optimal Rebalancingfor Institutional PortfoliosMinimizing costs using dynamic programming.

WALTER SUN, AYRES FAN, LI-WEI CHEN,

T O M SCHOUWENAARS, and MARIUS A. ALBOTA

WALTER S U N is a research

assistant at the MITLaboratory for liifomiation

and Decision Systems

in C'ambridge, MA.

waltsun@aluiTi.mit.edu

AYRES FAN is a research

assistant at the MITLaboratory for InfonTiaCionand Decision Systemsin Cambridge

L I - W E I C H E N IS a research

assistant at the MITLaboratory for Informationand Decision Systemsin Cambridge

TOM SCHOUWENAARS IS a

research assistant at the MITLaboratory tor Informationand Decision Systemsin Cambridge

MARJUS A . A L B O T A is a

research assistant at the MITResearch Laboratory

of Electronics in Cambridge

WtNTER 2006

Institutional money managers develop risk modelsand optimal portfolios to match a desired risk/rewardprofile. Utility functions express risk preferencesand implicitly reflect the views of fund trustees or

directors.Once a manager determines a target portfolio,

maintaining this balance of assets is non-trivial. A man-ager must rebalance actively because different asset classescan exhibit different rates of return. JManagers also mustrebalance if weights in the target portfolio are altered.This occurs when the model for expected returns ofasset classes changes or the risk profile changes.

Most academic theory ignores frictional costs, andassumes that a portfolio manager can simply readjustholdings dynamically without any problems. In prac-tice, trading costs are non-zero and affect the decision torebalance. Transaction costs involve commissions andmarket impact as well as cost of personnel and techno-logical resources. If the transaction costs exceed theexpected benefit fixim rebalancing, no adjustment shouldbe made, but without any quantitative measure for thisbenefit, we cannot accurately determine whether or notto trade.

Conventional approaches to portfolio rebalancinginclude periodic and tolerance band rebalancing (seeDonahue and Yip |20()3] and Masters [2003]). Withperiodic rebalancing, the portfolio manager adjusts to thetarget weights at a consistent time interval (e.g., monthlyor quarterly). The drawback with this method is thattrading decisions are independent of market behavior.

THE JOURNAL OF PORTFOLIO MANAGEMENT 3 3

. J

Thus, rebalancing may occur even it the portfolio is nearlyoptimal.

Tolerance band rebalancing requires managers torebalance whenever any asset class moves beyond somepredetermined tolerance band (e.g.. +5%). When thisoccurs, the manager fully rebalances to the target portfolio.While this method reacts to market movements, thethreshold for rebalancing is fixed, and the process ofrebalancing involves trading all the way back to the optimalporttolio.

Research on dynamic strategies for asset allocationhas established a so-called no-trade region around theoptimal target porttolio weights (see Perold and Sharpe11995] and Leland [1999|). If the proportions allocated toeach asset at any given time lie within this region, trad-ing is not necessary. If current asset ratios lie outside theno-trade region, though, Leland [1999] has shown thatit is optitnal to trade but only to bring the weights backto the nearest edge ofthe no-trade region rather than tothe target ratios.

The optimal strategy has been shown to reducetransaction costs by approximately 50%, but the full ana-lytical solution involves a complicated system ot partialditTerential equations in multiple dimensions.

Mulvey and Sinisek [20()2| model the problem ofrebalancing in the tace of transaction costs as a genera-lized network with side conditions and develop analgorithm for solving the resulting problem. Mitchelland Braun [2*.)02] describe a method for tending anoptimal portfolio when proportional transactions costshave to be paid.

Since then, Donohue and Yip [2003] have confirmedthe results of Leland [1999]. They characterize the shapeand size ofthe no-trade region, and compare the perfor-mance of different rebalancing strategies.

We present an approach that explicitly weighstransaction costs and portfolio tracking error. We assumewe are living in a CAPM world, which means that assetreturns are stationary, and mean and variance are theprimary portfolio statistics of interest (see Markowitz[ 1952]). Utility Rinctions coupled with asset return modelsthen yield a target portfolio that is a set of optimal weightsfor different asset classes. We also assume the portfolios areeither tax-free or tax-deferred, which is the case fbrendowments, charities, pension funds, and most individualretirement funds.

The main difficulty with reconciling transactioncosts and tracking error is that they are expressed in dif-t'erent units. Transaction costs are measured tangibly by

dollars. Tracking error h a more abstract concept. Becausethe optimal porttolio is the portfolio that maximizes ourgiven utility function, we can express tracking error asthe shortfall in utility from our current porttolio to theoptimal portfolio.

