Paper by MIT's Mark Kritzman c.s. on the Markowitz - van Dijk approach that plays an important role in the LMG Emerge Global Tactical Asset Allocation (GTAA) system.Kritzman did a set of Monte Carlo simulations and concluded that the approach added value.
Revere Street Working Paper Financial Economics 272-22 Portfolio Rebalancing: A Test of the Markowitz-van Dijk Heuristic First Version: October 31, 2006 This Version: February 6, 2007 Mark Kritzman Simon Myrgren Sébastien Page Windham Capital Management, LLC 5 Revere Street Cambridge, MA 02138 617 576-7360 [email protected]State Street Associates 138 Mt. Auburn Street Cambridge, MA 02138 617 234-9416 [email protected]State Street Associates 138 Mt. Auburn Street Cambridge, MA 02138 617 234-9462 [email protected]
Transcript
Revere Street Working Paper Financial Economics 272-22
Portfolio Rebalancing:
A Test of the Markowitz-van Dijk Heuristic
First Version: October 31, 2006 This Version: February 6, 2007
Institutional investors usually employ mean-variance analysis to determine optimal portfolio weights. Almost immediately upon implementation, however, the portfolio’s weights become sub-optimal as changes in asset prices cause the portfolio to drift away from the optimal targets. In an idealized world without transaction costs investors would rebalance continually to the optimal weights. In the presence of transaction costs investors must balance the cost of sub-optimality with the cost of restoring the optimal weights. Most investors employ heuristics that rebalance the portfolio as a function of the passage of time or the size of the misallocation. Sun et al [2006] employ dynamic programming to determine optimal rebalancing rules, and they demonstrate that their approach is significantly superior to standard industry heuristics. Their approach is seriously limited, however, because it does not scale beyond a few assets. It suffers from the curse of dimensionality. Markowitz and van Dijk [2004] present a quadratic heuristic for rebalancing a portfolio to capture shifting expectations. We apply the Markowitz-van Dijk heuristic to address the asset weight drift problem, and we compare it to the unscalable dynamic programming solution as well as to standard industry heuristics. Our tests reveal that the Markowitz-van Dijk heuristic provides solutions that are remarkably close to the dynamic programming solutions and far superior to solutions based on standard industry heuristics. In cases of five or more assets, in fact, it performs better than dynamic programming due to approximations required to implement the dynamic programming algorithm. Moreover, unlike the dynamic programming solution, the Markowitz-van Dijk heuristic is scalable to as many as several hundreds assets.
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Optimal Execution for Portfolio Transitions
Part I: Introduction
Institutional investors usually employ mean-variance analysis to determine
optimal portfolio weights. Almost immediately upon implementation, however, the
portfolio’s weights become sub-optimal as changes in asset prices cause the portfolio to
drift away from the optimal targets. In an idealized world without transaction costs
investors would rebalance continually to the optimal weights. In the presence of
transaction costs investors must trade off the cost of sub-optimality with the cost of
restoring the optimal weights. Most investors employ heuristics that rebalance the
portfolio as a function of the passage of time or the size of the misallocation. Sun et al
[2006] employ dynamic programming to determine optimal rebalancing rules, and they
demonstrate that their approach is significantly superior to standard industry heuristics.
Their approach is seriously limited, however, because it does not scale beyond a few
assets. It suffers from the curse of dimensionality. Markowitz and van Dijk [2004]
present a quadratic heuristic for rebalancing a portfolio to capture shifting expectations.
We apply the Markowitz-van Dijk heuristic to address the asset weight drift problem, and
we compare its solution to the unscalable dynamic programming solution as well as to
solutions based on standard industry heuristics. Our tests reveal that the Markowitz-van
Dijk heuristic provides solutions that are remarkably close to the dynamic programming
solutions and far superior to solutions based on standard industry heuristics. In cases of
five or more assets, in fact, it performs better than dynamic programming due
approximations required to implement the dynamic programming algorithm. Moreover,
unlike the dynamic programming solution, the Markowitz-van Dijk heuristic is scalable
to as many as several hundred assets.
