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JournalOfInvestmentManagement, Vol. 10, No. 4, (2012), pp. 518

JOIM 2012

PORTFOLIO MONITORING IN THEORY AND PRACTICE

Richard O. Michaud,a David N. Esch b and Robert O. Michaudc

The when-to-trade decision is a critical yet neglected component of modern asset man-

agement. Typical rebalancing rules are based on suboptimal heuristics. Rebalancing is

necessarily a statistical similarity test between current and proposed optimal portfolios.

Available tests ignore many real world portfolio management considerations. The first

practical test for mean-variance optimality, the Michaud rebalancing rule, ignored thelikelihood of information overlap in the construction of optimal and current portfolios.

We describe two new algorithms that address overlapping data in the Michaud test and

give examples. The method allows large-scale automatable noncalendar based portfolio

monitoring and quadratic programming extensions beyond portfolio management.

1 Introduction

Modern asset management requires effectiveportfolio monitoring. The rebalancing decision isnecessarily a statistical similarity test between thecurrent drifted portfolio and a proposed new opti-mal. However, portfolio rebalancing in practiceis typically based on suboptimal heuristics thatoften lead to trading in noise or ineffectivelyusing

aPresident, New Frontier Advisors, LLC, 10 High Street,Boston, MA,02110, USA.Tel.: (617) 482-1433; Fax: (617)482-1434; E-mail: [email protected] of Research, New Frontier Advisors, LLC.E-mail: [email protected] Director of Research and Investments,New Frontier Advisors, LLC. E-mail: [email protected].

useful information. Many managers routinelytrade on a fixed calendar basis. Others estimatehurdle rates that ignore the portfolio context andstatistical estimation issues (e.g., Masters, 2003).Other approaches include trade optimality (Dyb-vig, 2005), jumps in prices (Liu et al., 2003),trading frequency (Leibowitz and Bova, 2010),trading strategies (Perold and Sharpe, 1989),and dynamic programming (Sun et al., 2006;

Markowitz and van Dijk, 2003). None are basedon portfolio statistical similarity.

Statistically based portfolio similarity procedureshave been given by Shanken (1985), Jobson andKorkie (1985), and Jobson (1991) among others.However, the few statistical approaches requireunrealistic investment assumptions in order toensure familiar null distributions. Real-world

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6 RichardO. Michaud et al.

portfolio management requires linear inequal-ity constraints on portfolio weights, targetedrisk levels usually on some variation of theMarkowitz (1959) mean-variance (MV) efficient

frontier, trading costs, and asset manager stylecustomization.1 Practical decision rules requirecompute-intensive resampling methods to createoptimality tests which would be intractable usingtraditional analytical techniques.

The first practical procedure for statistical simi-larity is the Michaud rebalancing rule (Michaud,1998; extended in Michaud and Michaud, 2008a,b). It measures the statistical similarity of the

current portfolio relative to an associated MVrisk targeted portfolio on the Resampled EfficientFrontierTM (REF).2,3 In Michaudoptimality, eachportfolio on the REF is an average of prop-erly associated resampledMV optimal portfolios.Michaud rebalancing compares the tracking errorof a given portfolio to the target optimal rela-tive to the distribution of tracking errors asso-ciated with the target optimal created by theresampling process under the statistical similarity

null hypothesis.4

A high percentile, or rebalanceprobability, indicates that the current portfoliois statistically far from target optimal and thattrading is likely to be desirable. The thresholdpercentile level depends on the managers riskaversion; that is, whether or not the possibilityof trading in noise is less desirable than ineffec-tively using investment information. The tradingdecision depends on the level of confidence asso-ciated with the managers information, trading

costs, and numerous additional investment andstatistical issues.5

A critical consideration in portfolio monitoring isthat much of the information used to constructthe current portfolio may also be implicitly orexplicitly reused when defining the new optimal.This partial input match results in an overly con-servative Michaud rebalance signal. A portfolio

similarity test that properly uses overlappinginformation corrects the statistical significance ofthe percentile measure while often recommend-ing trading earlier or later than any fixed calendar

timetable.

We develop new critical values that are condi-tional on a specified amount of common informa-tion in the Michaud test. These new thresholdsdefine the appropriate critical ranges for theMichaud rule and boost its power. We describetwo procedures, one for purely historical data andthe other for the more common case of managedrisk-return estimates.6 The algorithms are illus-

trated with examples. We discuss customizationof the monitoring rule in the context of man-ager, investor, and marketing imperatives. Themethod allows large-scale automatable noncalen-dar based monitoring. More generally the pro-cedure provides a rigorous statistical context forquadratic programming estimation with potentialapplications beyond portfolio management.

