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Diversified reward-risk parity

Risk parity is a portfolio construction technique that seeks to equalize risk contributions from the different components of the portfolio. Risk parity with respect to uncorrelated risk sources maximizes diversification. Simple risk parity rules are based on the inverses of market beta, price standard deviation, or price variance. These methods can be combined with common reward risk metrics, such as the Sharpe ratio, Calmar ratio, STAR ratio, or Rachev ratio. The resulting diversified reward-risk parity allocations have not only outperformed equally-weighted risk portfolios and standard factor allocations but also provided enhanced risk management.

Choia, Jaehyung, Hyangju Kimb and Young Shin Kim (2021), “Diversified reward-risk parity in portfolio construction”.

The below are quotes of the above paper and some other sources which are linked next to the quote. Emphasis, headings, and text in brackets have been added for clarity.

What is risk parity?

“The risk parity approach defines a well-diversified portfolio as one where all asset classes [or underlying factors] have the same marginal contribution to the total risk of the portfolio. In this sense, a risk parity portfolio is an equally weighted portfolio, where the weights refer to risk rather than dollar amount invested in each asset.” [Kazemi]

“The idea is to equalize risk contributions from the different components of the portfolio. The risk contribution of a component is the share of total portfolio risk attributable to that component. It is computed as the product of the allocation in the component with its marginal risk contribution, the latter one being given by the change in the total risk of the portfolio induced by an infinitesimal increase…Dealing with risk contributions has become standard practice for institutional investors, under the label of ‘risk budgeting’.” [Maillard, Roncalli and Teiletche]

The maximum diversification portfolio is equivalent to a risk parity strategy with respect to the uncorrelated risk sources embedded in the underlying portfolio assets…This approach [is] a diversified risk parity strategy which turns out to be a reasonable alternative when it comes to risk-based asset allocation…Diversified risk parity has a built-in mechanism for tracking the prevailing risk structure thus providing a more robust way to achieve maximum diversification throughout time.” [Lohre, Oper and Orszag]

“Risk parity strategies have received increasing attention from both academia and industry. …This is because the risk parity portfolio focuses on the allocation of risk in order to construct a truly diversified portfolio.”

Types of risk parity

“[A range of] simple and heuristic risk parity approaches [have been developed and documented in academic literature]…

  • Equal risk contribution (ERC) strategies bet against market exposures via beta in the capital asset pricing model…The allocation weight of each asset in the ERC portfolio is given by [the inverse of its market beta divided by the sum of all inverses of market betas]…The ERC portfolio penalizes exposures to the market portfolio. By betting against the beta, it tries to reduce asset-level correlations to the benchmark.
  • Diversified risk parity (DRP) portfolios…use the standard deviation of each asset as the portfolio weights…The weights in the DRP portfolio are allocated with the inverse of volatility…In this portfolio construction, less volatile investment vehicles are more preferred to more volatile ones.
  • The diversified minimum variance portfolio allocates weights inversely proportional to variance.”

“In principle, the portfolio construction rules based on the heuristic risk parity simply assign more weights on assets with smaller risks…By favoring less risky assets over more risky assets, the allocation rules decide the extent of how much penalization is given to each asset based on risk measures.”

Reward-risk metrics

“Various reward-risk measures are employed for portfolio construction, performance assessment, and risk management.

