In braverock/PortfolioAnalytics: Portfolio Analysis, Including Numerical Methods for Optimization of Portfolios

% Constructing Portfolios of Dynamic Strategies using Downside Risk Measures % Peter Carl, Hedge Fund Strategies, William Blair & Co. % November 11, 2013

# R code here

Introduction

• Discuss the challenges of constructing hedge fund portfolios
• Offer a framework for considering strategic allocation of hedge funds
• Discuss various methods of evaluating portfolio objectives
• Show the relative performance of multiple objectives
• Discuss extensions to the framework

Objectives

• Identify several sets of objectives to establish benchmark and target portfolios
• Evaluate complex constraints, including some that equalize or budget risks using downside measures of risk
• Visualize portfolio problems to build intuition about objectives and constraints
• Use analytic solvers and parallel computation as problems get more complex <!--
• Rebalancing periodically and examining out of sample performance will help refine objectives -->

Strategic allocation

...broadly described as periodically reallocating the portfolio to achieve a long-term goal

• Understand the nature and sources of investment risk within the portfolio
• Manage the resulting balance of risk and return of the portfolio
• Apply within the context of the current economic and market situation
• Think systematically about preferences and constraints

Here we'll consider a strategic allocation to hedge funds

Selected hedge fund strategies

Monthly data of EDHEC hedge fund indexes from 1998

Relative Value

• Fixed Income Arb
• Convertible Arb
• Equity Market Neutral
• Event Driven

---

Directional

• Equity Long/Short
• Global Macro
• CTA

Index Performance

\includegraphics[width=1.0\textwidth]{../results/EDHEC-Cumulative-Returns.png}

Index Performance

\includegraphics[width=1.0\textwidth]{../results/EDHEC-RollPerf.png}

Index Performance

Add table of relevant statistics here

system('cat results/EDHEC-inception-cor.md')

Ex-post Correlations

\includegraphics[width=0.5\textwidth]{../results/EDHEC-cor-inception.png} \includegraphics[width=0.5\textwidth]{../results/EDHEC-cor-tr36m.png}

Investor preferences...

In constructing a portfolio, most investors would prefer:

• to be approximately correct rather than precisely wrong
• the flexibility to define any kind of objective and combine the constraints
• to define risk as potential loss rather than volatility
• a framework for considering different sets of portfolio constraints for comparison through time
• to intuitively understand optimization through visualization

... Lead to portfolio preferences

Construct a portfolio that:

• maximizes return,
• with per-asset position limits,
• with a specific univariate portfolio risk limit or target,
• defines risk as losses,
• considers the effects of skewness and kurtosis, and
• either limits contribution of risk for constituents or
• equalizes component risk contribution.

Risk budgeting

• Used to allocate the "risk" of a portfolio
• Decomposes the total portfolio risk into the risk contribution of each component position
• Literature on risk contribution has focused on volatility rather than downside risk
• Most financial returns series seem non-normal, so we want to consider the effects of higher moments

Return distributions

\includegraphics[width=1.0\textwidth]{../results/EDHEC-Distributions.png}

Return distributions

\includegraphics[width=1.0\textwidth]{../results/EDHEC-Distributions2.png}

Return autocorrelation

\includegraphics[width=1.0\textwidth]{../results/EDHEC-ACStats.png}

Return autocorrelation

\includegraphics[width=1.0\textwidth]{../results/EDHEC-ACStackedBars.png}

Measuring risk, not volatility

Measure risk with Conditional Value-at-Risk (CVaR)

• Also called Expected Tail Loss (ETL) and Expected Shortfall (ES)
• ETL is the mean expected loss when the loss exceeds the VaR
• ETL has all the properties a risk measure should have to be coherent and is a convex function of the portfolio weights
• To account for skew and/or kurtosis, use Cornish-Fisher (or "modified") estimates of ETL instead (mETL)

