Design of Robust Financial Products using System Dynamics and Multi-Objective Genetic Algorithms

Eder Oliveira Abensur

Centro de Engenharia, Modelagem e Ciências Sociais Aplicadas, Universidade Federal do ABC, Rua Santa Adélia nº166, Santo André, CEP:09210-170

Tel: 55.11. 9239-5021 e-mail: eder.abensur@uol.com.br

RESUMO

Produto robusto é definido como o estado onde o desempenho tecnológico, do produto ou do processo é minimamente influenciado pelos fatores causadores de variabilidade. Sob a ótica da engenharia, o desenho de um produto robusto possui o melhor desempenho possível sob os piores cenários. Esse conceito forma a base para o desenvolvimento de produtos industriais tangíveis. Seria possível aplicar o conceito de projeto robusto para o desenvolvimento de produtos financeiros intangíveis? O objetivo desse estudo é: (i) propor um método para desenvolvimento de novos fundos de investimento baseado nos conceitos de projeto industrial robusto e (ii) aplicar as técnicas de dinâmica de sistemas e algoritmo genético multi-objetivo para identificar a combinação mais favorável das variáveis para o lançamento do produto.

Palavras-chave: Dinâmica de Sistemas, Algoritmo Genético e Produto Robusto

Área Principal: MH - Metaheurística

ABSTRACT

Robustness is defined as the state where the technology, product or process performance is minimally sensitive to factors causing variability. From the engineering perspective, a robust product design has the best possible performance under worst-case variation scenarios. This concept forms the basis for the development of tangible industrial products. Should it be possible to apply robustness to intangible financial products? The aim of this research is: (i) to propose a framework for the development of new investment funds based on the concepts of robust industrial product design and (ii) apply system dynamics and a multi-objective genetic algorithm in order to identify the most favorable combination of variables for the product launch.

Keywords: System Dynamics, Genetic Algorithms and Robust Product

Main Area: MH - Metaheuristic

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1. Introduction

Many methodologies apply the Product Development Process (PDP), matching quality, time and productivity in order to achieve an efficient and effective PDP. Among them are the statistical methodologies that make descriptions, analysis, data production and market data correlations to solve problems and potential failures of PDP. The aim of these methodologies is to increase PDP reliability by eliminating potential failures. The consequence of a failure can be the total or partial loss of the product. The Taguchi method (Taguchi et al., 1989; Taguchi et al., 2000) is divided into three stages: (i) system project; (ii) parameters project and (iii) project of tolerances.

During the system project phase, a basic prototype of the product is developed to perform the desirable product functions with the minimum nominal values performance variation. In the following stage, the optimal levels of the parameters are defined in order to minimize the functional variations. Finally, the range of each parameter is defined and the quality-cost loss relationship is estimated.

According to Taguchi (1989), quality is defined as the deviation of a functional characteristic related to its expected value. There are three reasons for product characteristic variation: (i) environmental factors; (ii) deterioration and (iii) industrial process failures. These factors are called noises and cause loss of product quality, assessed by a loss function. The method proposes to define and optimize the loss function by using statistical techniques. The method minimizes the influence of the noise factors and was defined as a Robust Project.

From the engineering perspective, ignoring the variations in performance of the products under various usage situations and conditions may lead to malfunctions of the product and can possibly cause serious failures. From the marketing perspective, variations in performance of the product under different usage situations and conditions can have a significant impact on customer preferences for the product and on market share (Luo et al, 2005). Moreover, from the financial perspective, variations in product performance may cause negative impacts on manager and customer profitability.

The intangible characteristic of financial products has justified the difficulty encountered in applying many industrial PDP methods. Moreover, these products and customer behavior are influenced by macro-economic parameters such as the inflation rate, interest rate and exchange rate that can be considered as uncontrollable parameters.

Recently, parallel efforts have been made in the financial area in order to apply intelligent algorithms to financial decisions. These efforts have been focused on: (i) portfolio optimization (Crama; Schyns, 2003; Korczak; Lipinski, 2001); (ii) technical trading rules (Allen ; Karjalainen, 1999); (iii) financial predictions and insolvency risk (Mckee; Lensberg, 2002; Vareto, 1998) and (iv) investment recommendations (Li ;Tsang, 1999; 2000) and (v) new product development (Abensur, 2007). According to these studies, the genetic algorithm is the most used intelligent algorithm method applied to financial decisions.