We organize the paper as follows. In Part II we review dynamic programming by
presenting two simplified examples. Then we describe how we apply dynamic
programming to the portfolio rebalancing problem and discuss the curse of
dimensionality. In Part III we adapt the Markowitz-van Dijk heuristic to address the
asset weight drift problem. We present our results in Part IV and conclude in Part V.
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Part II: Dynamic Programming Bellman [1952] introduced dynamic programming in the same year that
Markowitz published his landmark article on portfolio selection. Dynamic programming
provides solutions to multistage decision processes and is used in a diverse set of
applications including automatic sign language recognition, hydropower optimization,
sequential bidding in auctions, ecological management, and robotics, to mention just a
few. A particularly intuitive illustration of dynamic programming is provided by Smith
[1997]. He demonstrates how dynamic programming can be used to find a soul mate.
Soul Mate Search
Imagine you have 10 years to find a soul mate and you meet one potential soul
mate each year. You rank each companion on a scale from 0 to 100 and assume that
scores are uniformly distributed. At the end of each year you must decide to marry your
current companion or ditch that person and continue searching. You are not allowed to
revert to previous companions, and if you have not found your soul mate by year ten,
your parents force you to marry the person you are with at that time.
Dynamic programming provides the optimal year-by-year decision policy by
working backwards from year ten. The expected score of your companion in year ten is
50; hence you should marry in year nine only if your companion at the time scores above
50. There is a 50% chance you will marry your companion in year nine. If you marry in
year 9, your companion’s expected score is 75 given that it must be above 50 in order for
your companion to be marriageable. There is also a 50% likelihood that you will not
marry in year nine and settle in year ten for a companion with an expected score of 50.
The expected score and therefore hurdle at year nine equals 62.5 (50% x 75 + 50% x 50).
In year eight you should marry your current companion only if he or she scores
above 62.5. The likelihood that your companion in year eight will score above 62.5 is
37.5%, and the expected score of marriageable candidates in year eight is 81.3, given that
his or her score must surpass 62.5. There is a 67.5% chance that you will continue your
search into year nine which has an expected score of 62.5. The expected score for year
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eight therefore equals 69.5 (37.5% x 81.25 + 62.5% x 62.5), and this score is the hurdle
for your companion in year seven.
By proceeding in this fashion we determine the scores for each year, as shown in
Exhibit 1. Your strategy for finding a soul mate is to marry if your current companion
scores higher than the expected score for the subsequent year.
Suppose that at each point in time we must either retain the incoming portfolio
weights or restore the portfolio to its optimal weights, and that our investment horizon
lasts for three periods. Just as in the soul mate problem, we start by working backwards
from the end of period 2. Following Sun et al [2006], we determine the cost of
misallocation by computing the decrease in expected utility associated with the
misallocation. We then compare it to the cost of rebalancing to the optimal weights. In
order to make this comparison, however, we must express the decrease in expected utility
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in the same units as trading costs, by converting expected utility to a certainty equivalent
value.1
Let us first consider the incoming portfolio resulting from two successive 26%
stock returns and 1% bond returns, resulting in a portfolio of 70% stocks and 30% bonds.
We determine the utility of this portfolio by substituting a 70/30 stock/bond portfolio for
the 60/40 portfolio in Exhibit 3, which yields expected utility of 6.8881%. The certainty
equivalent of the optimal 60/40 portfolio equals 1.071436% (e .069000 ), whereas the
certainty equivalent of a 70/30 portfolio equals 1.071308% (e .068881 ). Hence, the cost of
sub-optimality for the 70/30 portfolio equals 0.0127%.
How does this sub-optimality cost compare to the cost of rebalancing to the
optimal weights? Let us assume it costs 5 basis points to trade stocks and 7 basis points
to trade bonds; hence the cost of restoring the optimal weights equals 0.012% (0.10 x
0.0005 + 0.10 x 0.0007). Therefore, given a 70/30 stock/bond portfolio at the end of
period 2, we would choose to rebalance to the optimal mix because the cost of sub-
optimality exceeds the cost of rebalancing. We perform the same calculations to
determine the optimal decision for the eight other incoming portfolios at the end of period
2.
