Portfolio Construction

Portfolio construction can be perceived as an optimization problem, consisting of an objective function and a set of constraints. The objective function outlines the desired goal, such as maximizing a risk-adjusted return. Constraints, on the other hand, limit the actions that can be taken to achieve this goal. These may include restrictions on geographic, sector, industry, and single-security exposures, transaction costs, and exposure to specific factors. The nature of the objective function and constraints can reveal an investment manager’s philosophy and style.

Objective Functions

If risk is measured by predicted active risk, the objective function aims to maximize the information ratio, which is the ratio of active return to active risk. If risk is measured by predicted portfolio volatility, the objective function aims to maximize the Sharpe ratio, which is the ratio of return in excess of the risk-free rate to portfolio volatility. Ideally, these objective functions would specify net returns, adjusted for the costs associated with implementation.

Investment Objectives : Absolute and Relative

Investment objectives and constraints are pivotal in shaping the investment strategy. They can be expressed in absolute terms or relative to a benchmark, each having its unique constraints. For instance, consider a portfolio manager at a large investment firm like BlackRock. In an absolute approach, the manager might limit any single security position to no more than 2% of the portfolio and any single sector to no more than 20% of the portfolio. Conversely, in a relative approach, the manager might impose a constraint that a security must remain within \(\pm\)2% of its index weight and sector weights must remain within \(\pm\)10% of the index weights.

These approaches also impose other constraints:

• The absolute approach seeks to maximize the Sharpe ratio; the relative approach seeks to maximize the information ratio.
• The absolute approach might impose a portfolio volatility limit equal to 90% of the estimated benchmark volatility and a minimum weighted average security capitalization of $20 billion.
• The absolute approach limits any single security position to no more than 2% of the portfolio and any single sector to no more than 20% of the portfolio; the relative approach imposes a constraint that a security must remain within \(\pm\)2% of its index weight and sector weights must remain within \(\pm\)10% of the index weights.
• The absolute approach imposes a portfolio volatility limit equal to 90% of the estimated benchmark volatility and imposes a minimum weighted aver age security capitalization of $20 billion; The relative approach might impose a 5% active risk limit and the same capitalization constraint.
• It’s also possible to combine relative and absolute constraints in the same framework, such as limiting sector deviations against a benchmark while imposing absolute limits on security positions.

$$ \begin{array}{l|l|l|l} \textbf{Constraint} & {\textbf{Absolute} \\ \textbf{Framework}} & {\textbf{Relative} \\ \textbf{Framework}} & \textbf{Details} \\ \hline { \text{Individual} \\ \text{security} \\ \text{weights (w)} } & {w_i \leq 2\%} & {|w_{ip} – w_{ib}| \leq 2\%} & { \text{Limits overexposure to any} \\ \text{single security, promoting} \\ \text{diversification and} \\ \text{reducing idiosyncratic risk.}} \\ \hline {\text{Sectors} \\ \text{weights (S)}} & { S_i \leq 20\% } & {|S_{ip} – S_{ib}| \leq 10\%} &{ \text{Aims to manage systemic} \\ \text{risk by avoiding} \\ \text{overconcentration in any} \\ \text{particular sector of the} \\ \text{market.}} \\ \hline { \text{Portfolio} \\ {\text{volatility } (\sigma)} } & {\sigma_p < 0.9 \sigma_b}& — & {\text{Seeks to maintain a} \\ \text{portfolio volatility lower} \\ \text{than a benchmark to} \\ \text{control for overall} \\ \text{investment risk.}} \\ \hline {\text{Active risk} \\ \text{(TE)} } & — & {TE \leq 5\%} & {\text{Active risk is capped to} \\ \text{ensure the portfolio does} \\ \text{not deviate too much from} \\ \text{the benchmark.}} \\ \hline {\text{Weighted} \\ \text{average} \\ \text{capitalization} \\ \text{(Z)}} & {Z \geq 20bn} & {Z \geq 20bn} & { \text{Ensures the portfolio is} \\ \text{weighted towards} \\ \text{companies with a large} \\ \text{market capitalization,} \\ \text{indicating stability and} \\ \text{lower volatility.} } \end{array} $$

