Evaluating the influence of diversification strategies on portfolio optimization a case study in vietnam Evaluating the influence of diversification strategies on portfolio optimization a case study in vietnam
Problem statement and Methodology
Problem statement
Despite extensive research on diversification constraints in portfolio optimization, there is a significant lack of studies focusing on the Vietnamese stock market Constraints like Lower and Upper Bound (LBUB) and Sector Constraints (ST) can significantly affect portfolio performance by changing risk-return dynamics However, their specific effects within Vietnam's market context have not been thoroughly examined This study seeks to fill this gap by assessing the impact of these constraints through a Value at Risk (VaR) optimization model tailored to the Vietnamese stock market.
This study employs a two-stage methodological approach to investigate the influence of diversification constraints on portfolio optimization models within the Vietnamese market
The initial phase of model development aims to create a base model that reduces portfolio risk while achieving a desired return This involves incorporating various diversification constraints, which are essential in portfolio optimization to effectively distribute investments across multiple asset classes and manage risk Key steps in this development process include selecting suitable diversification constraints, utilizing Value at Risk (VaR) as the risk assessment measure, integrating these constraints into the model, and preparing a comprehensive set of models for evaluation in the subsequent stage.
The second stage of the study evaluates the impact of diversification constraints on portfolio performance by incorporating Lower and Upper Bound (LBUB) constraints and Sector Constraints (ST) into the base model, both individually and in combination To account for the fluctuations in market returns due to varying economic conditions, expected returns are adjusted according to different phases of the business cycle: expansion, peak, contraction, and trough This methodology ensures that the models effectively capture diverse risk-return profiles amid changing market dynamics Portfolio performance is assessed through key metrics such as Mean Return (MR), Sharpe Ratio (SR), Calmar Ratio (CR), and Stability Index (SI), with each model tested across multiple market scenarios for a comprehensive evaluation Data visualization will be facilitated using Tableau, while optimization will be conducted with the CPLEX solver, offering insights into how diversification constraints influence portfolio optimization within the Vietnamese stock market.
Experimental Framework Research
Note: Circle is input or output, Square is Process or model
My research employed a robust 7-elements experimental framework to investigate portfolio optimization strategies Here's how my process unfolded:
Data Training: I meticulously collected and prepared historical data on asset returns
This data was critical for training my simulation model and refining parameter estimations within my portfolio optimization model
Simulation Model: I developed a simulation model capable of generating plausible future scenarios of asset returns This model drew upon historical data and my understanding of asset market behavior
The simulation model produced 100 distinct scenarios for future returns of each asset in the portfolio, which were utilized as inputs for the optimization model This methodology encompassed a wide array of potential market outcomes, thereby strengthening the overall robustness of the analysis.
I developed an advanced portfolio optimization model that incorporates expected returns, risk assessments, asset correlations, and specific investment constraints to identify the optimal portfolio allocation The optimization process was executed using the CLPEX SOLVER.
Portfolio Formulation: This involves selecting the assets included in the portfolio and determining the percentage allocation of each asset
In my analysis of real datasets, I carefully computed key performance metrics including the Sharpe Ratio (SR), Calmar Ratio (CR), Sortino Ratio (SI), and Maximum Drawdown (MR) for both in-sample and out-of-sample periods.
In our comprehensive assessment of the portfolio optimization model, we meticulously analyzed its performance by comparing the risk-adjusted returns of the optimized portfolio with relevant benchmark portfolios, thereby evaluating the model's ability to deliver superior returns.
Key Takeaway: This experimental framework gave my research a structured approach to portfolio optimization, blending data training, simulation, and
10 optimization techniques The process empowered me to develop and rigorously evaluate portfolio optimization models with the goal of enhancing investment returns.
Models
1.3.1 Value at risk optimization development
The worst realization in Value at Risk (VaR) is grounded in the essential principle of evaluating potential investment losses under unfavorable market conditions It involves identifying and quantifying the worst-case financial loss scenarios within a defined timeframe and confidence level This worst realization is crucial for estimating VaR, as it offers valuable insights into the potential downside risks linked to an investment or portfolio.
Let \( r_i \) represent the random variable indicating the rate of return for the ith asset across s scenarios, while \( x_i \) denotes the proportion of total investment allocated to that asset The worst realization is defined as
𝑀(𝑥) = 𝑚𝑖𝑛 ∑ 𝑛 𝑖 =1 𝑟 𝑖𝑠 𝑥 𝑖 for s in S scenarios and is LP computable The portfolio optimization model with the worst realization as safety measure (the Minimax model) can be formulated as:
𝑥 ∈ 𝑄 where 𝜇 𝑖 is the mean return of asset 𝑖 𝑡ℎ , 𝜇 0 is the expected return of portfolio
A natural generalization of the measure M(x) is the statistical concept of quantile
In general, for given β∊[0,1], the β-quantile of a random variable R is the value q such that
For β in the range [0,1], the β-quantiles form a closed interval, and for a specific β value, the left endpoint of this interval is utilized to define the quantile measure in cases of non-uniqueness In this context, I represent the β-quantile as qβ(x), which signifies the rate of return associated with that quantile.
