**5. Portfolio "A" vs. Portfolio "B"**

This section recapitulates the main results of the new model comparing the returns of Portfolio "A" to the returns of Portfolio "B". The comparison is performed in terms of return and volatility for the observation period.

return = + ( - )+ + ( - ) k e k e return return k e e

0,5

Calculate portfolio return (controlled variable), return0, for the initial portfolio, given

 At each iteration n, the PID controller designates a controlled value for the portfolio return, called returnn given by equation [3]. The making of such rebalancing is necessary to minimize the error between current return (determined by current market conditions) and returnn and set-point. Since the objective is to reduce the error, en defined by the difference between current return, returnn, and the set-point or desired return defined as 0,005, each iteration contributes in reducing en. The error decrease is generally counteracted by the dynamics of the markets. Given ideal market conditions,

New market data acquisition and corresponding portfolio return, returnn, is calculated

 In this work, the parameters values were set according to an empirical criterion: under risk-free market conditions (portfolio with zero exposure to financial markets), the selection of a transient time domain response with a slight oscillatory response, exhibiting reasonable overshoot, and approaching set-point value within a small

 The objective of each iterations is to make returns as stable and consistent as possible given the contributions and interactions of the controller and the market dynamics influence. The change in asset mix is dictated by the controller indications and the

This section recapitulates the main results of the new model comparing the returns of Portfolio "A" to the returns of Portfolio "B". The comparison is performed in terms of return

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

 The previous items are iteratively re-executed until the end of the observation period. The PID parameters, chosen to be constant for all market conditions, are set to be:

en approaches zero after the transient system response has died out.

n

n n n- 1 n-1

Define set-point = Desired Return = 0,005.

current market conditions.

at end of each period (monthly).

number of iterations was adopted.

**5. Portfolio "A" vs. Portfolio "B"** 

and volatility for the observation period.

market behavior of the underlying securities.

 Kp = 0,5 Ki = 0,6 Kd = 0,5 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 = - -

e =(return 0,005) at time n e =(return 0,005) at time n-1 SetPoint = desired return = 0,005

n-1 p i d

where:

<sup>n</sup> p n i n n-1 n 1 d n n-1 -

(3)

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 and in cumulative data analysis.


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 observation: February 1999-February 2011.

After having analyzed the data in the observation period, it is considered interesting to analyze the data on a monthly basis.

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%, whereas the benchmark (Portfolio "B") exhibits a volatility of 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 Portfolio "B". Period of observation: February 1999-February 2011.

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 mentioned herein, that performances are also superior.

An Innovative Systematic Approach to Financial Portfolio Management via PID Control 241

numerator, divided by the standard deviation of the portfolio returns. In order to define the risk free rate the average of the Libor values in the 12 years (1999-2011) of observation period are calculated. This calculation has yielded a value equal to 2,80%. The results are

1999-2000 1,74 2,20 2000-2001 **0,38** Negative 2001-2002 **0,03** Negative 2002-2003 **0,22** Negative 2003-2004 0,46 1,79 2004-2005 Negative 2,05 2005-2006 1,90 2,39 2006-2007 Negative 0,26 2007-2008 **0,91** Negative 2008-2009 Negative Negative 2009-2010 2,52 3,56 2010-2011 **2,53** 1,76 Table 6. Sharpe Ratio of Portfolio "A" and Portfolio "B". This table presents the results of a risk adjusted return indicator, namely the Sharpe Ratio applied to the two portfolios for every year in the observation period. In bold are illustrated the cases in which Portfolio "A"

Fig. 3. Sharpe Ratio of Portfolio "A" and Portfolio "B". The chart presents, for each

2003-2004

observation period, the Sharpe Ratio values of the two portfolios. In particular, in black the values belonging to Portfolio "A" are represented. In grey, the corresponding values for Portfolio "B" are illustrated. The absence of a column shows that the indicator value is

2004-2005

Portfolio "A" Sharpe Ratio Portfolio "B" Sharpe Ratio

2005-2006

2006-2007

2007-2008

2008-2009

2009-2010

2010-2011

When in table 6, the word "Negative" is present, it means that for that specific year, it was not possible to record the indicator due to its negative value. The Sharpe Ratio is not

Portfolio "A" Sharpe Ratio Portfolio "B" Sharpe Ratio

depicted in the table below:

has outperformed Portfolio "B".