Optimization Approaches

Other optimization approaches focus on risk metrics, such as portfolio volatility, downside risk, maximum diversification, and drawdowns. These approaches do not explicitly integrate an expected return component but implicitly create an exposure to risk factors. For example, a portfolio built using a risk-based objective function might exhibit a Market beta below 1.0 and have a significant exposure to the Value factor and to the low-minus-high-\(\beta\) factor. This happens because an objective function that seeks to manage or minimize risk will tend to favor value and low-beta securities.

Not all objective functions are explicitly concerned with risk or returns. For instance, a quantitative manager might aim to maximize exposure to rewarded factors. This is represented by the equation:

$$ \text{MAX} \sum_{i=1}^{N} \frac{1}{3} \text{Size}_i + \frac{1}{3} \text{Value}_i + \frac{1}{3} \text{Momentum}_i$$

Where Size, Value, and Momentum are standardized proxy measures of Size, Value, and Momentum for security i. This implies that we have information about expected returns and expected risk. Some managers—typically discretionary managers—do not make explicit return and risk forecasts and instead seek to “maximize” their exposure to securities having specific characteristics. Embedded in their investment process is an implicit return-to-risk objective.

Discretionary Portfolio Management and Construction

• Discretionary portfolio managers typically operate under a specific investment strategy, often outlined in a mission statement. For example, a manager might specialize in large-cap US equity, focusing on deep value investment with a concentrated portfolio. This implies that they will target securities that exhibit strong value characteristics, according to their specific definition of value.

• The process of portfolio construction involves a delicate balance between security concentration and sector exposure, with the aim of maximizing returns while maintaining an acceptable risk level. The allocation process can be influenced by the manager’s judgment on risk-return trade-offs, a formal risk management protocol, or a feedback mechanism to ensure adherence to constraints during portfolio assembly or rebalancing.

• In cases where an explicit objective function is not utilized, heuristic methodologies can be employed to determine security weighting in a portfolio. These methodologies involve identifying securities with desired characteristics and weighting them based on their scoring or ranking on these characteristics. For instance, a security with a price-to-book ratio of 8 would have half the weight of a security with a price-to-book ratio of 4.

• Another approach involves identifying stocks with desired characteristics, ranking them based on their adherence to these characteristics, selecting the top x% of these stocks, and assigning them portfolio weights based on methodologies such as equal weight, equal risk, scoring, or ranking.

• However, these alternative methodologies may not allocate active risk as efficiently as a formal optimization framework. The constraints and objective function will reflect the philosophy and style of the manager. For example, a stock picker is likely to have fewer and more permissive constraints on security weights than a multi-factor manager seeking to minimize idiosyncratic risks.

Practice Questions

Question 1: An investment manager is constructing a portfolio and is considering the various measures and strategies to optimize the portfolio. The manager is considering the use of Active Share and active risk as part of their strategy. What does Active Share measure in relation to the manager’s investment strategy?

1. The standard deviation of the difference between the returns of the portfolio and the benchmark.
2. The percentage of portfolio holdings that differ from the benchmark index.
3. The ratio of return in excess of the risk-free rate to portfolio volatility.

Answer: Choice B is correct.

Active Share measures the percentage of portfolio holdings that differ from the benchmark index. It is a measure of the degree to which an investment manager’s portfolio differs from the benchmark index. Active Share is calculated by taking the absolute value of the differences in the portfolio weights and the benchmark weights, summing these differences, and dividing by two. A high Active Share indicates a high degree of deviation from the benchmark, suggesting a more active management style. This measure is useful for investors to understand how much of the portfolio’s performance can be attributed to the manager’s skill versus the performance of the benchmark index. It can also help investors assess the potential for outperformance or underperformance relative to the benchmark.