In finance, the Value-at-Risk (VaR) measure represents the maximum potential loss of a portfolio within a specified confidence interval This metric is derived from the cumulative distribution function, providing essential insights into risk management for investors and financial institutions.
Lower Bound & Upper Bound Constraints
Balancing risk and return is essential for effective portfolio optimization, as overly concentrated portfolios can lead to significant exposure to the volatility of a few assets.
12 assets To reduce this risk, a widely adopted strategy involves applying lower and upper bounds on individual asset weights (Antonios et al., 2021)
Antonios et al (2021) proposed a strategy that restricts individual asset weights to a maximum of 10% of the overall portfolio, while also limiting total investments in the banking, leasing, and insurance sectors to 30% This approach helps avoid over-concentration in specific areas, enabling portfolio managers to adhere to their sector preferences and investment guidelines.
Research shows that portfolio constraints not only reduce risk but also enhance portfolio efficiency, particularly in out-of-sample tests (Lin, 2013) Lin's study found that weight bounds and Lp-norm constraints significantly improved out-of-sample efficiency by mitigating estimation errors prevalent in historical return data, thereby preventing extreme allocations and fostering a balanced portfolio Furthermore, DeMiguel et al (2012) demonstrated that entropy constraints, which promote a more even distribution of weights, further boost out-of-sample performance.
Diversification is essential for portfolio optimization, as it entails distributing investments across various asset classes to mitigate risk through negative correlations Enhancing this strategy can be achieved by implementing sector constraints, which establish predefined limits on capital allocation to specific sectors or industries, thereby serving as protective measures (Black & Litterman, 1992).
Sector constraints are essential for reducing concentration risk, which can lead to increased portfolio volatility due to overexposure in a single sector For example, during the dot-com bubble burst, portfolios heavily invested in technology stocks experienced substantial losses By limiting allocations to specific sectors, these constraints foster greater diversification and help shield investors from such risks Research by Brodie et al (2002) indicates that implementing sector constraints can improve portfolio performance, especially in times of market stress.
My optimization model incorporates sector constraints that mandate investment in a minimum number of sectors, promoting diversification beyond traditional asset classes This approach not only reduces overall portfolio risk but also enhances resilience against sector-specific downturns Grounded in the principles of Modern Portfolio Theory (Markowitz, 1952), these sector constraints serve as an effective strategy for achieving diversification and mitigating risk within the investment framework.
Weight limits in investment sectors can be customized based on an investor's risk profile Conservative investors often impose stricter constraints to maintain a balanced portfolio, whereas risk-tolerant investors may adopt more flexible limits to seek higher returns from targeted sectors.
After narrowing down the stock universe using this data-driven filtering approach, my research group will proceed to the next phase: in-depth analysis using optimization models
When viewing losses as negative returns, Value at Risk (VaR) aligns with the quantile 𝑞^(𝑥) However, because of possible discontinuities in the cumulative distribution function, VaR cannot usually be calculated using linear programming (LP) Instead, the optimization of the portfolio is better approached through a Mixed Integer Linear Programming (MILP) model.
• 𝑝 𝑠 : the probability that scenario s happens
• 𝑟 𝑖𝑠 : return rate of asset i in scenario s
• LB, UB: lower and upper bounds of the allocation in assets
• 𝑒 𝑖𝑘 𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾 ∶ binary indicators, equal to 1 if stock i belongs to sector k, 0 otherwise
• 𝑧 𝑠 : binary variable, representing if scenario s is recognized as one of worst scenarios
• 𝑥 𝑖 : continuous variable representing the number of lots allocated to asset i
• 𝑦 𝑘 : binary variable, representing if sector k is selected for investing
• 𝑔 𝑖 : indicator variable, representing whether asset i is chosen
BASE MODEL (model 1) Portfolio optimization using VAR The problem is formulated as: max 𝑓 (1)
(1) The objective function is formulated to minimize (f) the Value at Risk
Due to inequality (2), binary variable 𝑧 𝑠 takes value 1 whenever variable 𝑓 is greater than the portfolio returns under scenario 𝑠 ( 𝑓 ≥ 𝑓 𝑡 = ∑ 𝑛 𝑖 =1 𝑟 𝑖𝑠 𝑥 𝑖 ) This constraint defines the return of the portfolio at 5% percentile
(3) guarantees that the probability of all scenarios such that 𝑓 ≥ 𝑓 𝑡 is equal α
Constraint (4) ensures that the summation of the percentages representing investor money allocation in the portfolio is constrained to be equal to 1
The base model establishes the groundwork for creating advanced models that include diversification constraints, which are designed to enhance portfolio diversification by restricting investment concentration in particular assets or sectors This article presents three additional models that are developed from the base model.
