A Stochastic Convergence Model

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A STOCHASTIC CONVERGENCE MODELFOR PORTFOLIO SELECTION

AMY V. PUELZ

[email protected]

(Received September 1998; revision received September 1999; accepted November 2000)

Portfolio selection techniques must provide decision makers with a dynamic model framework that incorporates realistic assumptions

regarding financial markets, risk preferences, and required portfolio characteristics. Unfortunately, multistage stochastic programming (SP)

models for portfolio selection very quickly become intractable as assumptions are relaxed and uncertainty is introduced. In this paper, I

present an alternative model framework for portfolio selection, stochastic convergence (SC), that systematically incorporates uncertainty

under a realistic assumption set. The optimal portfolio is derived through an iterative procedure, where portfolio plans are evaluated under

many possible future scenarios then revised until the model converges to the optimal plan. This approach allows for scenario analysis over

all stochastic components, requires no limitation on the structural form of the objective or constraints, and permits evaluation over any

length planning horizon while maintaining model tractability by aggregating the scenario tree at each stage in the solution process. Through

focused aggregation schemes, the SC approach allows for the implementation of a lower partial variance risk metric, which is preferred in

investment selection. In simulated tests, the SC model, with scenario aggregation, generated portfolios exhibiting performance similar to

those generated using the SP model form with no aggregation. Empirical tests using historical fund returns show that a multiperiod SC

decision strategy outperforms various benchmark strategies over a long-term test horizon under both asset-liability matching and return

maximization frameworks.

The selection of a portfolio whose performance issuperior to other portfolio alternatives is a complexproblem for which even a reasonable solution can resultin high potential payoffs. The gap between practice and

theory is small with there being a lucrative market fornew techniques for funds management (Bernstein 1995,

Kahn 1995). Whether the portfolio is designed to fund afirm’s pension plan, an insurer’s loss reserve, or a cor-

poration’s financial planning needs, the selection processshould incorporate the complexities inherent in financialmarkets and the decision maker’s risk preferences and port-

folio requirements. In short, there is a need for portfolioselection models that incorporate problem-specific uncer-

tainties while maintaining model validity and tractability.Scenario based stochastic programming (SP) models, where

uncertainty is embedded in a utility maximizing framework,have been shown to be a promising tool for financial deci-sion making (Koskosidis and Duarte 1997, Golub et al.

1995, Holmer and Zenios 1995, Cariño et al. 1994). Thispaper sets forth such a stochastic optimization technique,

stochastic convergence (SC).SC is based on stochastic programming (SP) models for

portfolio selection, and like SP, provides a robust modelingprocedure generating utility maximizing portfolios while

controlling performance variability. Unlike SP, SC elimi-nates the need for all simplifying assumptions necessaryfor decomposition and solution generation by employing

an iterative selection procedure of simulation evaluation of revised portfolios. The procedure is implemented through

the use of scenario aggregation at each step in the process

resulting in an optimization model whose size is indepen-dent of the number of scenarios used to represent uncer-tainty. The reduced size of the SC model is such that thesolution process can be implemented on virtually any plat-form, the objective can accommodate any form of discreteor continuous risk metric, and special form side constraintsunique to the decision-making environment can be incorpo-rated. In addition, through the use of focused aggregation

schemes, preferred lower partial variance based risk met-rics can be implemented.

Simulated and empirical tests reveal that SC generatessuperior performing portfolios. In simulated tests, SC withscenario aggregation generated portfolios exhibiting perfor-mance similar to those generated using SP with no aggrega-tion. Historical fund returns were used in empirical tests toexamine SC from a strategic perspective and compare it toindividual fund performance and benchmark strategies overan extended time horizon. Multiperiod SC portfolio selec-tion strategies consistently outperformed other strategies.

The next section provides background on portfolio selec-tion problems and stochastic programming techniques forsolution generation. The framework for the SC is then pre-sented followed by simulated and empirical tests of modelperformance.

1. BACKGROUND

Research into the development of models for portfolioselection under uncertainty dates back to the fifties withMarkowitz’s (1959) pioneering work on mean-variance effi-cient (MV) portfolios. Unfortunately, MV models have

Subject classifications: Finance, portfolio: stochastic optimization selection model. Programming, stochastic: convergence model for portfolio selection. Finance, investment:stochastic optimization for portfolio selection.

Area of review: Financial Services.

Operations Research © 2002 INFORMS

Vol. 50, No. 3, May–June 2002, pp. 462–476 4620030-364X/02/5003-0462 $05.00

1526-5463 electronic ISSN


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practical limitations, such as the requirement that stochasticparameters are symmetrically distributed or the investor’sutility function is quadratic. MV models, not easilyextended to a multiperiod planning horizon, provide a range

of efficient solutions rather than a single optimal solu-tion, and solutions have been shown to be extremely sen-

sitive to very small changes in model forecasts (Mulveyand Vladimirou 1989, Jai and Dyer 1996, Koskosidisand Duarte 1997). Stochastic programming (SP), whereexpected utility is maximized across a range of possible

future outcomes or scenarios provides a better frame-work for the class of portfolio selection problems. Cariñoet al. (1994) show significant savings can be realized byreplacing a MV portfolio selection approach with scenario-based stochastic programming. Holmer and Zenios (1995)discuss the importance of dynamic scenario-based deci-

sion models in integrated product management for financialintermediaries.

SP requires the specification of random parameter jointdistributions through discrete data realizations of possiblefuture states or scenarios. In the objective function, theprobability-weighted utility of performance across all sce-

narios is maximized. The most significant limitation in SPimplementation is that as the number of scenarios increasesto adequately represent the joint distribution of randomparameters the model becomes prohibitively large.

Research in SP for portfolio selection is focusedon models that incorporate realistic assumptions while

remaining computationally tractable. Various forms of SPhave been applied to a range of portfolio selection prob-

lems. Bradley and Crane (1972) discuss the need fordynamic, multistage models for bond selection and pro-vide a decomposition procedure for solution generation.Kallberg et al. (1982) develop a simple recourse linear

programming model allowing for asymmetric preferencesor differing penalty costs. In testing their model, they illus-trate that by incorporating randomness in cash require-ments they are able to extend the scope of portfolio mod-eling without a significant impact on model tractability.Kusy and Ziemba (1986) apply stochastic LP to bank

asset and liability management. Like the Kallberg, White,and Ziemba model, their model allows for randomness inexternal deposits and withdraws only while asset returnsare assumed deterministic.