Next we step back to the end of period 1. At this point there are only three
incoming portfolios to consider, but each portfolio leads to three additional portfolios at
the end of period 2; hence we must factor them in as well. Consider, for example the
65/35 portfolio at the end of period 1. There is a 25% chance that this portfolio will lead
to a 70/30 portfolio by the end of period 2. Given this outcome, we have just shown that
it is optimal to rebalance to the 60/40 portfolio; hence we must account for this
rebalancing cost. There is a 50% chance that the portfolio weights will remain at 65/35,
in which case we have shown is optimal to retain these weights; hence we must account
for the sub-optimality of this 65/35 portfolio. Finally, there is a 25% chance that the
portfolio will shift to a 60/40 portfolio. In this case there are neither sub-optimality costs
nor transaction costs to consider. We therefore calculate the weighted average of the 1 A certainty equivalent is a certain value that conveys the same amount of utility as the expected utility of a risky investment. Consider, for example a $100 investment that has an equal chance of increasing by 1/3 or decreasing by ¼. The utility of this investment for a log-wealth investor equals the natural logarithm of 133.33 times ½ plus the natural logarithm of 75.00 times ½, which equals 4.6052. The natural logarithm of 100.00 is also 4.6052; hence 100.00 is the certainty equivalent of this risky investment.
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transaction costs given the 70/30 portfolio and the cost of sub-optimality given the 65/35
portfolio, and discount this value back one period to derive the cost at the end of period 1
of the potential portfolios that could emanate from a 65/35 portfolio. We next calculate
the cost of sub-optimality of the 65/35 portfolio at the end of period 1 and add it to the
discounted cost associated with the probability weighted optimal choices at the end of
period 2 to derive the sub-optimality cost of remaining at the 65/35 portfolio at the end of
period 1. Equation 1 shows the cost of retaining the 65/35 portfolio.
We designed the previous example to illustrate dynamic programming and to
highlight the inter-temporal dependence of optimal decision making, but it is extremely
simplified in several important ways. It assumes that each portfolio yields only three
incoming portfolios, whereas in reality each portfolio yields a broad distribution of
incoming portfolios. Moreover, it assumes that the only choices are to rebalance fully to
the optimal weights or to retain the current weights, when in practice investors may
rebalance partially. It assumes only a three period investment horizon, when in practice
investors face the rebalancing decision much more frequently. Finally it assumes the
portfolio holds only two assets, whereas typical portfolios hold many assets.
Suppose instead that for the same two assets we rebalance over 12 decision
periods, and we use a granularity of 1%. This choice of granularity results in 101 possible
portfolios. For each portfolio we must derive the optimal decision in every time period.
Further suppose that, in contrast to the situation considered previously, we allow for
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partial rebalancing. In other words, for each of the 101 portfolios we evaluate 101
possible decisions, one decision associated with staying at the present allocation and then
another 100 possible decisions associated with moving to any of the other portfolios.
Thus, in every single time period we need to calculate the costs associated with 10,201
(101*101) possible decisions just to arrive at the 101 optimal decisions, one for each of
the 101 incoming portfolios.
As in the previous example we start at the terminal point and derive the 101
optimal decisions for period 12. There are no future decisions in period 12 and thus no
need to consider and discount the future costs associated with this period’s decision.
Consequently, as in the previous case for every incoming allocation (101) we calculate
the rebalancing cost and sub-optimality cost associated with each of the 101 possible
decisions. We then select the reallocation that results in the lowest total cost for each of
the 101 possible incoming allocations from end of period 11. Next we work backwards to
the beginning of period 11.