Measuring risk - directional strategies

\includegraphics[width=1.0\textwidth]{../results/EDHEC-BarVaR.png}

Measuring risk - non-directional strategies

\includegraphics[width=1.0\textwidth]{../results/EDHEC-BarVaR2.png}

ETL sensitivity

\includegraphics[width=1.0\textwidth]{../results/EDHEC-ETL-sensitivity.png}

Ex ante, not ex post

Ex post analysis of risk contribution has been around for a while

• Litterman ()

The use of ex ante risk budgets is more recent

• Qian (2005): "risk parity portfolio" allocates portfolio variance equally
• Maillard et al (2010): "equally-weighted risk contribution portfolio" or (ERC)
• Zhu et al (2010): optimal mean-variance portfolio selection under constrained contributions

We want to look at the allocation of risk through ex ante downside risk contribution

Contribution to downside risk

Use the modified CVaR contribution estimator from Boudt, et al (2008)

• CVaR contributions correspond to the conditional expectation of the return of the portfolio component when the portfolio loss is larger than its VaR loss.
• %CmETL is the ratio of the expected return on the position when the portfolio experiences a beyond-VaR loss to the expected value of the portfolio loss
• A high positive %CmETL indicates the position has a large loss when the portfolio also has a large loss

Contribution to downside risk

• The higher the percentage mETL, the more the portfolio downside risk is concentrated on that asset
• Allows us to directly optimize downside risk diversification
• Lends itself to a simple algorithm that computes both CVaR and component CVaR in less than a second, even for large portfolios

We can use CVaR contributions as an objective or constraint in portfolio optimization

Two strategies for using downside contribution in allocation

Equalize downside risk contribution

• Define downside risk diversification as an objective

Downside risk budget

• Impose bound constraints on the percentage mETL contributions

Start with some general constraints

Constraints specified for each asset in the portfolio:

• Maximum position: 30%
• Minimum position: 5%
• Weights sum to 100%
• Group constraints
• Rebalancing quarterly

Estimates

Table of Return, Volatility, Skew, Kurt, and Correlations by asset

Define multiple objectives

Equal contribution to:

• weight
• variance
• risk

Reward to risk:

• mean-variance
• mean-modified ETL

Minimum:

• variance
• modified ETL

Equal-weight portfolio

• Provides a benchmark to evaluate the performance of an optimized portfolio against
• Each asset in the portfolio is purchased in the same quantity at the beginning of the period
• The portfolio is rebalanced back to equal weight at the beginning of the next period
• Implies no information about return or risk
• Is the re-weighting adding or subtracting value?
• Do we have a useful view of return and risk?

Contribution of Risk in Equal Weight Portfolio

insert table

Equal Contribution to Risk

The risk parity constraint that requires all assets to contribute to risk equally is usually too restrictive.

• Use the Minimum Concentration Component (MCC) risk contribution portfolio as an objective
• Minimize the largest ETL risk contribution in the portfolio
• Unconstrained, the MCC generates similar portfolios to the risk parity portfolio
• The MCC can, however, be more easily be combined with other objectives and constraints

Constrained Risk Contribution

Risk Budget as an eighth objective set

• Drop the position constraints altogether
• No non-directional constituent may contribute more than 40% to portfolio risk
• No directional constituent may contribute more than 30% to portfolio risk, except for...
• ... Distressed, which cannot contribute more than 15%
• Directional, as a group, may not contribute more than 60% of the risk to the portfolio

Optimizers

Closed-form

• Linear programming (LP) and mixed integer linear programming (MILP)
• Quadratic programming

General Purpose Continuous Solvers

• Random portfolios
• Differential evolution
• Partical swarm
• Simulated annealing

Random Portfolios

From a portfolio seed, generate random permutations of weights that meet your constraints

• Several methods: Burns (2009), Shaw (2010), and Gilli, et al (2011)
• Covers the 'edge case' (min/max) constraints well
• Covers the 'interior' portfolios
• Useful for finding the search space for an optimizer
• Allows arbitrary number of samples
• Allows massively parallel execution

Sampling can help provide insight into the goals and constraints of the optimization

