Multi-Portfolio Optimization

Multi-Portfolio Optimization

Carisma 2008July 1, 2008

Pamela VanceAxioma

Multi-Portfolio Optimization

ConstraintsAggregated Portfolio

ConstraintsPortfolio

# 1


# 2


# 3

Individual Po

rtfolios

Objective #1 Objective #2Objective #3Joint

Individual Portfolios Joint

Aggregate

Constraints

Objective (for example Expected Return – Joint “cost”)

Applications

• Separately Managed Accounts– Rebalance Separately Managed Accounts Simultaneously

• Allows for the consideration of aggregate market impact

• Taxable Portfolio Management– Consider optimal tax management across several accounts held by the

same individual

• Mixing Portfolios– Mix “factor” portfolios in an integrated optimization framework (rather

than mixing the signals into a single alpha)

• Hedging– Aggregate hedging of multiple books

Separately Managed Accounts

Why use SMAs?– Provide the asset owner with flexibility for specifying special

circumstances

– Each account is managed “separately” with added flexibility

– Fees are charged accordingly

However– The rebalancing process does not scale well

– Market-impact is underestimated because accounts are traded together

Hence– Accounts are “spread” throughout the rebalancing cycle

– Similar accounts are rebalanced on the same day and traded together

Including Market Impact in Optimization

• Common market impact models used in portfolio optimization• Convex, increasing functions of the size of the trade (in $)

– Piece-wise linear– Quadratic– 3/2

• Incorporating a 3/2 market impact model results in the solution of a second order cone programming problem

• When trading multiple accounts, the market impact depends on the aggregate trades from all accounts

• See Almgren et al, “Equity Market Impact”, Risk, July 2005

• See Takriti and Tierens, “Incorporating Transaction Costs in the Construction of 130/30 Portfolios”, The Journal of Trading (Forthcoming)

Current Practice for Managing SMAs

Trading SMAs– Since trades are “similar” across accounts and rebalanced the same

day, they are pooled together for “efficient” execution

– Portfolio Manager communicates all trades together after rebalancing all the relevant accounts for the day

However– Trading accounts together creates additional market impact

– Trades need to be “allocated” across accounts that participate in the rebalancing

But– Naïve trade allocation can be “unfair” and introduce “bias”

– Optimal trades for each account would have been different if realistic trading assumptions had been incorporated for each individual rebalancing

Example: Optimizing Accounts Individually

• Two accounts start from cash• Portfolios are long-only with a 8% absolute risk constraint• Each account is rebalanced independently, and estimates of

market impact for each account are considered separately• Trades are then pooled together• Account 1 significantly underestimated market impact

Account 1 Account 2

Size $100M $10B

Expected Return (%) 11.22 9.48

Expected Market Impact (%) 0.0964 1.8098

Actual Market Impact (%) 3.1955 1.8338

Objective (%) 11.12 7.67

Actual Objective (%) 8.02 7.64

Change in Objective (%) -27.9 -0.4

Alternative Solutions with a Hidden Bias

• Optimize different accounts on different days– Alphas could be stale– Accounts traded on one day may (negatively) affect accounts

traded on subsequent days

• Optimize representative account and allocate trades pro-rata– Not always fair to allocate proportionally to size of account– Complexities arise with account-specific constraints

• Use an iterative heuristic approach to adjust market-impact estimates– Iterative algorithm may not converge– Algorithm may be stopped at a suboptimal (and biased) solution

Proposed Solutions that Aggregate Accounts

• Goals– Enforce compliance or specific constraints of each separate account– Maximize utility or expected return of each separate account– Consider the aggregate effects of trading all accounts together– Ensure that the process is “fair” and “unbiased”

• Collusive solution: First proposed by O’Cinneide, Scherer, and Xu (DBAM - 2005)– Solve the “multi-portfolio optimization” problem as an integrated optimization

problem• Aggregate objective is sum of individual account objectives • May contain additional constraints that go across accounts

• Cournot-Nash equilibrium: First proposed by Augustin Cournot, Researches into the Mathematical Principles of the Theory of Wealth (1897); John Nash, Non-cooperative games (1950)– Each account is optimized individually with knowledge of the trade amounts

amongst other accounts. Other trades are assumed to be fixed and are used to get better estimate of market impact.

Collusive Solution

Collusive Solution (Social Planner)O’Cinneide, Scherer, and Xu (DBAM 2006)

• Investors collude to maximize total welfare

• Total market-impact cost is included in objective

• Actual market-impact cost is exactly what is expected by each investor*– Maximizew jA j

Twj – (jA wj)T c(jA wj)

• Optimality conditions for each investor are not those that represent investors’ own interests– Creates potential of threat for accounts to leave collusion.