In period 11, as in periods 1 through 10, we must consider the impact of current
decisions on future decisions and costs. We determine the possible future cost resulting
from each decision as the discounted average cost across 50 potential allocations,
randomly selected through Monte Carlo sampling, which themselves result from the
optimal decision in period 12. 2 We continue working backwards from periods 10
through 1 always taking into account current period costs and discounted costs of the
optimal choices in subsequent periods. At the end of this process we know the exact
portfolio to rebalance to for each of the 101 possible portfolios each period assuming we
always select the optimal rebalancing option in subsequent periods.
The Curse of Dimensionality
As we add more assets, the computational challenge rises sharply. With three
assets to consider, assuming 1% granularity, there are 5,151 possible portfolios3, as
2 The Monte Carlo simulation might produce portfolio weights that do not match the weights of any of the assumed portfolios because their weights are rounded to 1% increments. In order to determine the sub-optimality and rebalancing costs of these randomly generated portfolios we interpolate the costs from the rounded portfolios on either side of the simulated portfolios. 3 The number of portfolios is given by the formula, N = (1/g + n-1)! ÷ ((n-1)! · (1/g)!), where g equals granularity and n equals number of assets.
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opposed to 101 with two assets. We must then analyze 26,532,801 (5,1512 ) decisions in
each period, and in periods 1 to 11 run 50 Monte Carlo paths for each possible decision,
leading to 14,619,573,351 (5,1512 x 50 x 11 + 5,1512) calculations. Exhibit 6 shows how
the number of portfolios and the number of calculations grow as we add more assets.
In our experiments we use a 28-processor grid computing platform. Grid
computing relies on parallel processing to allocate process execution efficiently, thus
enabling faster processing of large-scale computation problems. Even with access to a
grid computer, deriving the optimal decisions associated with a 10 asset portfolio and a
choice of 1% granularity is computationally intractable. On a regular workstation, for
example, the computing time required to solve this problem would be nearly 12,000
times of times the age of the universe4.
Part III: The Markowitz and van Dijk Heuristic Exhibit 6 underscores the limitations of dynamic programming when we wish to
consider more than a few assets. Markowitz and van Dijk (2004) propose an alternative
approach for determining optimal rebalancing rules. Although they apply their heuristic
to account for changing means in asset returns, we adapt it to address the asset weight
drift problem.
4 If, instead, you are one of the 80% of Americans who believe the universe was created only 6,000 years ago, the computing time would be 2.65x1010 times the age of the universe. In either case, grid computing would be of no use.
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As with the dynamic programming approach, we wish to minimize the combined
costs of sub-optimality and rebalancing, taking into account the current period’s costs as
well as the discounted expected costs of future choices. We start by defining the cost
function associated with a possible new portfolio given an incoming portfolio:
C = e ln(O) – e ln(N) (2)
+ ( |WSI - WSN| x TCS + |WBI – WBN| x TCB )
+ m((WSO - WSN)2 + (WBO – WBN)2)
Where,
C = cost function of possible new portfolio given incoming portfolio
e ln(O) = certainty equivalent of optimal portfolio
e ln(N) = certainty equivalent of possible new portfolio
WSI = stock weight of incoming portfolio
WSN = stock weight of possible new portfolio
WSO = stock weight in the optimal portfolio
WBI = bond weight of incoming portfolio
WBN = bond weight of possible new portfolio
WBO = bond weight in the optimal portfolio
TCS = transaction cost for stocks
TCB = transaction cost for bonds
m = coefficient chosen to best approximate the derived utility function.
The first line of equation (2) reflects the cost of sub-optimality for a possible new
portfolio. The second line reflects the rebalancing cost of shifting from the incoming
portfolio to a possible new portfolio. The third line is a quadratic function which
approximates the discounted cost of future optimal choices.
Next using Monte Carlo simulation we generate 200 possible incoming portfolios
given the expected returns, variances, and covariances of the component assets of the
initial optimal portfolio along with its weights. For a given coefficient m, we solve for a
new portfolio for each of the incoming portfolios such that we minimize cost as defined
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by equation 25. From these new portfolios, we again apply Monte Carlo simulation to
generate a new set of 200 incoming portfolios, and we solve for 200 cost-minimizing new
portfolios. We proceed forward through 12 periods and accumulate the costs. We then
calculate a figure of merit by taking the average of the 200 cumulative costs.
