Book 5. Risk and Investment Management

FRM Part 2

IM 4. Portfolio Construction

Presented by: Sudhanshu

Module 1. Portfolio Construction Process and Transaction Costs

Module 2. Practical Issues in Portfolio Construction

Module 1. Portfolio Construction Process and Transaction Costs

Topic 1. Portfolio Construction Process

Topic 2. Alpha-Refining Processes

Topic 3. Neutralization

Topic 4. Transaction Costs

Topic 1. Portfolio Construction Process

• Inputs to the Process

  • Current Portfolio: Assets and their weights. Most measurable input.

  • Alphas: Expected excess returns of portfolio stocks. Subject to forecast error and bias.

  • Covariances: Estimates of covariances. Subject to estimation error.

  • Transaction Costs: Estimated costs that increase with more frequent changes. Subject to estimation error.

  • Active Risk Aversion: Preference for lower volatility between actively managed portfolio returns and benchmark returns.

• Challenges with Inputs

  • Alphas, covariances, transaction costs, and active risk aversion are all subject to estimation error and potential bias.

  • Accurate measurement of these inputs is crucial for optimal portfolio construction.

Practice Questions: Q1

Q1. The most measurable of the inputs into the portfolio construction process is (are) the:
A. position alphas.
B. transaction costs.
C. current portfolio.
D. active risk aversion.

Practice Questions: Q1 Answer

Explanation: C is correct.

The current portfolio is the only input that is directly observable.

• Motivation: An alternative to directly imposing constraints (e.g., no short selling, maximum deviations from benchmark weights) in portfolio optimization.
• Method: Adjust estimated alphas in ways that effectively impose various constraints. This allows focusing on the effects of specific constraints on alphas, the key input for active portfolio construction. 2 methods are used:

  • Scaling: Adjusting the standard deviation (scale) of refined alphas to match the appropriate scale, especially after constraints are applied.

    • ​Alpha = (volatility) $\times$ (information coefficient) $\times$ (score)

    • The scale of alphas is proportional to the standard deviation of the score variable.

  • Trimming: Reducing large positive or negative alpha values to mitigate the impact of questionable data or extreme estimates.

Topic 2. Alpha-Refining Processes

Practice Questions: Q2

Q2. Which of the following is correct with respect to adjusting the optimal portfolio for portfolio constraints?
A. No reliable method exists.
B. By refining the alphas and then optimizing, it is possible to include constraints of both the investor and the manager.
C. By refining the alphas and then optimizing, it is possible to include constraints of the investor, but not the manager.
D. By optimizing and then refining the alphas, it is possible to include constraints of both the investor and the manager.

Practice Questions: Q2 Answer

Explanation: B is correct.

The approach of first refining alphas and then optimizing can replace even the most sophisticated portfolio construction process. With this technique, both the investor and manager constraints are considered.

Topic 3. Neutralization

• Definition: The process of removing biases and undesirable bets from alpha. The type of neutralization and strategy should be specified beforehand.
• Types of Neutralization

  • Benchmark Neutralization: Eliminates any difference between the benchmark beta and the beta of the active portfolio.

    • Ensures the active portfolio beta matches the benchmark portfolio beta, unless an active bet on market returns is intended.

    • Equivalent to adding a constraint on portfolio beta in a mean-variance optimization.

    • Calculation Example: Modified benchmark-neutral alpha = (alpha of active position) - (benchmark alpha $\times$ active position beta).

  • ​​Risk-Factor Neutralization: Addresses unintended risk exposures relative to the benchmark from other risk factors (e.g., small-cap returns minus large-cap returns, industry risk factors).

    • Alphas are adjusted so that the active portfolio's sensitivity to a specific risk factor matches that of the benchmark.

    • Reduces active risk by matching factor risks of the active portfolio to those of the benchmark.

  • Cash Neutralization: Adjusts alphas so that the portfolio has no active cash position.

• It is possible to make alpha values both cash-neutral and benchmark-neutral.

Topic 4. Transaction Costs

• Definition: Transaction costs are the costs changing portfolio allocations, primarily trading commissions and spreads.
• Transaction costs are uncertain and reduce active portfolio returns relative to the benchmark.
• Importance of Precision: Including transaction costs increases the importance of precise alpha estimates and the choice of scale.
• Rebalancing Decisions: Transaction costs can make it optimal not to rebalance, even with new information.
• Time Horizon Complication: Transaction costs occur at a point in time, while benefits are realized over the investment horizon. This complicates analysis and makes rebalancing decisions dependent on the estimated holding period of portfolio assets.
• With uncertain holding periods across portfolio holdings, determining the appropriate period for amortizing transaction costs becomes a key question.