More recently, Hiller and Eckstein (1993), Mulvey andVladimirou (1989, 1992), Zenios (1991) and Golub et al.

(1995) have developed SP network models that allowfor a broader range of parameter uncertainty. Hiller andEckstein’s (1993) stochastic dedication model for fixed-income portfolios employs an arbitrage-free pricing mech-anism in the scenario generation process. Notable in theirmodel is the risk measure, which reflects the probability

and severity of insolvency in a manner similar to thelower partial variance metric. This type of metric has beenshown by Bawa (1975) and Fishburn (1977) to provide a

better measure of risk when abnormally low returns areof concern. Before Hiller and Eckstein’s (1993) model,

the implementation of such a risk metric was consideredcomputationally intractable. However, to maintain a struc-ture suitable for model decomposition and solution genera-tion, their model evaluates the decision over a single-period

time horizon. Since rebalancing is not allowed in a single-period framework, the associated transaction costs are not

considered in their portfolio selection process. In the SCframework the variability of low returns can be measuredindependently of overall variability through focused aggre-gation schemes making it possible to implement a lower

partial variance metric allowing for control of downsiderisk.

Mulvey and Vladimirou (1989, 1992) develop a gener-alized network approach for multiperiod financial manage-ment, allowing for uncertainty in virtually all stochasticparameters. To maintain a structure necessary for decom-

position, all multiperiod investments are treated as zero-coupon bonds by forcing the reinvestment of cash spinoffs.

In other words, the model assumes cash flows from assetscannot be used to meet cash requirements until assetsmature or are liquidated. The assumption of dividend rein-vestment can be a limitation in asset-liability management

where in many cases assets are purchased because cashspin-offs can be used to meet future cash requirementswithout liquidation. Zenios (1991) extends the Mulvey andVladimirou model to include mortgage-backed securitiesand other fixed-income securities using an arbitrage termstructure model for scenario generation of bond prices.

Golub et al. (1995) develop a two-stage SP model for fixed-income securities and compare the performance of their

model to other strategies for money management usingmortgage-backed securities. They show through a simulatedtest procedure that a SP model strategy results in superiorperformance and is worth the added complexity in terms of

scenario and solution generation procedures.Robust optimization (RO) models, which are a class of

SP models, incorporate a nonlinear objective function togenerate portfolios that control for performance variability.RO models generate solutions that minimize performancesensitivity to data realizations. In a portfolio planning con-

text, the variability of the portfolio return is controlled in arisk-averse framework. Mulvey et al. (1995) provide a gen-eral RO framework. Bai et al. (1997) compare and contrastRO to other techniques for optimization under uncertaintyand provide a specific framework for financial planningproblems. Vladimirou and Zenios (1997) develop three SP

models with restricted recourse for RO and show effectivesolution generation experience for small to medium scaleproblems.

It is evident from the literature that superior performancecan be realized by using SP for financial decision making.However, it is also evident throughout the literature that

SP model size is a significant drawback to actual imple-mentation. Realistically sized, multiperiod SP models forportfolio selection must maintain a structure that is suit-

able for decomposition. This may require (1) viewing theproblem in a myopic framework, (2) limiting the number


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of stochastic components evaluated, (3) allowing cash spin-

offs only at maturity or liquidation, and/or (4) limitingspecial form side constraints unique to the problem at

hand. Even if a structure suitable for decomposition is

maintained, the task of solving the system of subproblemsbecomes significant when the number of stochastic com-

ponents, assets in the selection set and time periods in theplanning horizon, increases. In short, as Dahl et al. (1993)

point out, the “combinatorial explosion of scenarios” inSP financial planning models makes “the need to cap-

ture uncertainty in a systematic way of great importance”

(p. 34). It is this need to incorporate uncertainty whilemaintaining model validity and tractability that is addressed

in this research.

2. THE SC FRAMEWORK

SC is an iterative approach for addressing utility maxi-

mizing portfolio selection problems that allows for scenarioanalysis over all stochastic components, while maintaining

model tractability through scenario aggregation at eachstage. Aggregating scenarios into categories eliminates the

need for decomposition and associated model simplifica-

tions as the size of the optimization model is reduced to a

linear function of the deterministic model and the numberof aggregation categories. There are no limits on spe-

cial form side constraints as in network models requiring

decomposition. In addition, the risk metric can take anyfunctional form, discrete or continuous, depending on the

decision-maker’s risk preferences.

In SC, the optimal portfolio is derived through an iter-

ative procedure where portfolio plans are evaluated undermany possible scenarios and revised. The cycle of scenario

evaluation of revised portfolios continues until the model

converges to the optimal portfolio plan. One can also think of SC as a portfolio improvement model where at each iter-

ation revisions are made to the current portfolio to improve

its performance. The SC process is illustrated in Figure 1.

In the first stage of the selection process, an initial port-folio plan consisting of asset-liability allocations for each

period in the planning horizon is derived. The plan could be

derived from a deterministic model using expected param-eters or it could be based on a benchmark portfolio. 1

As in all stochastic models a set of scenarios is gen-erated that represents possible future data realizations of all random model parameters over the desired planning

horizon. In a financial planning context, random parame-

ters might include interest rates, asset returns, and liability

costs. The number and type of random parameters will varyfrom model to model as will the technique used to generate

random data realizations.2 The scenario specific random

parameters in a portfolio planning context are presented inTable 1.

The process of creating scenarios starts with the genera-

tion of stochastic parameter data realizations at the end of

the first period. The clock is then rolled forward one periodand second period data realizations are generated given the

Figure 1. Structure of the SC model.

Aggregation

Step

SCENARIO TREE

(See Figure 2a)

SIMULATION

NewPortfolio

Plan?

Yes …

No

Wealth

Category 1 …

Wealth

Category K

STOP

OPTIMIZATION

AGGREGATEDSCENARIO TREE

(See Figure 2b)

Determine Wealth

Time Period 1

INITIAL

PORTFOLIO PLAN

starting state described by each first period ending state.