Next we select a new value for the coefficient m and repeat the process. We
proceed in this fashion using a mesh approach to select new coefficients. We start with a
relatively coarse mesh and gradually refine its granularity until we identify the coefficient
which produces the best figure of merit. Computational intensity, which is low to begin
with, remains manageable as we add more assets6. Next we compare the efficacy of the
Markowitz-van Dijk heuristic (MvD heuristic) to dynamic programming.
Part IV: Results We test the relative efficacy of dynamic programming and the MvD heuristic with
data on domestic equities, domestic fixed income, non-US equities, non-US fixed
income, and emerging market equities. Exhibit 7 shows our returns, standard deviations,
and transaction cost assumptions.
Exhibit 7: Volatilities and Transaction Costs
Rebalancing Standard TransactionAsset Class Index Deviation Cost
Domestic Equities S&P 500 12.74% 0.40%Domestic Fixed Income Lehman US Aggregate 3.96% 0.45%Non-US Equity MSCI EAFE + Canada 13.41% 0.50%Non-US Fixed Income CGBI World ex US 8.20% 0.75%Emerging Market Equities MSCI EM 18.51% 0.75%
Exhibit 8 shows our correlation assumptions. We use monthly returns from
October, 2001 through September, 2006 to measure standard deviations and correlations.
To estimate expected returns we solve for the implied returns under the assumption that
the allocations in Exhibit 9 are optimal under mean-variance utility.
5 There are a variety of optimization algorithms to minimize this cost function. We use the fmincon function which is available in the optimization toolbox of MatLab. 6 For example, finding the best coefficient m for a 100 asset case would take slightly more than 10 days without grid computing.
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Exhibit 8: Correlations
Domestic Domestic Non-US Non-USEquities Fixed income Equities Fixed income
Domestic EquitiesDomestic Fixed Income -0.31Non-US Equity 0.84 -0.19Non-US Fixed Income -0.14 0.53 0.16Emerging Market Equities 0.77 -0.17 0.83 -0.05
We use a simple domestic stocks and domestic fixed income case for the two-
asset portfolio. We add Non-US equities for the three-asset case, Non-US fixed income
for the four-asset case, and emerging market equities for the five-asset case. We also use
individual stocks to test the MvD and other heuristics for portfolios of 10, 25, 50, and 100
assets.
Exhibit 9 shows the assumed optimal portfolio weights, which as stated before are
optimal under the standard mean-variance utility function.
Number of Search Number ofAssets Interval Discreetization Points
2 + or - 5% 5,0003 + or - 3% 604 + or - 3% 145 + or - 3% 7
* Because it is improbable that the portfolio will depart substantially from the optimal mix, we we sample with higher density around the optimal mix.
1,5082,174
Number ofPortfolios
5,0013,323
Part V. Conclusion
Portfolio allocations drift from their optimal weights as prices shift. Most
investors employ naïve heuristics to rebalance their portfolios. We describe how
dynamic programming can be used to identify an optimal rebalancing schedule, which
significantly reduces rebalancing and sub-optimality costs compared to naïve heuristics.
Unfortunately the curse of dimensionality prevents us from applying dynamic
programming to more than a few assets. As an alternative we examine the efficacy of a
more sophisticated heuristic called the MvD heuristic, which scales up to several hundred
assets. Our tests show that the MvD heuristic performs almost as well as dynamic
programming for up to four assets and better than dynamic programming for five assets.
In theory, of course, dynamic programming always yields the best result, but we cannot
observe these results beyond a few assets. Therefore, we have no way of determining
how the MvD heuristic would compare to the unobservable “correct” dynamic
programming solution. To the best of our knowledge, however, the MvD heuristic is the
superior alternative for rebalancing problems of more than a few assets.
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Appendix
Exhibit A1 shows the securities used to create the stock portfolios for the 10, 25,
50, and 100 asset cases.