This work shows that, despite the intangible characteristics of financial products, it is possible to identify operational variables that represent the performance of the financial product in order to use System Dynamics (SD) and Genetic Algorithms (GA), thereby improving knowledge of the financial variables and increasing the chances of success of the new product. The current study provides a method for reducing the time and cost of developing design prototypes of financial products, using the Robust Project, System Dynamics and Genetic Algorithms concepts, by integrating engineering, marketing and financial perspectives that represent cross-functional research.

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Three of the ten largest investment banks in Brazil were researched and their respective development of new investment funds was analyzed. The information was collected from interviews with managers and those responsible for the development of new investment funds. After 5 months of dialogue, a review of Brazilian investment fund legislation and field monitoring of the development process it was possible to understand and describe the main operational variables used to specify an investment fund and what makes the launch of a new investment fund successful. Finally, the product simulator, based on SD and a multi-objective genetic algorithm, was tested using 4 years of real data taken from the Brazilian financial market in order to develop a fictional investment fund. The professional GA software Evolver® was used to obtain the results.

This paper is structured as follow. Section 2 shows the main variables of new investment fund development process and how Robustness can be applied to new financial products development. In section 3, the System Dynamics framework is described. Section 4 shows an overview of the financial product design problem and in §5 the model is formulated and tested. In §6 the conclusions are presented.

2. Robust Project for Financial Products

A cross-functional approach takes into consideration a large number of factors that contribute to the design. Usually, these factors are interrelated and affect design decisions in many ways. The purpose of this section is to identify potential failures with the PDP of the financial product being studied, by including them in the model. In practical terms, it applies Robust Project concepts to the PDP of financial products.

According to the empirical evidence identified during investigation of the product development process, the characteristics of the variables in the investment funds’ specifications and other scientific studies (Elton et al., 2003; Sirri; Tufano, 1998), the main variables identified for the investment fund development are shown in table 1 below. From the financial product perspective and in accordance with Robust Project concepts, the following aspects should be included in the PDP of investment funds:

a) Multi-functional Aspects

Operational, financial and marketing aspects should be matched for an integrated and advanced analysis of the success of the launch. Operational issues are related to usual customer activities, such as investment and withdrawal. The functional characteristics of this group are: conversion period, withdrawal period and minimum period of investment. The financial aspects are divided into two parts: (i) manager’s profitability such as: management fee, performance fee and net equity (ii) investor’s profitability such as fund profitability.

The marketing aspects are related to commercial performance ratings and market share. Despite the fact that marketing aspects are not directly connected to functional fund characteristics, in an integrated analysis they are important for the product launch. The functional characteristics of this group are: an increase in net equity, an increase in new customers and the success rating per fund.

b) Potential Failures

The proposed tests will help managers minimize the influence of noises on the deviation of the product’s functional characteristics. Noises are market risks such as: interest rate, inflation rate, exchange rate and sovereign risk, summarized as the net flow of investments and withdrawals for each scenario. The potential failures are illustrated in figure 1 below. The tested functional characteristics are: conversion period, withdrawal period, minimum period of investment and net equity.

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c) Loss Function

Loss occurs when a quality functional characteristic (e.g.: net equity or profitability) varies from the expected value. There are many ways to obtain the total loss. Taguchi et al. (1989) proposed a quadratic loss function in which the loss A is equal to constant k multiplied by the squared deviation of expected value ∆ ( 2A k= ∆ ). By analogy, the financial market uses the Value at Risk (VAR) which means the biggest expected financial loss during a certain period of time in a confidence interval (Jorion, 1997). The proposed loss function is represented by the variation between the functional characteristics of the variables identified (management fee, performance fee, net equity, conversion period, withdrawal period, minimum period of investment) and their impacts on established goals (increase in net equity, manager’s profitability, customer’s profitability and risk).

d) Operational Limits

The results must represent the best values of the functional characteristics, with a minimum loss. By analogy with industrial tests, the new fund must be tested under usual and critical situations in order to define its operational limits. The range of functional characteristics can be obtained from the present market values, as shown in Table 2 below. Table 1 – Main Variables identified for the Investment Fund Specification used in the Model

Variable Description

Management fee The main remuneration received by the fund manager. Usually defined as a percentage of net equity.