Model 2: Lower and Upper Bound Constraints: This model integrates constraints that set minimum (LB) and maximum (UB) investment proportions for each asset
LBUB MODEL: Portfolio optimization using VAR The problem is formulated as
Constraint (9) and (10) imposes a lower bound on the allocation of investor funds to the selected asset, ensuring that at least LB is allocated
Constraint (11) enforces an upper bound on the allocation of investor funds to maintain all allocations below UB
Model 3: Sector Constraints: This model incorporates a constraint that restricts the number of chosen assets to a specific number (N) of pre-defined sectors or industries
SECTOR MODEL Portfolio optimization using VAR The problem is formulated as:
Constraint (6) ensures that the total number of selected sectors remains constant at N Meanwhile, Constraints (7) and (8) create a binary link between asset selection and sector affiliation, indicating that 𝑦 𝑘 is equal to 1 only when asset i is both selected and categorized under sector k.
Model 4: Comprehensive Constraints: This model combines both lower/upper bound and sector constraints, ensuring a diversified portfolio with controlled individual asset exposure and industry representation
FULL MODEL: Portfolio optimization using VAR The problem is formulated as
By systematically integrating these constraints, I create a series of portfolio optimization models that not only optimize risk (VAR) but also enforce desired levels of diversification within the portfolio
Data selection
The stock data is obtained from cafef.vn, a reputable website providing daily stock prices for Vietnamese investors from 2020 to 2022
The dataset consists of two primary tables:
● Table 1: Contains columns for date, stock_name, and close_price, recording daily closing prices for each stock
● Table 2: Maps stock_name to its respective sector_name
Table 1-1: Stock price by day
Table 1-2: Sector code for each stock sector code Mã CK
- Handling Missing Transactions: o The dataset only includes records for days with transactions To address this, prices from the previous trading day are carried forward to fill in missing days
The transformed data includes a newly created percentage change table that features columns for the date, stock name, and the percentage change in price relative to the next day Additionally, a sector dictionary is established, where keys represent pairs of assets and sectors, and the values indicate membership in the sector with a 1 for belonging and a 0 for not belonging.
Table 1-3: Percentage Change Table example
This transformation ensures the dataset is ready for optimization modelling.
Data simulation method
Simulation plays a vital role in portfolio optimization by effectively managing the uncertainties of future market conditions It addresses the limitations of historical data and allows for a comprehensive exploration of various potential portfolio performances This method is particularly beneficial when utilizing out-of-sample data, which consists of data points excluded during the model parameter estimation process Through simulation techniques, I can generate new insights and outcomes for portfolio management.
21 realistic datasets that go beyond historical observations, improving the robustness of my portfolio strategies
The Monte Carlo Method is a highly flexible and effective simulation technique in finance, utilizing repeated random sampling to produce numerical results It excels in addressing problems where theoretical calculations are challenging or impractical In portfolio optimization, this method can simulate thousands of potential future price paths for each asset, making it especially valuable for out-of-sample analysis By not relying solely on historical data, the Monte Carlo Method significantly enhances the reliability of portfolio optimization strategies.
The Geometric Brownian Motion (GBM) model is essential for portfolio simulation, especially when using the Monte Carlo Method It effectively captures the random fluctuations in asset prices over time through a continuous-time stochastic process, relying on two crucial parameters.
● Drift (μ): This represents the expected average return of the asset over a specific time horizon
● Volatility (σ): This captures the magnitude of random price fluctuations, essentially quantifying the level of risk associated with the asset
Gradient Boosting Machines (GBMs) are advanced machine learning models widely used in finance to analyze asset price dynamics and create various price scenarios This methodology is crucial for predicting asset behavior and assessing their risk profiles over specific time periods.
My methodology utilizes a core foundation of Geometric Brownian Motion (GBM) and enhances its capabilities through Gradient Boosting techniques Here's a breakdown of the process:
● Initial Price (S0): Baseline value of the asset at the present moment
● Maturity (T): The length of the simulated time horizon
● Number of Scenarios (N): The desired quantity of simulated asset price trajectories
● Drift (μ): Calculated as the average historical return of the asset, providing insight into its general price trend
● Volatility (σ): Derived from the standard deviation of historical returns, indicating the degree of expected price fluctuation
● Random Number Generation: Standardized normal random variables (mean 0, standard deviation = 1) are produced using robust methods (e.g., Box- Muller or inverse transform techniques) These simulate unpredictable market
● Brownian Path Construction: I construct a Brownian motion path by integrating the random shocks This path characterizes a drifting random walk, where the shocks dictate movement magnitude and direction
Simulating Asset Price Paths with GBM Enhancement
● GBM Formula: The classic GBM formula models potential price paths:
Gradient Boosting Integration enhances the modeling of asset price behavior by addressing the limitations of traditional GBM assumptions, which may oversimplify real-world market dynamics By training GBM models on historical data, we can predict drift and volatility parameters more accurately, resulting in improved simulations for financial markets.