The procedures we describe are addressed to theneed-to-trade, not how-to-trade, decision. How-

to-trade often requires consideration of manymarket, investment, and client factors and isbeyond the scope of this report. We note that thetrade-to portfolio may often be an intermediarypoint between current and target optimal.

A mathematical description of the portfolio mon-itoring framework is presented in Section 2.A computational procedure adjusting Michaudsrebalancing test for overlapping data is given in

Section 3 assuming optimization inputs are basedpurely on historical data. Later discussion forimplementation in themore practicalcase of man-aged inputs is presented. Section 4 illustrates theprocedures with applications to asset allocation.Section 5 discusses considerations of using therule in a practical monitoring scenario. Section6 provides some generalizations, a summary andsome conclusions.

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calculatePtat timet >0 uses much of the sameinformation used to construct P. Due to over-lapping data, rebalance probabilities may appearquite small, yet appropriately signal a need to

trade in spite of being substantially less than thenominal thresholdL.

Our overlap-adjusted threshold for the Michaudrebalance test is denoted as CL(k). We defineCL(k)as follows: let Xtbe composed of com-ponents Xt0and Xt\0, the intersection and setdifference of Xt with X0, respectively. ThenCL(k)is defined as the Lth percentile of the dis-tribution ofR(t)induced by replacing Xt\0with

a random draw from its predictive distributionwithin the calculation ofPt.

The next section describes general computationalprocedures for calculating CL(k). CL(k) has limitL for increasing k when P and Pt are asso-ciated portfolios on the REF. As t increases,R(t)increases and crosses the critical probabilityCL(k) at some time t. In the context of over-lapping data in the optimization inputs, CL(k)is the lower limit of a more suitable range for

interpreting a rebalance statistic as a positiverebalancesignal at confidence level L. CL(k) pro-vides a rigorous statistical benchmark for a givendata set and parameters Land k for determin-ing whether trading is desirable at a given timeperiod.

3 Computing CL(k)

We address two cases for calculatingCL(k). Thefirst, described in detail, assumes that the MVinputs associated with both P and Pt and forcomputing REF portfolio optimality are definedentirely by prior historical returns. This case isnot very practical but useful in clarifying the mainfeatures of the monitoring computation. The sec-ond case generalizes the framework for manyinvestment processes in practice.

3.1 MV optimization inputs from T periods

of historical returns

Return series X0 = [x1, . . . , xT] is used to

calculate optimization inputs associated with P.A sequence of simulated optimization inputs arecreated for time t = k, also based on T his-torical time periods. The following algorithmuses a partial resampling framework to simu-late overlapping datasets and obtain the empiricaldistribution of R(t). CL(k)is computed for thegiven data set and parameters k, T, L, and Z,whereZis the number of Monte Carlo simula-tions of rebalance probabilities. The definition ofa REF optimal portfolio at time talso requiresan assumption, called the Forecast ConfidenceTM

(FC) parameter that defines the number of sim-ulated observations used in computing simulatedrisk-return inputs and simulated mean-varianceefficient frontiers used in the portfolio averag-ing process. While the default assumption inthis report is to assume FC = T, we discussalternatives and applications later in the text.13

Algorithm A: Computing CL(k) from known

historical returns14

Compute P0, the optimal target portfolio attime 0, fromX0.

For i in{1, . . . , Z}:

Replace krandomly selected returns15 in X0with k returns simulated from the predic-tive distribution and compute the optimalportfolioPik

Compute the rebalance probabilityRi(k)of

P0with respect toPi

k

CL(k)isLth quantile ofRi(k)

3.2 General MV optimization

input estimation

Few, if any, asset managers rely solely onhistorical returns for computing MV estimates



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PortfolioMonitoring inTheory andPractice 9

for actual investment. Risk-return estimatesgenerally reflect information from a wide varietyof sources and aspects of the investment process.REF optimized portfolios P0 and Ptare based

directly on inputs calculated at times 0 and t. Inthis context there is no explicitvalueofTtodefinethe number of draws from the input distribution;consequently its value must be assumed.16 Forthis case,Trepresents the managers assumptionof theinformation cycleor information level asso-ciated with the number of draws of independent,identically distributed (iid) returns at time t.Theadjusted critical values CL(k)in this algorithm(B)arecomputedbyapplyingAlgorithmAtosim-

ulated datasets in place of the observed data X0.17Since some information is lost in reducing thedataset to optimization inputs, the two analysesmayproduce somewhatdifferent results. We haveindeed observed variability in the results from theabove algorithm on multiple datasets simulatedfrom a single set of inputs, as shown in Figure 1.

