Portfolio Optimization
University of Essex26th November June 2010
Dr Amadeo AlentornHead of Quantitative [email protected]
CF963, Autumn Term 2010-11Learning and Computational Intelligence in Economics and Finance
Part 1: Introduction to quantitative investing in hedge funds
Part 2: The problem of portfolio optimisation
Part 3: Application of heuristic portfolio optimization
3
OMAM at a glance
> Dynamic investment firm, focused on high performance and absolute returns
> Independent investment teams – no house view or investment style
> Expertise in
– Discretionary equities
– Fixed income
– Quantitative strategies
> Range of long only and alternative products designed to meet the needs of retail and institutional investors
> New product innovation
> Independent risk management team
> USD6.5 billion assets under management*
> Part of Old Mutual Group, a constituent of the FTSE 100 index.
* Source: OMAM, as at 30/10/09
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OMAM Quantitative Strategies Group
Investment team
Investment research
Portfolio construction and management
V-Lab: PhDand post-doctoral researchers
Systems development
Academic Advisory Board
5
Equity investing styles: discretionary/systematic spectrum
> Discretionary investing
– Mostly driven by fundamental analysis
– Analysis of company’s accounts, business plans, competitors, etc
– Meeting management and understanding business model
– Usually focus in one country/ sector/ industry
– Longer holding horizon
> Quantitative investing
– Factor models based on both fundamental and technical analysis
– Able to analyse and evaluate 1000s of companies
– Highly diversified portfolios
– Medium term holding horizon
> Statistical arbitrage
– Usually based on technical analysis, price driven (i.e. pairs trading)
– High frequency execution, usually intra-day trading
– Short holding horizon
> Across the industry, many different variations: “quantamentals”, quant overlays, etc.
Discretionary
Systematic
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Types of equity funds: passive/active spectrum
> Index tracking
– Tracking error TE < 0.25%
> Enhanced long only
– TE between 0.50 to 1.50%
> Active long only
– TE greater than 2.00%
> Short extension (130/30)
– Equivalent to index plus 30/30 market neutral
> Market neutral hedge funds
– Dollar neutral, i.e. 100/100
> Long/short hedge funds
– Variable net exposure
> Trade-off between risk and expected return.
Passive
Active
Long only funds
Hedge funds
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Types of equity funds: long, short and net positions
> Given $100 of capital, the structure of the portfolio will be very different, for different types of funds.
-150
-100
-50
0
50
100
150
200
Long only Shortextension
Marketneutral
Long short
Long book
Short book
Net position
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Performance of a market neutral fund vs. market> Market risk is the primary source of risk for a long only equity fund in absolute terms
– When equity markets fall by 30%, the best outcome for a LO fund is -30% + TE.
– Market neutral designed to deliver positive returns in all market environments
Jan94 Sep95 May97 Dec98 Aug00 Apr02 Nov03 Jul05 Feb07 Oct08
100
Return Risk Sharpe Sortino
MSCI World Index 3.1 15.5 0.20 0.28
Market neutral hedge fund 11.0 4.6 2.37 3.63
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Short-selling
> Traditionally investing (long only constraint) can create miss-pricings, because only positive views about stocks can be fully reflected into prices
> Hedge funds are able to reflect “buy” and “sell” ideas into the portfolios, by buying stocks with positive forecasts, and “shorting” stocks with negative forecasts
> How does it work?
– Broker/dealers on behalf of clients identify stock owners (i.e. pension funds)
– Hedge fund borrows stock, and sells it on the open market
– At a future date, hedge fund buys back the stock, and returns it to the lender
– Stock owner receives a lending fee for the service
– Note: “naked” short selling is banned!