Evaluation method
In the second stage, models undergo rigorous out-of-sample validation utilizing a rolling window technique that incorporates the business cycle phases of the Vietnamese economy and stock market for practical relevance This validation process employs data reflective of these phases, simulating real-world investment scenarios Expected returns for each model are adjusted based on the identified business cycle phase, capturing the dynamic nature of investor expectations The models are efficiently solved with the CPLEX solver, and their performance is assessed using key metrics, including Mean Return, Calmar Ratio, Sharpe Ratio, and Population Stability Index, providing a comprehensive evaluation of model performance.
The out-of-sample Mean Return (MR) indicates the average return an investor can anticipate from a portfolio based on data that was not used during its creation.
, where S is the number of scenarios s, and 𝑟 𝑠 is the return in scenario s
Calmar Ratio (CR) and Sharpe Ratio (SR) These ratios offer insights into risk- adjusted performance
Calmar ratio (CR) introduced by Terry W Young (1991), the CR reflects risk- adjusted performance It is calculated by dividing the average annual excess return
The Calmar ratio evaluates the return above a benchmark over a specified time frame, factoring in the maximum drawdown during that period, thereby measuring the return earned per unit of risk This ratio is closely related to the Sterling Ratio, with the main difference being that Calmar is calculated monthly while Sterling is calculated yearly In contrast, the Sharpe Ratio assesses the return per unit of volatility, emphasizing the additional excess return an investor gains for assuming more risk, and it operates under the assumption of a normal return distribution, favoring portfolios with higher expected returns in relation to their standard deviation.
𝑀𝐷𝐷 𝑜𝑣𝑒𝑟 [0, 𝑇] , where the Maximum Drawdown (MDD) is calculated as:
The Sharp Ratio define as:
- 𝑅 𝑝 : The multi-period return over period T,
- 𝑅 𝑓 : The base currency risk-free return over period T,
- 𝜎(𝑅 𝑝 ): The volatility of the portfolio return over period T
The Population Stability Index (PSI) emphasizes the importance of process stability, distinguishing it from traditional capability indices that primarily focus on performance Stability is defined as the lack of "special causes" that lead to unpredictable variations in a process over time While control charts are frequently used to visually assess stability, the PSI provides a quantitative measurement, offering a more precise evaluation of stability in processes.
● PSI < 0.1: Insignificant change— the optimization model is stable!
● 0.1 = 0.25: Significant change—the optimization model is unstable and needs an update
The SI is defined as:
- Let N be the total number of unique tickers in the dataset,
- Let T be the total number of iterations,
- Let 𝑤 𝑖,𝑗 be the Weight of ticker j at iteration i,
- ∑ 𝑁 𝑖=1 𝑤 𝑖,𝑗 : Sum over this unique ticker,
- ∑ 𝑇−1 𝑡=1 𝑤 𝑖,𝑗 : Sum over all iterations (from 1 to T − 1),
- Δ𝑤 𝑖,𝑗 : The absolute change in Weight for ticker j from iteration i to iteration i + 1
1.6.2 Evaluation time frame (business cycle)
To enhance the practical relevance of my research and connect theoretical models with real-world applications, I systematically integrate business cycle considerations By analyzing data from credible third-party sources, I identify the specific phases of the Vietnamese economy and stock market cycles.
The out-of-sample validation process utilizes data from identified phases that closely resemble real-world investment scenarios, highlighting how portfolio decisions are influenced by the current economic climate.
Figure 1-5: Economic Growth on Vietnam (General Statistic Office of Vietnam,
Economic expansion is the ideal phase characterized by consistent business growth, increased production and profits, low unemployment rates, and robust stock market performance During this period, consumer spending and investments rise, boosting demand for goods and services, which can lead to higher prices Key indicators of economic expansion include a GDP growth rate of 2% to 3%, inflation around the 2% target, unemployment rates between 3.5% and 4.5%, and a bull market in stocks.
The peak signifies the conclusion of economic expansion, typically arising from unsustainable growth driven by excessive business expansion, investor overconfidence, and inflated asset prices disconnected from fundamentals, leading to asset bubbles As production and prices hit their limits, the economy overheats, resulting in an impending downturn This peak serves as a crucial turning point before the onset of economic contraction.
