Multi-period Mean-Variance Interval Portfolio Selection Model with Cardinality Constraints

Investing is often imagined as a game of prediction, where the goal is to guess which stocks will rise and which will fall. For decades, the standard way to approach this has been to calculate an average expected return and a measure of how much that return might swing up or down. This method assumes that investors can pin down these numbers with a single, precise figure. But in the real world, the future is rarely that clear. Information is often incomplete, market conditions shift unexpectedly, and human emotions play a massive role in how people interpret data. When an investor looks at a stock, they might see a range of possible outcomes rather than a single point, and their own mood—whether they feel optimistic or pessimistic about the economy—can color how they weigh those possibilities.

This uncertainty is the central challenge tackled in a new study by Shili Dang, a researcher at the Guangdong University of Finance and Economics. The paper explores how to build a better investment plan when you cannot be sure of the exact numbers. Instead of relying on a single guess for how much a stock will earn, the study uses a range of values to represent the uncertainty. It also introduces a way to measure how an investor's feelings about the market affect their decisions. By combining these ideas with the practical realities of trading costs and the limits on how many different stocks a person can realistically manage, the research offers a fresh look at how wealth can grow over time. The goal is not just to find the mathematically perfect portfolio, but to find one that works for a human being navigating a messy, uncertain world.

The core of this work is a model that treats investment returns not as fixed points, but as intervals. Imagine a forecast that says a stock will return between 5 percent and 7 percent, rather than a flat 6 percent. This range acknowledges that the true outcome is unknown. The model then adds a "sentiment coefficient," a number that shifts the focus within that range based on how the investor feels. If an investor is extremely optimistic, the model leans toward the higher end of the range; if they are pessimistic, it leans toward the lower end. This allows the plan to adapt to the investor's psychological state, recognizing that two people looking at the same data might make different choices based on their outlook.

To make the model realistic, the researchers included several constraints that real investors face. They accounted for transaction costs, which are the fees paid every time a trade is made. They also considered borrowing limits and the fact that investors often cannot put all their money into a single asset. Perhaps most importantly, they included a rule on the number of assets held in the portfolio. This is known as a cardinality constraint. In practice, holding too many different stocks makes a portfolio difficult to manage and expensive to rebalance, while holding too few exposes the investor to unnecessary risk. The study sought to find the sweet spot where diversification is maximized without incurring excessive costs.

The researchers tested their model using data from thirty major Chinese stocks over a five-year period, simulating how a portfolio would perform across five distinct time stages. They ran the simulation with different limits on the number of stocks allowed in the portfolio, testing scenarios where an investor held anywhere from two to eight different assets. The results revealed a clear pattern: adding more stocks helped increase the final wealth, but only up to a point. When the portfolio contained five stocks, the final wealth reached its peak. Adding a sixth, seventh, or eighth stock did not improve the outcome; in fact, it slightly reduced the final wealth or left it unchanged. This suggests that there is a limit to the benefits of diversification. Once a portfolio has enough variety to spread out risk, adding more assets only introduces extra trading costs and complexity without generating additional profit.

The study also examined how the investor's sentiment influenced the growth of their wealth over time. When the sentiment coefficient was set to zero, representing a completely pessimistic outlook, the final wealth was lower than when it was set to one, representing extreme optimism. What was particularly striking was how this difference changed as time went on. In the first stage of the investment, the gap between the most optimistic and most pessimistic scenarios was relatively small. However, as the investment horizon extended through the five stages, this gap widened significantly. By the final stage, the difference in wealth between the optimistic and pessimistic scenarios had grown to nearly ten percent. This indicates that an investor's mindset does not just affect a single decision; its impact compounds over time, becoming a more powerful force as the investment period lengthens.

To understand the speed of this growth, the researchers analyzed the data to see if the wealth accumulation followed a predictable pattern. They found that the growth was approximately exponential, meaning the wealth grew at an accelerating rate rather than a steady, linear pace. This pattern held true regardless of whether the investor was optimistic or pessimistic, though the rate of growth was faster for those with a positive outlook. The study suggests that while the specific numbers depend on the market conditions and the investor's feelings, the underlying mechanism of wealth accumulation in this model is robust. The findings provide a quantitative way to see how a moderate level of diversification and a clear understanding of one's own market outlook can shape financial outcomes.

The implications of these findings are practical for anyone managing a portfolio. The research suggests that trying to own a vast number of different stocks is not always the best strategy. Instead, focusing on a moderate number of assets—around five in the context of this specific study—can balance the benefits of diversification against the costs of trading. Furthermore, the study highlights that the emotional component of investing is not merely a distraction but a variable that actively shapes the final result. An investor's expectations about the future, when combined with a disciplined approach to managing the number of assets, can lead to significantly different outcomes over a multi-year period. The model offers a framework for making these decisions more transparent, turning vague feelings about the market into a structured part of the planning process.

Ultimately, the study does not claim to predict the future or guarantee profits. It is a simulation based on historical data and specific assumptions about how markets behave. The researchers acknowledge that their data covers a specific decade and that real-world markets can change in unpredictable ways. However, the internal logic of the model holds up under scrutiny. The marginal gains from adding more stocks diminish quickly, and the influence of sentiment grows steadily over time. These are not just theoretical observations but patterns that emerged clearly from the data. By using ranges instead of single numbers and by accounting for the human element of sentiment, the study provides a more nuanced tool for navigating the complexities of long-term investing. It reminds us that in finance, as in many other fields, the path to a goal is often less about finding a single perfect answer and more about understanding the range of possibilities and the forces that shape them.