Risk Parity Portfolios with riskParityPortfolio Prof. Daniel P. - PowerPoint PPT Presentation

Risk Parity Portfolios with riskParityPortfolio Prof. Daniel P. Palomar (Joint work with Zé Vinícius) Hong Kong University of Science and Technology (HKUST) R/Finance 2019 University of Illinois at Chicago (UIC), Chicago, IL, USA 17 May 2019

Markowitz portfolio Let us denote the returns of N assets at time t with the vector r t . riskParityPortfolio D. Palomar (HKUST) 1 H. Markowitz, “Portfolio selection,” J. Financ. , vol. 7, no. 1, pp. 77–91, 1952. subject to w maximize 2 / 15 t Portfolio return is r portf The portfolio vector w denotes the normalized dollar weights of the Suppose that r t follows an i.i.d. distribution (not totally accurate but widely adopted) with mean µ and covariance matrix Σ , N assets ( 1 T w = 1). = w T r t . Markowitz proposed in his seminar 1952 paper 1 to fjnd a trade-ofg between the portfolio expected return w T µ and its risk measured by the variance w T Σ w : w T µ − λ w T Σ w w ≥ 0 , 1 T w = 1 , where λ is a parameter that controls how risk-averse the investor is.

History 1960s but the fjrst risk parity fund, called the “All Weather” riskParityPortfolio D. Palomar (HKUST) indication of its potential success. point to its performance during the fjnancial crisis of 2007-2008 as an Some portfolio managers have expressed skepticism but others fund , was pioneered by Bridgewater Associates LP in 1996. Some of its theoretical components were developed in the 1950s and Drawbacks of Markowitz portfolio : Markowitz’s portfolio has been allocation of risk rather than allocation of capital. Risk parity is an approach to portfolio management that focuses on only considers the risk as a whole and ignores the risk diversifjcation. portfolio is highly sensitive to parameter estimation errors, variance is not a good measure of risk, embraced by practitioners for many reasons: heavily critized for over half a century and has never been fully 3 / 15

From “dollar” to risk diversifjcation Equally weighted portfolio (aka uniform portfolio) vs risk parity portfolio: riskParityPortfolio D. Palomar (HKUST) 4 / 15 Portfolio allocation of EWP Relative risk contribution of EWP 0.4 0.09 0.3 dollars 0.06 risk 0.2 0.03 0.1 0.00 0.0 AAPL AMD ADI ABBV AEZS A APD AA CF AAPL AMD ADI ABBV AEZS A APD AA CF Portfolio allocation of RPP Relative risk contribution of RPP 0.15 0.09 0.10 dollars 0.06 risk 0.05 0.03 0.00 0.00 AAPL AMD ADI ABBV AEZS A APD AA CF AAPL AMD ADI ABBV AEZS A APD AA CF stocks stocks

Risk parity portfolio (RPP) From Euler’s theorem, the volatility can be decomposed as riskParityPortfolio D. Palomar (HKUST) allocate the risk according to the risk profjle determined by the More generally, the risk budgeting portfolio (RBP) attemps to contributions: The risk parity portfolio (RPP) attemps to “equalize” the risk 5 / 15 N RC i ∑ σ ( w ) = i = 1 where RC i is the risk contribution (RC) from the i th asset to the total risk σ ( w ) : RC i = w i ( Σ w ) i √ . w T Σ w RC i = 1 N σ ( w ) . weights b (with 1 T b = 1 and b ≥ 0 ): RC i = b i σ ( w ) .

Solving the RPP reduced to solving riskParityPortfolio D. Palomar (HKUST) subject to w minimize possible): 1 3 General nonconvex formulation (there are many reformulations 6 / 15 Vanilla convex formulation : suppose we only have the constraints 2 Naive diagonal formulation : pretend that Σ is diagonal and simply √ use the volatilities σ = diag ( Σ ) , obtaining: σ − 1 w = 1 T σ − 1 . 1 T w = 1 and w ≥ 0 , then after some change of variable the problem Σ x = b / x . ) 2 − F ( w ) ( ∑ N w i ( Σ w ) i − w j ( Σ w ) j i , j = 1 w ≥ 0 , 1 T w = 1 , w ∈ W .

