Expected Returns

This document explains the various methodologies used to calculate expected returns in the MantaCore system, specifically focusing on the mathematical approaches for portfolio and component-level expected returns estimation.

Overview

MantaRisk calculates expected returns using four different methodologies, which can be categorized along two dimensions:

1. Data Source Dimension:
   • Sample-based methods (using historical returns directly)
   • Factor model-based methods (using risk factor exposures)
2. Weighting Dimension:
   • Equally weighted methods (all observations have equal importance)
   • Exponentially weighted methods (recent observations are more important)

This results in four distinct methodologies:

1. Equally Weighted Sample
2. Exponentially Weighted Sample
3. Equally Weighted Factor Model
4. Exponentially Weighted Factor Model

Methodology Details

1. Equally Weighted Sample

This methodology calculates expected returns directly from historical returns data, giving equal weight to all observations in the time series.

Mathematical Formula:

The equally weighted expected return is given by the following formula

$$E[R] = \frac{1}{T} \sum_{t=1}^{T} r_t$$

Where:

• T is the number of observations in the time series
• r_t is the return at time t

Characteristics:

• Uses simple arithmetic mean of historical returns
• All observations in the time series have equal influence
• Less responsive to recent market changes
• More stable over time
• Useful for long-term strategic forecasting

2. Exponentially Weighted Sample

This methodology also uses historical returns directly but applies exponential weighting to give more importance to recent observations.

Mathematical Formula:

The expected exponentially weighted moving average (EWMA) return is given by the following formula

$$E^\mathrm{EWMA}[R] = \frac{\sum_{t=1}^{T} w_t \cdot r_t}{\sum_{t=1}^{T} w_t},\quad w_t=2^{-\frac{T-t}{h}}$$

Where:

• T is the number of observations in the time series
• r_t is the return at time t
• h is the halflife parameter (number of periods for weight to reduce by half)

Characteristics:

• Uses exponentially weighted moving average (EWMA)
• Recent observations have more influence than older ones
• The halflife parameter determines how quickly the weights decay. This parameter depends on the chosen horizon and will be faster for a daily horizon than for a monthly horizon
• More responsive to recent market changes
• Useful for tactical or short-term forecasting
• Better at capturing regime changes or trend shifts

3. Equally Weighted Factor Model

This methodology uses a risk factor model to calculate expected returns. It applies the portfolio's exposures (betas) to risk factors and multiplies them by the equally weighted expected returns of those risk factors.

Mathematical Formula:

The expected return for the equally weighted factor model is given by:

$$E[R_p] = \sum_{i=1}^{N} \beta_i \cdot E[R_i]$$

Where:

• N is the number of risk factors
• β_i is the exposure (beta) to risk factor i
• E[R_i] is the equally weighted expected return of risk factor i

Characteristics:

• Based on risk factor exposures rather than direct historical returns
• Uses equally weighted means for risk factor expected returns
• More stable and theoretically grounded
• Captures systematic risk exposures
• Useful for understanding return drivers
• Less influenced by idiosyncratic movements

4. Exponentially Weighted Factor Model

This methodology also uses a risk factor model but applies exponential weighting to the expected returns of the risk factors.

Mathematical Formula:

The expected returns for the exponentially weighted factor model is given by:

$$E^\mathrm{EWMA}[R_p] = \sum_{i=1}^{N} \beta_i \cdot E^\mathrm{EWMA}[R_i]$$

Where:

• N is the number of risk factors
• β_i is the exposure (beta) to risk factor i
• E^{EWMA}[R_i] is the exponentially weighted expected return of risk factor i

Characteristics:

• Based on risk factor exposures
• Uses exponentially weighted means for risk factor expected returns
• Balances theoretical grounding with responsiveness to recent data
• Captures systematic risk exposures while being more sensitive to recent factor performance
• Useful for medium-term forecasting

Application in Portfolio Analysis

These methodologies are applied at two levels in the portfolio analysis process:

1. Portfolio Level: Calculates expected returns for the entire portfolio as a whole
2. Component Level: Calculates expected returns for individual instruments within the portfolio

Both applications use the same core calculation methodologies, but they differ in how they obtain the inputs (returns and factor exposures) and how the results are utilized in subsequent analysis.

Choosing a Methodology

The choice of methodology depends on several factors:

• Investment Horizon: Longer horizons may benefit from equally weighted methods, while shorter horizons may benefit from exponentially weighted methods.
• Market Conditions: During stable periods, equally weighted methods may perform better. During changing regimes, exponentially weighted methods may be more appropriate.
• Analysis Purpose: Factor models provide more insight into return drivers, while sample-based methods may be more accurate for pure forecasting.

Conclusion

MantaCore's multiple methodologies for calculating expected returns provide flexibility and robustness. By considering both sample-based and factor model-based approaches, and by applying both equal and exponential weighting schemes, the system can generate a more comprehensive view of expected returns under different assumptions and for different purposes.