Performance fee Eventual payment received by the fund manager because of superior fund performance.

Net equity The sum of investments plus the value of the portfolio and other receivables, less other expenses.

Conversion period Period of time between the date of the investment request and the date when the investment is converted into fund quotas.

Withdrawal period Period of time between the date of the withdrawal request and the date funds are credited to the investor’s current account.

Minimum period of investment Minimum period of time before the first withdrawal may be made.

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Table 2 – Range of investment fund’s functional characteristics

Administration fee

(% per year)

Net Equity

(US$ billions)

Withdrawal Period (days)

Conversion Period (days)

Bank mín. max. mín. max. mín. max. mín. max.

Bank of Brasil 0.50 4.50 0.02 1.50 D+ 0 D+ 0 D+ 0 D+ 0

Boston 0.50 3.00 0.01 1.39 D+ 0 D+ 0 D+ 0 D+ 0

Bradesco 0.30 4.50 0.12 0.65 D+ 0 D+ 0 D+ 0 D+ 0

HSBC 0.50 3.50 0.01 1.15 D+ 0 D+ 0 D+ 0 D+ 0

Itaú 2.50 4.00 0.06 2.15 D+ 0 D+ 0 D+ 0 D+ 0

Santander 0.50 4.00 0.05 0.27 D+ 0 D+ 0 D+ 0 D+ 0

Unibanco 1.25 4.00 0.08 0.75 D+ 0 D+ 0 D+ 0 D+ 0

General 0.30 4.50 0.01 2.15 D+ 0 D+ 0 D+ 0 D+ 0

Source: Prepared by the author Source: Investment funds available on the Internet D = day of investor’s request

Figure 1 – Potential failure tree

Investment

or withdrawalrequest denied

Request below the minimum value

Request out of time limit

Request beyond the minimumperiod of investment

Insufficient Net Equity

Lack of Portfolio Liquidity

Excessive withdrawal or investmentrequests

Investment or withdrawalrequest denied

Request below the minimum value

Request out of time limit

Request beyond the minimumperiod of investment

Insufficient Net Equity

Lack of Portfolio Liquidity

Excessive withdrawal or investmentrequests

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3. System Dynamics Approach

System Dynamics focuses on the structure and behavior of systems composed of interacting feedback loops. The art of system dynamic modeling lies in discovering and representing the feedback process and other elements of the complexities that determine the dynamic of systems. The causal relationship diagram represents the cause and effect relationship between the identified variables. In a causal diagram, the arrows indicate the direction of influence and a ‘+’ or ‘-’ sign near the arrow implies increase/decrease in causal results. The dynamics arise from the interaction of two types of feedback loops, positive (self-reinforcing) and negative (self-correcting) loops. Positive loops tend to reinforce or amplify whatever is happening in the system. Negative loops counteract and oppose change. These loops all describe processes that create balance and equilibrium (Forrester, 1961; Roberts et al, 1983; Sharma et al, 2004; Narashima; Roy, 2007). Figure 4 presents the causal diagram for the studied system.

Figure 2 shows that an improvement in the design of the financial product increases banking sales. Consequently, an increase in sales implies an increase in total investments that increases the profitability of the fund (larger funds raise the bargaining power of managers) and investor profitability, defining a positive loop. An increase in the net equity of the fund increases the chances of achieving net equity and the manager’s profitability goals. However, an increase in net equity increases the volatility of the fund, thereby increasing its risk (VAR) and reducing the chances of achieving the risk goal, defining a negative loop. The management fee and the minimum period of investment present interesting effects. An increase in the minimum period of investment increases net equity but increases the management fee, because investors request a bonus for maintaining their investments in the fund otherwise they will invest in similar funds with no minimum period of investment. An increase in the management fee compensates for the bonus paid. A high management fee increases the chances of achieving the manager’s profitability goal but decreases the profitability of the fund and, consequently, the investor’s profitability. Some cause and effect causal relationships, such as the correlation between fund profitability and increase in net equity and the negative influence of fees on investor demand have been studied in other works (Sirri and Tufano, 1998). The relationships presented in the diagram are in accordance with the financial theory and were used in the simulation.