This process continues until data realizations are generated

for each period in the planning horizon. The general form

of the scenario tree representing these data realizations is

represented in Figure 2a. The tree nodes represent points

in time when new information about random parameters(such as asset, rate, and cash requirements) is available to

use in decision making. Each distinct branch in the tree

represents one of the J scenarios containing a unique set

of data realizations.

The initial portfolio plan is evaluated under each of

the J distinct scenarios and wealth determined at the end

of the first period. Scenarios are grouped together based

on first-period wealth into K categories and random data

realizations are averaged across all scenarios in a group

creating an aggregated scenario. The form of the aggre-

gated scenario tree is found in Figure 2b. Scenarios are

grouped by wealth at the end of the first period because of the dynamic nature of the multiperiod portfolio selection

Table 1. Simulation model variables.

Variable Description

r ht One-period interest rate time t under scenario (h = b for borrowing rate and l for lending rate)

phi t Market price (h = m), sales price (h = s) or purchaseprice (h = b) for one unit of asset i time t underscenario (sales and purchase prices include alltransaction costs)

ai t Cash flow for one unit of asset i at time t under

scenario .bt External cash flow time t under scenario


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Figure 2. Scenario tree aggregation.

problem.3 After the first period, new scenario-specific infor-mation is available and portfolio plans become scenariodependent. It is only through the first-period that nonantic-ipatory conditions require that portfolio decisions be iden-tical across scenarios.

Determining the ranges of the wealth aggregation cate-gories will depend on the decision-maker’s utility function.If the utility function is a continuous concave function,aggregation could be accomplished by assigning an equal

number of scenarios to each of the K aggregation cate-gories. This is the approach used in the simulated tests. In

the empirical tests, a different approach is used. In thesetests, scenarios with first-period wealth above the median(high wealth outcomes) are grouped together and scenarioswith wealth below the median (low wealth outcomes) areseparated into the remaining K −1 categories. This aggre-gation approach allows for risk assessment focused on theworst or below average outcomes resulting in a preferredlower partial variance related risk metric.4 If risk is to bemeasured by a discrete form metric such as value at risk,

VaR, grouping of scenarios would be dependent on thedecision-maker’s definition and acceptable probability of


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“worst case” performance. It is also possible for risk assess-ment to vary in functional form from category to category.

The aggregated scenario tree in Figure 2b is used to

create the optimization model where a revised portfolio willbe derived. Each time a new revised portfolio is derivedin the optimization process the aggregated scenario tree is

recreated for input to the model. The general form opti-mization model is:

Maxxkit y

hkt

Z =K

k=1

kU wk (1a)

St xki t =t−1 =0

xb ki −xs ki ∀k = 1 to K and

t = 1 to T (1b)

M i=1

aki txki t −p

b ki t x

b ki t +p

s ki t x

s ki t + y

b kt −y

l kt

− 1+ r b kt−1yb kt−1 + 1+ r

l kt−1y

l kt−1 = b

kt

∀k = 1 to K and t = 0 to T − 1, (1c)

M i=1

pm ki T +aki T x

ki T − 1+ r

b kT −1y

b kT

+ 1+ r l kT −1yl kT −wk = b

kT ∀k = 1 to K (1d)

xki 1 = xhi 1 x

l ki t x

b ki t y

l kt y

b kt 0

∀k = 1 to K and h = 1 to K , k = h (1e)

with the variables defined in Table 2.

The distribution of the random parameters in the modelgiven in Table 1, r ht p

hi tai t, and b t are rep-

resented in their aggregate form as r h kt , phki t , a

ki t , and

bkt . The objective function (1a) maximizes the sum of the

probability-weighted utilities of ending wealth, wk. Cumu-lative holdings of each asset in each time period are definedin (1b). The constraints in (1c) define cash flows for each

time period in the planning horizon t , for each category k .

Table 2. Optimization model variables.

Variable Description

M Number of assetsT Number of time periodsK Number of wealth categories in aggregation

xh ki t Units of asset i bought (h = b) or sold (h = s)time t in category k

xki t Cumulative holding of asset i time t in category

k = t−1

=0 xb ki −x

s ki

r h kt E r ht ∀ ∈ category k

ph ki t E phi t ∀ ∈ category k

aki t E ai t ∀ ∈ category kbkt E bt ∀ ∈ category k

yh kt One-period lending (h = 1) or borrowing (h = b)time t in category k

k Proportion of scenarios in category kwk Ending wealth in category k

The constraints in (1d) define ending wealth for each of the

K categories. Nonanticipatory requirements that first-period

decisions be identical across all wealth scenario categories

are modeled in (1e). Any other situation-specific side con-

straints, such as maximum or minimum allocations, can be

added to the general form model.

The optimization model uses new aggregated asset, rate,and cash requirement information from the initial port-

folio to select a revised portfolio with higher expected

utility. In a general risk-averse framework, the revised port-

folio, when compared to the initial portfolio used to aggre-

gate, will have higher (lower) allocations to assets that

have relatively high (low) overall returns during periods

when fund performance is below (above) average. Once a

new revised portfolio is derived, the aggregated scenario

tree must be recreated, based on first-period wealth of the

revised portfolio across scenarios. New expected parame-

ters for aggregated scenarios are derived, the optimization

model re-executed, and the portfolio revised again. The SCmodel converges to the optimal plan by shifting the port-

folio at each revision to better balance cash flows across

each aggregation category and increase utility. This cycle of

scenario tree aggregation and portfolio revision continues

until the optimal portfolio plan emerges. The model con-

verges to the optimal solution when the portfolio plan does

not change from the previous simulation.

To understand the convergence process, consider a one-

period planning horizon and two assets, A and B, with

negatively correlated returns. Assume that the initial port-

folio is allocated 100% to asset A. Ending wealth will

be perfectly correlated with asset A’s returns and nega-tively correlated with asset B’s returns. Thus, for example,

aggregation categories representing relatively low wealth

outcomes will be those where asset B has relatively high

returns. In a risk-averse framework, the revised portfolio

will shift a portion of the portfolio holdings to asset B,

increasing overall utility. When scenarios are reaggregated

based on the revised portfolio, asset B will be less attrac-

tive, since ending wealth will not be as negatively corre-

lated with asset B’s returns as in the initial portfolio. When

the portfolio is revised a second time, a portion of asset B’s

holdings will be shifted to asset A. This process of shifting

a portion of holdings between A and B will continue untilsteady state is reached and the allocations to A and B do

not change in the revision step.