Our tirst contribution is applying the concept ofcertainty-equivalents to create risk-adjusted returns that allowus to convert tracking error into a dollar-denoniinated cost(see Bernoulli ] 1954[). Note that we are not restricted toquadratic utility but can use arbitrary utility functions.Once we have a dollar cost, we can directly compare thetransaction costs for rebalancing with the suboptimalitycosts for not rebalancing.

Yet this is leaving ont an essential piece of thepuzzle: Our actions this period also atfect outcomesand decisions in future periods. Our secoiid contribu-tion is to then apply the method o'i dynamic progranimitigto minimize a cost function that explicitly models thispoint. Thus our optimal policy trades only when theexpected cost of trading is less than the expected cost ofdoing nothing, evaluating costs over the next periodand all future periods.

In addition, we search over the rebalancing spacetrom 0% (no rebalancing) to 100% rebalancing (fullrebalancing) and the points in between. In most cases,partial rebalancing can provide nearly the same utility asfull rebalancing while saving on transaction costs.

Our third contribution is a framework to evaluaterebalancing strategies quantitatively. We show that ourmethod performs better than traditional methods ofrebalancing and is robust to model error.

OPTIMAL REBALANCINGUSING DYNAMIC PROGRAMMING

We consider a multiasset problem where we aregiven an optimal portfolio consisting of a set of target port-folio weights w* = {Wj, w-, " '^- j , where jVis the totalnumber of assets. The optimal strategy should be to main-tain a portfolio that tracks the optimal portfolio as closelyas possible while minimizing the transaction costs.

The model allows us to observe the contents ofthe portfolio w^ at the end of each month. At this point,we have the option of rebalancing the portfolio (i.e..apply our policy, or control, u^). Thus, the portfolio atthe beginning ofthe next month is w^ + u^. Assumingnormal returns in the process noise are subject to n^,we use a simple multiplicative dynamic model so that

34 OlMlMAL RpllAi ArMCiNC RIR [NST[1"UTK>NAI Poi< I H WiNTkR 2(106

I'Vi ~ (1 + ",''f"'r + ",J^ although in general n' _ | can bean arbitrary function of ic , N , and n^.

The decision to rebalance should be based on a con-sideration of three costs: the tracking error associatedwith any deviation in our portfolio from the optimalportfolio; the trading costs associated with buying orselling any assets during rebalancing; and the expectedfuture cost from next month onward, given our actionsin the current month. The optimal strategy; determinedthrough dynamic programming, minimizes the sum ofthese three costs.

Dynamic programming is an optimization tech-nique that tuids the policy that minmiizes expected cost,given a cost function and a dynamic model of statebehavior (see Bellman |1957], Bellman and Dreyfus[1962], and Bertsekas [2000]). The cost at any givenperiod is the expected cost from itot + l along with theexpected cost from t + 1 onward.

Assuming convergence, this recursion approachesa fixed point known as the cost-to-go value. Once thesevalues are known, the optimal rebalancing decision is tochoose the policy that achieves this mmimum.

To apply dynamic programming, we specify thecost function as a sum of trading and suboptimality costswhere the trading costs include tangible fl-es such as com-missions and market impact, but also indirect costs suchas employee labor, while the suboptimality cost representsthe cost of not having an optimal portfolio.

Note diat the cost-to-go values, and hence the optinuilstrategy, will depend on the cost functions chosen. In thecertainty-equivalence approach, we model the investor'spreferences using a utility function (see Luce [2000]).

Because future returns are unknown, we need to useexpected utility to create an optimal portfolio or to decicieon a rebalancing policy. Lev>' and Markowitz |1979] haveshown that for most relevant utility functions this expectedutility U can be approximated using truncated Taylorseries expansions to be a function of mean and standarddeviation, U(fi, &j.

Exhibit 1 lists the three utility functions and the cor-responding expected utilities that we use (see Cremers,Kritzman. and Page |2004]). For each utility, f.(x) for( - {q, /, p] (where q indicates quadratic, / indicateslogarithmic, and;j indicates power) represents the utilityin Htils given a return .v, which we also refer to as theempirical utility. (_/(>, oj for i ^ {q, I, p] is the expectedutilit\'. Details and derivations may be found in Sun et al.[2004].