4 Examples

We provide a comprehensive illustration of thenew procedure with respect to a data set usedin prior REF optimization studies. This data set

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Algorithm AAlgorithm B

Figure 1 Michaud (1998) data. Michaud rebalance probabilities are simulated, replacing kmonths of 120 ineach simulation, with T0= December 1990. Each combination of parameters was run with 1,000 simulations.Boxplots (Tukey, 1977) show distribution summaries of simulated Michaud rebalance probabilities for valuesofkfrom 1 to 24. Algorithms A and B are juxtaposed for each value ofk. The three horizontal rules in the boxesmark the three quartiles of the distribution, and the dotted line whiskers extend to 1.5 times the interquartilerange, or the limit of the data if it is closer. Values beyond the whiskers are marked with + symbols.

represents anasset allocation case foreight capitalmarket indices as described in Michaud (1998,Ch. 2).18 The main results are shown as distribu-tion plots and percentile curves in Figures 1 and

2, and detailed results of a few modifications tothe inputs appear in the appendix.

The Michaud (1998, Ch. 2) data consists of 18years of monthly historical total returns for eightcapital market indices, six equities and two fixedincome, from January 1978 to December 1995.For our main analysis we use T =120 monthlyperiods; we also treat T = 60 in the appendix.Unless indicated we also assumeFC= T. Ptforthis case would be the REF optimized portfo-lio estimated at tmonthly periods subsequent toDecember 1990. For example, ift =12, then Ptis the REF optimized portfolio at December 1991calculated using the previous 120 months of data.Adjusted critical values CL(k)are computed forvalues ofkfrom 1 to 24.

Figures 1 and2 display two differentpresentationsof the critical rebalance probability as a functionofkfor the middle portfolio on the REF based on10 years of monthly historical returns for the indi-cated confidence levels.19 The middle of the box

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Figure 2 CL(k)as a function ofkfrom the Algorithm B data in Figure 1. Here we show the 25th, 50th, 75th,and 90th percentiles of the distributions.

plots in Figure 1 represent the 50% value ofL andthe extremes the 25% and 75% level, for CL(k)foreachvalueofk for theindicatedcases.Thedis-play also shows that the need-to-trade probabilityrequired with a 1-year rebalance cycle (k =12)at the 90% confidence level for T = 120 isroughly 20%. Figure 2 simply traces CL(k)asa function ofkfor different levels ofL. Theoreti-callyCL(k)is monotone increasing as a functionof k; Monte Carlo estimation error is responsi-ble for deviations.20 In effect, CL(k)is simply a

recalibration of the need-to-trade probability forgivenL.

The simulated rebalance probabilities shown inFigures 1 and 2 have several noteworthy proper-ties. Both figures vividly demonstrate the impactof overlapping data on the Michaud rebalancingrule. Even after a year, the critical probabil-ity required for trading to optimal is far lessthan confidence level L. Figure 1 also shows

that the critical probability, or benchmark formeasuring optimality relative to estimation errorand assumptions, has less variation in AlgorithmA than B due to more specificity in startingassumptions.

The appendix provides further insights into thestatistical character of the adjusted critical valuesCL(k). Rather surprisingly, there is little variation

acrossthedifferentrisk levels of theefficientfron-tier. Although the rebalance tests are comparinggreater tracking errors for greater target risk lev-els on the frontier, the relative positions of thesimulated aged portfolios among the simulatedstatistically equivalentportfolios remainsremark-ably consistent for otherwise equivalent cases.Variation of the number of simulated periods ofoverlap, via the parameter Tchanges the rate atwhich the CL(k) converges to its limiting value askincreases. The overlapping data effect vanishes

more quickly for increasingkasTincreases.

Other changes in input can affect the limitingdistribution to which the simulations converge.Changing target rank between times 0 and torusing a number ofFCperiods different from Twill affect the value of CL(k). Details of theseeffects appear in the appendix.

