Risk Parity Portfolio Prof. Daniel P. Palomar ELEC5470/IEDA6100A - - PowerPoint PPT Presentation

Risk Parity Portfolio Prof. Daniel P. Palomar ELEC5470/IEDA6100A - Convex Optimization Hong Kong University of Science and Technology (HKUST) Fall 2019-20

Outline 1 Introduction 2 Warm-Up: Markowitz Portfolio Signal model Markowitz formulation Drawbacks of Markowitz portfolio 3 Risk Parity Portfolio Problem formulation Solution to the naive diagonal formulation Solution to the vanilla convex formulation Solution to the general nonconvex formulation 4 Conclusions

Outline 1 Introduction 2 Warm-Up: Markowitz Portfolio Signal model Markowitz formulation Drawbacks of Markowitz portfolio 3 Risk Parity Portfolio Problem formulation Solution to the naive diagonal formulation Solution to the vanilla convex formulation Solution to the general nonconvex formulation 4 Conclusions

Motivation it only considers the risk of the portfolio as a whole and ignores the Risk Parity Portfolio D. Palomar (HKUST) 2019. 1 Z. Zhao, R. Zhou, D. P. Palomar, and Y. Feng, “Portfolio optimization,” submitted , assets by properly redefjning the portfolio formulation. We will address here the risk diversifjcation among the portfolio . was observed in the 2008 fjnancial crisis): solution is the risk parity risk diversifjcation (i.e., concentrates risk too much in few assets, this 3 Markowitz’s portfolio has never been fully embraced by practitioners, robust optimization and improved parameter estimation , it is highly sensitive to parameter estimation errors (i.e., to the 2 is to use alternative measures for risk , e.g., VaR and CVaR, both the unwanted high losses and the desired low losses: the solution variance is not a good measure of risk in practice since it penalizes 1 4 / 81 among other reasons (Zhao et al. 2019) 1 because covariance matrix Σ and especially to the mean vector µ ): solution is

Outline 1 Introduction 2 Warm-Up: Markowitz Portfolio Signal model Markowitz formulation Drawbacks of Markowitz portfolio 3 Risk Parity Portfolio Problem formulation Solution to the naive diagonal formulation Solution to the vanilla convex formulation Solution to the general nonconvex formulation 4 Conclusions

Outline 1 Introduction 2 Warm-Up: Markowitz Portfolio Signal model Markowitz formulation Drawbacks of Markowitz portfolio 3 Risk Parity Portfolio Problem formulation Solution to the naive diagonal formulation Solution to the vanilla convex formulation Solution to the general nonconvex formulation 4 Conclusions

Returns months, 5-min intervals, etc. Risk Parity Portfolio D. Palomar (HKUST) Engineering . Foundations and Trends in Signal Processing, Now Publishers, 2016. 2 Y. Feng and D. P. Palomar, A Signal Processing Perspective on Financial Let us denote the log-returns of N assets at time t with the vector covariance matrix denoted as (Feng and Palomar 2016) 2 The time index t can denote any arbitrary period such as days, weeks, Note that the log-returns are almost the same as the linear returns 7 / 81 r t ∈ R N (i.e., r it = log p i , t − log p i , t − 1 ). R it = p i , t − p i , t − 1 , i.e., r it ≈ R it . p i , t − 1 F t − 1 denotes the previous historical data. Econometrics aims at modeling r t conditional on F t − 1 . r t is a multivariate stochastic process with conditional mean and µ t ≜ E [ r t | F t − 1 ] [ ] ( r t − µ t )( r t − µ t ) T | F t − 1 Σ t ≜ Cov [ r t | F t − 1 ] = E .

i.i.d. model (which is not very innacurate in general). That is, both the conditional mean and conditional covariance are constant : Very simple model, however, it is one of the most fundamental assumptions for many important works, e.g., the Nobel prize-winning Markowitz portfolio theory (Markowitz 1952) 3 . 3 H. Markowitz, “Portfolio selection,” J. Financ. , vol. 7, no. 1, pp. 77–91, 1952. D. Palomar (HKUST) Risk Parity Portfolio 8 / 81 For simplicity we will assume that r t follows an i.i.d. distribution µ t = µ , Σ t = Σ .