E X H I B I T 1Utility Functions with CorrespondingApproximate Expected Utilities

Utility Fund ion Jixpcctcd l.tility_QuadraticLog wealth

Power

fq(x)

fp(x)

= X - a/2 (x - Xo) U l

= log(l+x) Ui{

- 1 - 1 / ( 1 + x ) U (

H.u) = (.i - iilla-

^,a)= 1 - l/(l + Li)-ci^/(l +M)^

For any portfolio with weights w, we observe thatthere exists a risk-free rate, which we will denote asr^j:(u'}, that produces an identical expected utihty. We callthis the certainty-equivalent return for the weights ir.'

One interpretation ofthe certainty-equivalent thenis a risk-adjusted rate ot return, given the risk preferencesembedded in the utihty function.

If we hold a suboptimal portfolio w, the utility ofthat portfolio U(u') will be lower than U(u>'), with a cor-respondingly lower certainty-equivalent return. We caninterpret this as losing a risk-free return (equal to thedifference between the two certainty-equivalents) over oneperiod, corresponding to the penalty paid for trackingerror. The difference between the certainty-equivalentsof a non-optimal portfolio and the optimal portfolio isdefined as the cost of not being optimal.

We use a certainty-equivalent because the tradingand suboptimality costs must have commensurate values.We know that the cost will be in terms of dollars orbasis points or some other absolute measure. It is morestraightforward then to convert the portfolio trackingerror into a similar absolute measure using certainty-equivalents rather than try to express the trading costs interms of diminished expected utility (although this canbe done as well).

Assume we have a portfolio ic, and we want to goto another portfolio u\^. The most basic model for tran-saction costs is simply to assume a linear cost. This isjust one specitic choice ot transaction cost. Alternativetransaction cost functions can be chosen as appropriate.For example, price impact models can be used, or aproportional plus fixed cost model can be used to dis-courage frequent trading.

EXPERIMENTAL RESULTS

We use tive asset classes: hedge funds, developedmarkets equity, emerging markets equity. U.S. equity,and private equity'. Exhibit 2 shows the mean and stan-dard deviation of each asset class for historical monthly

WINTER 2006 THEJCIURNAL of l'tiKfTHLio MANAGEMENT 3 5

E X H I B I T 2Annual Mean Returns and Standard Deviationsfor Asset Classes

E X H I B I T 3Correlation Coefficients

Hedge Funds

Developed Markets

Emerging Markets

US Equily

Private Equity

Index as Proxy (Source)

HFR Mkt Neutral (Bloomberg)

MSCI EAFE - Canada (Datastream>

MSCI EM(Datastrcam)

Rus.sell 3000 (Daiastream)

Wilshire LBO (Bloomberg)

Mean Return(%)5.28

6.65

7.88

6.84

12.76

Std Dev(%)

10.16

16.76

23.30

14.99

44.39

returns over the last decade. The correlation matrix isshown in Exhibit 3.

Ofthe different assets, private equity provides thehighest expected return, but it also has the greatestamount ofrisk. At the other extreme, hedge tunds haveboth the lowest expected return and the least amountof variability. (The mean returns were provided byState Street Associates, and the variances and correla-tions are computed empirically from data acquiredfrom Datastream and Bloomberg.)

To introduce the problem of portfolio rebalancing,we first consider an example involving only two assetclasses: developed markets and hedge funds. The benefitsof the two-risky asset model are that: the optimal port-folio can be computed in closed form (see Brealey andMyers [1996] for a derivation); we can visually examinethe changes in portfolio weights since a suigle asset'sweight represents the full description ofour portfolio;and there are few enough parameters that we can easilyperform sensitivity analyses. We later carry out extensivesimulations with a multiasset model.

At first we consider only quadratic utility with riskaversion parameter a = 1.5. Using this assumption, theoptimal portfolio balance is 51% in developed marketsand 49% in hedge funds. To illustrate the behavior ofourrebalancing method, we simulate the returns ofthe twoasset classes over a single 20-year realization.

Exhibit 4 shows how the portfolio weight of devel-oped markets moves over one 240-month sample path.With no rebalancing (Panel A), the weight drifts from theoptimal amount of 49% to under 25%, resulting in highsuboptimality costs.

The optimal rebalancing strategy rebalances onlywhen necessary (Panel B). During months 110-120 and170-215. the portfolio partially rebalances nearly everymonth to handle sharp changes in the portfolio, while formonths 120-160, the lack of strong market movements

Hedge Funds

Developed Markets

Emerging Markets

US Equity

Private Equity

Hedge

Funds

1.00

0.09

0.21

0,29

0.36

Developed

Markets

0.09

1.00

0.42

0.46

0.38

Emerging

Markets

0.21

0.42

1.00

0.45

0.40

US

Equity

0.29

0.46

0.45

1.00

0.64

Private

Equity

0.36

0.38

0.40

0.64

1.00

in either direction allows us to avoid any transaction costs.The market movement during these times can be seen byexainining the change in portfolio weights in Panel A,where there is no rebalancing.