5 Implementing the Michaud monitoringrule

Implementation of the monitoring rule requiresappropriate settings of the parameters. Propertuning ofL, FC, k,and Tallows customizationof the procedure relative to a managers invest-mentprocess, clientobjectives, outlook, andotherconsiderations.



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The parameterLquantifies trading risk aversion.A large value ofLreduces trading frequency byavoiding the danger of trading in noise but mayalso ignore performance opportunities. Alterna-

tively, a small Lincreases trading frequency bymore closely tracking to target but may result instatistically ineffective trades. A default value ofL=0.5 may often be useful.

The appropriate value of L also depends ona managers assessment of the level of infor-mation in risk-return estimates at given time t,which may vary depending on various investmentissues. These considerations are also reflected in

the choice of the forward looking FCparame-ter which is embedded within the posterior andposterior predictive distributions f(| Xt) andf(X | Xt). Higher FClevels create less disper-sion in theresampleddatasets, andgenerallymoreactive use of investment information in the REFportfolios. The choice ofFCmay interact with L.For example, a highFCvalue may be associatedwith a lower level of Land vice versa. This isbecause the manager may decide that the infor-mation is very reliable and tracking to optimalityis more important than the likelihood of tradingineffectively.

Setting the value of T is straightforward whenthe only source of information is historical dataas in Algorithm A. However, because risk-returnestimates are typically managed, practical appli-cation will often require Algorithm B and anestimate ofT. In this caseTdoes not necessarilyreflect a historical time period but more appro-

priately the denominator in the fraction k/T ofnew information in the analysis. Alternatively, Treflects the time it takes for information to com-pletely cycle out of the analysis. As Tbecomessmaller, the rebalancing threshold CL(k) willincrease more rapidly as kincreases.

Once parameters have been set and CL(k)com-puted, the implementation of the monitoring rule

is straightforward. The manager calculates theMichaud rebalance test R(t) for the monitoredportfolio Prelative to thenew REFoptimal Ptandcomparesthevalueto CL(k). Rebalanceprobabil-

ities greater thanCL(k)indicate that rebalancingmay be desirable. Note that CL(k) is simplya newscale forL. It may often be convenient to recali-brate theCL(k)distribution onto theLscale. Theparameters defining the monitoring rule provideinteresting opportunities to dynamically managethe trading decision in real time.21

In applications the set of monitored portfoliosmay differ by risk level. The Michaud monitoring

rule can be accommodated by associating eachportfolio with one of several pre-defined targetREF portfolios. Once the critical probabilities arecomputed, the monitoring rule requires only atable lookup. Consequently automated customiz-able portfolio monitoring may be practical evenfor very large-scale applications.

6 Summary and extensions

The when-to-trade decision is a critical func-tion of professional asset management. Tradingon noise is costly and unproductive but tradingtoo little uses investment information subopti-mally. Trading effectiveness impacts investmentperformance and manager competitiveness. At amacro level, trading effectiveness affects efficientallocation of capital in global markets.

Yet current practice is largely characterized

by suboptimal calendar rules and simple pointestimateheuristics. Portfolio monitoring is essen-tially a statistical test of the similarity of two port-folios. Until recently, statistical similarity testsrequired unrealistic assumptions. The Michaudrebalancing testaddressedpractical portfolio con-struction and investment management consid-erations but is overly conservative if the twoportfolios are based on overlapping data. We



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describe two new algorithms for properly deter-miningwhen trading is statistically desirablewithoverlapping information.

The procedures we present have a number ofgeneralizations. In particular there is no needfor the target portfolio to reside on the REF orany MV efficient frontier. Since the ability toresample datasets enables computation of statisti-cal equivalence regions, any portfolio calculationmethod, along with a measure of portfolio simi-larity, can produce a rebalance test statistic whichcould be calibrated by the procedure in this paper.Note also that can parameterize any returndistribution, not just mean-variance based.

While our methods and illustrations are inspiredand guided by the demand characteristics of mod-ern asset management in practice, they are notlimited to this context. The procedures are appli-cable to a wide variety of process monitoringapplications where inequality or other constraintsrequire computational methods for estimationand inputs include overlapping data. In particu-lar the procedures can be used for data analysismonitoring applications that uses multivariatelinear regression with overlapping data and lin-ear inequality constraints. More generally, theprocedure may be viewed as providing a statis-tical context for many quadratic programmingapplications.22

Our portfolio monitoring framework presents

novel concepts and challenges for managers andinvestors. However, the effort to quantify themonitoring decisionwithproperlydefined param-eters may be helpful in clarifying and controllinginvestment process and intuition. Investors mayfind such a manager likely to have enhanced per-formance over traditional ones. The potential foradding investmentvalue and enhancingreliabilityseems a challenge worth the effort.

