Finance
May 22, 2025
Risk-return and investor profile: a performance analysis of the investment portfolio
DOI: 10.22167/2675-6528-2024063
E&S 2025, 6: e2024063
Júlia Pereira Colobiale e Ricardo de Souza
The relationship between risk and return became central to the construction of investment portfolios following the publication of Portfolio Selection, by Harry Markowitz[1], which inaugurated the modern portfolio selection theory. By breaking with the logic of prioritizing only assets with higher returns, the author highlighted the importance of diversification as a strategy to optimize investments, incorporating risk as an essential element. Based on this approach, this article analyzes how diversification influences the performance of portfolios that were created and adjusted for different investor profiles, considering their distinct risk tolerances for a given expected return.
According to data from the Brazilian stock exchange (B3)[2], the quarterly result of the study on investor evolution showed a significant increase in the number of market participants between June 2022 and June 2023. When segmenting the data by investment type, a 34% growth is noted in the volume invested in fixed income, compared to a 23% increase in variable income.
B3 attributes these results to Brazil’s economic, political, and cultural scenario[2]. In recent years, the basic interest rate (SELIC) has shown frequent variations in short periods of time. According to the Central Bank of Brazil (BCB)[3], the annual Selic target in January 2021 was 2.0%. After successive increases, the rate ended the year at 9.25%, reached 13.75% in August 2022, and remained at this level until June 2023, when it began a cycle of reductions, closing 2023 at 11.75%[4]
Besides interest rates, other factors influenced the preference for fixed income, such as the political instability arising from the 2022 elections and a cultural shift, driven by the dissemination of investment content on communication channels and social media. Furthermore, B3[5] highlights that maintaining the Selic at high levels increases the opportunity cost, which contributes to the attractiveness of fixed income. This cost can be associated with more conservative investor behavior, seeking greater security in the face of economic risks and uncertainties, as evidenced by B3[5] data.
According to Markowitz[1], diversifying investments among securities that provide the maximum expected return is essential for efficient portfolio management. However, the author emphasizes that the portfolio with the highest expected return is not always the one that will allow for minimum variance. Thus, the investor must seek a balance between minimizing risk and their expected return, considering their risk tolerance profile.
Thus, understanding the formation of an investment portfolio from the perspective of risk-return theory and its suitability to the investor’s profile is fundamental to supporting decision-making. In this context, the present study aims to compose a portfolio aligned with risk and return theory, considering the presented Brazilian scenario.
Regarding the data collection and analysis methodology of this study, the information was obtained through a systematic approach. Information regarding the Selic rate, the interbank deposit certificate (CDI), and the Ibovespa index was extracted from the BCB website, to enable the comparison of the hypothetical portfolio’s performance with the main market indicators. Subsequently, the closing price data for the analyzed stocks were collected through the Refinitiv platform[6].
The sample was composed of 118 stocks listed on B3. The daily prices of these stocks were exported to spreadsheets and, based on this set, the six most profitable stocks in the analyzed period were selected. The selection criterion was based on the average quotation prices, covering the period between November 06, 2020, and April 09, 2024. After selecting the stocks, their respective risk and return levels were compared.
Additionally, the CDI rate was adopted as the risk-free rate in the calculations, as it is widely used as a profitability benchmark in investment funds. It is noteworthy that the present study does not include the analysis of transaction costs or issues related to the liquidity of assets included in the portfolio.
Markowitz and the risk-return theory
Markowitz[1] founded the modern portfolio selection theory and explained how rational investors use the principle of investment diversification to optimize their portfolios. The consensus at the time suggested that assets with the highest returns should be chosen for investment. However, Markowitz[1] demonstrated that when the highest-return asset is exclusively concentrated in the portfolio, the investor can achieve both high gains and excessive losses, which highlights the existing risk.