• Precision in scale is important in addressing the tradeoff between alphas and transaction costs.

• Annual transaction costs = (cost of a round-trip trade)/(holding period in years).

Practice Questions: Q3

Q2. Which of the following statements most correctly describes a consideration that complicates the incorporation of transaction costs into the portfolio construction process?
A. The transaction costs and the benefits always occur in two distinct time periods.
B. The transaction costs are uncertain while the benefits are relatively certain.
C. There are no complicating factors from the introduction of transaction costs.

D. The transaction costs must be amortized over the horizon of the benefit from the trade.

Practice Questions: Q3 Answer

Explanation: D is correct.

A challenge is to correctly assign the transaction costs to projected future benefits. The transaction costs must be amortized over the horizon of the benefit from the trade. The benefits (e.g., the increase in alpha) occur over time while the transaction costs generally occur at a specific time when the portfolio is adjusted.

Module 2. Practical Issues in Portfolio Construction

Topic 1. Active Risk Aversion

Topic 2. Proper Alpha Coverage

Topic 3. Portfolio Revisions and Rebalancing

Topic 4. Portfolio Construction Techniques

Topic 5. Screening

Topic 6. Stratification

Topic 7. Linear Programming

Topic 8. Quadratic Programming

Topic 9. Portfolio Return Dispersion

Topic 1. Active Risk Aversion

• Determination of Appropriate Risk Aversion​

  • Portfolio managers often have intuition about their information ratio and acceptable active risk, rather than an intuitive idea of optimal active risk aversion.

  • Equation:

  • Example: If Information Ratio = 0.8 and desired Active Risk = 10%, so

    •  

  • The utility function for the optimization is:

    • ​

  • Accuracy of active risk aversion estimate depends on the accuracy of the inputs, information ratio and preferred level of active risk.

• Aversions to Specific Factor Risk

  • Addressing Large Potential Losses: Helps managers address risks associated with positions having large potential losses (e.g., sector risks not matching the benchmark).

  • Reducing Dispersion: High risk aversion values for specific factor risks will reduce dispersion of holdings and performance across multiple managed portfolios, leading to greater similarity in client portfolios.

Topic 2. Proper Alpha Coverage

• Proper Alpha Coverage: Refers to addressing situations where the manager has forecasts of stocks that are not in the benchmark or where the manager does not have alpha forecasts for stocks in the benchmark.
• Non-Benchmark Stock Coverage: When managers have forecasts for stocks outside the benchmark, assign zero benchmark weight but allow active weights to generate alpha
• Missing Alpha Forecasts: For benchmark stocks lacking alpha forecasts, adjusted alphas can be inferred from the alphas of stocks for which there are forecasts.
• Step 1: Calculate average alpha for the stocks with forecasts as below:
  • Value-weighted fraction of stocks with forecasts = Sum of active holdings with forecasts
  • ​
• Step 2: Subtract this measure from each alpha for which there is a forecast and set alpha to zero for assets that do not have forecasts. ​This provides a set of benchmark-neutral forecasts where assets without forecasts have alphas of zero.

Practice Questions: Q1

Q1. A manager has forecasts of Stocks A, B, and C, but not of Stocks D and E. Stocks A, B, and D are in the benchmark portfolio. Stocks C and E are not in the benchmark portfolio. Which of the following is correct concerning specific weights the manager should assign in tracking the benchmark portfolio?
A.
B.
C.
D.

Practice Questions: Q1 Answer

Explanation: A is correct.

The manager should assign a tracking portfolio weight equal to zero for stocks for which there is a forecast but that are not in the benchmark. A weight should be assigned to Stock D, and it should be a function of the alphas of the other assets.

Topic 3. Portfolio Revisions and Rebalancing

• With zero transaction costs, portfolios should be revised whenever new information arrives.
• In practice, trading depends on expected active return, active risk, and transaction costs.
• Managers should be conservative due to uncertainty in these measures.

  • Underestimating costs → excessive trading.

  • Frequent trading + short horizons → highly uncertain alpha estimates.

  • Optimal horizon = point where alpha certainty justifies trading costs.

• Rebalancing weighs transaction costs vs. value added.
• A no-trade region exists where rebalancing costs exceed benefits.

