereas the benchmark (Portfolio "B") exhibits a volatility of 2,02%.

Portfolio "B". Period of observation: February 1999-February 2011.

mentioned herein, that performances are also superior.

Portfolio "A" Portfolio "B"

Portfolio "A" Portfolio "B"

and in cumulative data analysis.

observation: February 1999-February 2011.

analyze the data on a monthly basis.

<sup>n</sup> p n i n n-1 n 1 d n n-1 n n-1 p i d return = + ( - )+ + ( - ) k e k e return return k e e where: return =output= return at time n u =output= return at time n-1 k =Proportional Constant = 0,5 k =Integral Constant = 0,6 k =Derivative Constant = - n n n- 1 n-1 0,5 e =(return 0,005) at time n e =(return 0,005) at time n-1 SetPoint = desired return = 0,005 (3)


$$\mathbf{K\_{P}} = 0.5$$

```
 Ki = 0,6
```

```
 Kd = 0,5
```

The main results of this methodology are illustrated in the following paragraph.