In this sense, Markowitz[1] was a pioneer in introducing risk as a primary concept in investment portfolio construction. For him, the ideal portfolio is one that, through asset diversification, simultaneously seeks to minimize inherent risk and maximize expected return. Furthermore, the author developed a mathematical model that stipulates infinite combinations of portfolio assets and determines the ideal allocation percentages for each. However, as each investor has a distinct risk profile regarding risk tolerance and return expectation, there is no single ideal portfolio for everyone. The portfolio composition should be suited to each investor’s preferences and risk appetite.
Sharpe Ratio (SR)
According to Ferreira and Costa[7], the function of the Sharpe ratio (SR) is to serve as a performance indicator to relate the return of an investment to the risk assumed. Developed by William Sharpe, the SR is part of Portfolio Selection Theory through the Capital Asset Pricing Model (CAPM), being used to identify, on the Capital Allocation Line (CAL), the portfolios considered optimal.
Furthermore, the authors Ferreira and Costa[7] elaborate that the IS is calculated based on the arithmetic mean of the excess returns provided by the asset in a given time interval, divided by the standard deviation of these returns, which represents variability. Thus, the higher the IS, the better the risk-adjusted performance of the investment.
Efficient Frontier (EF)
Markowitz[1] published an article that grounds the decision-making process for choosing optimal portfolios, reiterating the importance of portfolio diversification to compose a quality portfolio that balances expected return and risk, according to individual investor preferences. Hieda and Oda[8] state that, for the construction of the efficient frontier (EF) curve of risky assets, it is necessary to model the portfolio based on logical-mathematical constraints. The authors explain this process based on Sharpe’s[9] dominance principle, according to which the investor selects the portfolio that offers the lowest possible risk for each level of expected return, and the highest possible return for each level of risk.
Still according to Hieda and Oda[8], each point on the EF is equivalent to an efficient portfolio, that is, with the highest return given a certain degree of risk. It is through this frontier that the optimal portfolio is identified.
Capital Allocation Line (CAL)
According to Bodie et al.[10], each investor seeks to allocate the available assets in their portfolio in order to match the choice of these resources to their tolerated risk level and expected return, which results in the formation of their investment opportunity set. The greater the risk aversion, the lower the exposure to high-risk investments and the greater the preference for risk-free alternatives tends to be. The authors Bodie et al.[10] illustrate this opportunity set through a straight line called the Capital Allocation Line (CAL), which represents all possible risk and return combinations available to the investor. The CAL serves as a reference for identifying the best portfolio composition alternatives, considering the individual risk profile.
Optimal wallet
The optimal portfolio, as described by Markowitz[1], is the one that presents the lowest variance among an infinite series of portfolios that offer the same expected return. The investor, according to their degree of risk aversion, seeks to maximize this return.
Investor profiles can be classified as conservative, moderate, or aggressive. The conservative profile prioritizes risk minimization, concentrates investments in risk-free assets, and limits exposure to more volatile assets. The aggressive profile, on the other hand, has a higher risk tolerance and, consequently, adopts bolder strategies to achieve higher returns. The moderate profile is positioned between the two, tolerating a certain level of risk in exchange for gains superior to those obtained by conservative investors.
To determine the optimal portfolio, Bodie et al.[10] introduce the concept of Utility Function, which expresses the utility an investor obtains from a portfolio. Utility increases as the expected rate of return grows, and decays as the risk rate increases. The same authors also describe the indifference curve (IC), which is the graphical representation of all points that form a certain degree of utility, given the compensation demanded by the investor between risk and return. The higher the curve, the greater the utility. The juxtaposition of the IC with the CAL allows identifying the highest IC that is tangent to the CAL. The point of tangency represents the equalization between risk and expected return, which characterizes the optimal complete portfolio.
The six stocks with the best performance, according to risk and return theory, used in portfolio analysis, are represented in Table 1 below.
Table 1. Most profitable stocks between November 6, 2020, and August 3, 2023, and their B3 codes
| Company share code | Company name |
| BAUH4.SA | Excelsior Alimentos Corporation |
| RSUL4.SA | Metalúrgica Riosulense Corporation |
| LAND3.SA | Holy Land Agricultural Properties SA |
| LOGN3.SA | Log-in Intermodal Logistics SA |
| MRSA5B.SO | MRS Logística Corporation |
| BAZA3.SA | Amazon Bank SA |
The proportional portfolio, built on the basis of risk and return theory, displays the risks represented by standard deviation and the returns, expressed by the average profitability in the analyzed period, in percentage terms. The analysis considers each of the six selected stocks, as well as the total portfolio performance, with the addition of the risk-free rate, represented by the daily CDI of the considered period.