Acknowledgements

The authors wish to acknowledge the helpfulcomments of the discussant Denis Chaves and

other participants at the Journal of InvestmentManagement Fall Conference, Boston, 2010. Inhis discussion of our presentation, Dr. Chavesreviewed and presented his own summary of ourmathematical methods.

Appendix

Detailed results for various modifications to

the inputs

Consistency across the Frontier

Figure 3 displays the simulations of Figure 1 forlow to high risk on the entire efficient frontierbroken out by single values ofkon each panel.23

Although there is a slight increase in the proba-bilities needed for trading at the lower and upperends of the efficient frontier, the distribution ofsimulated probabilities is remarkably consistentacross different risk levels of the frontier. Most

if not all of our experimentation has shown thesame result: the target risk level on the frontierdoes not greatly affect the distribution of simu-lated rebalance probabilities, and hence CL(k),despite the wide variation in the distribution oftracking errors used to calculate each rebalancetest.

Number of input time periods

Figure 4 displays an analysis identical to Figure 1

except that it is based on T = 60 monthly his-torical returns rather than 120. Comparison of thetwo figures demonstrates that an increase in esti-mation error in risk-return estimates significantlyincreases thecritical probability required for trad-ing to optimal. This increase occurs because eachadditional simulated return as k is incrementedhas twice the information relative to the wholedata set. Note that even in this case, the critical



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Figure 3 The Algorithm B simulations are shown for givenkacross the entire range of frontier risk levels ineach panel. This Chart shows surprisingly little variation across different risk levels in the rebalance statistic,except at the ends of the frontier. There is variation across the frontier within resampled datasets, as can beseen by the paths of some outliers, but the aggregated simulations occupy a remarkably stable range across thefrontier.

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Algorithm AAlgorithm B

Figure 4 Boxplots for Michaud (1998) data, Algorithms A and B, T = 60. Note the much more rapidconvergence of the distributions towards uniform coverage of the[0, 1]interval.



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probability is small relative to confidence level atshort monitoring periods reflecting the dominanteffect of overlapping data. On the other hand, theoverlapping data effect tends to diminish rapidly

much beyond a year and at k =24 the distribu-tion of rebalance probabilities is much closer touniform than in the T =120 case.

Nonassociated portfolios at t =0 and t = k

Risk preferences may change over time. In suchcases target portfolio risk may vary from oneperiod to another, and the analyst would choose adifferent set of associated portfolios to averagewhen computing the REF. Changes in portfo-lio risk alone are often enough to indicate thatrebalancing may be desirable. When a differentfrontier point is targeted, the analyst should usethe value ofCL(k)computed such thatP0andPkare associated withPt. This applies a uniformly

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P0Middle Risk; PkLower Target Risk

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Figure 5 Algorithm B, lower target risk level. The data in panel 2 exactly matches the Algorithm B data fromFigure 1. When the target risk differs from initial, the simulated rebalance statistics are typically greater thanwhen they match, and converge to a nonuniform distribution as kincreases. The boxplots for largekin the firstpanel can be seen to have quartiles at values greater than 25%, 50%, and 75%, the quartiles of the uniformdistribution.

strict standard to the rebalance test regardless ofprevious status.

While not useful as strict criteria for rebalancing,

the distributions of overlap-simulated rebalanceprobabilities from different parts of the efficientfrontier are useful in understanding the behaviorof the rebalance test and how changing target riskaffects the need to rebalance. The upper panel ofFigure 5 illustrates the distributions of rebalanceprobabilities for lower target riskPkrelative to amiddle REF optimized portfolio P0.24 For com-parison, the lower panel of Figure 5 shows theidentical analysis when bothP0and Pkare mid-dle target risk, and is identical to the AlgorithmB data of Figure 1. In this figure the distribu-tion of rebalance probabilities is shifted towardsgreater values in the upper panel. When targetportfolio risks are different the Lth quantile ofthe distribution of overlap-simulated rebalance