  • Volatility measures return fluctuations…We compute the volatility as either of two different definitions: the standard deviation and the conditional volatility from ARMA-GARCH models…
  • Sharpe ratio is defined as the ratio of expected return [above a risk-free rate] to standard deviation of a time series [of that excess return]…Two different Sharpe ratio definitions are used for portfolio construction and performance assessment. One is the traditional Sharpe ratio definition that is the ratio of empirical average return to historical standard deviation. Another is conditional Sharpe ratio calculated as the ratio of conditional mean to conditional volatility computed by ARMA-GARCH models.
  • Maximum drawdown is the worst consecutive loss in a specified time period…It is the worst realized performance since the inception of a portfolio [over a certain time interval]…Maximum drawdown encodes the information on time evolution of a return series opposite to other quantile-based risk measures. For example, even when we use a given return series for risk calculation, different maximum drawdowns are obtained by different sequential orders of the time series.
  • Calmar ratio is the ratio of realized cumulative return to maximum drawdown…within a given investment time horizon… In the original definition of Calmar ratio, the time window for return and maximum drawdown was given to 36 months. In this paper, the Calmar ratio is computed from the realized return and the maximum drawdown in the six-month window for portfolio construction.
  • Value-at-Risk (VaR) is the loss at a given quantile of a performance distribution…VaR describes quantile risks of return distributions as a single number computed from historical data or simulated data…Not satisfying sub-additivity for VaR implies that portfolio diversification could bring worse VaR values than the weighted sum of VaRs from its components. It is obviously counter-intuitive to investment insights that diversification can reduce investment risks.
  • Conditional Value-at-Risk (CVaR)…is the average loss of extreme losses worse than a given quantile loss…When a portfolio suffers from severe losses worse than a threshold, a CVaR value indicates the average of such extreme losses…Opposite to the characteristics of VaR, CVaR fulfills not only the sub-additivity but also other properties of coherent risk measures.
  • Stable tail adjusted return ratio (STAR)…enhances the concept of Sharpe ratio. Instead of using the standard deviation [as denominator of the ratio of expected excess return to risk] a portfolio is penalized by CVaR which represents genuine downside risks…Assets with less downside tail risks exhibit larger STAR ratios.
  • Rachev ratio…is defined as the ratio of expected upward tail gain to downward tail loss. It is the ratio of two CVaRs [with] sign-inverted return time series in the numerator [upside risk] [and the] CVaR from an original return in the denominator [downside risk].”

Diversified reward-risk parity strategies

“We introduce alternative reward-risk parity strategies based on the diversification of reward-risk measures. The diversified reward-risk parity is the two-folded generalization of applying reward-risk measures and more generic allocation rules to traditional diversified risk parity portfolios.”

“The various allocation weights mentioned above can be generalized into the following form:

weight of asset i =
inverse of risk-reward measure of i / sum of inverses of risk reward measures across assets”

“In order to capture autocorrelation, volatility clustering, asymmetry, and heavy tails of financial time series, the reward-risk measures are calculated from the ARMA(1,1)-GARCH(1,1) model with CTS innovations.”

Empirical lessons

“In order to test diversified reward-risk parity allocation across various asset classes, we cover global equity benchmarks, larges mutual funds and ETFs, SPDR U.S. sector ETFs, Dow Jones Industrial Average components. Adjusted daily price datasets of these asset classes were downloaded.”

“We empirically test advanced reward-risk parity strategies and compare their performance with an equally-weighted risk portfolio in various asset universes…The portfolio construction based on diversified reward-risk parity not only outperforms the equally-weighted portfolio but also provides enhanced risk management…[producing] higher average returns, Sharpe ratios, and Calmar ratios.”

“Regardless of financial markets, the new heuristic reward-risk parity portfolios outperform the equally-weighted portfolio. The diversified reward-risk parity allocations are more profitable in annualized return than the benchmark. In particular, the diversification rules by Calmar ratio and Sharpe ratio are the best way of allocating capital to increase portfolio profits. Additionally, the reward-risk portfolios constructed from Rachev ratio and STAR ratio are also improved in profitability with respect to the benchmark.”

“The outperformance of the diversified reward-risk parity strategies is achieved by taking less downside risks. Many alternative portfolios in diverse asset classes are consistently less exposed to various risks represented by standard deviation, VaR, CVaR and maximum drawdown than any other portfolios including the equally-weighted portfolio. Moreover, the skewness and kurtosis of the portfolio performances are improved. Sharpe ratios and Calmar ratios.”

“The Carhart four-factor analysis also supports the outperformance of the diversified reward-risk parity portfolios. After controlling the Carhart factors, the portfolio performances still remain more profitable than the equally-weighted portfolio. With high R2 values, the larger regression intercepts are obtained for the diversified reward-risk allocations. The inverse risk parity portfolios achieve the largest factor-adjusted returns among the alternative portfolios, and the raw Rachev ratio portfolios also mark stronger alphas than the benchmark.”


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