The calculations were performed with the “Solver” tool from Microsoft Excel, used to determine the optimal allocation of each asset’s weights in the portfolio. Furthermore, the IS was calculated, which assesses the average excess return of an asset over a given period, adjusted for its risk.
The proportional portfolio according to risk and return theory is presented in Table 1 below. The weights assigned to each asset were distributed with the aim of achieving the most balanced proportion between risk and return. The analysis of Table 1 allows us to infer that the portfolio shows positive profitability, given that the IS obtained is greater than zero.
Additionally, the coefficient of variation (CV) was included, an indicator that measures the variation of risk against the return of each asset and the portfolio as a whole. The higher its value, the greater the relative variability of returns in relation to the average. In this case, the portfolio is revealed to be composed of assets with reasonably homogeneous variation.
Table 1. Risk-return theory portfolio analysis
| Indicators | BAUH4.SA | RSUL4.SA | BAZA3.SA | LAND3.SA | LOGN3.SA | MRSA5B.SO | Free of risk | Wallet |
| ————————————————————In percentage (%) | ||||||||
| Return | 0,11 | 0,16 | 0,11 | 0,04 | 0,18 | 0,12 | 0,02 | 0,14 |
| Risk | 1,85 | 2,63 | 2,32 | 1,13 | 3,39 | 2,51 | 0,00 | 1,26 |
| Weights | 29,00 | 25,00 | 9,00 | 1,00 | 13,00 | 23,00 | – | 100,00 |
| Coefficient of variation | 16,81 | 16,43 | 21,09 | 28,25 | 18,83 | 20,91 | – | 9,33 |
| Sharpe Ratio | 0,09 | |||||||
Note: The weight fields were left blank for the “Risk-free” and “Portfolio” items for methodological reasons.
The FE analyzes the points where all viable investment portfolios are formed, focusing on the best possible risk-return relationship. As presented in Table 2 below, 20 distinct portfolios were simulated, built based on risk-return theory, which resulted in the formation of the FE curve for the analyzed scenario.
It is observed that, from the first point to the last in Table 2, the distribution ranges from the highest return viable portfolio to the lowest risk one. The analysis indicates that the interval between points 9 and 10 presents the best balance between risk and return; this is the point where the IS most closely approaches the return value, which characterizes the most viable portfolio among the simulated options.
Table 2. Efficient frontier
| Number | –—————-Return (y) | ——————-Risk (x) |
| ——————————————In percentage (%) | ||
| 1 | 0,181 | 3,386 |
| 2 | 0,176 | 2,701 |
| 3 | 0,170 | 2,253 |
| 4 | 0,165 | 2,012 |
| 5 | 0,159 | 1,817 |
| 6 | 0,154 | 1,650 |
| 7 | 0,148 | 1,505 |
| 8 | 0,143 | 1,390 |
| 9 | 0,138 | 1,300 |
| 10 | 0,132 | 1,236 |
| 11 | 0,127 | 1,177 |
| 12 | 0,121 | 1,122 |
| 13 | 0,116 | 1,070 |
| 14 | 0,111 | 1,024 |
| 15 | 0,105 | 0,982 |
| 16 | 0,100 | 0,947 |
| 17 | 0,094 | 0,918 |
| 18 | 0,089 | 0,897 |
| 19 | 0,084 | 0,884 |
| 20 | 0,078 | 0,879 |
Table 3 below presents the study of the portfolio with and without risk in the period between November 6, 2020, and August 3, 2023. The CAL reveals all risk and return combinations available to investors in constructing an investment portfolio.