Contraction spans the period from the peak to the trough and reflects declining economic activity It is characterized by rising unemployment, weak stock market
During a bear market, performance declines and GDP growth falls below 2%, prompting businesses to reduce operations A recession is typically defined by two consecutive quarters of GDP decline, yet recovery can remain sluggish even after the recession officially concludes.
The trough signifies the lowest point in the economic cycle, marking the shift from contraction to expansion Recovery from this stage is typically slow and inconsistent In the Vietnamese market, the latest trough was observed in August 2023.
Types of phases Corresponding period
I carefully modify the expected return assumptions for each model according to the current phase of the business cycle, acknowledging the changing nature of investor expectations and the potential for market fluctuations in various economic conditions.
This model aims to minimize risk while achieving targeted expected returns, evaluated at levels of 5%, 10%, 15%, 20%, 25%, and 30% Analyzing the model across these varying return targets offers valuable insights into its ability to balance risk and return under diverse conditions The results will help assess the model's adaptability and efficiency in optimizing investment portfolios for different types of investors.
Investors can choose different expected return targets based on their risk preferences; conservative investors may opt for lower returns to minimize risk, while aggressive investors might seek higher returns despite increased risk This framework facilitates a comprehensive understanding of the risk-return trade-offs inherent in the model Additionally, evaluating outcomes against various benchmarks demonstrates the model's robustness and reveals its optimal and suboptimal performance scenarios Such analysis not only confirms the model's effectiveness but also provides essential insights for practical portfolio management, ensuring it caters to the diverse needs of investors while aligning with modern portfolio theory principles.
Result
Model performance
The table compares four optimization models—Base, ST, LBUB, and Full—focusing on their structural components and computational performance Each model is assessed based on the number of binary variables, continuous variables, linear constraints, and runtime in seconds.
The Base model, featuring 134 binary variables, 34 continuous variables, and 104 linear constraints, offers the fastest runtime of 0.00049 seconds due to its minimal complexity, establishing it as a foundational benchmark for the optimization problem In contrast, the ST model retains the same number of binary and continuous variables as the Base model but significantly increases the number of linear constraints.
The model's complexity has increased to 30 constraints, totaling 355, which results in a longer runtime of 0.00166 seconds This rise in constraints indicates that the model includes more structural elements, potentially improving its real-world applicability, albeit at the expense of computational efficiency.
The LBUB model, similar to the Base and ST models, incorporates 134 binary variables and 34 continuous variables, while utilizing 176 linear constraints With a runtime of just 0.00082 seconds, it strikes a balance between increased complexity and computational efficiency, offering an intermediate level of sophistication.
The Full model is the most comprehensive version, featuring 135 binary variables, 35 continuous variables, and 440 linear constraints Consequently, it has the longest runtime of 0.00206 seconds, reflecting its complexity The inclusion of these additional variables and constraints enhances the model's detail and robustness in tackling the optimization problem.
The evaluation of diversification constraint's impacts on the model's
After thorough data processing and model development, I achieved results based on specific metrics for comparing models These metrics were assessed following meticulous integration and weight adjustments for various influencing factors.
Figure 2-1: Mean return by model
All models indicate an average positive return, ranging from a median of 0.083 to 0.107 Notably, LBUB presents the possibility of much higher returns, with an upper whisker of 0.710, but it also entails the risk of more substantial losses, as reflected in its lower whisker of -0.213.
Base and ST offer a good balance (upper whiskers exceeding 0.64) with a moderate range of returns (IQR around 0.29)
The Full model offers a lower risk option, demonstrating a narrow interquartile range (IQR) of 0.20 around its median return of 0.083 However, it presents limited opportunities for exceptionally high returns, with an upper whisker of 0.496.
From a risk-reward perspective, Base emerges as the leader with its median Sharpe Ratio of 0.355, signifying it generates the highest excess return (risk premium) per unit of volatility
Figure 2-2: Sharp Ratio by model
While Base and ST offer significant advantages, they also carry a risk of considerable negative performance, as shown by their upper whisker values exceeding 2 In contrast, risk-averse investors looking for capital protection may prefer Full and LBUB.
While their median Sharpe Ratios are lower (around 0.25), the upper whisker values hover below 2, suggesting a lower chance of experiencing significant losses
Figure 2-3: Calmar Ratio by model
The Calmar Ratio provides valuable insights into risk-adjusted returns, particularly for risk-averse investors focused on capital protection With upper hinges of 1.738 for Full and 1.559 for LBUB, these options present appealing opportunities, as they offer the potential for modest positive excess returns that surpass benchmark performance while minimizing the risk of maximum drawdowns.
Investors aiming for a favorable 2:1 return-to-risk ratio may discover opportunities in Base (upper whisker: 3.891), ST (upper whisker: 3.962), and LBUB (upper whisker: 3.729), as these assets have the potential to yield returns that significantly exceed their maximum drawdowns.