It is possible that the SC model might converge at a

local optimum that differs from the global optimum. In

Appendix A, the conditions necessary for model conver-

gence to a global optimum are set forth. The most restric-

tive of these conditions are that (1) the impact that a change

in an asset’s allocation has on expected utility is uniform

across all aggregation categories (A.4), and (2) an asset’s

diminishing marginal impact on expected utility, given an

increase in that asset’s holding is not entirely accounted for

by the joint impact associated with changes in allocationsof the other assets in the selection set (A.6).


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The first required condition is violated when an asset’s

return behavior varies across aggregation categories. Forexample, consider a callable bond with returns and cash

flows that are a discontinuous function of interest rates, and

wealth which is correlated with interest rates. Increasing

the amount held of the callable bond may increase overall

expected utility but the change in expected utility willvary across aggregation categories based on first-period

wealth. In this case, the SC model might converge at alocal optimum which is inferior to the global optimum. By

altering the manner in which scenarios are aggregated, this

problem can be solved and the SC technique used given

a selection set containing derivative securities with discon-tinuous return functions.5

The second required condition for global optimum con-

vergence can be attributed to scenario aggregation. If thereturns on an asset and the set of alternative assets are

highly correlated in either direction, aggregation might well

average out pricing or cash flow characteristics that withoutaggregation would have led to the selection of a new supe-rior solution. This problem will be reduced as K (the

number of aggregation categories) increases. Another solu-

tion to this problem would be to run the model severaltimes, placing restrictive constraints on different sets of

highly correlated assets to more fully search the feasible

solution space. Sensitivity analysis could also be used to

revise the aggregation scheme. For example, if the objec-tive function coefficients of decision variables in a partic-

ular category have a relatively small optimality range the

category could be divided into two or more categories for

further analysis. The model could also be used in its cur-rent form as a portfolio improvement model to evaluate and

improve upon candidate portfolio plans. A Tabu-type search

technique (Glover 1989, 1990) could also be employed toinsure the attainment of the global optimum.

Another consideration in SC is the potential for portfolio

plans to reappear and looping to ensue. This problem is

remedied by implementing a modified branch and boundalgorithm. The algorithm is summarized in Appendix B.

Finally, if first-period return used for aggregation is not cor-

related with ending wealth, multiperiod SC will not con-verge. However, given the nature of the portfolio selection

problem this is not the case (see Endnote 3).

3. MODEL VALIDATION AND

COMPARATIVE ANALYSIS

Simulated and empirical tests were conducted to examine

the performance of different SC model structures and tocompare them to other strategies for portfolio structuring.

The test results to follow illustrate the impact on model

derived portfolio performance of (1) the level of aggrega-tion in the model, (2) the decision maker’s level of risk-

aversion, (3) the length of the planning horizon, and (4)

the nature of external liabilities. In both simulated and

empirical tests, one unit of wealth was allocated to variousinvestment alternatives with the decision-maker’s utility

function quadratic in form. Other forms of utility couldbe implemented, such as those belonging to the isoelasticfamily. Quadratic utility was selected for the model testsbecause efficient solution algorithms are available and withfocused aggregation in SC (not possible in SP) the limita-tions of quadratic utility in portfolio selection can actually

be reduced.6 The initial portfolio plan input into SC forthe tests was derived from the deterministic model whereexpected values based on historical returns for all parame-ters were used.

3.1. Simulated Tests

The first set of tests conducted using simulated assetreturns, illustrates the impact aggregation has on SC per-formance. Single-period SC models with varying levels ofscenario aggregation were compared to SP models withno aggregation. The first step in the simulated tests wasthe generation of 250 scenarios, each representing the cash

flows of three assets and one liability which are describedin Table 3.

Using these scenarios, SC models with different num-bers of aggregation categories (K = 1, 3 and 6) and anSP model with no aggregation (K = 250) were executed.Mean-variance efficient (MV) portfolios were also gener-ated for three target returns of 3.0%, 3.5%, and 4.0%. Theperformance of SC- and SP-generated portfolios were com-pared to illustrate the impact of the number of aggregationcategories on performance. The robustness of SC, SP, andMV model performance was also compared using a holdout sample of 250 scenarios not used in model creation.

In Figure 3, the expected minimum and maximumportfolio returns across all scenarios used in model cre-ation (in-sample scenarios) were averaged over 30 sim-ulation trials.7 Clearly, SC models with complete aggre-gation (K = 1), where performance variability cannot becontrolled, are inferior to those models which incorpo-rate scenario specific performance. Also evident is that forlower levels of risk-aversion, fewer categories are neces-sary in SC to mimic the spread between best and worstcase performance realized using SP. The same relationshipsare found in comparing expected minimum and maximumreturn performances across a hold out sample of 250 sce-

narios not used in model creation (out-sample scenarios).

Table 3. Description of asset/liability cashflows forsimulated tests.

Correlation Matrix

Asset 1 Asset 2 Asset 3 Liability

Asset 1 100 009 −093 038Asset 2 009 100 −008 092Asset 3 −093 −008 100 −032Liability 038 092 −032 100

Asset 1 Asset 2 Asset 3 Liability

Mean 1.0291 1.0325 1.0301 0.0314Standard deviation 0.0284 0.0529 0.0031 0.0072


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Figure 3. Simulation test results averaged over 30 trialsusing single-period SC models with 1, 3, and6 aggregation categories (K = 1, 3, and 6),and SP models with no aggregation. Mean,

maximum, and minimum ending portfoliovalues across 250 scenarios used to create the

model (in-sample scenarios).