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Figure 6 An illustration of the effect of decoupling the Forecast ConfidenceTM from the total informationparameterTused in computingCL(k). The data are taken from Michaud (1998), using 120 months of data. Theleft series of boxplots show the same (Algorithm B) data of in Figure 1, for which both FC and total informationTwere set to 120 time periods. The right series shows the effect of changing the FC to 60 time periods. In thiscase CL(k) is converging to values less than L since replacing a set of 120 observations results in less dispersionabout the optimal portfolio than does replacing only 60 observations.

probabilities in fact tends toward a value greaterthan Las k approaches T. Only a small prob-ability mass of the distributions in the upperpanel of occur below the medians in the lowerpanel.This fact is consistent with theintuitionthatchanging portfolio target risk alone often requiresrebalancing.

Variations in FC

In prior examples it wasconvenient to assumethattheTandFCperiods are the same. However, inapplications, each parameter has a separate roleto play in defining the monitoring process. Theirroles areclearest when considering historical dataas in Algorithm A. In this case T is the num-ber of periods in the data set while FC is the

number of resampling periods representing theinvestors level of certainty in investment infor-mation. WhileTis fixedFCmay vary dependingon market outlook, manager style, investmenthorizon, and other considerations.

When Tand FCdiffer, CL(k) no longer necessar-ily converges toLaskincreases. This is becausethe two parameters imply different amounts of

information and levels of dispersion about the tar-get portfolio. The net effect is that CL(k)tendstoward values smaller or larger thanLdependingon whether the number ofFCresampling periodsis less or greater than T,respectively. Figure 6shows the effect of changing the FC windowfrom 120 to 60 time periods with Tfixed at 120time periods. Since 120 time periods yield moreinformation and less dispersion of simulatedport-folios than 60 periods, the simulated rebalancetest statistics are small and CL(k)converges torelatively small values.

Notes

1 Markowitz (2005) demonstrates the critical impor-tance of realistic linear inequality as well as equality

constraints on financial theory as well as applications.2 Michauds REF is a generalization of the linear equal-ity and inequality constrained Markowitz MV efficientfrontier that includes estimation error in inputs andresampling to define portfoliooptimality (Michaud andMichaud 2008a, Ch.6; 2008b). TheMichaudoptimiza-tion and rebalancing test are protected by US patentsand patents pending.

3 Michaud (andMarkowitz) efficiency has beencritiquedas inconsistent with the rationality axioms of the von



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Neumann and Morgenstern (1944) Expected UtilityHypothesis (EUH). However the EUH is itself sub-ject to numerous serious critiques. As a consistentformal axiom system, EUH is neither consistent with

observed investor behavior (Allais, 1953, Kahnemanand Tversky, 1979), unique (Quiggin, 1993, Luce,2000), nor complete (Gdel, 1931, Church, 1936, Tur-ing, 1936), andcannot claim definitive characterizationof rational decision makingunder uncertainty. Analter-native rational framework of investor behavior basedon evolutionary principles (Farmer and Lo, 1999; Lo,2004) can be viewed as fundamentally consistent withMichaud optimality. REF optimal portfolios properlydefined enhance the likelihood of investor prosperityor survival in controlled statistical experiments relativeto classical MV optimization (Michaud, 1998, Ch. 6;

Markowitz and Usmen, 2003, Michaud and Michaud,2008a, Chs. 6, 9) in the investment period. In our casepsychological hypotheses of species behavior or ecol-ogyareunnecessary. Inpracticalterms, REFoptimalityis simply a convenient framework for constructingenhanced linear constrained MV efficient portfoliosthat address estimation error uncertainty endemic inapplied finance and asset management.

4 Tracking error may be replaced by other measures ofportfolio dissimilarity. For example, a utility-function-based metric could be used. We use tracking error asthe most straightforward of these measures.

5 We will discuss some of these issues further below.6 US patent pending.7 Return distributions are well known to be nonstation-

ary. However, for many practical situations there is notenough data to support estimation of a more complexprobability model, so the more parsimonious station-ary model is preferred, and the input data is limited toa relevant time period to the analysis or weighted todecrease in importance for less relevant time periods.See Esch (2010) for further discussion.

8 We use the traditional notations for Bayesian analysisbut intentionally leave model choice at the discretion

of the manager. Many types of information can beincluded in the analysis.

9 Managers typically use linear inequality constraintswhen defining optimality in anytime period.This prac-tice is one of the main reasons analytical calculationof the rebalance test is intractable, and Monte Carlomethods must be used.

