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Not Equal: A Comparison of “Risk Parity” and “Equal Risk Contribution”

July 19, 2012

The term “Risk Parity” is often confusing because it is defined and applied differently depending on the firm.  With the advent of multiple variations in tactical asset allocation and indexing strategies, it is perhaps not a stretch to claim that Risk Parity is as vague and elusive  as using the term “hedge fund” to describe a style of investment management. I have created a table below to clarify two different variations that are often used interchangeably and are not the same.   A good history of the origins of risk parity can be found here: http://en.wikipedia.org/wiki/Risk_parity. Systematic Investor– an excellent blog-  provides some useful R code and testing here: http://systematicinvestor.wordpress.com/2012/03/19/backtesting-asset-allocation-portfolios/. Meb Faber– always a great resource for new ideas- provides an application of Risk Parity to asset allocation as well as a good list of articles here: http://www.mebanefaber.com/2012/03/22/risk-parity-vs-endowment-model-vs-permanent-portfolio/

In my opinion, it is best to simply consider Risk Parity as a broad class of risk-budgeting schemes where the risk of each asset in the portfolio is leveraged (if necessary) to have the same volatility. Essentially, each asset has “equal risk”, and thus the portfolio is considered to be more diversified since returns and risk are less dependent on any one asset. In contrast, Roncalli http://www.thierry-roncalli.com/riskparity.html  coins the term “Equal Risk Contribution” (ERC)  which is a distinct sub-class of Risk Parity that seeks to equalize risk contributions from each asset to the portfolio. If both methods sound the same after reading various articles, that is because for the often cited two-asset case  they have the same mathematical solution and explanation for their validity. Nevertheless, not all Risk Parity portfolios are the same-  it is necessary to understand the differences between the equal risk and equal risk contribution approach to avoid improper application or interpretation. Below is a table that helps to define the similarities and differences of the two approaches and the relative advantages of each in the context of portfolio allocation.

Features and Qualities Shared By Equal Risk Contribution (ERC) and Risk Parity (RP)

Algorithm Characteristics Equivalency Equal Risk Contribution (ERC) Risk Parity (RP)
Requires Historical/Expected Asset Returns No No
Considered Heuristic Algorithms That Lack Strong Theoretical Support Yes Yes
Generally Uses Leverage Yes Yes
Generally Uses a Target Portfolio Risk Yes Yes
Allocation Across All Assets in the Universe Selected Yes Yes
Low Turnover Relative to Traditional Models Yes Yes
Comparatively Less Sensitive to Estimation Error than Traditional Models Yes Yes
Generally Superior Return and Sharpe to Equal Weight Portfolios Yes Yes
Better Returns and Sharpe than 60/40 Balanced Stock/Bond Portfolios Yes Yes
Solution for a Two-Asset Portfolio inverse volatility weighted inverse volatility weighted
Conditions for Being Equivalent to Equal Weight Portfolios Equal Asset Volatility and Constant Correlation Equal Asset Volatility and Constant Correlation
Conditions for Being Optimal on the Efficient Frontier Equal Sharpe Ratios and Constant Correlations Equal Sharpe Ratios and Constant Correlations



Features and Qualities That Favor Equal Risk Contribution (ERC)

Algorithm Characteristics Equivalency Equal Risk Contribution (ERC) Risk Parity (RP)
Objective Function volatility of risk contributions=0 inverse volatility weighted
Uses Beta/Covariance Data Yes No
Always Assumes Constant Correlation Matrix No Yes
Holdings Have Equal Risk Contributions to Portfolio Volatility Yes No
Performance Highly Sensitive to the Universe of Assets Selected No Yes (ie. 5 equities and 1 bond create imbalance)
Flexibility of Use for Risk Budgeting and Factor Tilts winner loser
Out of Sample Risk (Given the same target) winner loser
Out of Sample Return (Given the same target or leverage constrained) winner loser
Out of Sample Drawdowns winner loser
Out of Sample Sharpe Ratios winner loser
Out of Sample Diversification (Concentration and Average Correlation) winner loser
Favorable Hybrid Between Minimum Variance and Equal Weight Yes No
Can Be Equivalent to Minimum Variance When Correlation Matrix is Minimized Not Possible



Features and Qualities That Favor Risk Parity (RP) or Equal Risk

Algorithm Characteristics Equivalency Equal Risk Contribution (ERC) Risk Parity (RP)
Simple to Calculate for the Mathematically Challenged (Napkin Factor) loser winner
Intuitively Easy to Understand and Use in Most Software Packages No Yes
Best Sounding Name/Cool Factor loser winner
Practical Use for Dynamic Allocation With Long-Term Monthly Data loser winner
Sensitivity to Covariance Estimation Error loser winner
Generally Requires a Sophisticated Solver for Optimization Yes No
Speed of Calculation for Large Universe Slow Fast
Performance on very large data sets with limited sample size loser (too many covariances relative to sample size winner (fewer variables to estimate relative to sample size)
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11 Comments leave one →
  1. Victor permalink
    August 1, 2012 6:45 am

    This is a very helpful comparison & summary. Indeed, the term “risk parity portoflio” has been used widely but inconsistently. In my opinion, ERC is a superior approach in comparison. However, one thing I have been struggling to figure out is that what happen when you have negative MRC in one or some of your assets within the portfolio (in the ERC approach). I was hoping by varying the weight allocation (which would also change the ERC of course) and eventually a solution could be found that yield equal RC (all positive). However it seems the optimiser is always struggling to overcome this problem. Whenever there happen to be negative ERC, the optimiser will push down the weight of that asset to virtually zero. So far I have not come across any literature that addresses this problem either. Any comments on this please, David?

  2. Wouter permalink
    August 6, 2012 12:02 pm

    Do we really need to estimate full covar matrices to compute the ERC solution? What about this simple iteration
    1. Compute volats v per asset and assume start weights w=1/N per asset (equal weights)
    2. Compute the porto using these w and the correlation c per asset with this porto
    3. Compute new weight per asset w = 1/(c*v), normalize so that 0<w<1 and sum w =1, and repeat step 2.

    If I got this right, calculation should be simple, even for large universes / data sets since it looks of order N instead of NxN. I have not tried this so I dont know if it solves Victors problem.

  3. May 3, 2013 12:28 am

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