We evaluate different rebalancing algorithms usinga Monte Carlo simulation process. Each month is sam-pled independently from the others, so we do not modeleffects such as trends, momentum, or mean reversion. Eoreach sample path, we simulate the various rebalancingmethods by generating a return value for each month thatis net of transaction costs.

Exhibit 5 shows the annualized costs of differentrebalancing strategies. Trading costs are 40 basis pointsfor buying or selling developed markets and 60 bp forbuying or selling hedge funds.

We show two metrics to evaluate performance. Inboth cases, the numbers are in terms of shortfall from anidealized rebalancing strategy that is allowed to rebalanceto the optimal portfolio for free every month.

The first metric is to aggregate expected cost shownin the third column. This is simply the sum ofthe trad-ing costs in the first column and the suboptimality costsin the second column. The suboptimality cost is computedat the end of each month as the difference between theexpected risk-adjusted returns for the optimal portfolioand the current portfolio after the rebalancing policyhas been applied. Note that this is precisely the metricthat our dynamic programming approach is designed tominimize.

The second metric is what we call empirical utilityshortfall. For each month, we compute the return net oftransaction costs and then compute the utils associated withthat return using our empirical utility function f. (seeExhibit 1). This metric is similar to aggregate cost, butworks with actual utility rather than the expected utilityused to compute certainty-equivalents. We expect thesample average and the expected value to converge, whichthey do in our case."

36 Oi' l lMAL REi3ALAN(:iNG FOR INSTITUTIONAL PORTFOLIOS WINTER 200f.

E X H I B I T 4Developed Market Weights in Two-Asset Example—Different Rebalancing Models

No Rebalancing Opiimal RebalarKing 5% Tolerance Band Rebalancing

so 100 150 200Annual Rebalancing

50 100 150 200Monthly Rebalancing

fiO 100 1S0 200 SO 100 150 ZOO 50 100 150 200

Vcrtiiii! lines indicatf rebalaiiciu^ eventi {omiuvd in monthly rvhalimdn^ since it occurs at ci'cr)' time point).

E X H I B I T 5Annualized Costs and Utility Shortfall—Six DifferentRebalancing Strategies—Two Risky Assets

IdealOptimal DPNo Trading5% ToleranceMonthlyQuarterlyAnnual

Trading Cosi (bps)

0,001.570.003.68

12,927.463.71

Suboptimality Cost (bps)

0,000.464.740.130.000,050.26

Aggregate Cosi (bps)

0,002.034.743.81

12.927,513.97

Utility Shonfall (uiils x 10*}

0.002,154,563.65

12.947.464.05

Rcsutti iirc an average of 10,000

From Exhibit 5 we observe that our methodminimizes the aggregate cost. Assuming a portfolio ofSI billion, the aggregate annual cost by our algorithmis $203,000. The cost tor rebalancing using 5% tolerance,the next-least expensive method, is $381,000 annually.

The results for each rebalancing method makeintuitive sense. Monthly rebalancing produces no devia-tion from optimality, but at the cost ot high tradingfees. Intrequent trading yields lower trading costs, but

higher suboptimality costs. Ourmethod of rebalancing whenever thecost of non-optimality exceeds thetrading costs allows us to adequatelytrade off the cost of non-optimalitywith that of trading.

So far we have assumed that themodel for each asset is known. Inpractice, the mean and variance ofeach assets returns as well as the cor-relation between the assets must beestimated (using historical observa-

tions). Errors in the parameter estimate, such as deviationfrom the true unknown value, will cause inaccuracies inthe costs-to-go obtained from the dynamic program,resulting in suboptimal rebalancing.

To investigate the impact of errors in each ot theseparameters on the rebalancing strategy, we consider themeans, standard deviations, and correlation parameters.For each simulation, only one ofthe three parameters isvaried to isolate the effects of changes to each of these

WINTER 2006 THEJOURNAL OF PORTFOLIO MANAGEMENT 37

E X H I B I T 6Sensitivity Analysis for Two-Asset Example

15

o 5

Sensitivity for error in mean return Sensitivity tor error in standard deviation

\ "°5% tol ~^.,

[_-——mon

' • • - ^

My

•*~~ quarterly

annual-.. no rebalance

,-'

• ' ' /

DP

6 6.5 7 7.5Annual return for developed markets

14

Sensitivity for error in correlation

15

1(01 5

monthly '

S%tol " ~ .