-25%

-20%

-15%

-10%

-5 %

0%

5%

10%

15%

20%

25%

S C , K = 1

S C , K = 3

S C , K = 6

S P

S C , K = 1

S C , K = 3

S C , K = 6

S P

R e t u r n

Ra=10 Ra=25

Figure 4 compares SC, SP, and MV portfolios in a

risk-return framework plotting average return relative to

standard deviation of returns across the 250 out-samplescenarios. As seen in Figure 3, the lower the level of risk-

aversion the more closely SC performance matches SP per-

formance. In general, the greater the number of aggregation

categories in SC, the greater the level of risk control. It is

interesting to note, however, that none of the SC portfolios

are dominated by the comparable SP portfolio. By contrast,

MV portfolios are dominated by both SC and SP portfolios

where asset-liability matching is imbedded in the frame-

work. The exception is the MV portfolio with a high target

return of 4.0% which dominates the deterministic form SC

portfolio with a single aggregation category.

3.2. Empirical Tests

The empirical tests used historical returns from an asset

selection set of 8 funds listed in Table 4. These funds were

selected to represent a broad range of investment alterna-

tives. In addition, all funds paid dividends on a monthly,

quarterly, semiannual, or annual basis. Quarterly price and

dividend data were collected from Bloomberg’s Financial

Markets service for the period from March 1987 to March

1998 with the test period being the most recent 28 quarters

starting March 1991.

For each quarter in the test period the previous 16quarters of historical returns were used to estimate future

prices and dividends for the planning horizons tested; four-quarters and one-quarter. The technique used to estimatereturns was the principal component analysis techniqueused in Mulvey and Vladimirou (1989).8 Dividends wereestimated based on historical percentage payouts and sea-

sonal patterns. Transaction costs for all models were based

on published load figures with the minimum transactioncosts (for no-load funds) set at 1% . At no point was futureinformation used for price or dividend estimation.

In validating and testing SC performance, the rollinghorizon approach used in Mulvey and Vladimirou (1992)was implemented. In the first quarter of the test period,

March 1991, the model is run assuming the investor has oneunit of wealth to allocate. The model form reflects the plan-ning horizon being tested, either one or four quarters. Theresulting first quarter portfolio decisions are implemented,the time horizon rolled ahead one quarter and new infor-mation incorporated. At this point, the model is re-executed

with the starting portfolio equal to that which would havebeen implemented from the previous quarter regardless of

whether a planning horizon of one or four quarters is beingtested. This process of quarterly rebalancing of the port-folio continues throughout the test period. In this manner,the test replicates the strategy of an investor who on a quar-terly basis assimilates new information and rebalances theportfolio, based on the model-derived optimal plan regard-less of the length of the planning horizon.

The actual performances of SC strategies were comparedto the performances of individual funds, several bench-mark strategies suggested in Mulvey and Vladimirou (1989,

1992) as representative portfolio strategies and a range of MV strategies. The allocations associated with differentbenchmark strategies are presented in Table 5. The MVstrategy involved deriving mean-variance efficient portfo-

lios using the same historical information used in SC.The MV portfolios were rebalanced annually, allowing fora portfolio that reflected current return information whilelowering transaction costs relative to quarterly rebalancing.During periods when rebalancing did not occur, excesscash was allocated to funds based on the MV percentagesderived at the last rebalancing. Likewise, cash deficits werepaid by liquidating funds based on these same percentages.

Figure 5 illustrates the different forms of SC scenarioaggregation used in each test. Wealth categories wereequally distributed below the 50th percentile of expectedwealth at the end of the first period. Only scenarios whereperformance was below average were categorized basedon wealth while scenarios where performance was aboveaverage were aggregated into one category. This type of

focused categorization, where only poorly performing sce-narios are aggregated into separate categories, allows for arisk measure similar to the preferred lower partial variance.

The utility of ending wealth, modeled in the objectivethrough a quadratic utility function, was defined as

K k=1

U wk =K

k=1

wk −w2k (2)


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Figure 4. Simulation test results averaged over 30 trials using single-period SC models with 1, 3, and 6 aggregationcategories (K = 1, 3, and 6), SP models with no aggregation and MV models. Mean return relative to standarddeviation of return over 250 scenarios not used in model creation (out-sample scenarios).

K=1

K=3

K=6

K=1

K=3

K=6

r=3.0%

r=3.5%

r=4.0%

-0.7%

-0.6%

-0.5%

-0.4%

-0.3%

-0.2%

-0.1%

0.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0%

Standard Deviation of Returns

M e a n R e t u r n

SC, Ra=10 SC, Ra=25 SP, Ra=10 SP, Ra=25 MV

(Ra=10 & 25)

where wk is ending wealth in category k as defined in(1d).9 The decision-maker’s absolute risk-aversion (Ra) wasassumed to be constant and decreasing relative to wealthat the beginning of the quarter, wo, over the 28-quarter testperiod. Ra is defined as

Ra =2

1−2wo (3)

Table 4. Fund selection set.

Fund Name Transaction Cost

FPA Capital (Mid-cap) 6.5%Fidelity Real Estate Investment Portfolio 1%Principal Preservation Portfolio 4.5%

S&P 100 Plus FundTempleton Growth Fund (global stock) 5.75%T Rowe Price International Bond Fund 1%Vanguard Fixed-Income Long-Term 1%

US Treasury PortfolioVangard Fixed-Income Long-Term 1%

Corporate PortfolioVanguard Fixed-Income GNMA 1%

Note. Transaction costs are based on reported load figures plus

1% processing costs. All fund information was gathered from the

Bloomberg Financial Markets service.

The portfolios were generated for several risk-aversion

levels ranging from Ra = 0 (risk neutral) to Ra = 5.10

Short sales and short-term borrowing were not allowed.

The portfolio was entirely invested in the eight fund alter-

natives or in the short-term risk-free asset.11 These invest-

ment restrictions, which are not requirements in the model

framework, were adopted for testing purposes as they allow

for comparative analysis based on known fund perfor-

mance.

The first set of tests were conducted to compare perfor-

mance in an asset-liability management context. In these

Table 5. Benchmark strategies.

Portfolio S&P LT Govt LT Corp Cash

BP1 40% 40% 20%BP2 20% 30% 30% 20%BP3 45% 20% 25% 10%BP4 55% 15% 20% 10%BP5 60% 40%BP6 65% 10% 15% 10%BP7 75% 15% 10%BP8 90% 5% 5%

Note. Benchmark strategies are the same as those suggested in

Mulvey and Vladimirou (1989, 1992).