Aggressive investors with a high risk tolerance may find the ST model appealing due to its potential for significant upside, as indicated by an upper whisker of 3.962 However, it is crucial for these investors to recognize that all models carry the risk of substantial drawdowns, which should be carefully evaluated before making investment decisions.
Figure 2-4: Stability Index by model
All portfolio optimization models, including Full, LBUB, Base, and ST, demonstrate stability with a median PSI of less than 0.1, indicating statistically insignificant performance changes from the training data and confirming their ongoing effectiveness.
Risk-tolerant investors should monitor Full, LBUB, and ST, as their upper whisker values around 0.58 suggest a potential for model instability in real-world markets The Base model presents a slightly higher upper whisker of 0.622, indicating an even greater likelihood of instability.
Risk-averse investors focused on capital preservation can take solace in the low PSI values observed across all models, with a lower whisker of 0 and a lower hinge ranging from 0.016 to 0.021, indicating minimal downside risk.
The meaning of Business Cycle Models for Portfolio Optimization
This section explores the performance of four essential metrics—Mean Return (MR), Sharpe Ratio (SR), Calmar Ratio (CR), and Population Stability Index (PSI)—to highlight the significance of diversification constraints for investors and those constructing portfolio optimization models.
My analysis is enriched by highlighting statistically significant numerical evidence and extracting actionable insights for informed decision-making
2.3.1 Mean Return and the Trough Period
Our research group has created various portfolio optimization models, and understanding their performance across different economic cycles is essential for maximizing their utility This analysis explores the out-of-sample mean return potential of these models during contraction, expansion, peak, and trough phases, providing valuable insights for investors with varying risk profiles.
Figure 2-5: Mean return by business period
Let's delve deeper into the potential performance implications for each phase of the economic cycle:
● Overall Direction: My research group's models indicate modest potential for positive returns during contraction However, losses are still possible, as indicated by the median MR of 0.060
Risk-tolerant investors may find that certain higher-risk assets in model portfolios could outperform during economic contractions, as indicated by the upper whisker of 0.208 However, it is crucial to recognize the heightened risk of losses, reflected by the lower whisker of -0.059.
● Conservative Investors: Prioritizing stability remains critical in this phase Though potential losses remain, risk-averse investors should focus on assets offering reliable income or capital preservation
● Overall Direction: The expansion phase signals a challenging environment, with most models indicating a negative mean return (median MR -0.044)
Risk-tolerant investors may find that even high-risk asset classes face challenges in generating positive returns during economic expansion, as indicated by a modest upside potential of 0.193 This phase carries significant risks, highlighting the possibility of substantial losses, with a lower range of -0.256.
● Conservative Investors: Capital preservation should be the absolute priority during expansion Even conservative strategies may experience larger losses compared to contraction
● Overall Direction: Economic peaks offer the most significant return potential (median MR 0.264) This phase appears the most promising for my research group's models
● Risk-tolerant Investors: Growth-oriented assets may perform exceptionally Ill at the economic peak The potential for very high returns (upper whisker 0.768) corresponds with an increased risk profile
● Conservative Investors: A diversified approach combining stable assets with moderate-risk holdings could offer both growth potential and a degree of downside protection in this phase
● Overall Direction: Models predict positive returns during a trough, though generally smaller than during the peak (median MR 0.269)
● Risk-tolerant investors: Opportunities for solid gains exist during a trough (upper whisker 0.478) However, this phase still carries greater risk compared to the peak
● Conservative Investors: A balanced approach focused on steady growth with prudent risk mitigation may be advisable during a trough
● No Universal Model: My research confirms that the ideal portfolio changes with the economic cycle
● Understanding Timing: Accurately predicting economic phases is vital for success when utilizing these models
● Risk Tolerance is Key: Investors must carefully consider their risk tolerance as downside risk varies significantly across phases
Here are the potential performance implications of the Sharpe Ratio for each phase of the economic cycle:
Figure 2-6: Sharp ratio by business period
● Overall Direction: My research group's models indicate modest potential for positive risk-adjusted returns during contraction (median SR 0.148) However, losses remain possible (lower whisker -0.676)
Risk-tolerant investors may benefit from incorporating higher-risk assets into their model portfolios, as this strategy could enhance risk-adjusted performance, evidenced by an upper whisker of 1.061 However, it is essential to remain mindful of the increased potential for losses associated with such investments.