0,00 0,30 0,60 0,90 1,20 1,50 1,80 2,10 2,40 2,70 3,00 3,30

negative, hence non-interpretable.

1999-2000

2000-2001

2001-2002

2002-2003


Table 5. Portfolio "A" and Portfolio "B" annual returns and volatilities. This table presents annual returns and annual standard deviations of Portfolio "A" and Portfolio "B" for each observed year. Period of observation: from February 1999 - February 2011.

The following chart illustrates, graphically, the dynamics of volatility of Portfolios "A" and "B". It can be noticed that the continuous line representing the volatility of Portfolio "A" is often higher than that of the benchmark. However, it is interesting to underline the stabilization effect starting from 2004 and becoming evident under the PID control action. As it is well known, this instrument needs a history before it can enable its efficient control action and make it functional.

Fig. 2. Annual Volatility of Portfolio "A" and Portfolio "B". This chart presents the annual volatility dynamics of the two portfolios in the observation period February 1999-February 2011. The continuous line represents Portfolio "A"; the dotted line represents Portfolio "B".

After having calculated the return and risk of the two portfolios, a comparison of the two portfolios is performed by using a risk adjusted return indicator, the Sharpe Ratio. This indicator is defined as the ratio of the difference between return and risk free return at the

1999-2000 16,05% 19,25% 7,62% 7,49% 2000-2001 6,07% 2,08% 8,67% 6,38% 2001-2002 2,97% -1,25% 6,11% 6,90% 2002-2003 4,54% -8,71% 7,90% 7,63% 2003-2004 8,27% 13,21% 11,94% 5,82% 2004-2005 -1,32% 8,06% 4,70% 2,57% 2005-2006 18,28% 14,74% 8,13% 4,99% 2006-2007 2,19% 3,55% 5,20% 2,84% 2007-2008 9,98% 1,99% 7,91% 4,83% 2008-2009 -16,27% -24,22% 6,21% 9,94% 2009-2010 19,85% 22,83% 6,77% 5,62% 2010-2011 16,34% 10,13% 5,35% 4,16% Table 5. Portfolio "A" and Portfolio "B" annual returns and volatilities. This table presents annual returns and annual standard deviations of Portfolio "A" and Portfolio "B" for each

Portfolio "B" Return

Portfolio "A" Annual Volatility

Portfolio "B" Annual Volatility

Portfolio "A" Return

observed year. Period of observation: from February 1999 - February 2011.

action and make it functional.

0,00% 2,00% 4,00% 6,00% 8,00% 10,00% 12,00% 14,00%

1999-2000

2000-2001

2001-2002

2002-2003

2003-2004

The following chart illustrates, graphically, the dynamics of volatility of Portfolios "A" and "B". It can be noticed that the continuous line representing the volatility of Portfolio "A" is often higher than that of the benchmark. However, it is interesting to underline the stabilization effect starting from 2004 and becoming evident under the PID control action. As it is well known, this instrument needs a history before it can enable its efficient control

Fig. 2. Annual Volatility of Portfolio "A" and Portfolio "B". This chart presents the annual volatility dynamics of the two portfolios in the observation period February 1999-February 2011. The continuous line represents Portfolio "A"; the dotted line represents Portfolio "B". After having calculated the return and risk of the two portfolios, a comparison of the two portfolios is performed by using a risk adjusted return indicator, the Sharpe Ratio. This indicator is defined as the ratio of the difference between return and risk free return at the

2004-2005

Portfolio "A" Annual Volatility Portfolio "B" Annual Volatility

2005-2006

2006-2007

2007-2008

2008-2009

2009-2010

2010-2011

numerator, divided by the standard deviation of the portfolio returns. In order to define the risk free rate the average of the Libor values in the 12 years (1999-2011) of observation period are calculated. This calculation has yielded a value equal to 2,80%. The results are depicted in the table below:


Table 6. Sharpe Ratio of Portfolio "A" and Portfolio "B". This table presents the results of a risk adjusted return indicator, namely the Sharpe Ratio applied to the two portfolios for every year in the observation period. In bold are illustrated the cases in which Portfolio "A" has outperformed Portfolio "B".

Fig. 3. Sharpe Ratio of Portfolio "A" and Portfolio "B". The chart presents, for each observation period, the Sharpe Ratio values of the two portfolios. In particular, in black the values belonging to Portfolio "A" are represented. In grey, the corresponding values for Portfolio "B" are illustrated. The absence of a column shows that the indicator value is negative, hence non-interpretable.