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Figure 5. Scenario aggregation scheme used in empir-ical tests.

Wealth

U

K=1

Wealth

U

K=2

K=4 K=6

Wealth

U U

Wealth

tests, the quarterly external liability to be funded was set

equal to the return on a high-yield bond fund offered by

Seligman. This fund was selected to represent external

liabilities because of its excess return kurtosis resulting

in a flat tailed return distribution. Duffie and Pan (1997)

show that the return distribution characteristics of equi-

ties, exchange rates, commodities, and interest rates are

flat tailed, meaning that unusual returns at both ends of

the distribution are more likely than would be predicted

by a comparable normal distribution of returns. This isillustrated in Figure 6, where the distribution of quarterly

returns on the external liability with kurtosis equal to 0.26

is compared to that under return distribution normality.

Figure 6. Flat-tailed distribution of actual liability costpercentage versus comparable distributionform if normal.

0.00

0.05

0.10

0.15

0.20

0.25

Return

Actual Probability Probability if Normal

-5% -2% +1% +4% +7% +10% +13%

Mean portfolio return relative to the standard deviationof portfolio returns allows comparison of different port-folio selection strategies over the 28-quarter test period.Figure 7 presents the performance of SC strategies, based

on a multiperiod (4 quarter) planning horizon, and dif-ferent scenario aggregation plans (K = 1 through K = 6)

to the performance of individual funds. When scenarioswere completely aggregated (K = 1), risk-averse behaviorcannot be modeled and the resulting SC strategy perfor-mances were almost identical to that of highest risk-return

fund, mid-cap. As would be expected, the higher level of absolute risk-aversion the lower the standard deviation of returns over the test period. Also revealed in these compar-isons is that the number of aggregation categories (K ) isnegatively correlated with the standard deviation of returnsover the test period. This is because as more categories

are added below the 50th percentile of returns, abnormallylow returns become more of a factor in deriving expected

utility. In terms of overall variability as measured by returnstandard deviation, none of the SC strategies are dominatedby individual fund performance, and with the exception of the mid-cap fund, the mean quarterly returns of individual

funds are less then zero, while those for SC strategies arepositive.

In Figure 8, additional results are presented givenfunding for the same external liability tied to the returnon a high-yield bond fund. Comparisons are made betweenindividual fund performance, benchmark strategies, MV

strategies, and myopic and multiperiod SC strategies.12 Allthe multiperiod SC strategies resulted in higher returns

than other candidate strategies with the exception of themid-cap fund. This can be explained by the fact thatSC incorporates future expected cash spin-off and liabilityinformation along with transaction costs associated with the

future liquidation and rebalancing of assets necessary tofund liabilities. In addition, the aggregation scheme allowedfor fund selection based on risk measured by low-end returnvariability with no risk penalty attached to extremely highreturns. Myopic SC strategies did not perform as well asmultiperiod strategies because of their inability to incor-

porate transaction costs associated with future portfolioliquidation and rebalancing decisions, which is particularlycritical when external liabilities are to be funded.

In this set of comparisons, where a risky external liabilityis funded, there exists a potential for significant losses. Tocompare performance in a preferred context that accounts

for poor performance (or loss) risk rather than overall vari-ability, the value at risk (VaR) metric is employed. VaR is apopularly embraced technique for measuring downside risk in a portfolio, and in the context of losses is defined as thepth percentile of portfolio quarterly loss. For high p values(e.g. 99, 95, or 90) it can be thought of as identifying the

“worst case” outcome of portfolio performance. Figure 9presents a comparison of the same strategies as in Figure 8,but with mean returns compared relative to a VaR metric

rather than return standard deviation. In this case VaR isdefined as the 90th percentile of portfolio losses over the


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Figure 7. Standard deviation relative to mean portfolio quarterly return comparison across number of aggregation cate-gories (K ) for multi-period SC strategies, absolute risk-aversion (Ra) equal to 2 and 4.

S&P

GNMA

glob stock

real estate

LT corp

LT govtglob bond

mid cap

k=6k=4

k=2

k=1 (Ra=2 and 4)

k=6k=4

k=2

-4%

-3%

-2%

-1%

0%

1%

2%

3%

3% 4% 5% 6% 7% 8%

Standard Deviation of Portfolio Quarterly Return

M e a n P o r t f o l i o Q u a r t e r l y R e t u r

n

individual funds Ra = 2 Ra = 4

28-quarter test period. Viewed in this framework, multi-period SC strategies and most myopic SC strategies exhibitsuperior performance to benchmark and MV strategies. Allindividual funds with the exception of the mid-cap fund aredominated by most SC strategies. Relative to the high risk-return mid-cap fund, several SC strategies provide severalsimilar returns with lower VaR.

In the second set of tests external liabilities are elimi-nated and performance of various strategies examined in

Figure 8. Standard deviation relative to mean portfolio quarterly return comparison across absolute risk-aversion (Ra)for SC strategies (K = 6), individual funds, benchmark strategies, and mean-variance efficient (MV) strategies.

mid cap

glob bond

LT govt

LT corp

real estate

glob stock

GNMA

S&P

bp1

bp2

bp3bp4

bp5bp6

bp7

bp8

Ra=5Ra=4

Ra=3

Ra=2

Ra=1Ra=0

Ra=0

Ra=1

Ra=2

Ra=3

Ra=4

Ra=5

-4.0%

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0%


M e a n P o r t f o l i o Q u a r t e r l y R e t u r n

individual funds benchmarks MV SC, multi-period SC, myopic

a pure return maximizing framework rather than in anasset-liability matching framework as in the last set of tests.

All other model parameters were identical to those usedin the first set of empirical tests. The results of this set oftests are found in Figure 10, where mean return is com-pared relative to standard deviation of returns. In this set ofpure return maximizing models, the multiperiod SC strate-

gies outperformed individual funds, benchmarks, and MVstrategies. The relative improvement is not, however, as


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Figure 9. VaR relative to mean portfolio quarterly return comparison across absolute risk-aversion (Ra) for SC strategies(K = 6), individual funds, benchmark strategies, and mean-variance efficient (MV) strategies.