4. The Financial Product Design Problem

Traditionally, the multi-attribute product design problem is divided into two groups: (i) a simple product, which involves the launch of a unique product and (ii) a line of products, when many products are launched simultaneously. The simple product design problem incorporates the definition of the ideal levels of its attributes in order to maximize a target function. As examples of target functions there are the buyer’s welfare problem and the seller’s welfare problem. The multi-attribute product design problem has been researched for many years. There are many approaches to the subject, showing increasing performance due to the use of intelligent algorithms (Kohli; Krishnamurti 1987, 1989; Kohli; Sukumar, 1990; Balakrishnan; Jacob, 1996; Shi et al, 2001; Gruca ; Klemz, 2003). The financial product design can be understood as a set of K attributes (the main variables identified in §2)

1, 2,..... K K∈Ω = with a set of Jk levels 1,2,.... k kJ J∈Φ = (e.g.: management fee between 0.3% and 4.5% per year). According to the manager’s decisions, the appropriate fitness function (objective function) is the following equation (Hillier; Lieberman, 1980):

1 1

n n

jk j k

k j

minimizeZ c x g−

= =

= ∑ ∑ (1)

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kg =goal (increase of net equity, manager’s return, financial risk, investor’s profitability) =coefficient jkc

jx = variable of decision (net equity, management fee, performance fee, conversion period, withdrawal period, minimum period of investment)

The above equation, a multiple-objective function, represents the minimum sum of variations in relation to the defined goals.

+ Banking Sales

Figure 2 – Causal Diagram of the Studied System

5. The Multi-objective Genetic Algorithm

Almost all real world problems require the simultaneous optimization of multiple conflicting objectives. Different from single objective optimization, the solution to this kind of problem is a set of points. GA has been recognizing as an attractive technique for solving multiple objective problem. The GA has several advantages such as: (i) the search is conducted at a population of points rather than at a single point, thus increasing the chances of success; (ii) direct use of the objective function, thus the candidate product profiles are evaluated, based on the specified objective(s); (iii) it fully evaluates specified candidate solutions, unlike other techniques (dynamic programming), which evaluate profiles sequentially, one attribute at a time. Moreover,

Improvement of the Design of the Fund

Total of Investments

Investor’s profitability

Profitability of the Fund

Net Equity

Net Equity Goal

+ +

Manager’s Profitability Goal Minimum period

of investment

Fees (management + performance)

Risk Goal

+ + +

VAR

+

+

+

+

+

+

+

Withdrawal Period _

+ _ _ _

Conversion Period +

_

+

_

+ Investor’s Profitability Goal

+

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GA and other intelligent algorithms (simulated annealing, tabu search) are appropriate techniques for NP-Hard problems.

The analyzed situation, the investment funds’ specifications of robust financial product, is a multi-objective problem with conflicting criteria such as the traditional duality profitability versus risk. The multiple-objective function represents the minimum sum of variations in relation to goals such as the increase in net equity, manager’s profitability, investor’s profitability and risk management. The proposed model evaluates the most favorable combination of identified variables (management fee, performance fee, net equity, conversion period, withdrawal period, minimum period of investment) in order to minimize the variations and reduce potential failures.

The proposed simulator is divided into two major parts: (i) a product simulator, which involves the main variables and the goals identified during investigation of the product development process and (ii) a market simulator which includes historical scenarios, in extreme and divergent situations, including the product’s life-cycle divided into the launch and maturity periods and with economic turbulence (financial crises) or without it.

