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