> Risks
– Stock recall
– Short squeezes: when prices go up, rush to cover positions
– Potential of infinite loss
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Short-selling: Northern Rock case study
Northern Rock Plc
0
10
20
30
40
50
60
70
80
90
100
Jun0
7
Aug
07
Oct
07
Dec
07
Feb
08
Apr
08
Jun0
8
Aug
08
Oct
08
Dec
08
Feb
09
Apr
09
Jun0
9
Aug
09
Oct
09
Dec
09
Feb
10
Apr
10
Jun1
0
0
200
400
600
800
1000
1200
Utilisation Price
Source: Data Explorers www.dataexplorers.com
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Types of equity hedge funds
> Hedge fund indices have become a useful tool for monitoring the performance of the different hedge fund strategies
– But issues with survivorship bias, self-selection bias (voluntary reporting), etc
> Useful indication of risk associated with each strategy
Source: Hedge Fund Research www.hedgefundresearch.com
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Our investment approach and philosophy
> Market inefficiencies result from behavioural biases and structural anomalies
> Investment insights, not statistics, drive research
– Many of the investment strategies are not different than those used by fundamental fund managers (i.e. valuation, earnings quality, etc)
> Rigorous backtesting of data to validate investment criteria
– One of the key advantages over fundamental managers, we are able to test the historic performance of an investment strategy
> Continuous research to maintain and enhance information ratio
– To mitigate risk of imitation and of crowded trades
> Multi factor models designed to perform throughout the business cycle
– Factor modelling lends itself to market neutral investing, generating both buys and sells
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Investment strategies based on behavioural finance
> Financial markets are complex systems
– Prices are the result of complex interactions between market participants
> Academic literature on behavioural finance aims to provide an understanding of how human behaviour influence prices, and helps to explain some of the observed miss-pricings and anomalies:
– Overconfidence: “all traders are above average”, over-trading
– Prospect theory: people value gains and losses differently
– Anchoring
– Under-reaction/ overreaction / herding
– Naïve extrapolation
> A solid understanding of the rationale for an observed market “anomaly” from a behavioural finance point of view, when researching investment strategies, helps to avoid falling into “data mining” traps.
> Quantitative investment allows to remove emotion from the investment process
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Academic research vs. industry research
> Liquidity
– Many academic studies use the CRSP database as the universe of stocks on which to test a hypothesis, up to 26,000 stocks.
– In practice, only around 3,500 stocks worldwide with enough liquidity to invest with negligible market impact.
> Survivorship bias
– Important to include companies that are no longer in the universe (impact of bankruptcies/corporate actions)
> Reporting lags
– Point in time databases.
> Backfilling
– See “Rewriting history”
> Transaction costs
– Specially for high frequency strategies
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Factors: academic debate alpha vs. risk? (Fama French)
1930 1940 1950 1960 1970 1980 1990 2000 2010
100
101
102
Rtrn Rsk IR Sortino
Market 5.6 18.9 0.30 0.45
Size 2.3 11.5 0.20 0.34Value 4.2 12.4 0.34 0.59
RF 3.6
Source: Kenneth French website
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Quantitative investment process
> Fully systematic investment process.
Universe: top 95% of market capitalisation in each country: around 3,500 stocks1
Stocks ranked by
attractiveness
Optimise risk and return profile of portfolio
Finalportfolio
Multi factor stock
selection criteria
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Portfolio optimisation process
Trade list
Trade cost
model
Relative return model
Risk model
Optimisation and portfolioconstruction
Continuous research on all three proprietary models and process
Part 1: Introduction to quantitative investing in hedge funds
Part 2: The problem of portfolio optimisation
Part 3: Application of heuristic portfolio optimization
19 19
What’s portfolio optimisation?
> Given a universe of stocks, and a set of expectations about the future performance of these stocks, the problem is how to construct a portfolio by selecting how to allocate the capital.
> The classic model for portfolio construction is the mean-variance optimisation introduced by Markowitz in 1952.
> A portfolio optimization problem typically consists in maximising return, minimising risk or maximising utility by finding the optimal set of stock weights (i.e., percentages of invested capital) that satisfies a set of constraints.
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The mean-variance frontier
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The traditional optimisation problem
> The traditional portfolio optimisation problem with no shorting allows only positive (long) stock positions:
> The mean-variance objective function represents a trade-off between expected return and expected risk, weighted by a risk aversion parameter λ:
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Long vs short stock positions
> Long: a long position means the investor has gone to the market and
bought some shares in a company. The proportion of the value of these
shares of the total portfolio is the portfolio weight of that company in the
investor’s portfolio, represented by a positive portfolio weight.
> Short: a short position is achieved when an investor “sells” shares that
does not own, and this is represented by a negative portfolio weight.
23 23
The optimisation problem for a market neutral hedge fund
> For a market neutral hedge fund, we remove the no-shorting constraint to allow negative weights:
> Objective functions:
> Mean Variance
> Mean Value-At-Risk
24 24
Why heuristic optimisation?
> Stock returns exhibit non-normal characteristics, including skewness and fat tails, and most investors are loss averse
> These invalidate the assumptions in the traditional approach.
> Heuristic optimisation methods provide a more flexible toolset where no simplifying assumptions are needed.
> Some of the applications have successfully achieved:
> The use of non-quadratic risk measures such as Value at Risk
> Incorporation of integer constraints, such as cardinality constraints.