The distribution between risky and risk-free assets considers different investor profiles. The most aggressive profile is represented by the arrangement of 100% in risky assets and 0% in risk-free assets. The moderate profile chooses a balanced distribution, with 50% in risky assets and 50% in risk-free assets. In turn, the conservative profile selects the portfolio with 0% of risky assets and 100% of risk-free assets.
Based on these proportions, the investor adjusts resource allocation according to their degree of risk aversion to maximize expected return within their profile. Conservative investors focus on minimizing risk, while aggressive investors are more willing to assume greater risks in pursuit of superior returns. See Table 3 below:
Table 3. Portfolio with and without risk – 2020/2023
| Investor profile | ————–Participations | —–Return —–(daily) | ——Risk —–(daily) | ——Return ——-(annualized) | ——Risk —(annualized) | |
| —–Risk | —–No Risk | |||||
| ———————————————————-In percentage (%) | ||||||
| Risky | 100 | 0 | 0,14 | 1,26 | 40,34 | 22,64 |
| 95 | 5 | 0,13 | 1,20 | 38,39 | 19,16 | |
| 90 | 10 | 0,12 | 1,14 | 36,48 | 16,23 | |
| 85 | 15 | 0,12 | 1,07 | 34,59 | 13,73 | |
| 80 | 20 | 0,11 | 1,01 | 32,73 | 11,58 | |
| 75 | 25 | 0,11 | 0,95 | 30.89 | 9,75 | |
| 70 | 30 | 0,10 | 0,88 | 29,08 | 8,19 | |
| 65 | 35 | 0,10 | 0,82 | 27,29 | 6,85 | |
| 60 | 40 | 0,09 | 0,76 | 25,53 | 5,71 | |
| 55 | 45 | 0,09 | 0,69 | 23,79 | 4,71 | |
| Moderate | 50 | 50 | 0,08 | 0,63 | 22,08 | 3,88 |
| Conservatism principle | 45 | 55 | 0,07 | 0,57 | 20,39 | 3,17 |
| 40 | 60 | 0,07 | 0,51 | 18,72 | 2,56 | |
| 35 | 65 | 0,06 | 0,44 | 17,08 | 2,04 | |
| 30 | 70 | 0,06 | 0,38 | 15,45 | 1,59 | |
| 25 | 75 | 0,05 | 0,32 | 13,85 | 1,21 | |
| 20 | 80 | 0,05 | 0,25 | 12,28 | 0,89 | |
| 15 | 85 | 0,04 | 0,19 | 10,72 | 0,61 | |
| 10 | 90 | 0,04 | 0,13 | 9,19 | 0,37 | |
| 5 | 95 | 0,03 | 0,06 | 7,68 | 0,17 | |
| 0 | 100 | 0,02 | 0,00 | 6,19 | 0 | |
| Wallet great complete | 13,90 | 85,10 | 0,04 | 0,18 | – | – |
Note: The annualized values of the complete optimal portfolio were not calculated at this stage as they are not directly used in the presented simulations, serving only as a theoretical reference point for the CA.
In order to update this study on the best investment portfolios for each investor profile, the period of one year, from April 10, 2023, to April 09, 2024, was analyzed. During this interval, the average daily CDI was calculated, an indicator used as the risk-free rate, as presented in Table 4 below.
The CAL represents all combinations of risk and return available to investors in a portfolio. The analysis of Table 4 should follow the same criteria adopted in Table 4, regarding the distribution of investor risk profiles.