● Conservative Investors: Prioritizing stability during contraction is critical Risk-averse investors should focus on assets with lower volatility, even though the possibility of negative returns persists
● Overall Direction: The expansion phase signals a challenging environment, with most models indicating negative Sharpe Ratios (median SR -0.382) This implies that risk outweighs potential rewards
● Risk-tolerant Investors: Even riskier asset classes could face difficulties delivering positive risk-adjusted returns (upper whisker only 0.626) This phase presents substantial downside risk on a risk-adjusted basis
● Conservative Investors: Capital preservation should be the absolute priority during expansion, with minimal risk exposure advisable
Economic peaks present the greatest opportunity for strong risk-adjusted returns, evidenced by higher Sharpe Ratios, with a median of 0.722 This phase is particularly advantageous for the models developed by my research group.
● Risk-tolerant Investors: Growth-oriented assets may excel in a peak environment Higher risk might translate to very substantial returns on a risk- adjusted basis (upper whisker 2.066)
● Conservative Investors: A more balanced approach with moderate risk exposure could offer the potential for growth, while providing a degree of downside protection at a peak
● Overall Direction: Models predict positive Sharpe Ratios during a trough, although generally smaller magnitudes than during the peak (median SR 1.090) This phase presents favourable conditions for risk-adjusted returns
● Risk-tolerant investors: Opportunities for strong risk-adjusted returns exist (upper whisker 1.995) However, this phase still carries greater risk compared to the peak
● Conservative Investors: A balanced strategy focused on moderate growth with prudent risk management may be advisable during a trough
2.3.3 Calmar Ratio and Risk Management
Our research group's models reveal a notable variation in Calmar ratios (CR) throughout different phases of the economic cycle, underscoring differences in their capacity to produce risk-adjusted returns.
Figure 2-7: Calmar ratio by business period
During the contraction phase, the median capital return (CR) is notably low at 0.148, highlighting limited returns in comparison to potential drawdowns This cautious approach emphasizes the importance of capital preservation, utilizing models that demonstrate even modest positive returns.
During the expansion phase, the prevalent negative CR values, with a median of -0.382, highlight the significant challenges faced, as losses often outweigh gains To navigate this phase effectively, it is advisable to prioritize damage control by identifying models that exhibit the least negative CRs.
At the economic peak, significantly elevated capital ratios (median of 0.722) indicate robust returns that exceed potential drawdowns, presenting a prime opportunity to leverage favorable market conditions by focusing on models with high capital ratios.
The median CR of 1.090 during a trough indicates significant potential for appealing risk-adjusted returns, highlighting opportunities to 'buy low.' Models showing higher CRs in this phase suggest favorable conditions for investment, setting the stage for gains as the economy begins to recover.
Insights for Different Investor Profiles
○ The contraction phase, with its emphasis on capital preservation, offers suitable models with positive CRs—even if returns are moderate
○ For those willing to accept some volatility, models with higher CRs in peak and trough phases can offer the potential for enhanced returns
Balanced investors can benefit from peak and trough phases, which demonstrate capital returns (CRs) in the 2-3 range, effectively balancing rewards with moderate risk For those seeking a more cautious investment strategy, models based on contraction phases present a lower-risk alternative.
○ High-risk, high-reward opportunities lie in the peak and trough phases, where models often boast CRs exceeding 3 Investors embracing volatility can target these models for potentially significant returns
● Contraction: Prioritize models with positive Calmar ratios, ensuring any returns outweigh potential drawdowns, safeguarding capital during a downturn
● Expansion: Damage control is key Prioritize models with the least negative CRs to minimize losses during this challenging phase
● Peak: Seize maximum returns during this prosperous period by aiming for models with the highest CRs
● Trough: Investing strategically 'at the bottom' is enabled by models with higher CRs This positions investors to benefit from the subsequent upswing
2.3.4 Population Stability Index and Uncertainty Management
Figure 2-8: Stability Index by business period
The elevated PSI values, particularly the Upper Whisker at 0.506, indicate significant instability, revealing a marked change in the structure of underlying data such as asset prices and company valuations compared to the original training of the models.
● Unreliable Predictions: Models created during prior phases are likely to produce inaccurate forecasts or risk assessments during a contraction This could lead to misinformed investing and potentially severe losses
Despite a decrease in PSI values (Upper Whisker 0.304), model instability remains a concern, indicating that these models may struggle to accurately reflect the dynamic changes characteristic of an Expansion phase.
To achieve optimal performance, it is essential to refine and recalibrate models regularly This process may include updating them with the latest data and integrating economic indicators specifically designed for periods of expansion.
● Moderately Concerning: PSI values signal a risk of model instability, a major concern during a peak period characterized by heightened market volatility
Key remarks
This section summarizes the key findings derived from the previous three stages of analysis Each key observation has been carefully identified based on the data patterns
45 and results obtained from the optimization models and economic phase evaluations
To validate observed trends and gain deeper insights, I will perform hypothesis testing on each remark This statistical method will ascertain whether the identified patterns are significant or merely coincidental, thereby strengthening the foundation for decision-making in portfolio optimization The outcomes of this testing will enhance portfolio strategies and promote data-driven investment decisions.
