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Feasible set

What Is Feasible Set?

The feasible set, in the context of portfolio theory, represents all possible combinations of assets that an investor can choose from, given a defined set of constraints and investment opportunities. It encompasses every portfolio that an investor could theoretically construct based on their available capital and specific investment rules. This concept is fundamental to portfolio optimization within the broader field of portfolio theory, providing the universe of choices from which an optimal portfolio can be selected. The boundaries of the feasible set are determined by various factors, including the types of assets available, the investor's total capital, and any regulatory or personal financial constraints.

History and Origin

The concept of the feasible set is intrinsically linked to the development of modern portfolio theory (MPT), pioneered by Harry Markowitz. In his groundbreaking 1952 paper, "Portfolio Selection," Markowitz introduced a mathematical framework that allowed investors to quantify and manage the trade-off between risk and expected return10. Before Markowitz, investment decisions often relied on rules of thumb rather than a scientific approach. His work marked a "Big Bang Theory in finance," laying the foundation for almost all subsequent portfolio research9.

Markowitz's framework implicitly defined the feasible set as the collection of all portfolios whose weights sum to one (representing the total investment capital) and satisfy any other specified constraints, such as non-negativity of weights (no short selling). This theoretical construct allowed for the visual representation of potential portfolios in a risk-return space, making it possible to identify which portfolios offered the best risk-adjusted returns.

Key Takeaways

  • The feasible set comprises all possible portfolios an investor can construct given their available assets and specific constraints.
  • It serves as the foundation for portfolio optimization, defining the universe of choices.
  • Constraints, such as budget limitations, regulatory rules, and investment policies, define the boundaries of the feasible set.
  • Within the feasible set, investors aim to identify portfolios that offer the highest expected return for a given level of risk or the lowest risk for a given expected return.

Formula and Calculation

The feasible set is not defined by a single formula but rather by a system of inequalities and equalities that represent the constraints on portfolio weights. For a portfolio consisting of (n) assets, where (w_i) is the weight (proportion) invested in asset (i), the primary constraints typically include:

  1. Budget Constraint: The sum of all portfolio weights must equal 1 (or the total capital invested). ∑i=1nwi=1\sum_{i=1}^{n} w_i = 1
  2. Non-Negativity Constraint (No Short Selling): Unless short selling is allowed, the weight of each asset must be non-negative. wi≥0for all i=1,…,nw_i \ge 0 \quad \text{for all } i = 1, \dots, n
  3. Holding Constraints: Specific limits on individual asset weights or asset classes. For example, a maximum allocation to a single stock or industry. Li≤wi≤Uifor all i=1,…,nL_i \le w_i \le U_i \quad \text{for all } i = 1, \dots, n where (L_i) is the lower bound and (U_i) is the upper bound for asset (i).
  4. Leverage Constraints: Limits on borrowing to finance investments, which can affect the sum of weights. ∑i=1n∣wi∣≤Leverage Limit\sum_{i=1}^{n} |w_i| \le \text{Leverage Limit}

The feasible set, therefore, is the geometric region in an (n)-dimensional space (or a 2-dimensional risk-return space after mean-variance analysis) that satisfies all these conditions simultaneously. It is typically a convex set, meaning that any combination of two portfolios within the set will also result in a portfolio that is part of the set.

Interpreting the Feasible Set

Interpreting the feasible set involves understanding the full range of investment possibilities available to an investor under specific conditions. In a two-dimensional graph plotting expected return against standard deviation (as a proxy for risk), the feasible set appears as a region or shape, often curved, representing all achievable risk-return combinations. Every point within or on the boundary of this region corresponds to a valid portfolio.

For example, a point near the bottom-left of the feasible set would represent a portfolio with low expected return and low risk, while a point towards the top-right would indicate higher expected return and higher risk. The boundaries of the feasible set are defined by the most extreme combinations of assets that satisfy the constraints. An investor's risk tolerance and return objectives guide their choice of an optimal portfolio from within this set. Understanding the shape and extent of the feasible set helps investors visualize the trade-offs involved in their investment decisions.

Hypothetical Example

Consider an investor with $10,000 to allocate between two assets: Stock A and Stock B.

  • Stock A has an expected return of 8% and a standard deviation of 15%.
  • Stock B has an expected return of 12% and a standard deviation of 25%.
  • The correlation between Stock A and Stock B is 0.30.

The investor's constraints are:

  1. Budget: The entire $10,000 must be invested (sum of weights = 1).
  2. No Short Selling: Weights must be non-negative.

To define the feasible set, we can consider various allocation percentages (weights) for Stock A and Stock B, such as:

  • 100% Stock A, 0% Stock B
  • 0% Stock A, 100% Stock B
  • 50% Stock A, 50% Stock B
  • Any combination where $w_A + w_B = 1$ and $w_A, w_B \ge 0$.