The increase in net equity C during a certain period of time n due exclusively to the result of daily investments and withdrawals ∆m from an initial net equity P0 at t0 under the influence of a minimum period of investment (factor Fi assumes values 0 or 1) can be formulated as:

(2) 0 *

1

[ (1 )] - i

n

i C

i

n P m FC=

+= ∏ M

Usually, when the investor requests a withdrawal his profitability is influenced by the annual management fee ta, and by the withdrawal period dr. Assuming Rpi as the fund profitability at i (this information can be obtained by past data from similar funds launches), du as the number of working days per year and ib as the benchmark rate (or the opportunity cost during the withdrawal period, the investor’s profitability Rinv is given by the following equation:

Rinv = [ Rp∑=

n

i 1i - (1 + ta) (1/du) ] - (1 + ib) (dr/du) (3)

According to the category of the fund, there is a performance fee representing the superior performance, when compared to a benchmark rate (e.g.: interest rate), achieved by the manager’s fund strategy. Considering that the performance fee tp occurs only if the fund’s profitability exceeds the benchmark rate ib , then the full investor’s profitability equation is:

Rinv = [ Rp∑=

n

i 1i - (1 + ta) (1/du) – ] - (1 + i

n

i 1max( - , 0)p i bt Rp i

=∑ b) (dr/du) (4)

The fund manager’s profitability L is a result of the sum of the contributions of the management fee ta, and the performance fee tp minus the bonus Ca paid to the investors who did not withdraw during the minimum period of investment dca, minus the total cost CT (fixed plus variable) and minus taxes IR as given by the following equation:

L = (( (1/du)

1

(1 ) n

a

i

t=

+∑ + ( )) P ) - (Ca ( ) - CT – IR) (5) n

i 1max 0p i bt (Rp - i , )

=∑ i 1

ca

ca ca

ca

n d

i d i di d

P P+

− + −

=

−∑

According to the Basle II Agreement and the risk management procedures adopted by the banks we studied, the value at risk (VAR) of the fund’s daily incomes with a standard deviation σ , an initial investment W0 during a period of time t∆ is given by the following equation (Jorion, 1997):

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VAR = 1,65W0σ t∆ (6)

The aim of the proposed model is to achieve a good solution, by balancing the defined multiple objectives. The goals were defined in accordance with those of similar funds that have been launched and from the comments of managers. Considering MV, ML , MR as the investor’s goal, the manager’s return goal and the fund’s risk goal, respectively, then, the final expression of equation (1) , in accordance with (2), (4), (5) and (6):

Min Z

= 01

((1 ))1

i i CC

nP m F

M i+ −∏

=

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

M +

n(1/ ) ( / )

1 i 1

1( (1 ) ) ( max 0) (1 )u r u

nd d

i a p i b bV i

Rp t t (Rp - i , i MM = =

− + − − + −⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦∑ ∑ d

V +

n(1/ )

1

1 i 1

1( (1 ) ) ( ( max 0))) ( ( ) )

cau

ca ca

ca

n n dd

i a i p i b i d i d LL i i d

P t P t (Rp - i , Ca P P CT IR MM

+

− + −

= = =

+ − − − − − −⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦∑ ∑ ∑

+ 01

1, 65 - RR

W t MM

σ ∆⎡ ⎤⎣ ⎦

subject to:

P0 P≤ 0max (maximum initial net equity, in accordance with similar funds launched and the chosen scenario)

dr ≤ 5 (withdrawal period in days, in accordance with the Brazilian legislation)

dc d≤ r (withdrawal period longer than conversion period)

dca ≤ 4 (minimum market period of investment less than 4 months)

ta t≤ a max (maximum market management fee in accordance with to table 2)

tp ≤ tp max (maximum market performance fee in accordance with to table 2. Fixed income funds have no performance fee)

min ( )≥ P90

n

iPi=

min (minimum legal net equity required, in accordance with Brazilian legislation)

The selected historical scenarios contain a diversity of situations including a financial crisis such as: the Asian crisis (1997), the Russian crisis (1998) and the Real crisis (devaluation of the Brazilian currency in January 1999) and non-turbulent periods (October 2002 to February 2004). Furthermore, the scenarios were divided according to product life-cycle and market volatility. The product’s life-cycle was divided into two parts: birth (from the beginning to the third month) and maturity (from the forth month on). The volatility was divided into with or without economic turbulence. The four groups of test scenarios are: (a) birth, under favorable conditions, with no economically turbulent maturity phase; (b) birth, under favorable conditions with an economically turbulent maturity phase; (c) birth, under unfavorable conditions with no economically turbulent maturity phase and (d) birth, under unfavorable conditions with an economically turbulent maturity phase.