Parameter estimation T Risk Parity Portfolio D. Palomar (HKUST) estimators, Black-Litterman estimators, etc. Many more sophisticated estimators exist, namely: shrinkage 1 Consider the i.i.d. model: sample covariance matrix : 9 / 81 sample mean : The simplest estimators are the sample estimators: practice based on T observations. r t = µ + w t , where µ ∈ R N is the mean and w t ∈ R N is an i.i.d. process with zero mean and constant covariance matrix Σ . The mean vector µ and covariance matrix Σ have to be estimated in ∑ T µ = 1 ˆ t = 1 r t ˆ ∑ T Σ = t = 1 ( r t − ˆ µ )( r t − ˆ µ ) T . T − 1

Parameter estimation and a portfolio design (e.g., Markowitz mean-variance) based on Risk Parity Portfolio D. Palomar (HKUST) covariances on optimal portfolio choice,” Journal of Portfolio Management , 1993. 5 V. Chopra and W. Ziemba, “The efgect of errors in means, variances and 4 A. Meucci, Risk and Asset Allocation . Springer, 2005. used by practitioners. Indeed, this is why Markowitz portfolio and other extensions are rarely those estimates can be severely afgected (Chopra and Ziemba 1993) 5 . As a consequence, the estimates contain too much estimation error lack of stationarity of data. unavailability of data or In practice, T cannot be large enough due to either: estimator, with very noisy estimates (Meucci 2005) 4 . For instance, the sample mean is particularly a very ineffjcient otherwise the estimation error is unacceptable. 10 / 81 µ and ˆ The parameter estimates ˆ Σ are only good for large T ,

Outline 1 Introduction 2 Warm-Up: Markowitz Portfolio Signal model Markowitz formulation Drawbacks of Markowitz portfolio 3 Risk Parity Portfolio Problem formulation Solution to the naive diagonal formulation Solution to the vanilla convex formulation Solution to the general nonconvex formulation 4 Conclusions

Portfolio return B Risk Parity Portfolio D. Palomar (HKUST) respectively. N N Suppose the capital budget is B dollars. 12 / 81 assets). R p Then the portfolio return is The portfolio w ∈ R N denotes the normalized dollar weights of the N assets such that 1 T w = 1 (so B w denotes dollars invested in the For each asset i , the initial wealth is Bw i and the end wealth is Bw i ( p i , t / p i , t − 1 ) = Bw i ( R it + 1 ) . ∑ N i = 1 Bw i ( R it + 1 ) − B ∑ ∑ t = = w i R it ≈ w i r it = w T r t i = 1 i = 1 The portfolio expected return and variance are w T µ and w T Σ w ,

Performance measures Treasury bill). Risk Parity Portfolio D. Palomar (HKUST) expected value of the loss above some quantile. ES (Expected Shortfall) or CVaR (Conditional Value at Risk) : VaR (Value at Risk) : quantile of the loss. Drawdown : decline from a historical peak of the cumulative profjt 13 / 81 Sharpe Ratio (SR) : expected return per unit of risk Volatility : Expected return : w T µ √ w T Σ w SR = w T µ − r f √ w T Σ w where r f is the risk-free rate (e.g., interest rate on a three-month U.S. Information Ratio (IR) : SR with r f = 0. X ( t ) : D ( T ) = max t ∈ [ 0 , T ] X ( t ) − X ( T )

Practical constraints Capital budget constraint : Long-only constraint : Dollar-neutral or self-fjnancing constraint : Holding constraint : positions, respectively. D. Palomar (HKUST) Risk Parity Portfolio 14 / 81 1 T w = 1 . w ≥ 0 . 1 T w = 0 . l ≤ w ≤ u where l ∈ R N and u ∈ R N are lower and upper bounds of the asset

Practical constraints Leverage constraint : Cardinality constraint : Turnover constraint : Market-neutral constraint : D. Palomar (HKUST) Risk Parity Portfolio 15 / 81 ∥ w ∥ 1 ≤ L . ∥ w ∥ 0 ≤ K . ∥ w − w 0 ∥ 1 ≤ u where w 0 is the currently held portfolio. β T w = 0 .

Risk control the average benefjt. However, in practice, the average performance is not enough to characterize an investment and one needs to control the probability of going bankrupt . Risk measures control how risky an investment strategy is. The most basic measure of risk is given by the variance (Markowitz 1952) 6 : a higher variance means that there are large peaks in the distribution which may cause a big loss. There are more sophisticated risk measures such as downside risk , VaR , ES , etc. 6 H. Markowitz, “Portfolio selection,” J. Financ. , vol. 7, no. 1, pp. 77–91, 1952. D. Palomar (HKUST) Risk Parity Portfolio 16 / 81 In fjnance, the expected return w T µ is very relevant as it quantifjes

Mean-variance tradeofg multi-objective optimization problem. Risk Parity Portfolio D. Palomar (HKUST) agressive or risk-averse the investor is. The choice of a specifjc point in this tradeofg curve depends on how curve). They defjne a fundamental mean-variance tradeofg curve (Pareto Thus, we are faced with two objectives to be optimized: it is a vice-versa. Usually, the higher the mean return the higher the variance and important performance measures. the standard deviation or volatility 17 / 81 The mean return w T µ and the variance (risk) w T Σ w (equivalently, √ w T Σ w ) constitute two

Mean-variance tradeofg D. Palomar (HKUST) Risk Parity Portfolio 18 / 81