10 See Michaud and Michaud (2008a, Ch. 6, App. A) andEsch (2012).

11 The Michaud procedures by default use tracking error,that is, relative variance, as the discrepancy function,butanysuitable distance metriccanbe substituted. Anywell-defined norm on portfolio space induces a dis-

crepancy function when evaluated on the difference ofportfolios.12 The Michaud procedures use the empirical form of

the distribution of D(P, Pt) obtained by simula-tion. This is an approximation of the true continuousdistribution which has no closed form for generalinequality-constrained optimization problems.

13 The patented FCprocedure can be conveniently for-mulated as an index with values from one to ten toheuristically reflect increasing certainty in risk-returnpoint estimates independent ofT. FCcan be associ-ated with numerous practical investment management

issues. See further discussion in Michaud andMichaud(2008a, b).

14 Chaves (2010) suggests a computationally efficientapproximation to our algorithm. This approach mod-ifies the calculation of R(t)but has some importantlimitations. In our method, each scenario has its ownupdated posterior and discrepancy function. Simulat-ing separate rebalance tests provides a more realisticcalibration to overlapping data, as well as providingadditional flexibility with settings of various parame-ters of theoptimizationprocess,asdetailedinSection5.Managers may also need flexibility to use differ-

ent predictive distributions conditioned on differentinformation sets in the simulation process from therebalance test so as not to reflect unusual events in thecontrafactual simulations.

15 See Davison and Hinkley (1997) for discussions andgeneralizations on simulation schemes.

16 In practice, asset managers use many techniques formodifying MV inputs and enhancing their forecastvalue. See Michaud and Michaud (2008a, Chs. 8 and11) for further discussion and references.

17 The computational algorithm and further technicaldetails are given in Michaudet al.(2010).

18 Weusedthesamedatasetthathasbeenusedinkeyarti-cles written on Michaud optimization. These includeMichaud (1998), Michaud and Michaud (2008a),Markowitz and Usmen (2003), Harvey, Leichty, andLeichty (2008), and Michaud and Michaud (2008c).This data set is sufficiently complex to illustrate ournew methods. Complete tables of means, standarddeviations, correlations, and many efficient fron-tier portfolios are available in the above references.



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Our qualitative results hold for a wide range ofdatasets.

19 Due to compute-efficiency and statistical stability con-siderations, the return-rank algorithm is the recom-

mended procedure in Michaud (1998), Michaud andMichaud (2008a, b) for computing REF optimal port-folios. Return ranks identify increasing risk levels ofportfolios on the REF. As in Michaud and Michaud(2008a, b), we compute 51 return-rank portfolios, fromlow to high, on the REF. We identify portfolio 26 asthe middle portfolio on the REF. More recent research(Esch, 2012) has shown that dividing the efficient fron-tier into segments of equal arc length provides furtherimprovement to out-of-sample performance over thereturn-rank algorithm. There is no loss in generality ifquadratic utility, stock/bond ratios, tracking error rel-

ative to a benchmark, or other procedures are used toidentify different risk levels on the REF. As we willshow, we have found that the statistical character ofCL(k) maynotvarymuchfordifferentrisklevelsacrossthe frontier.

20 CalculatingCL(k)for many values ofkis particularlycomputationally intensive. In applications managersmay be interested inCL(k)at only one value ofk. Forour purposes, the computation provides a comprehen-sive illustration of the procedure.

21 For example, the VIX index may indicate changesin market volatility that may impact need-to-trade

decisions.22 Theil (1971, pp. 347354) notes that analysts have orshould have a view of the range of values of coeffi-cients in multivariate linear regression estimation thatshould be included in the procedure. He also notes thatquadratic programming may have important applica-tions in proper data analysis if a statistical basis for theprocedure were available.

23 As discussedearlier, the results are computed from lowto high risk computed from the return-rank algorithmfor 51 points on the frontier.

24 The lower risk target optimal portfolio relative to

the middle portfolio is computed for rank 16 of 51portfolios with equally spaced expected returns.

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Keywords: Asset allocation; mean-variance opti-mization; Markowitz optimization; Michaudoptimization; Monte Carlo simulation; resam-pled efficient frontier; overlapping data; portfolio

monitoring; quadratic programming; rebalanc-ing; resampling; computational finance


portfolio monitoring (joim)

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