Package riskParityPortfolio Some R packages contain functions to compute the RPP, e.g., PortfolioAnalytics, FRAPO, cccp, and FinCovRegularization. But they are based on general-purpose solvers and may not be effjcient. riskParityPortfolio is the fjrst package specifjcally devised for the computation of difgerent versions of RPP in an effjcient way: https://CRAN.R-project.org/package=riskParityPortfolio Published on Christmas of 2018 and somehow was well-received by the community (600 downloads in 2 days). Authors: Zé Vinícius and Daniel P. Palomar. D. Palomar (HKUST) riskParityPortfolio 7 / 15

Using riskParityPortfolio AMD riskParityPortfolio D. Palomar (HKUST) R>> 0.156 0.068 0.125 0.133 0.045 0.129 0.158 0.085 0.101 CF AA APD A AEZS ABBV ADI AAPL Load Package: R>> print (rpp_vanilla $ w, digits = 2) "risk_contribution" R>> [1] "w" names (rpp_vanilla) rpp_vanilla <- riskParityPortfolio (Sigma) The simplest use is for the vanilla RPP: # to get help for the function ?riskParityPortfolio library (riskParityPortfolio) 8 / 15

Using riskParityPortfolio Naive diagonal formulation: riskParityPortfolio D. Palomar (HKUST) w_ub = 0.16) mu = mu, lmd_mu = 1e-3, rpp_mu <- riskParityPortfolio (Sigma, subject to 9 / 15 minimize w and box constraints: Unifjed nonconvex formulation including expected return in objective formulation = "diag") rpp_naive <- riskParityPortfolio (Sigma, ) 2 − λ w T µ ( ∑ N w i ( Σ w ) i − w j ( Σ w ) j i , j = 1 w ≥ 0 , 1 T w = 1 , l ≤ w ≤ u .

Risk concentration terms b i riskParityPortfolio D. Palomar (HKUST) b i Many formulations included in the package: b j 10 / 15 ( ) 2 R ( w ) = ∑ N w i ( Σ w ) i − w j ( Σ w ) j i , j = 1 R ( w ) = ∑ N i = 1 ( w i ( Σ w ) i − θ ) 2 ( w i ( Σ w ) i ) 2 R ( w ) = ∑ N w T Σ w − b i i = 1 ( ) 2 w j ( Σ w ) j w i ( Σ w ) i R ( w ) = ∑ N − i , j = 1 ( ) 2 R ( w ) = ∑ N w i ( Σ w ) i − b i w T Σ w i = 1 √ ( w i ( Σ w ) i ) 2 R ( w ) = ∑ N w T Σ w w T Σ w − b i √ i = 1 ( w i ( Σ w ) i ) 2 R ( w ) = ∑ N − θ i = 1

Using riskParityPortfolio D. Palomar (HKUST) riskParityPortfolio 11 / 15 Risk contribution 0.4 Markowitz RPP (naive) RPP (vanilla) RPP + mu 0.3 risk 0.2 0.1 0.0 AAPL AMD ADI ABBV AEZS A APD AA CF stocks

Using riskParityPortfolio Illustration of the expected return vs risk concentration trade-ofg: riskParityPortfolio D. Palomar (HKUST) 12 / 15 1.0 0.9 Expected return 0.8 0.7 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Risk concentration

Using riskParityPortfolio Illustration of the volatility vs risk concentration trade-ofg: riskParityPortfolio D. Palomar (HKUST) 13 / 15 0.63 0.62 Volatility 0.61 0.60 0.59 0 2 4 6 8 Risk concentration

References algorithm for computing high-dimensional risk parity portfolios,” riskParityPortfolio D. Palomar (HKUST) Signal Process. , vol. 63, no. 19, pp. 5285–5300, 2015. mization methods for risk parity portfolios design,” IEEE Trans. Y. Feng and D. P. Palomar, “SCRIP: Successive convex opti- Unifjed formulation and advanced algorithms: SSRN , 2013. T. Griveau-Billion, J.-C. Richard, and T. Roncalli, “A fast Standard textbooks: 2013. F. Spinu, “An algorithm for computing risk parity weights,” SSRN , Berman , 2012. H. Kaya and W. Lee, “Demystifying risk parity,” Neuberger Vanilla formulations: E. Qian, Risk Parity Fundamentals . CRC Press, 2016. Press, 2013. T. Roncalli, Introduction to Risk Parity and Budgeting . CRC 14 / 15

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