The daily flow of investments and withdrawals for each scenario can be obtained by past data for similar funds launched by the institution, or competitors. These flows reflect the investor’s behavior under similar conditions for each test scenario. (Brazilian Investment Bank Association – Anbid, 2004). The tests are done using the described product simulator in order to achieve the optimal values of the variables in accordance with the chosen scenarios. This procedure is in

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accordance with many industrial product development processes (automobile industry, petroleum industry) and other studies (Sirri ; Tufano, 1998; Thomke, 1998; Elton et al. 2003).

The objective of the proposed model is to evaluate the most favorable combination of variables in order to minimize the variations in relation to the defined goals. According to the GA terminology, if there are three attributes A1 (withdrawal period), A2 (conversion period) and A3 (minimum period of investment) and each one has 6 levels (0, 1, 2, 3, 4, 5), a possible representation (chromosome) would be 5 1 3 which means a withdrawal period of 5 days, a conversion period of 1 day and a minimum period of investment of 3 months. Once the product characteristics have been defined by attributes and levels, an initial population M was generated randomly. After many attempts a convergence of the results was achieved with the following GA parameters: (i) initial population M = 1,000; (ii) uniform crossover rate = 0.5; (iii) mutation rate = 0.1; (iv) stop condition = 50,000 iterations. The results are presented in Table 3 below and show the possibilities of success of the new product.

As expected, unfavorable scenarios B and D require higher values of the decision variables than favorable scenarios A and C. The inclusion of legal restrictions enlarges the boundaries of analysis, because the model is not only influenced by market conditions. The withdrawal period of one day in all scenarios shows that it is a useful legal condition to balance the conflicts that exist. From the same perspective, the minimum period of investment of three months is recommendable in scenario B. The simulated management fees are lower than the market management fees, because the model balances manager’s and investor’s interests with equal weights. The efficiency of the GA can be proved by comparing the simulated initial net equity (13,984.61) with the real case that occurred in the market (13,372.00), a difference of 4.5%.

The robust financial product design can be understood as a set of K attributes (the main variables identified) 1, 2,..... K K∈Ω = with a set of Jk levels 1,2,.... k kJ J∈Φ = (e.g.: management fee between 0.3% and 4.5% per year). As shown in table 3, the investment funds’ specifications for scenario C are: management fee (2.38%), performance fee (0.08%), net equity (US$ 13,984.61) conversion period (0 days), withdrawal period (1 day) and minimum period of investment (0 months).

6. Conclusions

The new product development process is a crucial worldwide activity, which separates successful companies from unsuccessful ones. This research presented a new general optimization framework for constructing new financial products, based on System Dynamics, Genetic Algorithms (product simulator) and using Robust Project concepts in order to achieve good solutions for the product design problem. Robust Project concepts helped to define the test characteristics, in order to identify and avoid potential failures. Simultaneity is a particularly important aspect of the proposed tests. System Dynamics represents the dynamic behavior of the studied system, using the feedback process between the identified variables and incorporating simultaneity in the proposed tests. Genetic Algorithms provided an efficient decision-making tool to be used to find a good solution from the millions of possible alternatives. Finally, the presence of a structured test stage supported by a simulation tool reinforces the team’s commitment and team work, thus increasing the synergy between the areas from the beginning (idea) to the end of the development process (launch). This synergy allows for discussion and the improvement of conditions for the launch of the new fund. A simulation tool, as used in industrial development, would improve debate, through a rapid assessment of any possible impact caused by the proposals.

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Table 3 – Best Results found by GA

Scenario A Scenario B Scenario C Scenario D

Increase of Net Equity (%) -0.45 -37.40 -11.29 -57.89

Manager’s profitability variation (%) -0.02 0.03 -0.02 -5.40

VAR (%) -0.01 0.22 0.12 0.95

Investor’s profitability variation (%) -0.58 -1.81 -0.84 -3.24

Objective Function (%) 1.07 39.46 12.27 67.49

Initial Net Equity (US$) 15,693.59 20,812.70 13,984.61 14,000.00

Management fee (% per year) 2.11 3.38 2.38 4.50

Performance fee (% per year) 0.04 0 0.08 1.03

Withdrawal period (days) 1 1 1 1

Conversion period (days) 0 0 0 0

Minimum period of investment

(months)

0 3 0 0

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