• Let MCAR = marginal contribution to value added and MCAR = marginal contribution to active risk

• The benefit of adjusting the number of shares of a given portfolio asset is given by:

  •  

• So, the no-trade region is defined as:

• Rearranging this relationship with respect to alpha gives a no-trade region for alpha:

  •  
  •  

• The size of the no-trade region is determined by transaction costs, risk aversion, alpha, and the riskiness of the assets.

Practice Questions: Q2

Q2. An increase in which of the following factors will increase the no-trade region for the alpha of an asset?
I. Risk aversion.
II. Marginal contribution to active risk.
A. I only.
B. II only.
C. Both I and II.
D. Neither I nor II.

Practice Questions: Q2 Answer

Explanation: C is correct.

This is evident from the defiinition of the no-trade region for the alpha of the asset.

[2 × (risk aversion) × (active risk) × (MCAR)] − (cost of selling) < alpha of asset < [2 × (risk aversion) × (active risk) × (MCAR)] + (cost of purchase)

Topic 4. Portfolio Construction Techniques

• Four main institutional portfolio construction techniques—screens, stratification, linear programming, and quadratic programming—all share the same goal: high alpha, low active risk, and low transaction costs.
• An active manager's value depends on generating returns relative to the benchmark that exceed the penalty for active risk and additional transaction costs of active management.
  • ​​

Topic 5. Screens

• Screens filter stocks based on specific criteria; for example, selecting the 60 benchmark stocks with the highest alphas to construct equal- or value-weighted portfolios.
• Another screening method assigns buy, hold, or sell ratings based on alpha rankings (e.g., buy for top 60 alphas, hold for next 40, sell for remainder).
• Portfolio rebalancing involves purchasing stocks on the buy list not currently held and selling portfolio stocks on the sell list, with turnover adjusted by modifying category sizes.

Topic 6. Stratification

• Stratification divides stocks into mutually exclusive categories (e.g., large/medium/small-cap across industry sectors) to match benchmark portfolio weights for size and sector coverage.
• Stratification serves as a risk control method; effective risk control depends on whether the chosen categories adequately capture the benchmark's risk dimensions.
• This approach reduces bias in estimated alphas across firm size and sector categories but eliminates the ability to add value by deviating from benchmark size-sector weights.
• Stratification sacrifices potential value from information about actual alphas beyond their category and any sector-specific alpha opportunities.

• Linear programming improves on stratification by using multiple risk characteristics (e.g., firm size, volatility, sector, beta) without requiring mutually exclusive categories.
• The methodology selects assets to ensure active portfolio category weights closely resemble benchmark weights while incorporating transaction costs and position size limits.
• Linear programming's strength is creating portfolios that closely match benchmarks, but may differ significantly in the number of assets included and unincluded risk dimensions

Topic 7. Linear Programming

• Quadratic programming incorporates alphas, risks, transaction costs, and constraints to efficiently produce optimal portfolios, making it theoretically the best optimization method.
• For a 400-stock universe, quadratic programming requires estimates of 400 volatilities and 79,800 covariances, making estimation error a critical consideration.
• Small estimation errors in covariances may not significantly reduce value added, but moderate errors (e.g., 5%) can dramatically decrease or even eliminate value added.
• Accurate estimates of inputs, especially covariances, are essential for successful quadratic programming optimization.

Topic 8. Quadratic Programming

Topic 9. Portfolio Return Dispersion

• Dispersion refers to the variability of returns across client portfolios, commonly measured as the difference between maximum and minimum returns for separately managed accounts over a period.
• Managers can reduce dispersion by minimizing differences in asset holdings and portfolio betas through better supervision and control, though some causes like differing client constraints (e.g., derivatives restrictions) are unavoidable.
• Zero dispersion is unachievable with transaction costs present, as there exists an optimal level of dispersion that balances transaction costs against gains from rebalancing to optimal weights.
• When acquiring new clients, managers face a tradeoff: matching new portfolios to existing ones sacrifices returns from new information, while rebalancing existing portfolios incurs transaction costs that reduce overall returns.
• Higher transaction costs lead to greater optimal dispersion, and for a given number of portfolios, dispersion is proportional to active risk.
• A greater number of portfolios and higher active risk both increase optimal dispersion, and as long as alphas and risk are not constant, dispersion will decrease over time toward convergence, though the rate of convergence remains uncertain.