S&P

GNMA

glob stock

real estate

LT corpLT govt glob bond

mid cap

bp8

bp7

bp6bp5

bp4bp3

bp2

bp1

Ra=5Ra=4

Ra=3

Ra=2 Ra=1

Ra=0

Ra=5

Ra=4Ra=3

Ra=2

Ra=1

Ra=0

-4.0%

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0% 4.5% 5.0% 5.5% 6.0% 6.5% 7.0% 7.5% 8.0%

VaR - 90th Percentile of Portfolio Quarterly Loss

M e a n P o r t f o l i o Q u a r t e r l y R e t u r

n


significant as that seen in the previous tests where liabilities

were funded. This is because the SC model’s incorporation

of cash spin-offs does not have the same impact on perfor-

mance when external liabilities are not being funded. As in

the last set of tests, myopic SC strategies did not perform as

well compared to benchmark strategies or individual funds.

Again, this can be attributed to the fact that myopic models

Figure 10. Standard deviation relative to mean portfolio quarterly return comparison across absolute risk-aversion (Ra)with no external liabilities for SC strategies (K = 6), individual funds, benchmark strategies, and mean-variance efficient (MV) strategies.

mid cap

glob bond

LT govtLT corp

real estateglob stock

GNMA

S&P

bp8

bp7

bp6bp5

bp4bp3

bp2

bp1

Ra=5

Ra=4

Ra=3

Ra=2

Ra=1

Ra=0

Ra=0

Ra=1Ra=2

Ra=3Ra=4

Ra=5

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

5.0%

5.5%

6.0%

2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0%


M e a n P o r t f o l i o Q u a r t e r l y R e t u r n


fail to account for transaction costs of future rebalancingand investment of cash spin-offs.

3.3. Computational Issues

The SC model was implemented in MATLAB on a UNIX5/300 Alpha processor with 512 MB RAM. The Numeric

Algorithms Group (NAG) quadratic solver was used.


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Figure 11. A comparison of average CPU time formultiperiod SC model portfolio generation.CPU time is divided into the average totaltime to run NAG quadratic optimization

model and test for convergence and theaverage total time to generate and aggregate

scenarios.

0.00 100.00 200.00 300.00 400.00

Optimization and convergence testing Scenario generation and aggregation

CPU time in Seconds

Ra=2

K=1

K=4

K=2

Ra=1

Ra=0

K=6

Ra=5

Ra=3

Ra=2K=6

800.00

The average CPU times required to derive optimal port-folios using the multiperiod model over the empirical testperiod are presented in Figure 11 for various model char-acteristics. In the empirical tests, the number of scenarios

generated in the model ranged from 3,000 to 6,000. In addi-tion, when the portfolio asset allocations changed by lessthan one-percent from the previous cycle, optimal portfolioconvergence was assumed.

The average CPU total time is divided into two cate-gories, (1) the time required to run the optimization modelsusing NAG’s quadratic solver and test for portfolio conver-

gence, and (2) the time required to generate and aggregatescenarios using the Matlab software. Times are presentedfor different levels of absolute risk-aversion given 6 aggre-gation categories (K = 6). Times are also presented fordifferent numbers of aggregation categories given absolute

risk-aversion, Ra, equal to two. Figure 11 reveals that thetime required to run the SC model is positively correlatedwith K and Ra.

The model form that required the longest average CPUtime per quarter to converge was the multiperiod modelwhere Ra was set to 5 and K to 6. This model took onaverage 803 seconds to converge to the optimal portfolio.

In this model, the average number of cycles before conver-gence over the test period was 17, looping occurred in 29%of the model runs, and when looping did occur, an averageof 1.75 branches were necessary to end the loop. This par-ticular model was also the largest in size with 337 variables

and 185 constraints. The comparable SP model with no

aggregation given 3,000 scenarios would be 156,025 vari-ables and 84,017 constraints.

4. CONCLUSIONS

SC provides a new, dynamic and reliable decision frame-

work of practical value for those involved in portfolio selec-tion. The technique permits the evaluation of the problemunder a realistic assumption set with minimal technologicalrequirements. SC is appropriate not only in deriving newportfolio plans but in evaluating current portfolio plans andidentifying revisions that would lead to improved perfor-

mance. The inclusion of uncertainty in all stochastic param-eters and the maximization of performance over any lengthplanning horizon are both feasible in SC. In addition, thedecision maker is not limited in the form of risk metricused or the types of side constraints necessary to modelspecial portfolio requirements.

Simulated tests show that SC, with scenario aggregation,generates portfolios exhibiting performance similar to thosegenerated using SP with no aggregation. Single-period SCmodels with as few as two aggregation categories comparefavorably to SP models using both in-sample and holdoutout-sample scenarios. A comparison of model forms in

a risk-return framework reveals that SC portfolio riskapproaches that realized using SP as the number of aggre-gation categories increases. SC portfolios with aggregationare, however, not dominated by SP portfolios and clearlydominate mean-variance efficient (MV) portfolios.

Empirical tests using historical fund returns over an

extended test horizon illustrate that using a multiperiod

SC strategy to select portfolio plans results in performancethat is superior to that of holding individual broad-basedfunds or of employing various benchmark or MV strategies.This is particularly evident when asset-liability manage-ment models forms are compared relative to value at risk

(VaR). Multiperiod models also clearly dominate myopicmodels in the rolling time horizon tests. The inability toincorporate rebalancing costs in a single-period frameworkresults in portfolio plans the exhibit substandard perfor-mance. Computational requirements of the SC models areminimal.

Finally, there are two important practical features of SC.

First, SC permits focused aggregation on low end perfor-mance allowing for control of risk in a preferred lower par-tial variance context; this is not feasible in nonaggregatedformats making the implementation of models controllingfor only low end variability difficult, if not impossible, in

SP. Second, SC model size is independent of the number ofscenarios and, therefore, not a factor in solution generation.This implies that SC can be implemented on virtually anycomputing platform and the model’s structural form neednot adhere to requirements necessary for decomposition asin SP. This has not been the case in the past and has been

a significant drawback in the implementation of stochastic

models for portfolio selection.


