Overview

Using marginal risk analytics offers deeper insight into the impact on risk from changing a security position size to the portfolio's risk profile. This "bang-for-your-buck" metric is accompanied by the "X-Sigma-Rho" framework, which decomposes risk across intuitive risk drivers — Exposure/weighting, Risk, and Correlation — along with marginal contribution analytics.

Available as a view within Analyze.Assets, the "X-Sigma-Rho" framework calculates marginal contribution, risk & risk contribution, and correlation and is available in absolute (MCAR) and active space (MCTE) — resulting in slightly different analytic interpretations:

Absolute, non-Active

Absolute Volatility
The standalone risk of the asset itself
Absolute Correlation
The correlation of the standalone asset with the portfolio
Marginal Contribution
The expected increase in total risk given an increase in holding of the asset. Also known as Marginal Contribution to Total Risk
Risk Contribution
The contribution to total risk in terms of standard deviation

Relative, Active

Relative Volatility
The active risk of the asset itself against the index
Relative Correlation
The correlation of the asset against the index
Relative Marginal Contribution
The expected increase in active risk given an increase in the active holding of the asset. Also known as Marginal Contribution to Tracking Error
Active Risk Contribution
The contribution to active risk in terms of standard deviation.

MCTE Interpretation

MCTE reflects relative volatility * relative correlation, isolating the directional and scale sensitivity of each asset, meaning
- A higher MCTE value means that increasing the asset's weight will increase the portfolio's tracking error more significantly
- A lower or negative MCTE suggests the asset either diversifies the portfolio (negative correlation with active risk), or has low volatility relative to the benchmark.
MCTE is a first-order (linearized) approximation, valid for small changes and non-linear effects are ignored.

MCTE in Trading

Assume you are increasing the asset's weight by 1% (0.01), and the asset's MCTE is known.
- ex. MCTE for MSFT is 0.60
Then MCTE * ∆(asset weight) = ∆(tracking error)
- ex. 0.60 * 0.01 = 0.006 = 60 bps increase to tracking error
- Increasing MSFT's weight by 1% is expected to increase tracking error by 60 bps

For a detailed discussion on X-Sigma-Rho and MCAR vs. MCTE, please see the calculation methodology below, and sources at the bottom of this document.

Using the X-Sigma-Rho Risk Decomposition via API

The API provides options to generate risk decomposition analytics as granular as needed using the methodology most appropriate for one’s investment process:

Absolute, non-active
→ no base, attributionMethod: MCAR
Relative, active
→ base specified, attributionMethod: MCTE
Absolute, active
→ base specified, attributionMethod: MCAR. This option is available via API only, where:
- Absolute Volatility
  The standalone risk of the asset itself
- Absolute Correlation
  The correlation of the standalone asset with the index
- Marginal Contribution
  The expected increase in active risk given an increase in holding of the asset
- Risk Contribution
  The contribution to active risk in terms of standard deviation. Also known as Marginal Contribution to Active Risk

Risk Contributors

As before, the riskContributors endpoint returns asset level risk decompositions. Historically Omega Point only provided ‘% of Risk’ / ‘% of Active Risk’ on the Total, Specific, and Factor level as a percentage of the total/active risk of the portfolio. Using the exposureRiskCorrelation endpoint, one can now get the % of Risk / % of Active Risk broken down into it’s drivers: Volatility and Correlation, as well as Marginal Contribution.

{
  model(id: "AXWW4-MH") {
    simulation(
      positionSet: {type: PORTFOLIO, id: "portfolio_2017"}
      base: {type: INDEX, id: "IDX-SPY"}
      from: "2017-06-30"
      to: "2017-06-30"
    ) {
      risk {
        date
        total
      }
      riskContributors(attributionMethod: MCTE) {
        id
        total
        exposureRiskCorrelation {
          volatility
          correlation
          marginalContribution
          riskContribution
        }
        attribution {
          summary {
            factors
            specific
          }
          exposureRiskCorrelation {
            factors {
              volatility
              correlation
              marginalContribution
              riskContribution
            }
            specific {
              volatility
              correlation
              marginalContribution
              riskContribution
            }
          }
        }
      }
    }
  }
}

Grouped Risk Contributors

groupedRiskContributors now supports Absolute and Relative methods for % of Risk and risk contribution on the group level for total, factor, and specific contributions.

{
  model(id: "AXWW4-MH") {
    simulation(
      positionSet: {type: PORTFOLIO, id: "portfolio_2017"}
      base: {type: INDEX, id: "IDX-SPY"}
      from: "2017-06-30"
      to: "2017-06-30"
    ) {
      risk {
        date
        total
      }
      groupedRiskContributors(
        groupBy: CLASSIFICATION
        classificationTier: "SECTOR"
        groupById: "GICS"
        attributionMethod: MCTE
      ) {
        id
        totalPercentEquity
        exposureRiskCorrelation {
          riskContribution
        }
        attribution {
          summary {
            factors
            specific
          }
          exposureRiskCorrelation{
            factors{
              riskContribution
            }
            specific{
              riskContribution
            }
          }
        }
      }
    }
  }
}

Deriving Marginal Contribution to Tracking Error (MCTE)

Implementation

This section outlines the methodology for computing Marginal Contribution to Tracking Error (MCTE) and its related analytics. This feature enhances risk attribution by introducing MCTE calculations, providing deeper insights into how each asset contributes to tracking error.

Calculation Methodology

Step 1: Compute Portfolio Active Risk

Use existing logic to compute the active risk of the portfolio.

Step 2: Construct Dummy Portfolios

For each asset, create a dummy portfolio where that asset is held at 100%, and all others are 0%. Compute the active weight against the real index.

Example:
This creates two dummy portfolios, one for each security in the original portfolio.

Asset	Dummy Port Weight	Index Weight	Dummy Active Weight
MSFT	1	0.5	0.5
AAPL	0	0.5	-0.5

MSFT	0	0.5	-0.5
AAPL	1	0.5	0.5

Step 3: Compute Active Risk for Dummy Portfolios

Use existing active risk calculations to compute the active risk of each dummy portfolio:

MSFT Dummy Active Risk
AAPL Dummy Active Risk

Step 4: Compute Active Covariance

Calculate the covariance between each dummy portfolio’s active weights and the true active weights:

MSFT Dummy Active Covariance with True Active
AAPL Dummy Active Covariance with True Active

Step 5: Compute MCTE and Related Metrics

Using the results from previous steps:

Relative Volatility:
Relative Volatility = Dummy Active Risk
Relative Correlation:
Relative Correlation = (Dummy Active Covariance with True Active) / (Dummy Active Risk * Portfolio Active Risk)
MCTE Calculation:
MCTE = Dummy Active Risk * Relative Correlation
% of Active Risk:
% of Active Risk = True Active Weight * MCTE
Active Risk Contribution:
Risk Contribution = % of Active Risk * Portfolio Active Risk

Additional Notes:

As these calculations make use of the factor’s covariance matrix, the horizon built-into the marginal contribution calculation follows the horizon of the model’s covariance matrix.

Sources

Jose Menchero and Ben Davis. Risk Contribution is Exposure times Volatility times Correlation: Decomposing Risk Using the X‐Sigma‐Rho Formula. The Journal of Portfolio Management, Winter 2011, pp. 97‐106.

Phil Durand, Kim Jensen, and Jose Menchero. Different Ways to Measure Marginal Contribution to Risk: Demystifying MCAR versus MCTE. MSCI Research Insight, September 2013.

Marginal Contributors Risk Decomposition