For each combination, the portfolio's expected return (E_p) and standard deviation (\sigma_p) can be calculated:

Ep=wAEA+wBEBE_p = w_A E_A + w_B E_B σp=wA2σA2+wB2σB2+2wAwBσAσBρAB\sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B \rho_{AB}}

where (\rho_{AB}) is the correlation coefficient.

By calculating (E_p) and (\sigma_p) for all possible weights (e.g., from 0% to 100% in 1% increments), one can plot these points on a graph. The resulting curve or region represents the feasible set of portfolios. For two assets, this typically forms a curve in the risk-return space. If more assets are introduced, the feasible set becomes a multi-dimensional region, and its projection onto a two-dimensional risk-return plane forms a larger, more complex shape, reflecting the benefits of diversification.

Practical Applications

The concept of the feasible set is central to real-world portfolio management and asset allocation. Financial professionals utilize it to:

  • Portfolio Construction: Identify all viable investment portfolios that meet a client's specific requirements, such as minimum return targets, maximum risk limits, or liquidity constraints.
  • Risk Management: Understand the full spectrum of risk-return trade-offs available, allowing for informed decisions about how much risk to take for a desired level of return.
  • Regulatory Compliance: Ensure that investment strategies adhere to regulatory guidelines (e.g., limits on certain asset classes for institutional investors) or self-imposed policies like those concerning short selling or maximum holdings in a single security8. For instance, pension funds often face stringent rules on permissible investments, which effectively limit their feasible set.
  • Strategic Asset Allocation: Guide long-term investment planning by mapping out the universe of possible outcomes under various market conditions and investor objectives. This allows for a robust selection of an optimal portfolio that balances growth potential with stability.
  • Addressing Economic Conditions: Recognize how broader economic factors and financial constraints can influence the investment behavior of individuals and firms, thereby impacting the practical boundaries of their feasible investment choices7.

Limitations and Criticisms

While the feasible set is a foundational concept in finance, its practical application, particularly in the context of traditional modern portfolio theory (MPT), faces several limitations:

  • Reliance on Historical Data: The calculation of expected returns, variances, and correlations to define the feasible set heavily relies on historical data, which may not accurately predict future market behavior6. This can lead to a "garbage in, garbage out" problem, producing suboptimal portfolios5.
  • Assumptions of Normality and Rationality: MPT assumes that asset returns are normally distributed and that investors are rational and seek to maximize their utility function. However, real financial markets exhibit "fat tails" (more extreme events than a normal distribution predicts), and investors often display behavioral biases like overconfidence or loss aversion4.
  • Static Covariance Matrix: MPT typically uses a static covariance matrix, implying constant relationships between assets. In reality, these relationships are dynamic and can change rapidly, especially during market stress, potentially undermining the benefits of diversification3.
  • Real-World Constraints: While the feasible set is defined by constraints, some practical limitations such as transaction costs, taxes, and liquidity issues are often simplified or overlooked in basic MPT models, complicating real-world implementation2. For example, minimum order sizes or difficulty in quickly selling large positions can restrict actual portfolio choices1.

These criticisms highlight the need for investors and portfolio managers to temper theoretical models with practical considerations and a dynamic understanding of market conditions.

Feasible Set vs. Efficient Frontier

The terms "feasible set" and "efficient frontier" are closely related but distinct concepts in portfolio theory. The feasible set represents all possible portfolios that an investor can construct given their available assets and defined constraints. It is the entire region or space of achievable risk-return combinations.

In contrast, the efficient frontier is a subset of the feasible set. It consists only of the optimal portfolios within the feasible set. Specifically, for any given level of risk, an efficient portfolio offers the highest possible expected return, and for any given expected return, an efficient portfolio offers the lowest possible risk. Graphically, the efficient frontier forms the upper-left boundary of the feasible set. Investors typically aim to select a portfolio from the efficient frontier, as any portfolio below the efficient frontier is considered suboptimal—meaning a better portfolio (either higher return for the same risk, or lower risk for the same return) exists within the feasible set.

FAQs

What defines the boundaries of the feasible set?

The boundaries of the feasible set are defined by the various constraints placed on an investment portfolio. These can include the total capital available, regulatory restrictions, limitations on short selling, specific limits on asset weights, and any other rules that dictate how funds can be allocated.

Why is the feasible set important in investment?

The feasible set is important because it illustrates the entire universe of possible investment decisions an investor can make. By understanding this range, investors and portfolio managers can systematically identify and select portfolios that best align with their risk tolerance and return objectives, leading to more informed portfolio optimization strategies.

Can the feasible set change over time?

Yes, the feasible set can change over time. This can happen due to shifts in market conditions, such as new assets becoming available or existing assets changing their risk and return characteristics. Additionally, changes in regulatory environments or an investor's personal financial constraints can alter the boundaries of the feasible set.