• Does have the advantage of reducing account dispersion

Market-Impact Modeling

• Possible methods for aggregating trades and computing aggregate market impact

– Net Buys and Sells– Buys Plus Sells– Max of Buys and Sells

• Decision should be dependent on how trades are executed

– Are accounts allowed to cross internally?– Are buys and sells sent to same broker for potential cross?– Are buys and sells sent to different brokers?

Collusive Approach - Properties

• Critical to have unique solution– Market-impact term in collusive case isn’t strictly convex– Maximizing sum of utilities introduces strict convexity– Can generate unfair trades when maximizing alpha (no risk included in the

objective) if constraints do not force uniqueness

• Effects of Market-impact modeling– Using net trades can have “Marxist effect” – “From each according to his

ability, to each according to his need.”– Independent buy and sell impact introduces fairness, but may overestimate

total cost of trading

• Identical initial accounts remain identical after rebalancing

• Pareto-optimal

• Infeasibilities may create unfairness– Forced trades increases entry cost for other accounts

• Same issues under Average Daily Volume constraints

Collusive Solution

• Collusive solution provides the best aggregate objective value• However, Account 1 can produce a better objective acting alone• Hence, Account 1 is negatively impacted in order to increase total welfare• Accounts are identical in the collusive solution

Individual Solution Collusive Solution

Account 1 Account 2 Account 1 Account 2

Size $100M $10B $100M $10B

Expected Return (%) 11.22 9.48 9.46 9.46

Actual Market Impact (%) 3.1955 1.8338 1.8081 1.8081

Actual Objective (%) 8.02 7.64 7.65 7.65

Aggregate Objective (%) 7.6469 7.6562

Collusive Approach: Non-Unique Solution

• Collusive approach maximizes aggregate objectives but is not guaranteed to be fair in the presence of multiple optimum solutions

• Example maximizes alpha less market-impact of net trades s.t. asset bounds of [25,75]. Each of the solutions has the same collusive objective.

• Considering independent Buy and Sell Impact in a collusive approach also does not guarantee uniqueness of solution

Collusive Solutions

Solution 1 Solution 2 Solution 3

Account 1 Account 2 Account 1 Account 2 Account 1 Account 2

Asset 1 50 50 25 75 75 25

Asset 2 50 50 75 25 25 75

Infeasibility Issues – Net Trade Impact

• If initial holdings are infeasible then rebalancing in a collusive approach with market impact computed on net trades can be unfair

• Account 1 moves away from optimal solution in order to reduce impact

Collusive with Net Trade

Initial Holdings Final Holdings


Scenario 1 (Optimal) (Infeasible) (Feasible) (Feasible)

Asset 1 75 0 65 65

Asset 2 25 100 35 35


Scenario 2 (Feasible) (Feasible) (Feasible) (Feasible)

Asset 1 75 25 70 70

Asset 2 25 75 30 30

Infeasibility Issues – Buy/Sell Impact

• If initial holdings are infeasible, then rebalancing in a collusive approach with independent Buy/Sell impact doesn’t negatively impact the other account

Collusive with Buy/Sell



Scenario 1 (Optimal) (Infeasible) (Optimal) (Feasible)

Asset 1 75 0 75 62.5

Asset 2 25 100 25 37.5


Scenario 2 (Optimal) (Feasible) (Optimal) (Feasible)

Asset 1 75 25 75 69

Asset 2 25 75 25 31

Threat to Collusion

• Individuals may wish to act alone rather than participate in “collusion” or “social planning”

• Suppose portfolio managers in the same group are competing for AUM– Manager may get superior results treating small account independently– Example showed that return of smaller investor is reduced significantly

for minor improvement for larger investor

• Is there something in between collusion and independence?

Cournot-Nash Solution

Investors Compete for Liquidity Cournot-Nash Equilibrium

• Each investor assumes the trades of the other investors are fixed

• Each investor’s problem:– Maximize expected return – own impact cost (including influence of

others)– Maximizewi

iTwi – wi

T c(jA wj)

– Objective is exactly what each investor would write as his own objective and thus is what is used to compare solutions

• Actual market-impact cost is exactly what is expected by each investor

• Solution process involves an iterative approach

• In some cases, problem can be written as a single optimization problem that accommodates joint constraints and can be solved in one go to find the equilibrium

Cournot-Nash Approach - Properties

• Market-impact is strictly convex– Generates fair trades even when maximizing net alpha

• Effects of Market-impact modeling– Using net trades can create “illusion” that net alpha is increased in one

account is selling something that all other accounts are buying.– Independent buy and sell impact introduces fairness, but overestimates

• Identical initial accounts may not remain identical after rebalancing– Smaller account may trade more illiquid assets

• Not Pareto-optimal

• Infeasibilities can create unfairness– Forced trades in one account increases entry cost for other accounts