> Heuristic algorithms already used to tackle this problem:
– Genetic Algorithms, and
– Threshold Accepting algorithms
> In the last part of the lecture, we will look at a new heuristic algorithm: the GNMA
Part 1: Introduction to quantitative investing in hedge funds
Part 2: The problem of portfolio optimisation
Part 3: Application of heuristic portfolio optimization
26 26
Introduction
> This work is based on a recent paper presented at UKCI
– “Heuristic Portfolio Optimisation for a Hedge Fund Strategy using the Geometric Nelder-Mead Algorithm”
> We present a framework for implementing a heuristic portfolio optimisation framework for a market neutral hedge fund investment strategy.
> We also illustrates the application of the recently developed Geometric Nelder-Mead Algorithm (GNMA) in solving this real world optimization problem, and compare it with a Genetic Algorithm (GA) approach.
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Nelder-Mead Algorithm (NMA)
> Numerical optimization method published in 1965 by Nelder and Mead!
> Heavily used by practitioners in optimisation
> Cousin of Particle Swarm Optimization (PSO) and Differential Evolution (DE):
- it does not use derivative information
- it can be seen as a population based evolutionary algorithms with special operators
> Its search is explicitly geometric and attempts to adapt to the fitness landscape
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Operations in geometric transformation
> NMA applies these 4 operations until convergence:
> Worst solution reflected about the centre
> If R is the best, expand in this direction
> If R is the worst, we may have “overshoot”
> If nothing has worked, shrink towards best
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Geometric Generalization & Specification
> Recently, Moraglio and Johnson (2010) generalised the NMA from continuous to
combinatorial spaces.
> Reflection, expansion, contraction, shrinking operations can be naturally defined in
terms of distance relations among points and generalized to spaces equipped with a
formal notion of distance
> By plugging a new distance in their definitions, we can derive the same operators for a
new space: by using the Hamming distance on binary strings, we can derive these
operators for binary strings
A. Moraglio and C. Johnson. Geometric generalization of the nelder mead algorithm. In Proceedings of the 10th European Conference on Evolutionary Computation in Combinatorial Optimization, 2010.
31 31
Binary Encoding of Portfolio
> Fixed set of reference stocks
> The composition of the portfolio is represented by a binary string
> Each stock is associated with two bits:
> Presence bit: stock included or not
> Sign bit: long (0) or short (1) position
> Quantity of stock: budget equally partitioned on shorts and longs to have a neutral portfolio
0 0 1 0 0 0 0 0 1 1
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Data
> Daily stock returns for the 100 names of the FTSE100 index between Jan 2007 and Dec 2009
> Covariance matrix based on full covariance with a rolling window of 250 days
> VaR based on the historical method, the 99% VaR is the third worst point over the last 250 days
> Stock return forecasts are such that by construction reflect some level of predictive power (5%), with 95% pure noise.
33 33
Experiments
> Two optimisation algorithms:
> GA vs GNMA
> Interested in fitness and execution time
> Two objective functions (strategies):
> Mean-variance vs mean-VaR
> Risk aversion parameters were calibrated so that the risk levels of the two strategies were similar
> Interested in differences in investment performance
> A total of 4 strategies
> Portfolios optimised over 250 days, average 10 runs.
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Results: quality of solutions
> Similar fitness levels achieved by both algorithms
> Similar execution time taken to achieve best fitness by both algorithms
> The new GNMA algorithm is comparable with the established GA approach.
> Realised risks across strategies is comparable.
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Results: investment performance
36 36
Findings of the paper
> Portfolios optimised for VaR deliver higher returns, as they take advantage of the asymmetry in the risk penalty function
> Better reflects investor preferences about loss aversion and downside risk
> The results validate the models chosen:
> Return: we have shown that a low level of information content in return forecasts (95% of noise) delivers realistic positive returns
> Risk: forecasted vs. realised risks are in line
> We have shown for the first time how the recently developed GNMA method is suitable for tacking a real world problem, delivering a performance in line with the GA approach.
> Optimising using VaR instead of variance as the risk measure for a market neutral portfolio limits downside risk and is able to deliver higher returns, while variance limits both downside and upside potential and therefore results in lower returns.
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Conclusion of the lecture
> Overview of the world of quantitative investing in practice
> Issues encountered in portfolio optimization:
– Size of problems: hard to move away from traditional frameworks
– Number and types of constraints
– Inputs are estimates: how to incorporate errors?
> Companies doing interesting work in portfolio optimisation:
– Market leader in portfolio optimisation software: www.axioma.com
– Robust portfolio optimisation software: http://bitarisk.cor-fs.com/
– Heuristic methods for random portfolio analysis: www.portfolioprobe.com
Thank you! Questions?
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