Feasible set of portfolios: Meaning, Criticisms & Real-World Uses

What Is a Feasible Set of Portfolios?

The feasible set of portfolios, also known as the attainable set or portfolio opportunity set, encompasses all possible portfolios that an investor can construct from a given collection of available financial assets. It represents every combination of assets and their respective weights that an investor could theoretically hold, subject only to the constraints of the available assets and the total investment capital. This concept is fundamental to Portfolio Theory, providing the universe from which investors select their optimal holdings.²⁶,²⁵,²⁴

Within this set, each unique portfolio is characterized by a specific expected return and a corresponding level of risk, typically measured by standard deviation. The feasible set of portfolios can be visualized graphically as a region in a risk-return plane, with every point within and on the boundary of this region representing a unique portfolio achievable by combining the given assets.²³ Understanding the feasible set is the first step in constructing an investment strategy aimed at balancing risk and return.

History and Origin

The concept of the feasible set of portfolios is inextricably linked to the development of Modern Portfolio Theory (MPT). This groundbreaking theory was introduced by economist Harry Markowitz in his seminal 1952 paper, "Portfolio Selection," published in The Journal of Finance.²², Markowitz's work revolutionized investment management by proposing a mathematical framework for assembling a portfolio of assets that maximizes expected return for a given level of risk, or minimizes risk for a given expected return.

Before MPT, investors often focused solely on the risk and return of individual securities. Markowitz demonstrated that an asset's risk and return should not be assessed in isolation, but rather by how it contributes to the overall risk and return of the entire portfolio. The feasible set emerged as a core element of this framework, providing the comprehensive landscape of all possible asset allocation combinations from which to derive optimal portfolios.²¹ His work, for which he later received the Nobel Memorial Prize in Economic Sciences in 1990, laid the foundation for quantitative portfolio analysis and the systematic approach to diversification that remains central to finance today.²⁰

Key Takeaways

The feasible set of portfolios represents all possible combinations of assets an investor can create, given a specific group of available securities.
Each portfolio within the feasible set has a unique risk-return profile.
This concept is foundational to Modern Portfolio Theory, which introduced a systematic way to analyze portfolio risk and return.
The feasible set provides the universe of investment opportunities from which an investor can identify the most desirable portfolios.
It highlights the importance of considering how individual assets interact within a portfolio, rather than in isolation, to manage overall risk.

Formula and Calculation

The calculation of a feasible set of portfolios involves determining the expected return and standard deviation for every possible combination of assets within a given universe. For a portfolio consisting of (n) assets, with each asset (i) having an expected return (R_i) and a weight (w_i) (representing the proportion of total capital invested in that asset), the portfolio's expected return ((E(R_p))) is:

E(R_p) = \sum_{i=1}^{n} w_i R_i

The portfolio's variance ((\sigma_p^2)), which is a measure of its risk, depends not only on the individual asset variances but also on the correlation between the assets. For a portfolio of two assets (Asset 1 and Asset 2), the variance is:

\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2

Where:

(w_1), (w_2) = weights of Asset 1 and Asset 2, respectively ((w_1 + w_2 = 1))
(\sigma_1^{2), (\sigma_2}2) = variances of Asset 1 and Asset 2
(\rho_{12}) = correlation coefficient between Asset 1 and Asset 2

For portfolios with more than two assets, the formula expands to account for the covariance between all pairs of assets:

\sigma_p^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \text{Cov}(R_i, R_j)

Where:

(\text{Cov}(R_i, R_j)) = covariance between the returns of asset (i) and asset (j). If (i=j), then (\text{Cov}(R_i, R_i) = \sigma_i^2) (the variance of asset (i)).

By iterating through all possible combinations of weights ((w_i)) that sum to 1, while considering any constraints like no short selling (i.e., (w_i \geq 0)), one can plot all achievable risk-return points, thereby mapping out the entire feasible set of portfolios.

Interpreting the Feasible Set of Portfolios

Interpreting the feasible set of portfolios involves understanding the range of investment outcomes available to an investor given a specific set of asset classes or individual securities. Each point within the boundaries of the feasible set represents a distinct investment portfolio with a unique combination of expected return and risk.

The shape of the feasible set reveals important insights about the diversification benefits among the assets. If assets have low or negative correlation, the feasible set will curve more sharply inward, demonstrating that combining such assets can significantly reduce overall portfolio risk without sacrificing much expected return. This curvature underscores a key tenet of Modern Portfolio Theory: diversifying across assets that do not move in perfect lockstep can lead to superior risk-adjusted returns.