DP

' • • . ^

quarisriy

• *

no rebalance

-0.4 -0.2 0.2 0.4 0.6Annual standard deviation for developed markets Correlation between developed markets 8 hedge funds

A. Mean return. B. Standard deviation. C. Corrclatioti coefficient.

I'crtical line indicates actual rate used hy the rehabnciii}; stratei^ics.

factors. We then compute annualized aggregate cost,while the true value ofthe third parameter varies aroundthe assumed value. This cost is computed relative to anidealized portfolio that knows the correct value andrebalatices tor tree to the true optimal portfolio everymonth.

Most rebalancing strategies are inherently model-based because tiie target portfolio is determined by themodel. When the target portfolio is determined withincorrect parameters, the target portfolio is itself subop-tuiial. Therefore we would expect all the rebalancingmethods to be sensitive to model error. The exception tothis among the rebalancing methods we examine is thestrategy of no-rebalancing.

For our testing methodology, the assumed modelplays no role except in determining the initial portfolio.Theretore we would expect no rebalancing to be leastsensitive to model error. Our algorithtn should be evetitnore strongly influenced by the model because incorrectassumptions can also result in suboptimal rebalancingdecisions from the dynamic program. Other methods suchas calendar and tolerance band strategies are heuristic,and do not directly rely on the model when fortning therebalancing policy.

The results for mean sensitivity are shown in Exhibit 6,Panel A. For each point, 10,000 sequences of 20 yearsof monthly returns are generated, and the performance ofeach rebalancing strategy is averaged over each sequence.

Our dynamic rebalancing strategy is as robust astolerance band and calendar rebalancing. We performbetter than no-rebalancing until the annual return exceeds7.6%.

This is because the annual return for developedmarkets is higher than the return for hedge funds. So,without rebalancing, as the annual return for developedmarkets increases, the average weight tor developed mar-kets tends to increase as well. At the same time, it becomesmore advantageous to hold developed markets due to thehigher return, so the true optimal portfolio will includea higher percentage of developed markets. Thus, in thiscase, a passive strategy' will typically let the portfoHo moveinadvertently toward the optimal portfolio.

In Exhibit 6, Panel B, we see that the dynamic pro-gramming approach again outperforms the other approachesdespite large errors in estimating the standard deviation—it still outperforms the other methods even for inaccura-cies in the standard deviation of several percentage pointsper year. In this instance, the strategy of no-rebalancingperforms better for lower standard deviations.

As in the mean sensitivity case, because developedmarkets have a higher average return than hedge funds,their weight tends to increase with tmie. So as the stan-dard deviation for developed markets falls, the true opti-mal portfolio will again have a higher weight in developedmarkets, and the portfolio will again tend to drift towardthe optimal portfolio.

Finally, in Exhibit 6, Panel C, we observe that thedynamic programming approach is relatively insensitiveto errors in estimation ofthe correlations between theassets. As the cost does not change much for ditTerenttrue correlations, we conclude that correlations neednot be accurately estimated for the purposes of ourapproach.

38 HEHALANCING FOU INSTITUTIONAL PfRTForrcis WINTER 2006

E X H I B I T 7Efficient Frontier and Optimal Portfoliosfor Different Utility Functions Discussed

10 124 6 8Annual Standard Deviation (%)

fo) indicates optiiiuil portfolio lor pouvr utilily. (+) iiitiicalcs oplimalportfolio for quadratic utility with a - 1.5. (') indicates optimal port-folio for /o ' wealth utilily.

In the general case of Nrisky assets, we examine fiverisky assets, and assert that another choice ot N > 2 wouldproceed similarly; the main difference is computationtime. According to Cremers, Kritzman, and Page [2()03|,standard mean-variance portfolio optimization producesoptimal portfolios only if returns are normally distributedor if quadratic utility is assumed. Otherwise, full-scaleoptimization must be performed to compute optimalportfolios when one is using more advanced utility func-tions such as log wealth or power utility.

Later work by Cremers. Kritzman, and Page [20()4|indicates that except when returns are highly non-normal,it is sufficient to perform mean-variance optimizationon a Markowitz-st^'le approximate expected utility func-tion in terms ofjust the mean and standard deviation. Theyshow that the resulting portfolios and portfoHos generated

from tiiU-scale optimization do not perform significantlydifferently.