Table 4. Portfolio with and without risk – 2023/2024
| Investor profile | ————–Participations | —–Return ––—(daily) | ——Risk ——(daily) | —–Return ——–(annualized) | ——-Risk —(annualized) | |
| ——Risk | —–No Risk | |||||
| ———————————————————In percentage (%) | ||||||
| Risky | 100 | 0 | 0,01 | 2,14 | 2,62 | 207,69 |
| 95 | 5 | 0,01 | 2,04 | 3,10 | 159,24 | |
| 90 | 10 | 0,01 | 1,93 | 3,57 | 122,01 | |
| 85 | 15 | 0,02 | 1,82 | 4,05 | 93,40 | |
| 80 | 20 | 0,02 | 1,71 | 4,53 | 71,43 | |
| 75 | 25 | 0,02 | 1,61 | 5,02 | 54,55 | |
| 70 | 30 | 0,02 | 1,50 | 5,50 | 41,60 | |
| 65 | 35 | 0,02 | 1,39 | 5,99 | 31,65 | |
| 60 | 40 | 0,03 | 1,29 | 6,48 | 23,95 | |
| 55 | 45 | 0,03 | 1,18 | 6,98 | 18,12 | |
| Moderate | 50 | 50 | 0,03 | 1,07 | 7,47 | 13,65 |
| Conservatism principle | 45 | 55 | 0,03 | 0,96 | 7,97 | 10,21 |
| 40 | 60 | 0,03 | 0,85 | 8,47 | 7,59 | |
| 35 | 65 | 0,03 | 0,75 | 8,97 | 5,57 | |
| 30 | 70 | 0,04 | 0,64 | 9,48 | 4,03 | |
| 25 | 75 | 0,04 | 0,54 | 9,98 | 2,85 | |
| 20 | 80 | 0,04 | 0,43 | 10,49 | 1,93 | |
| 15 | 85 | 0,04 | 0,32 | 11,00 | 1,24 | |
| 10 | 90 | 0,04 | 0,21 | 11,52 | 0,71 | |
| 5 | 95 | 0,05 | 0,11 | 12,03 | 0,31 | |
| 0 | 100 | 0,05 | 0,00 | 12,55 | 0,00 | |
In order to ascertain whether the previously constructed portfolio had outperformed the Ibovespa market index in the period from April 10, 2023, to April 09, 2024, the returns and risks of each of the six assets, the portfolio as a whole, and the indicator itself were calculated. The results are presented in Table 6 below. It is observed that, in the analyzed interval, the index surpassed the portfolio proposed in this study and presented daily and accumulated returns throughout the year superior to those obtained by the portfolio.
Table 6. Portfolio and Ibovespa (IBOV) Comparison
| Assets | Ibovespa | BAUH4.SA | RSUL4.SA | BAZA3.SA | LAND3.SA | LOGN3.SA | MRSA5B.SO | Wallet |
| ————————————————————–In percentage (%) | ||||||||
| Internal Rate of Return (12 months) | 22,88 | 6,33 | 27,19 | 55,77 | -70,07 | -86,11 | 0,79 | 2,55 |
| Return (daily) | 0,09 | 0,03 | 0,11 | 0,22 | -0,28 | -0,35 | 0,00 | 0,01 |
| Risk | 0,96 | 0,97 | 2,05 | 1,45 | 2,16 | 7,14 | 1,27 | 2,14 |
| Weights | – | 29,00 | 25,00 | 9,00 | 1,00 | 13,00 | 23,00 | – |
Note: The weight fields for ‘Ibovespa’ and ‘Portfolio’ were omitted as they represent the consolidated allocation result and an external to the portfolio.
As observed, until August 2023 the portfolio showed high profitability, with returns that surpassed the degree of risk assumed. From September 2023 onwards and throughout 2024, however, its performance began to decline, mainly due to internal factors of the component companies, which resulted in a performance inferior to the Ibovespa in the period from April 2023 to April 2024.
The main reason for this drop, especially compared to the excellent performance between November 6, 2020, and August 3, 2023, was the decline of the LOGN3.SA asset – then the most profitable in the portfolio. According to Table 6, this asset suffered a significant drop in the period. According to Valor Econômico magazine, Log-In Logística Intermodal SA, responsible for LOGN3.SA, reported a millionaire net loss in September 2023, which justifies the abrupt decline[11].
It is recalled that Markowitz’s Risk-Return Theory[1] guides the investor to diversify among assets that maximize expected return and minimize risk, according to their profile. The portfolio followed these guidelines, based on the theory and the Brazilian economic scenario of the period.