When in table 6, the word "Negative" is present, it means that for that specific year, it was not possible to record the indicator due to its negative value. The Sharpe Ratio is not

An Innovative Systematic Approach to Financial Portfolio Management via PID Control 243

This situation is interesting and it is visible in figure 4. It illustrates the stabilization effect of Portfolio "A" on Down Side Risk. The continuous line (new model) tends visibly to smooth

Fig. 4. Comparison between the Down Side Risk of Portfolio "A" and Portfolio "B". In the figure, the continuous line illustrates the DSR Portfolio "A". The dotted line serves for

2004-2005

Portfolio "A" Annual DSR Portfolio "B" Annual DSR

2005-2006

2006-2007

2007-2008

2008-2009

2009-2010

2010-2011

After having calculated the value of the Down Side Risk, it is possible to calculate the risk adjusted return indicator defined above, Sortino ratio. Differently from Sharpe, this indicator has at the denominator, not the standard deviation, hence the volatility of the portfolio, but rather uses the DSR, hence the volatility defined for the returns below the risk free rate. As it can be observed from table 8, Portfolio "A" obtains better results than Portfolio "B" in 6 years out of 11 (the year 2008-2009 is not considered since both portfolios

1999-2000 1,50 2,01 2000-2001 **0,29** Negative 2001-2002 **0,02** Negative 2002-2003 **0,15** Negative 2003-2004 0,40 1,26 2004-2005 Negative 0,67 2005-2006 **1,79** 1,62 2006-2007 Negative 0,08 2007-2008 **0,71** Negative 2008-2009 Negative Negative 2009-2010 2,36 3,48 2010-2011 **1,91** 0,93 Table 8. Sortino ratio for portfolios "A" and "B". This table presents the results of the riskadjusted return Sortino, applied to the two portfolios, for the whole observation period. In bold, the cases when Portfolio "A" over performs Portfolio "B" are highlighted. The indication "Negative" shows the fact that for a negative numerator, the indicator is not

Portfolio "A" Sortino Portfolio "B" Sortino

out the extreme values better than the movement of the benchmark.

Portfolio "B".

0,00% 5,00% 10,00% 15,00% 20,00% 25,00%

1999-2000

2000-2001

2001-2002

2002-2003

2003-2004

defined.

defined for negative values. Hence, the Sharpe Ratio becomes meaningless since a return net of the risk free is negative. The case of a negative numerator in the Sharpe Ratio formulation can occur in two situations: when the portfolio return for that period is negative, or when the portfolio return for that period is positive but inferior to the risk free rate of return.

The analysis of table 6 allows the reader to notice that Portfolio "A" is able to obtain better results than Portfolio "B" in 5 instances out of 11 (the observation for year 2008-2009 is eliminated since both portfolios have negative Sharpe Ratios). The consistent returns of Portfolio "A" in many cases, allow the overcoming of the risk free return when Portfolio "B" is not able to do so; hence, Portfolio "B" presents negative Sharpe Ratios (examples in the range 2000-2003).

Figure 3 represents the trend of Sharpe Ratios of the 2 portfolios.

When a column of one of the two portfolios is not visible, it means that one of the two values is negative.

It was considered interesting to investigate another risk adjuster return indicator: Sortino. This indicator of risk adjusted return, is defined as the ratio of the difference between the return and the risk free return, and, at the denominator, a risk measure defined as the Down Side Risk (DSR). The Down Side Risk is a measure of risk that considers only the volatility of the returns inferior to the risk free return. By calculating the Down Side Risk, we investigated the type of reduced risk, up or downside risk. We have analyzed if the new model acts more successfully in decreasing positive risk or downside risk.


Table 7. The Down Side Risk of Portfolios "A" and "B". The table represents for every year in the observation period the comparison between the Down Side Risk of the two portfolios. The DSR is calculated considering the volatility of returns inferior to the risk free rate relative to the risk free rate itself.