Exhibit A1: Securities used for stock portfolios
MICROSOFT SLM SIGMA ALDRICH MORGAN ST ANLEYIBM GOLDEN WEST FINANCIAL GENERAL DYNAMICS GOLDMAN SACHSCISCO SYSTEMS PFIZER DANAHER FANNIE MAEDELL JOHNSON & JOHNSON CENDANT US BANCORPORACLE AMGEN GENERAL ELECTRIC WASHINGTON MUTUALEBAY UNITEDHEALTH GROUP UNITED TECHNOLOGIES PRUDENTIAL FINL.YAHOO MEDT RONIC BOEING LEHMAN BROTHERSFIRST DATA ELI LILLY 3M MET LIFEADOBE SYSTEMS WYET H TYCO INTL. ALLSTAT EHOME DEPOT CARDINAL HEALT H UNITED PARCEL SER. SAINT PAUL T RAVELERSLOWE'S COMPANIES GILEAD SCIENCES CAT ERPILLAR SUNTRUST BANKST ARGET SCHERING-PLOUGH HONEYWELL INT ERNATIONAL BANK OF NEW YORK STARBUCKS GUIDANT EMERSON ELECTRIC FRANK.RES.BEST BUY CAREMARK RX LOCKHEED MARTIN HARTFORD FINANCIAL SERVICESSEARS HOLDINGS STRYKER FEDEX INTELNIKE VALERO ENERGY BURLINGT ON NORTHERN SANTA FE CORPORATION HEWLETT-PACKARDAMAZON.COM BURLINGTON RES ILLINOIS TOOL WORKS QUALCOMMKOHLS DEVON ENERGY UNION PACIFIC APPLE COMPUTERSCLEAR CHANNEL COMMUNICATIONS ANADARKO PETROLEUM CITIGROUP MOT OROLAOMNICOM GROUP PROCTER & GAMBLE BANK OF AMERICA TEXAS INST RUMENTSHARLEY-DAVIDSON WAL MART ST ORES AMERICAN INT ERNAT IONAL GROUP CORNINGYUM! BRANDS PEPSICO JP MORGAN CHASE & COMPANY EMCAMERICAN EXPRESS WALGREEN WELLS FARGO & COMPANY APPLIED MAT ERIALSFREDDIE MAC ANHEUSER-BUSCH WACHOVIA AUTOMAT IC DAT A PROCESSINGCAPIT AL ONE FINANCIAL ECOLAB MERRILL LYNCH & COMPANY ADVANCED MICRO DEVICES
For example, the first 10 securities in column one constitute the 10 asset portfolio,
the securities in the first column constitute the 25 asset portfolio.
We determine the risks and correlations of the securities in Exhibit A1 based on
monthly historical returns from January, 2005 through January, 2006 and estimate the
expected returns as the implied returns under the assumption that the equally weighted
portfolio is optimal under mean-variance optimization.
Exhibits A2 through A9 show the trading cost and a sub-optimality cost
components for the various rebalancing algorithms.
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Exhibit A2: Performance Comparison - Two Assets(5,000 Monte Carlo Simulations)
Costs Costs CostsRebalancing Trading Sub-optimality Total
Bellman, R.E. 1952. On the theory of dynamic programming. Proceedings of the National Academy of Sciences, 38. pp.716-719 Markowitz, Harry. “Portfolio Selection.” Journal of Finance, 7 (1952), pp.77-91 Markowitz, Harry and Erik L. van Dijk. “Single-Period Mean–Variance Analysis in a Changing World (corrected).” Financial Analysts Journal, Mar 2003, Vol. 59, No. 2, pp. 30-44 Smith, David K., Dynamic programming: an introduction, PASS Maths, http://plus.maths.org/issue3/dynamic/, September 1997 Sun, W., A. Fan, L. W. Chen, T. Schouwenaars, and M. Albota. “Optimal Rebalancing for Institutional Portfolios.” The Journal of Portfolio Management. Winter 2006, pp. 33-43