Therefore, the study’s objective was achieved: the portfolio was structured according to Markowitz’s principles and showed significant profitability in a relevant part of the observed period. The analysis also allowed for the identification of downturns and understanding of their causes, highlighting the importance of continuous performance monitoring. The significance of long-term studies on investment portfolios is emphasized, as only with continuous monitoring is it possible to measure results solidly and reduce biases. The analysis showed that, although profitable in the first period, the portfolio requires strategic adjustments to maintain performance in future scenarios.
A limitation is recognized in the asset selection criterion, based exclusively on past profitability. Other methods could complement the analysis. For future studies, it is recommended to include robustness tests, variations in asset weights, transaction costs, and liquidity as additional criteria. Therefore, it is reinforced that past profitability does not guarantee future results. The portfolio’s performance was calculated based on historical data, without considering projections or expectations. This demonstrates the importance of combining quantitative and qualitative analysis. Finally, when comparing portfolios in different periods, it is essential to go beyond the numbers. It is not enough to sort data to draw conclusions: it is necessary to investigate fundamentals, contextualize results, and project future scenarios with critical and informed analysis.
References
[1] Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.
[2] B3 – Bolsa de Valores. Empresas listadas. São Paulo, Brasil. Disponível em: https://www.b3.com.br/pt_br/produtos-e-servicos/negociacao/renda-variavel/empresas-listadas.htm
[3] Banco Central do Brasil (BCB). Dados diários do Selic: banco de dados. Brasília, Distrito Federal, Brasil. Disponível em: https://www.bcb.gov.br/estabilidadefinanceira/selicdadosdiarios
[4] B3 – Bolsa de Valores. Número de investidores na B3 cresce 34% em renda fixa e 23% em renda variável em 12 meses. São Paulo, Brasil. Disponível em: https://www.b3.com.br/pt_br/noticias/numero-de-investidores-na-b3-cresce-34-em-renda-fixa-e-23-em-renda-variavel-em-12-meses.htm
[5] Banco Central do Brasil. Sistema Gerenciador de Séries Temporais: CDI banco de dados. Brasília, Distrito Federal, Brasil. Disponível em: https://www3.bcb.gov.br/sgspub/localizarseries/localizarSeries.do?method=prepararTelaLocalizarSeries
[6] Refinitiv – Thomson Reuters. London Stock Exchange Group. Disponível em: https://www.lseg.com/en
[7] Ferreira, E.C.; Costa, S.R.R. (2016). A utilização do índice de Sharpe como ferramenta para comparação de risco/retorno em fundos de investimento. Revista Espacios, 37(15).
[8] Hieda, A.; Oda, A.L. (1998). Um estudo sobre a utilização de dados históricos no modelo de Markowitz aplicado à Bolsa de Valores de São Paulo. In: Seminários de Administração – III SEMEAD, 3, out. 1998, São Paulo. Anais do III SEMEAD. Faculdade de Economia, Administração e Contabilidade da USP, São Paulo, Brasil.
[9] Sharpe, W.F.; Alexander, G.J.; Bailey, J.V. (1995). Investments. Prentice Hall, New Jersey, USA.
[10] Bodie, Z.; Kane, A.; Marcus, A. (2014). Fundamentos de investimentos. AMGH Editora, Porto Alegre, RS, Brasil.
[11] Meneses, V. (2024). Log-In reverte lucro e tem prejuízo de R$ 35,9 milhões no 4º trimestre. Valor Econômico. Disponível em: https://valor.globo.com/empresas/noticia/2024/03/06/log-in-reverte-lucro-e-tem-prejuizo-de-r-359-milhoes-no-4o-trimestre.ghtml
Como citar
Colobiale, J.P.; Souza, R. Risco-retorno e perfil de investidor: uma análise de desempenho da carteira de investimentos. Revista E&S. 2025; 6: e2024063
About the authors
Júlia Pereira Colobiale – Specialist in Finance and Controlling from the MBA USP/Esalq. Campinas, São Paulo, Brazil
Ricardo de Souza – Advisor professor. Rua Cezira Giovanoni Moretti, 580, Santa Rosa, CEP: 13414-157, Piracicaba/SP, Brazil