The Down Side Risk (DSR) of Portfolio "A" and of Portfolio "B" was calculated and analyzed for this purpose. The main results of this study on downside risk are depicted in Table 7. As illustrated in this table, the new model exhibits a DSR lower than the benchmark in 5 cases out of 12.

defined for negative values. Hence, the Sharpe Ratio becomes meaningless since a return net of the risk free is negative. The case of a negative numerator in the Sharpe Ratio formulation can occur in two situations: when the portfolio return for that period is negative, or when the portfolio return for that period is positive but inferior to the risk

The analysis of table 6 allows the reader to notice that Portfolio "A" is able to obtain better results than Portfolio "B" in 5 instances out of 11 (the observation for year 2008-2009 is eliminated since both portfolios have negative Sharpe Ratios). The consistent returns of Portfolio "A" in many cases, allow the overcoming of the risk free return when Portfolio "B" is not able to do so; hence, Portfolio "B" presents negative Sharpe Ratios (examples in the

When a column of one of the two portfolios is not visible, it means that one of the two

It was considered interesting to investigate another risk adjuster return indicator: Sortino. This indicator of risk adjusted return, is defined as the ratio of the difference between the return and the risk free return, and, at the denominator, a risk measure defined as the Down Side Risk (DSR). The Down Side Risk is a measure of risk that considers only the volatility of the returns inferior to the risk free return. By calculating the Down Side Risk, we investigated the type of reduced risk, up or downside risk. We have analyzed if the new

1999-2000 8,84% 8,19% 2000-2001 11,44% 11,06% 2001-2002 **10,74%** 12,20% 2002-2003 **11,43%** 14,40% 2003-2004 13,81% 8,27% 2004-2005 11,12% 7,81% 2005-2006 8,64% 7,38% 2006-2007 10,45% 9,13% 2007-2008 **10,18%** 10,33% 2008-2009 **15,68%** 19,42% 2009-2010 7,21% 5,76% 2010-2011 **7,09%** 7,87% Table 7. The Down Side Risk of Portfolios "A" and "B". The table represents for every year in the observation period the comparison between the Down Side Risk of the two portfolios. The DSR is calculated considering the volatility of returns inferior to the risk free rate

The Down Side Risk (DSR) of Portfolio "A" and of Portfolio "B" was calculated and analyzed for this purpose. The main results of this study on downside risk are depicted in Table 7. As illustrated in this table, the new model exhibits a DSR lower than the benchmark

Portfolio "A" Annual DSR Portfolio "B" Annual DSR

Figure 3 represents the trend of Sharpe Ratios of the 2 portfolios.

model acts more successfully in decreasing positive risk or downside risk.

free rate of return.

range 2000-2003).

values is negative.

relative to the risk free rate itself.

in 5 cases out of 12.

This situation is interesting and it is visible in figure 4. It illustrates the stabilization effect of Portfolio "A" on Down Side Risk. The continuous line (new model) tends visibly to smooth out the extreme values better than the movement of the benchmark.

Fig. 4. Comparison between the Down Side Risk of Portfolio "A" and Portfolio "B". In the figure, the continuous line illustrates the DSR Portfolio "A". The dotted line serves for Portfolio "B".

After having calculated the value of the Down Side Risk, it is possible to calculate the risk adjusted return indicator defined above, Sortino ratio. Differently from Sharpe, this indicator has at the denominator, not the standard deviation, hence the volatility of the portfolio, but rather uses the DSR, hence the volatility defined for the returns below the risk free rate. As it can be observed from table 8, Portfolio "A" obtains better results than Portfolio "B" in 6 years out of 11 (the year 2008-2009 is not considered since both portfolios


Table 8. Sortino ratio for portfolios "A" and "B". This table presents the results of the riskadjusted return Sortino, applied to the two portfolios, for the whole observation period. In bold, the cases when Portfolio "A" over performs Portfolio "B" are highlighted. The indication "Negative" shows the fact that for a negative numerator, the indicator is not defined.

An Innovative Systematic Approach to Financial Portfolio Management via PID Control 245

Portfolio "A" presents, in 6 years out of 11, a risk adjusted return value for the Down Side Risk better than the benchmark. These initial results confirm that the PID based asset allocation technique seems to be a good instrument, adapt for adverse market conditions. It effectively controls and bounds negative volatility. At the light of the current results herein achieved, the authors desire to further and develop the model in the attempt to seek and understand relations, functions and interacting factors among the managed portfolio characteristics and intrinsic and endogenous parameters of the model, such as the set-point,

The authors will further the model verifying and testing its applicability on various financial market indices and diversified portfolios, including the impact of transaction costs. The goal is to confirm broad-spectrum negative volatility controllability, steadiness and performance