Under this approximate mean-variance optimiza-tion, the optimal portfolio lies on the etEcient frontier (seeMarkowitz [1952]).^ Therefore, to construct optima!portfolios for different utility' functions, we first computethe etTicient frontier by solving a quadratic programmingproblem and then search those portfolios to find the onewith the highest expected utiHty.

Exhibit 7 displays the efficient frontier for the fiveasset classes when short sales are not allowed. Searchingover this frontier for each ofthe utility functions resultsin the optimal portfolios with the weights shown inExhibit 8. These weights are the optimal weights weuse throughout the analysis. Note that power utility isthe most risk-averse utility, and log wealth is the risk-seeking Lit

Monte Carlo Simulations

Exhibit 9 shows the results of various rebalancingalgorithms for the tive-asset case. The results are generatedin a manner analogous to the two-asset case with MonteCarlo simulations over 20 years for 10,000 sample paths.Trading costs are 60 bp for hedge funds, emerging mar-kets, and private equity; 40 bp for developed markets; and30 bp for U.S. equity.

For the quadratic utility case, we see that our optimaldynamic programming (DP) method performs approxi-mately 30% better than the next-best method, which is5% tolerance bands. If we examine the costs, we see, asexpected, that monthly trading incurs no suboptimaHtyat the expense of high trading costs. The other extremeof no-trading incurs an extremely high suboptimalitycost because over a 20-year period assets can becomequite unbalanced if unadjusted.

For power utility, our expected loss is 24% less thanthe runner-up. 5% tolerance band rebalancing. The

E X H I B I T 8Optimal Portfolio Weights for Different Utility Functions

Hedge Funds

Developed Markets

Emerging MarketsUS Equity

Private Equity

Quadratic (a = 1.5)

0.243

0.222

0.1850.194

0.156

Logarithmic

0.033

0.2400.275

0.160

0.292

Power

0.341

0.2130.143

0.210

0.093

W I N T E R 2*H)6 T H E JOURNAL OF PORTFOI.KT MANA(;F.MENT 3 9

E X H I B I T 9Quadratic, Power, and Log Wealth Utility Costs and Shortfall—Six Different Trading Strategies—Five Risky Assets

Quadratica= 1.5

Power

Log wealth

IdealOptimal DPNo Trading5% ToleranceMonthlyQuarterlyAnnualIdealOptimal DPNo Trading5% ToleranceMonthlyQuarterlyAnnualIdealOptimal DPMo Trading5% ToleranceMonthlyQuarterlyAnnual

TradingCost (bps)

0.003.970.007.29

23.6713.696.840.003.390.005.19

20.05!l.595.810.004.800.00

11.9528.1516.278.05

SuboptimalilyCost (bps)

0.001.49

30,180.700.000.281.550.001.15

25.990.820.000.181.020.001.93

30.520.430.000.402.17

AggregateCost (bps)

0.005.47

30.187.99

23.6713.968.390.004.54

25.996.01

20.0511,786.840.006.72

30.5212.3828.1516.6710.22

Utility Shortfall(utiisx 10 )

0.005.23

32.897.88

23.7314.218.430.004.32

26.045.95

19.9611.876.800.006.57

32.0112.7228.1916,8010.65

Simulated over a 20-year period 10,000 liines.

benefits for this method are reduced from the quadraticutility case primarily because power utility is more risk-averse, so the optimal portfolio has a lower variance.Therefore less rebalancing is needed overall, because theportfolio simply does not move as much. This can beseen in the quarterly and annual rebalancing methods,which trade less and thus suffer lower suboptimalitycosts.

For the log wealth utilitv" case, our algorithm resultsin an expected loss that is 35% less than the best alterna-tive, annual rebalancing in this case. The 5% tolerancemethod tails short in this case simply because the logwealth portfoho is a high-variance portfoho. In thequadratic case, the trading costs are only marginallyhigher than the annual rebalancing method. But in the logwealth case, they are 48% higher because the tolerancebands are breached more often.

As a general rule, we see it is more important to getthe rebalancing right when dealing with higher-varianceportfolios simply because many more rebalancing oppor-tunities arise.

Note that even though tolerance bands do betterthan annual rebalancing for quadratic utility and power

utility and worse tor log wealth utility,it is not a valid conclusion that toler-ance band methods perform better forlow-variance portfohos and calendarrebalancing methods perform better forhigh-variance portfolios. There is adegree of freedom in each algorithm.

For tolerance bands, the bandscan be loosened or tightened, depend-ing on the variance ofthe optimal port-folio. For example, in the log wealthcase, it is clear that the tolerance bandsare too tight because of the sheerimbalance between trading costs andsuboptimality costs.