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APPENDIX A: CONDITIONS NECESSARY

FOR SC MODEL CONVERGENCE TO A GLOBAL

OPTIMUM SOLUTION

The Hessian of the SC model objective function, Z =K

k=1 kU k (where U k = U wk), is

H =

K k=1

k 2

U k2x11 1

+U k 2

k2x11 1

+

2 kx11 1

U kx11 1

· · ·

K k=1

k 2

U kx

j i tx

11 1

+U k 2

kx

j i t x

11 1

+

k

xj i t

U kx11 1

+ k

x11 1

U k

xj i t

· · ·

K k=1

k2U k

xj i tx

11 1

+U k2 k

xj i tx

11 1

+

k

xj i t

U kx11 1

+ k

x11 1

U k

xj i t

· · ·

K k=1

k2U k

2xj i t

+U k2 k

2xj i t

+

2 k

2xj i t

U k

2xj i t

· · ·

H is a square R by R symmetric matrix where R is thenumber of asset allocation decision variables. For sim-plicity elements in H are referred to as aij where aij isthe element in the ith row and j th column. The objectivefunction is assumed to have continuous second-order par-tial derivatives for each X = x 11 1 · · ·x

K M T ,) which is an

element of the feasible set S. The objective can be shownto be a concave function on the feasible set S if the nthprincipal minor of H has the same sign as −1n.

The first principal minors (aiis or principal diagonal ele-ments) of H are negative if the following conditions hold:

2U k

2xj i t


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2. The asset with the largest absolute value difference is

selected as the branching asset.3. A less than constraint is added to the model for the

selected branching asset. The RHS value is the maximum

value that will result in no change in the branching decisionvariable from cycle to cycle. The model is restarted.

4. If the model converges after the addition of the con-straint in Step 3, then the constraint is multiplied by −1

and the model restarted.5. If the model does not converge at either Step 3 or

Step 4, then a new branching constraint is added in Step 1

and the process continues.Branches are pruned (terminated) if the objective func-

tion value for any solution in the loop is less than the

current best converged solution objective function value.

Pruning branches will not eliminate superior solutions,as the converged solution resulting from the addition a

branching constraint will always be inferior to those solu-

tions in the loop prior to the addition of the branching con-straint. Branches are added and pruned in this manner until

an exhaustive search of the solution space is accomplished.

The converged solution with the highest objective function

value is the optimal solution.

ENDNOTES

1In some cases it might be appropriate to run SC with

several initial portfolio plans. This is the case when theconditions necessary for the model convergence to a global

optimum are violated. In these cases SC can be consid-

ered a portfolio improvement model and will generate port-

folios that improve upon the initial portfolio plan. SeeAppendix A for a detailed description of conditions neces-

sary for convergence to a global optimum.2The SC approach can be implemented using any tech-

nique for the generation of random returns. For example,

if fixed-income securities alone comprise the asset selec-

tion set, an arbitrage-free term structure model would beappropriate for scenario generation (Zenios 1991, Hiller

and Eckstein 1993). If non fixed-income securities are to

be included in the selection set, the use of principal compo-

nent analysis could be employed (Mulvey and Vladimirou1989).

3As long as the first derivative of first-period returnrelative to ending wealth is different from zero, aggregatingbased on first-period wealth in SC will result in a portfolio

that maximizes the utility of ending wealth given that the

general conditions set forth in Appendix A are met. This

is true regardless of the level of mean reversion of returnsover the planning horizon.

4In empirical tests the median is used to divide sce-

narios into above and below average performance. In gen-eral, however, to focus on poor performing outcomes, most

aggregation categories should be used to separate low

wealth outcomes where the variability of returns is critical

and fewer categories are used to separate high wealth out-comes.

5For example, consider again a callable bond. The first

aggregation step would be to divide the scenarios into two

categories, those where the bond would be called and those

where it would not be called. Within each of these ini-

tial categories additional aggregation would be conducted

given wealth under each scenario. Under this scheme the

condition specified in (A.4) and (A.5) are satisfied. Thesame approach could be applied to any instrument that

exhibits different return behavior determined by external

factors such as interest rates.6Two limitations of applying quadratic utility are that

U wk wmax and Rawk > 0.

7Negative returns over the planning horizon result

when the cost of the liability over the planning horizon is

greater then the portfolio return.8Certainly other procedures could be employed to gen-

erate data realizations for uncertain returns. PCA was

chosen for these tests because it reduces the dimensionality

of the forecasting problem and is a reasonable technique for

capturing the correlation between uncertain parameters over

a short planning horizon. Mulvey and Vladimirou (1989)

provide a detailed description of the process for generating

returns using PCA.9Two limitations of applying quadratic utility are that

U wk wmax and Rawk > 0. The first lim-

itation U wk < 0 for wk > wmax is actually less of a

problem with scenario aggregation when high level wealth

categories are aggregated into a single large category (see

Figure 5). The more scenarios aggregated into the high

wealth category the greater the probability that wk < wmaxand U wk > 0 for all wk. The second limitation is that

Rawk > 0 rather than Ra being a decreasing function of

w as plausible utility theory would suggest (Pratt 1964).

Kallberg and Ziemba (1983), however, provide strong

empirical evidence that over time horizons of less than one

year different forms of utility (including quadratic) result

in similar optimal portfolio choices. They suggest that for

purposes of tractable solution procedures, quadratic utility

is a reasonable surrogate for more plausible forms of utility.

It is reasonable to assume the performance results provided

in these tests are robust across utility function forms.

10Applying a constant Ra relative to wo over timeimplies that the cash equivalent for risk at initial wealth

is the same over time rather than increasing (decreasing)

as initial wealth increases (decreases) as utility theory sug-

gests if utility remains constant over time. Because the out-

come of these tests is a comparison of portfolio perfor-

mance in a general risk/return framework and no attempt

is made to compare mean return across set levels of Ra,

this assumption has no impact on the conclusions drawn

in these tests. Holding Ra constant eliminates the need to

arbitrarily redefine absolute risk-aversion at the beginning

of each period in the test.

11The short-term risk-free asset (cash) has a returnequal to the three-month t-bill.


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12All SC models in this set of comparisons used 6aggregation categories (K = 6).

13The determinant of a symmetric matrix is defined asnj =1 pjj using “Forward Doolittle’s Scheme.”

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A Stochastic Convergence Model