Sampled portfolios

\includegraphics[width=1.0\textwidth]{../results/RP-EqWgt-MeanSD-ExAnte.png}

Sampled portfolios

\includegraphics[width=1.0\textwidth]{../results/RP-Assets-MeanSD-ExAnte.png}

Sampled portfolios with multiple objectives

\includegraphics[width=1.0\textwidth]{../results/RP-BUOY-MeanSD-ExAnte.png}

Modified ETL instead of volatility

\includegraphics[width=1.0\textwidth]{../results/RP-BUOYS-mETL-ExAnte.png}

Ex-ante results

\includegraphics[width=1.0\textwidth]{../results/Weights-Buoys.png}

Risk contribution

\includegraphics[width=1.0\textwidth]{../results/mETL-CumulPerc-Contrib-Buoys.png}

Conclusions

As a framework for strategic allocation:

• Component contribution to risk is a useful tool
• Random Portfolios can help you build intuition about your objectives and constraints
• Rebalancing periodically and examining out of sample performance can help you refine objectives
• Differential Optimization and parallelization are valuable as objectives get more complicated

R Packages used

PortfolioAnalytics

• Unifies the interface across different closed-form optimizers and several analytical solvers
• Implements three methods for generating Random Portfolios, including 'sample', 'simplex', and 'grid'
• Preserves the flexibility to define any kind of objective and constraint
• Work-in-progress, available on R-Forge in the ReturnAnalytics project

PerformanceAnalytics

• Returns-based analysis of performance and risk for financial instruments and portfolios, available on CRAN

Packages for Mathematical Programming Solvers

ROI

• Infrastructure package by K. Hornik, D. Meyer, and S. Theussl for optimization that facilitates use of different solvers...

RGLPK

• ... such as GLPK, open source software for solving large-scale linear programming (LP), mixed integer linear programming (MILP) and other related problems

quadprog

• ... or this one, used for solving quadratic programming problems

Packages for Generalized Continuous Solvers

DEoptim

• Implements Differential Evolution, a very powerful, elegant, population based stochastic function minimizer

GenSA

• Implements functions for Generalized Simulated Annealing

pso

• An implementation of Partical Swarm Optimization consistent with the standard PSO 2007/2011 by Maurice Clerc, et al.

Packages for more iron

foreach

• Steve Weston's parallel computing framework, which maps functions to data and aggregates results in parallel across multiple CPU cores and computers.

doRedis

• A companion package to foreach by Bryan Lewis that implements a simple but very flexible parallel back end to Redis, making it to run parallel jobs across multiple R sessions.

doMPI

• Another companion to foreach that provides a parallel backend across cores using the parallel package

Thanks

• Brian Peterson - Trading Partner at DV Trading, Chicago
• Kris Boudt - Faculty of Business and Economics, KU Leuven and VU University Amsterdam
• Doug Martin - Professor and Director of Computational Finance, University of Washington
• Ross Bennett - Student in the University of Washington's MS-CFRM program and GSOC participant

References

Figure out bibtex links in markup

http://www.portfolioprobe.com/about/random-portfolios-in-finance/

Appendix

Slides after this point are not likely to be included in the final presentation

Differential Evolution

All numerical optimizations are a tradeoff between speed and accuracy

Differential evolution will get more directed with each generation, rather than the uniform search of random portfolios

Allows more logical 'space' to be searched with the same number of trial portfolios for more complex objectives

doesn't test many portfolios on the interior of the portfolio space

Early generations search a wider space; later generations increasingly focus on the space that is near-optimal

Random jumps are performed in every generation to avoid local minima

Insert Chart

Other Heuristic Methods

GenSA, SOMA,

Ex-ante vs. ex-post results

scatter plot with both overlaid

Turnover from equal-weight

scatter chart colored by degree of turnover

Degree of Concentration

scatter chart of RP colored by degree of concentration (HHI)

Scratch

Slides likely to be deleted after this point

braverock/PortfolioAnalytics documentation built on April 18, 2024, 4:09 a.m.