For calendar algorithms, the periodbetween rebalancing can be adjusted.For instance, setting the rebalancing timeto two years tor the power utility caseresults in an expected loss of 6.32 bpper year, a savings of 0.52 bp over theannual strategy. This is achieved byaccruing more than twice as muchexpected tracking error (2.21 bp versus1.02 bp) but also reducing trading costs

— by 29% (4.11 bp versus 5.81 bp). A moreexhaustive search of possible fixed-

interval rebalancing strategies could presumably yield aneven better result.

Optimal Heuristic Algorithms

Our 5% tolerance and periodic rebalancing periodsof monthly, quarterly, and annually are a subset ofgeneral tolerance band and calendar-based rebalancingalgorithms. We make no claims that the parameters chosenare optimal but rather that the settings seem to be conunonin the literature.

Our work can be thought of in two ways. First,we show that our method is superior to any toleranceband or calendar-based rebalancing method. It shouldbe noted that our method may be thought of as a dynamictolerance band approach. Fixed-tolerance methods area subset ofthe controls available to our algorithm, so theycan never do better.

Second, the tull dynamic programming algorithmis computationally intensive, especially with many assets.If we instead confine the search to a smaller subset ofacceptable policies, we have a strategy that is not neces-sarily optimal, but perhaps good enough.

40 OHTIMAI REBALANCING FOR INSI nui io FORTFODOS WINTER 2006

E X H I B I T 1 0Annualized Strategy Costs for Different Calendar-Based and Tolerance Band Settings—Quadratic Utility

10

Best periodic rebalancing frequency

\

9 \

7-

Best tolerance band

Optimai rebalancing Optimai rebalancing

10 15 20 25 30 35 40 45 4

Rebalancing Frequency (months)

A. Plot of costs (IS funciioii of different periodic rebalancing periods.

B. Plot of costs as function of different tolerance band windows.

6 8 10 12rolerance Band (%)

14

Exhibit 10 graphs the results for the quadraticutility case. Each point is obtained by performing MonteCarlo simulations over 20 years for 10,000 realizations.In Panel A. we plot the total costs as a function ofthe number of months between rebalancing. The lackof smoothness in the curve, particularly between lessfrequent rebalancing, arises because the amount ofrebalancing in a 240-month period must necessarilychange in discrete steps.

We see that the best performance occurs for 18-monthperiods at a cost of 7.53 bp. Note this is still over 2 bpworse than our algorithms performance {shown as thehorizontal dashed line) but nearly 1 basis point better thanannual rebalancing.

In Panel B, we vary the tolerance setting that trig-gers rebalancing. We fmd that the best setting occurs fora tolerance band of 9% and results in a cost of 6.25 bp,again worse than the 5.47 bp obtained by our algorithm.

Computational Complexity

To provide some information regarding the com-putational complexity of our approach, note we allow onthe order of 15 possible weights for each asset. For fiveassets, we have an observation space of approximately750,000 points m which we must develop the optimalpolicy. Our current implementation processes around600,000 points per hour on a single PC, although valueiteration can be easily parallelized, so the total processmgtime also depends on the number of machines available.

For a non-parallelized implementation, the run-time estimate for five assets is 75 minutes. If we assumethe possibility of M ditFerent weights for an additional asset.

addition of this asset to our N-asset model would increasecomputation time by a factor oi'M. Memory requirementsincrease by a similar amount.

Note this includes detailing the computation forlearning the optimal policy. Once that is done, actuallyapplying the policy is very fast.

Alternative Cost Functions

Some may wonder whether the results would bedifFerent for alternative trading costs. Exhibit 11 showsthe results when we cut the trading costs in half andapply this to the quadratic utility strategy. 5% tolerancebands remain the next-best strategy, but our advantage hasnarrowed from 30% to 20%. The reason is that the otheralgorithms trade too much, and now they are penalizedless tor it.

Transaction costs for the other methods are cut inhalf, while suboptimality remains the same. This pro-duces reductions m aggregate cost ranging from 41%for annual rebalancing to 50% for monthly rebalancing(ignoring no-rebalancing, which obviously does notbenefit).

Our algorithm actually reduces trading costs by lessthan half This means that we are sensibly trading more,now that transaction costs have been lowered—so ouralgorithm automatically trades otFa little extra trading toreduce the suboptimahty costs by a greater amount.