Amenc, N., Malaise P. & Martellini, L. (2004). Revisiting Core-Satellite Investing. A dynamic

Amman, M., Kessler, S. & Tobler, J. (2006). Analyzing Active Investment Strategies. Using

Anson, M. (2004). Strategic versus Tactical Asset Allocation. Beta versus alpha drivers. *The* 

Arnott, D. R. & Fabozzi, F.J. (eds) (1988). Asset *allocation: A Handbook of Portfolio Policies, Strategies and Tacties*, Probus Professional Publishers, ISBN 1557380139, USA. Arshanapalli, B., Switzer, N. L. & Hung, T. S. L. (2004). Active versus Passive Strategies for

Carhart M. (1997). On persistence in mutual funds performance. *The Journal of Finance,* 

Da Silva, S., A., Lee, W. & Pornrojnangkool, B. (2009). The Black-Litterman Model for Active

Don, P. & Lee, J. (1989) Current issue: Tactical Asset Allocation. *Financial Analyst Journal*

Faff, R., Gallagher, R. D. & Wu, E. (2005). Tactical Asset Allocation: Australian Evidence.

Fama, F. E. (1965). The Behaviour of Stock Market Prices. *The Journal of Business*, Vol. 38,

Fama, F. E. (1970). Efficient Capital Markets: review of theory and empirical work. *The Journal of Finance*, Vol. 25, No. 2, (May, 1970), pp. 383-417, ISSN 0022-1082. Gandolfi, G., Sabatini, A. & Rossolini, M. (2007) PID feedback controller used as a tactical

Vol.52, No. 1, (March, 1997), pp. 57-82, ISSN 0022-1082.

No.1, (January, 1965), pp. 34-105, ISSN 1573-0697.

(September, 2007), pp. 71-78, ISSN 0378-4371.

Vol. 45, No. 2, (March-April, 1989), pp. 14-16, ISSN 0015-198X.

model of relative risk management. *The Journal of Portfolio Management,* Vol 31, No.

tracking error variance decomposition. *The Journal of Portfolio Management,* Vol. 33,

*Journal of Portfolio Management*. Vol.30, No. 2, (Winter, 2004), pp. 8-22, ISSN

EAFE and the S&P500. *The Journal of Portfolio Management*. Vol. 30, No. 4, (Summer,

Portfolio Management. *The Journal of Portfolio Management*, Vol.35, No. 2, (Winter,

*Australian Journal of Management,* Vol. 30, No. 2, (December, 2005), pp. 261-282, ISSN

asset allocation technique: The G.A.M. model. *Physica A*, Vol. 383, No. 1,

aiming to maximize returns' stabilization effects.

stabilization for financial portfolio managers.

1, (Fall, 2004), pp- 64-75, ISSN 00954918.

2004), pp. 51-60, ISSN 00954918.

2009), pp. 61-70, ISSN 00954918.

No.1, (Fall, 2006), pp. 56-67, ISSN 00954918.

**7. References** 

00954918.

1320-5161.

exhibited negative values). This indicates that, selecting the criterion of the most negative of the risk factors, the DSR, (that is the returns inferior to the risk free rate) the new model is bale to guarantee a better performance in comparison to the benchmark.

The following chart allows the visualization of the comparison of the two portfolios. It is to be remembered that when a column is missing, it indicates that its corresponding value is negative. In year 2001-2002, the column of portfolio A since its value is negligible. However, Sortino's value in that year is relevant.

Fig. 5. Sortino ratio of portfolios "A" and "B". The chart represents per each year of observation, Sortino values for the two portfolios. In particular, in black the results for Portfolio "A" are represented. In grey, the results of Portfolio "B" are illustrated. The absence of a column indicates that the indicator value is negative, hence non-interpretable.

If Sortino and Sharpe Ratio results are compared it is evident the ability of Portfolio "A" to better perform in comparison of Portfolio "B". Since the difference between Sortino and Sharpe resides in the definition of the denominator portion of the formula, it is apparent that Portfolio "A" acts more efficiently on the DSR than on the total volatility. Hence, this selectivity capability of the model is a good feature. The PID control action on financial portfolios seems to function as a stabilizer of returns. Above all, it diminishes the worst component of the returns, namely the ones inferior to the risk free rate.