• Same issues under Average Daily Volume constraints

Cournot-Nash Equilibrium Example

• Market-impact costs are those expected by each investor• Aggregate objective value is greater than for individual solutions• Equilibrium solution has greater objective value for each account

Individual Solution Equilibrium Solution


Size $100M $10B $100M $10B

Expected Return (%) 14.54 11.72 10.97 9.46

Expected Market Impact (%) 0.0964 1.8098 2.7783 1.8159




Solutions of a 3-Asset Example

• Expected returns and market-impact costs are greater for Asset 1, then Asset 2, followed by Asset 3

• Individual and Cournot-Nash equilibrium solutions have higher net expected return because of larger investment in Asset 1

Individual Solutions Collusive Solution Equilibrium Solution

Account 1 Account 2 Account 1 Account 2 Account 1 Account 2

Asset 1 10.1% 3.1% 3.1% 3.1% 7.4% 3.1%

Asset 2 51.2% 49.2% 48.9% 48.9% 55.1% 49.0%

Asset 3 38.6% 47.8% 48.0% 48.0% 37.5% 47.9%

Non-Pareto Optimality of Cournot-Nash

• Cournot-Nash Solution is not Pareto optimal• Each investor can improve their objective by trading less• “Improved” solution below improves objective for both accounts• Cournot-Nash doesn’t consider how other react to own actions

Equilibrium Solution Improved Solution


Size $100M $10B $100M $10B

Expected Return (%) 10.97 9.46 10.94 9.45




Infeasibility Issues – Net Trade Impact

• If initial holdings are infeasible, then rebalancing in a Cournot-Nash equilibrium with impact on net trades is not fair

• Account 1 moves away from optimal solution to reduce impact for Account 2

Cournot-Nash with Net Trade



Scenario 1 (Optimal) (Infeasible) (Feasible) (Feasible)

Asset 1 75 0 73.5 62.75

Asset 2 25 100 26.5 37.25



Asset 1 75 25 75 69

Asset 2 25 75 25 31

Infeasibility Issues – Buy/Sell Impact

• If initial holdings are infeasible, then rebalancing in a Cournot-Nash approach with independent buy and sell impact is fair

Cournot-Nash with Buy/Sell



Scenario 1 (Optimal) (Infeasible) (Optimal) (Feasible)

Asset 1 75 0 75 62.5

Asset 2 25 100 25 37.5



Asset 1 75 25 75 69

Asset 2 25 75 25 31

Summary: Cournot-Nash vs. Collusive

Cournot-Nash• Not Pareto-optimal

• Strictly convex

• Self-interest is optimized

• Generates different portfolios

Collusive• Pareto optimal

• Not strictly convex

• Total welfare is optimized

• Generates similar accounts – little dispersion

Solution Algorithm

• A direct solution of the problem through an “integrated” optimization algorithm

• Complete flexibility in modeling for each individual account

• Ability to set “joint” constraints (for example, aggregate ADV) and “joint” objective (aggregate market impact)

• Capable of solving either Collusive problem or Cournot-Nash equilibrium

• Direct solution of the problem through a tailored implementation of Axioma’s proprietary optimization algorithm

Test Results of Direct Approach

• Using 500 asset universe, approximately 60 accounts can be optimized in an hour

• Solution times increase quadratically in the number of accounts

• Method does not need to be modified for any particular strategy

Number of Accounts

Solution Time (seconds)

10 137

20 411

30 913

40 1626

50 2538

60 3616

Taxable Overlay Management

• Manage taxable holdings for single investor

• Each “portfolio” is a sub-portfolio for the single investor

• Each sub-portfolio has own objective and constraints– For example, limit tracking error to a model portfolio

• Taxes are managed across all sub-portfolios

• Assets can be transferred from one sub-portfolio to another for tax greater tax efficiencies

Mixing Optimized Portfolios

• No clear relationship between performance of portfolios based on multi-factor alpha models and performance of individual factors

• Alternatively, mix optimized portfolios– Compute optimal or target portfolio for each single factor– Solve multi-portfolio problem minimize trading costs– For each sub-portfolio, constrain tracking error to target, number of names, etc.

• Efficiencies gained– Trading costs are reduced because of crossing trades– Performance is a linear combination of performance of individual accounts

• “Portfolio Construction Through Mixed-Integer Programming at Grantham, Mayo, Van Otterloo and Company”, Interfaces, 29:1, Jan-Feb 1999, pp. 49-66.

Future Development

• Decomposition Approach (v 2.0)– Iterative approach

• Each account is optimized individually based on estimate of costs

• Estimate of costs are modified based on aggregation of individual account trades

• Formal decomposition method assures convergence

– Method is highly scalable (almost linear on the number of accounts)

– Method can be parallelized

Questions?

Documents

Multi-Portfolio Optimization