CONCLUSION

The ad hoc methods of periodic and toleranceband rebalancing provide simple but suboptimal ways to

WlNTI^R 2006OURNAL OP [ '(IRTFOLK) MANAGEMENT 4 1

E X H I B I T 1 1Quadratic Utility Costs and Shortfall—Transaction Costs Halved

IdealOpiimal DPNo Trading5% ToleranceMonthlyQuarterlyAnnual

Trading Cost (bps)0.002.600.003.64

11.836.843.42

Suboptimality Cost (bps)0.000.85

30.180.700.000.281.55

Aggrepate Cost {bps)0.003.45

30.184.34

11.837.124.97

Utility Shortfall (utils X lO')0.003.75

32.894.53

11.86

7.185.15

rebalance portfolios. Calendar-based approaches rely on

the fact that, on average, we expect a portfolio to become

less and less optimal as time goes on, but they do not use

any knowledge about the actual state of tbe portfolio. The

tolerance band approach uses the current portfolio to

make a decision, but tbis method has no sense of tbe

proper tolerance band setting, or even how wide this

band should be.

We have shown that by formulating tbe rebalanc-

ing problem as an optimization problem and solving it

using dynamic programming, we reduce the overall costs

of portfol io rebalancing. T b e reduced costs bold for

different investor risk preferences, wbetber quadratic, log

wealth, or power utility functions.

The costs of transactions are mucb more tangible

than tbe cost of being suboptimal. Tbrough the use ot

certaint^'-equivalents. however, we have provided a method

tbat quantifies tbe cost of being suboptimal. O u r simula-

tions confirm that this optimal method provides gains over

the best of tbe traditional techniques of rebalancing.

Note that the analysis assumes returns are indepen-

dent across different intervals. T b e literature observes that

mean reversion may exist. Unde r sucb circumstances, we

expect our metbod to perform even better than periodic

rebalancing, because our algorithm would likely rebalance

even less often.

Tbere are several possible extensions o f o u r work.

First, one may want to consider proportional plus fixed

transaction costs. This model is appropriate if one believes

there is a fixed cost in making each and every transaction.

Such an adjustment would likely favor dynamic trading

methods over periodic rebalancing.

Next , we might want to examine rebalancing over

taxable portfolios. Asset managers of such funds have tbe

additional consideration of tax consequences when they

decide to transact.

Tbe relaxation ofthe short sales constraint is anotber

possible extension. Although many tax-deferred funds

do not allow short sales, several either explicitly allow

short-selling or implicitly participate in short sales tbrough

investments m hedge hinds.

We also assume an instantaneous rebalancing at tbe

end of each montb . O n e could incorporate more general

trading models that consider tbe effects of price impact.

Finally, for the multiasset case, we search a o n e -

dimensional policy space representing portfolios tbat are

a Imear combina t ion of the current portfolio and tbe

target portfolio. We ideally want to search over the entire

space of possible portfolios around tbe optimal portfolio.

Tbis would be particularly usefiil when trading costs have

a fixed component . In tbis case, it may be better to trade

on only a subset of assets rather than a portion of all asset

classes.

ENDNOTES

The authon thank Sebastien Page and Mark Kritznian fortheir guidance and advice and Edward Freyfogle and JoshuaGrover for their input. This article originated in a course pro-ject at the Massachusetts Institute of Technology's Sloan Schoolof Management.

'The condition for this is U{r^.,j,, 0) = U(^Ju', w'^'Aw). Thecertaint\'-equivalents for the three expected utility functions thatwe are using are:

Quadratic: r^jrfu'j = U (^^w, u''^Au')Log wealth: r^^M = expfU,f/x^[i', w'Aw))- 1Power: v^^Jw) = 1/(1 - U^^i^'^w,w^Au>)) - 1-The units on mils multiplied by 10'' in the last column

of Exhibit 5 are similar to the basis points in che first threecolumns. This is clear for the quadratic case where the certainty-equivalent is equal to che empirical ucility. For the other twocases, caking a linear approximacion around .v = i) showsthat the ucilitics are proportional Co .x. So utih times 10"' is

42 OPTIMAL REISALANCINC; FOR INSTITUTUINAL WrNTER 200ft

reasonably commensurate with basis points and explains whythe numbers in the bsc cwo columns are similar.

'Risk-averse expected ucility functions increase mono-tonically in tenns of retum and decline monotonically in Cermsof risk. Hence, if a portfolio is not on the efficient frontier, thereis a portfolio with equivalent retum and less risk or more retumand the same risk; therefore, this portfolio cannot be optimal.

iq. Princeton. NJ: Princeton

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