Investors use the feasible set to identify the subset of portfolios that lie on the upper-left boundary, known as the Efficient Frontier. These are the "efficient" portfolios, offering the highest expected return for any given level of risk, or the lowest risk for any given expected return. A rational investor, exhibiting risk aversion, would aim to select a portfolio that lies on this frontier, tailoring it to their individual risk tolerance.

Hypothetical Example

Consider an investor, Alex, who has identified two potential assets for a portfolio: Stock A and Bond B.

Stock A: Expected Return = 10%, Standard Deviation = 15%
Bond B: Expected Return = 4%, Standard Deviation = 5%
Correlation between Stock A and Bond B ((\rho_{AB})): 0.30

Alex wants to explore the feasible set of portfolios by combining these two assets in various proportions (weights).

Let (w_A) be the weight in Stock A and (w_B) be the weight in Bond B, such that (w_A + w_B = 1).

Scenario 1: 100% Stock A, 0% Bond B

Expected Return = (1.00 \times 10% + 0 \times 4% = 10%)
Standard Deviation = (1.00 \times 15% = 15%)

Scenario 2: 50% Stock A, 50% Bond B

Expected Return = (0.50 \times 10% + 0.50 \times 4% = 5% + 2% = 7%)
Standard Deviation ((\sigma_p)) calculation:
- (\sigma_p^2 = (0.50)^2 (0.15)^2 + (0.50)^2 (0.05)^2 + 2 (0.50)(0.50)(0.30)(0.15)(0.05))
- (\sigma_p^2 = 0.25 \times 0.0225 + 0.25 \times 0.0025 + 0.50 \times 0.30 \times 0.0075)
- (\sigma_p^2 = 0.005625 + 0.000625 + 0.001125 = 0.007375)
- (\sigma_p = \sqrt{0.007375} \approx 0.08589), or approximately 8.59%

Scenario 3: 0% Stock A, 100% Bond B

Expected Return = (0 \times 10% + 1.00 \times 4% = 4%)
Standard Deviation = (1.00 \times 5% = 5%)

By calculating many such combinations (e.g., 10% increments: 10% Stock A, 90% Bond B; 20% Stock A, 80% Bond B, and so on), Alex could plot these risk-return points on a graph. The resulting curve connecting all these points would represent the feasible set for these two assets. The upper left portion of this curve would constitute the Efficient Frontier, showing the most efficient combinations of risk and return. This illustrates how the feasible set allows an investor to visualize all available portfolio options.

Practical Applications

The concept of the feasible set of portfolios has several practical applications in investment management and financial planning:

Portfolio Construction and Optimization: Financial advisors and asset managers use the feasible set as the foundational space for identifying optimal portfolios for their clients. By understanding all possible risk-return combinations, they can then apply Modern Portfolio Theory principles to select portfolios that align with an investor's risk aversion and financial goals. This involves navigating the trade-off between higher potential returns and increased market risk.¹⁹,
Strategic Asset Allocation: The feasible set helps in determining the optimal asset allocation across different asset classes (e.g., stocks, bonds, real estate). By analyzing the correlations between these classes, investors can construct diversified portfolios that achieve a desired level of return for a given risk appetite, as advocated by the U.S. Securities and Exchange Commission (SEC).¹⁸,¹⁷
Performance Evaluation: While not directly used for evaluation, the feasible set (and its efficient frontier) serves as a benchmark. Investment professionals can assess if a client's current portfolio is "efficient" by checking if it lies on or near the efficient frontier within the feasible set for the available assets.
Risk Management: By mapping out the feasible set, investors can clearly see the range of risks associated with different portfolio compositions. This visualization aids in understanding the impact of various asset weightings on overall portfolio risk and helps in making informed decisions about diversification to mitigate specific risks. As Reuters notes, building a profitable portfolio often involves considering how different assets perform under various market conditions.¹⁶,
Product Development: Financial product developers, such as those creating mutual funds or exchange-traded funds (ETFs), implicitly consider the feasible set. They aim to offer funds that provide investors with access to efficient portfolios or components thereof, allowing them to easily construct their own diversified holdings.¹⁵

Limitations and Criticisms

While the concept of the feasible set of portfolios is fundamental to Modern Portfolio Theory, it is built upon certain assumptions that have faced criticisms and reveal limitations in real-world application:

Reliance on Historical Data: The calculation of expected returns, variances, and correlation coefficients for constructing the feasible set typically relies on historical data. Critics argue that past performance is not necessarily indicative of future results, and market dynamics can change, rendering historical relationships less relevant.¹⁴,¹³ This means the feasible set derived from historical data may not accurately represent future investment opportunities.
Assumptions of Rational Investors and Efficient Markets: MPT, and by extension the feasible set, assumes that investors are rational, risk-averse, and make decisions solely based on expected return and standard deviation (risk). It also assumes markets are efficient, meaning all available information is immediately reflected in asset prices.¹² However, behavioral finance challenges these assumptions, demonstrating that investor behavior can be irrational, influenced by emotions and cognitive biases, and markets may not always be perfectly efficient.¹¹,¹⁰
Risk Measure Definition: MPT defines risk primarily as the standard deviation of returns, treating both positive and negative volatility as equally "risky." Many investors, however, view upside volatility (returns greater than expected) favorably and are primarily concerned with downside risk (losses). This narrow definition of risk can lead to portfolios that optimize for a metric that doesn't fully align with an investor's true concerns.⁹,⁸
Computational Complexity: As the number of assets in the investment universe increases, the number of possible portfolio combinations grows exponentially. Calculating the feasible set becomes computationally intensive, particularly when considering various constraints (e.g., limits on asset weights, transaction costs, taxes).
Lack of Real-World Constraints: The theoretical feasible set often ignores practical constraints faced by investors, such as liquidity needs, transaction costs, taxes, and specific investment mandates or ethical considerations. Including these factors makes the practical feasible set smaller and more complex to define.
Stability of Correlations: The benefits of diversification within the feasible set depend heavily on the stability of correlation between assets. During periods of market stress or crisis, asset correlations tend to increase (move towards 1), reducing the diversification benefits that the feasible set initially suggests. This is a significant challenge to the practical application of MPT.⁷

Despite these limitations, the feasible set remains a crucial conceptual tool in Portfolio Theory, providing a clear visual representation of investment possibilities and forming the basis for more advanced portfolio optimization techniques. As the CAIA Association notes, while MPT has limitations, many alternative methodologies also have drawbacks, highlighting the ongoing challenge of accurate data and assumptions in asset allocation.⁶

Feasible Set of Portfolios vs. Efficient Frontier

The terms "feasible set of portfolios" and "Efficient Frontier" are closely related but distinct concepts within Modern Portfolio Theory.

The feasible set of portfolios (or attainable set) refers to all possible portfolios that can be constructed from a given collection of available assets. It represents every single combination of weights for those assets, and each combination yields a specific expected return and risk level. When plotted on a graph with risk (standard deviation) on the x-axis and expected return on the y-axis, the feasible set forms a region or cloud of points, encompassing all achievable portfolios.⁵,⁴

The Efficient Frontier, on the other hand, is a subset of the feasible set. It is the upper-left boundary of the feasible set, representing the set of optimal portfolios. Specifically, for any given level of risk, an efficient portfolio on the frontier offers the highest possible expected return. Conversely, for any given expected return, an efficient portfolio offers the lowest possible risk.³ Rational, risk-averse investors will always seek to hold a portfolio that lies on the Efficient Frontier, as any portfolio below the frontier is suboptimal (either offering less return for the same risk, or more risk for the same return).²

In essence, the feasible set is the entire universe of what can be done, while the Efficient Frontier highlights what should be done by an investor aiming to maximize return for a given risk level.

FAQs

What does "feasible" mean in this context?

In the context of the feasible set of portfolios, "feasible" means that a portfolio can actually be constructed given the available assets and any constraints. It implies that the asset combinations are mathematically possible and allowed within the investment universe.¹

How does diversification relate to the feasible set?

Diversification is the core principle that shapes the feasible set. By combining assets that are not perfectly positively correlated, an investor can achieve portfolios within the feasible set that offer a better risk-return trade-off than individual assets. This creates the curved shape of the feasible set and defines the Efficient Frontier, allowing for risk reduction without necessarily sacrificing return.

Can the feasible set change over time?

Yes, the feasible set of portfolios is dynamic. It can change if the universe of available assets changes, if the expected return or risk characteristics of individual assets change, or if the correlation between assets shifts. Market conditions, economic outlooks, and new investment opportunities constantly influence these parameters, thereby altering the shape and boundaries of the feasible set.