# Remark 1 While all models predict positive returns on average (median between
LBUB presents the opportunity for significantly higher returns, with an upper whisker of 0.710, but also entails the risk of larger losses, indicated by a lower whisker of -0.213 In contrast, the Base and ST models provide a balanced approach, featuring upper whiskers above 0.64 and a moderate interquartile range (IQR) of approximately 0.29 For investors prioritizing lower risk, the Full model offers the tightest return range, with an IQR of 0.20 around a median of 0.083, although it has limited potential for exceptionally high returns, capped at an upper whisker of 0.496.
From a risk-reward perspective, Base stands out with a median Sharpe Ratio of 0.355, indicating it offers the highest excess return per unit of volatility However, this advantage is tempered by the potential for significant negative performance, as shown by its upper whisker values exceeding 2 In contrast, risk-averse investors may prefer Full and LBUB, which, despite having lower median Sharpe Ratios around 0.25, present upper whisker values below 2, suggesting a reduced likelihood of substantial losses.
Examining risk-adjusted returns through the Calmar Ratio provides valuable insights for different types of investors For those prioritizing capital protection, Full (1.738) and LBUB (1.559) are appealing choices, as they offer the potential for modest positive excess returns that outweigh possible maximum drawdowns Meanwhile, balanced investors aiming for a favorable 2:1 return-to-risk ratio may find Base (3.891) and ST (3.962) to be worthwhile options.
Aggressive investors with a high tolerance for volatility may find models like ST (upper whisker: 3.962) appealing due to their potential for significant returns, exceeding double their worst drawdown However, it is crucial for these investors to recognize that all models carry the inherent risk of substantial drawdowns, which should be carefully evaluated before making investment decisions.
All portfolio optimization models, including Full, LBUB, Base, and ST, demonstrate stability with a median PSI below 0.1, indicating statistically insignificant performance changes compared to training data For risk-tolerant investors seeking higher returns, monitoring is advised for Full, LBUB, and ST due to upper whisker values around 0.58, suggesting a potential for model instability in real-world markets The Base model shows a slightly higher upper whisker at 0.622, indicating a marginally greater instability risk In contrast, risk-averse investors can take comfort in the low PSI values across all models, with lower whisker values at 0 and lower hinge values between 0.016-0.021, signifying minimal downside risk.
Model performance varies significantly throughout the economic cycle, with contraction providing modest gains (median MR 0.060) but also posing loss risks (lower whisker -0.059) During expansion, returns are largely negative (median MR -0.044), impacting even high-risk assets (upper whisker 0.193) The peak phase presents the highest potential for returns (median MR 0.264, upper whisker 0.768), while troughs yield smaller but positive returns (median MR 0.269) Investors need to align their risk tolerance with these economic phases, as downside risks fluctuate considerably Adopting a dynamic model selection approach based on the current economic phase is essential for optimizing investment returns.
My analysis indicates that the Sharpe Ratio of portfolio models fluctuates significantly throughout the economic cycle During contraction, the median Sharpe Ratio is 0.148, reflecting modest gains, while the potential for upside reaches 1.061, alongside a risk of loss that dips to -0.676 In contrast, expansion phases are characterized by higher risk, predominantly showing negative Sharpe Ratios.
Peaks demonstrate the highest potential for risk-adjusted returns, with a median Sharpe Ratio (SR) of 0.722, while troughs also show favorable conditions with a median SR of 1.090 However, it is crucial to recognize the limitations of the Sharpe Ratio Investors should select models that align with their risk tolerance and the prevailing economic phase to achieve optimal investment outcomes.
My portfolio models demonstrate notable differences in risk-adjusted performance throughout the economic cycle During contraction phases, the median Calmar ratio is 0.148, suggesting modest returns with limited drawdowns, making them ideal for conservative investors Conversely, expansion phases show a median CR of -0.382, indicating that losses exceed gains The peak and trough phases present the most attractive Calmar ratios, with medians of 0.722 and 1.090, respectively, highlighting stronger returns relative to risk Investors seeking a balanced approach may prefer peak and trough models, while those with a higher risk tolerance could pursue the substantial potential gains associated with increased volatility in these phases It is essential to consider individual risk tolerance and diversify across models aligned with various economic conditions for optimal portfolio performance.
The Population Stability Index (PSI) analysis indicates significant instability in my models throughout the economic cycle, with extreme instability during Contraction (Upper Whisker: 0.506) suggesting unreliability based on past data Although there is some improvement in the Expansion phase (Upper Whisker: 0.304), adjustments are still necessary for optimal performance The Trough phase reveals the highest instability (Upper Whisker: 0.774), indicating a need for major model revisions None of the models achieve ideal stability (PSI