Choice A is incorrect. The standard deviation of the difference between the returns of the portfolio and the benchmark is a measure of active risk, not Active Share. Active risk, also known as tracking error, measures the volatility of the difference between the returns of the portfolio and the benchmark. It is a measure of the risk associated with active management and is used to assess the potential for the portfolio to deviate from the benchmark.

Choice C is incorrect. The ratio of return in excess of the risk-free rate to portfolio volatility is a measure of the Sharpe ratio, not Active Share. The Sharpe ratio is a measure of risk-adjusted return and is used to assess the performance of an investment considering its risk. It is not a measure of the degree to which a portfolio differs from its benchmark index.

Question 2: In investment portfolio management, different approaches can be used to set investment objectives and constraints. These approaches can be absolute, relative, or a combination of both. If a portfolio manager is using an absolute approach, they might limit any single security position to a certain percentage of the portfolio. Similarly, a relative approach might impose a constraint that a security must remain within a certain range of its index weight. Considering these approaches, which of the following statements is correct about the constraints imposed by the absolute and relative approaches?

1. The absolute approach limits any single security position to no more than 2% of the portfolio and any single sector to no more than 20% of the portfolio, while the relative approach imposes a constraint that a security must remain within \(\pm\)2% of its index weight and sector weights must remain within \(\pm\)10% of the index weights.
2. The absolute approach limits any single security position to no more than 5% of the portfolio and any single sector to no more than 25% of the portfolio, while the relative approach imposes a constraint that a security must remain within \(\pm\)5% of its index weight and sector weights must remain within \(\pm\)15% of the index weights.
3. The absolute approach limits any single security position to no more than 10% of the portfolio and any single sector to no more than 30% of the portfolio, while the relative approach imposes a constraint that a security must remain within \(\pm\)10% of its index weight and sector weights must remain within \(\pm\)20% of the index weights.

Answer: Choice A is correct.

The statement in Choice A correctly describes the constraints imposed by the absolute and relative approaches in portfolio management. The absolute approach is a type of investment strategy that seeks to achieve specific returns regardless of the performance of the broader market or a specific benchmark index. In this approach, the portfolio manager might limit any single security position to no more than 2% of the portfolio and any single sector to no more than 20% of the portfolio. This is done to manage risk and ensure diversification in the portfolio. On the other hand, the relative approach is a type of investment strategy that aims to outperform a benchmark index. In this approach, the portfolio manager might impose a constraint that a security must remain within \(\pm\)2% of its index weight and sector weights must remain within \(\pm\)10% of the index weights. This is done to ensure that the portfolio’s performance closely tracks the performance of the benchmark index.

Choice B is incorrect. The percentages mentioned in this choice are arbitrary and do not accurately represent the constraints typically imposed by the absolute and relative approaches in portfolio management. While the exact percentages can vary depending on the specific investment strategy and risk tolerance of the investor, the percentages mentioned in this choice are generally higher than what is typically used in practice.

Choice C is incorrect. Similar to Choice B, the percentages mentioned in this choice are arbitrary and do not accurately represent the constraints typically imposed by the absolute and relative approaches in portfolio management. The percentages mentioned in this choice are generally higher than what is typically used in practice, which could lead to a higher level of risk in the portfolio.

Glossary

• Active Share: The percentage of portfolio holdings that differ from the benchmark index.
• Active Risk: The standard deviation of the difference between the portfolio and benchmark returns.
• Information Ratio: The ratio of active return to active risk.
• Sharpe Ratio: The ratio of return in excess of the risk-free rate to portfolio volatility.
• Absolute Approach: An investment strategy that aims to achieve returns over a set period of time independent of the market environment.
• Relative Approach: An investment strategy that aims to outperform a benchmark index.
• Price-to-Book Ratio: A financial ratio used to compare a company’s current market price to its book value.

Portfolio Management Pathway Volume 1: Learning Module 3: Active Equity Investing: Portfolio Construction;

LOS 3(d): Discuss the application of risk budgeting concepts in portfolio construction.