Optimal Rebalancing: A Scalable Solution
Mark Kritzman
Windham Capital Management, LLC, 5 Revere Street, Cambridge, MA 02138, [email protected]
617-234-9410
Simon Myrgren
State Street Associates, 138 Mt. Auburn Street, Cambridge, MA 02138, [email protected]
617-234-9416
Sébastien Page
State Street Associates, 138 Mt. Auburn Street, Cambridge, MA 02138, [email protected]
617-234-9462
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. We apply a quadratic heuristic to address the optimal rebalancing problem, and we compare it to a dynamic programming solution as well as to standard industry heuristics. The quadratic heuristic provides solutions that are remarkably close to the dynamic programming solution. Moreover, unlike the dynamic programming solution, the quadratic heuristic is scalable to as many as several hundreds assets.
This version: June 30th, 2007
We thank Harry Markowitz for helpful comments.
Optimal Rebalancing: A Scalable Solution
Abstract: 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. We apply a quadratic heuristic to address the optimal rebalancing problem, and we compare it to a dynamic programming solution as well as to standard industry heuristics. The quadratic heuristic provides solutions that are remarkably close to the dynamic programming solution. Moreover, unlike the dynamic programming solution, the quadratic heuristic is scalable to as many as several hundreds assets.
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.
2
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 views about the mean returns of portfolio assets. It has been shown previously that we can closely approximate a variety of utility functions with quadratic functions (see, for example: Levy and Markowtiz (1979), Kroll, Levy and Markowtiz (1984), Cremers, Kritzman and Page (2003), Cremers, Kritzman and Page (2005)).
We adapt 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 for those cases in which dynamic programming is feasible and far superior to solutions based on standard industry heuristics. In the case 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.
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II. The General Portfolio Rebalancing Problem
We begin by assuming an investor with log-wealth utility wishes to select a set of portfolio weights that maximize expected utility over a forthcoming period. The expected utility E(U) of the portfolio is written as the weighted sum of the n security expected returns under m scenarios, each with associated p probability
m⎛n⎞
(1) E(U)=∑p ln⎜1+ i
where
∑
i=1 ⎝j=1 ⎠
⎡μ11 μ12 Kμ1n⎤ μ = ⎢μ21 μ22 K μ2n ⎥
⎢ M ⎥ ⎣μm1 μm2 Kμmn⎦
is the matrix of expected returns, X = [X 1 ,K, X n ] are the current portfolio weights, and p = [p ,K, p ] are the probabilities associated with the m scenarios. Let X opt ,
1m
X opt = [X opt ,…, X opt ], denote the optimal portfolio weights. E(U) is then maximized
when X = X opt and denoted E(U*). With the passage of time asset prices change, and X deviates from Xopt resulting in a loss of expected utility. For a given sub-optimal E(U) we quantify the loss in expected utility as the certainty equivalent cost (CEC), which for the log wealth investor is given by:
(2) CEC=eE(U*) −eE(U)
Doing so conveniently converts the portfolio’s sub-optimality cost into monetary units that are directly comparable to transaction costs.
1n
X μ ⎟= pln(1+μX′) j ij
4
The transaction costs (TC) at period t are written as n
The general portfolio rebalancing problem is therefore to minimize the combined costs associated with deviations from X opt as defined in (2) while also minimizing transaction costs as defined in (3). Rebalancing decisions in the current period will influence future costs and decisions, which must be accounted for in the optimal solution.
III. The Dynamic Programming Solution
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.1
Following Sun et al. (2006), we define the dynamic programming solution to portfolio rebalancing as the recursive minimization of the cost function
(4) Jt(Xt,Xt−1)=CECt +TCt +Jt+1(Xt+1,Xt)
where the total cost for the current period J t (X t , X t −1 ), is a function of the current CEC
and TC, but also of future costs J t +1 (X t +1 , X t ). In our experiments, we estimate the
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(3)
weights X jt .
TCt= CX−X
∑ i=1
j jt jt−1
whereCj is the cost of trading security j from the previous weights Xjt−1 to the new
potential future cost of each decision as the discounted average cost across 50 potential allocations randomly generated by Monte Carlo sampling.
Unfortunately, this approach suffers from the curse of dimensionality. To rebalance a portfolio among three assets in increments of 1%, for example, we must consider 5,151 possible portfolios2 and analyze 26,532,801 (5,1512 ) rebalancing decisions for each period. Moreover, to solve this problem recursively we must generate at least 50 Monte Carlo paths for each possible decision at each time step. For a one-year horizon with monthly monitoring (12 time steps), we must therefore perform 14,619,573,351 (5,1512 x 50 x 11 + 5,1512) calculations. Table 1 shows how the number of portfolios and the number of calculations grow as we add more assets.
[Insert Table 1 here]
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 universe.
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IV. The Markowitz and van Dijk Heuristic
Table 1 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.
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. However, we replace
J t (X t +1 , X t ) in (4) by a quadratic function of the current and optimal portfolio weights. In general, a quadratic approximation Q to J t (X t +1 , X t ) has the following form:
(5) Q=∑aX +∑b2X2 +∑∑c X X
n
( 6 ) Q = ∑ d ⎜⎝ X i − X ⎟⎠
ii
ii
i j>i
ij i j
To simplify our experiments, however, we conjecture that Q is separable and is minimized by the target portfolio, so that it is proportional to the squared deviations (the “drifts”) multiplied by a coefficient d:
The cost function (4) then becomes
(7) Jt(Xt,Xt−1)=CECt +TCt +Qt
⎛ opt′⎞2 i=1
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To determine the value of coefficient d in (6) we use Monte Carlo simulations. We generate 200 return paths. We minimize cost as defined in (7)3 at each decision point during the simulation. We continue to run simulations and change d until we find its best performing value. Computational intensity, which is low to begin with, remains manageable as we add more assets4.
V. 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. For these portfolios the expected portfolio return is
(8) Ep = and the expected portfolio variance is
where X =[X1,K,Xn]isthesetofassetweights, μ=[μ1,K,μn]isthesetofexpected returns on the n assets, σ ij is the covariance between assets i and j, and C is the covariance matrix (σij ).
(9) Vp =
∑∑ i=1 j=1
XXσ =XCX’ i j ij
n
∑ i=1
X μ = Xμ’ ii
nn
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Table 2 shows our returns, standard deviations, and transaction cost assumptions. [Insert Table 2 here]
Table 3 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 Table 4 are optimal under mean-variance utility and a fully invested budget constraint:
E(U)= Xμ’−λ XCX’ (10) 2
s.t. X1′ =1 N
Here λ is the risk aversion parameter (7.5) and 1N is a vector of ones. We thus calculate the implied returns as follows:
(11) μ =λCX’+−λ+1N C−1μ’1′ impl 1N C−11N ‘ N
[Insert Table 3 here]
We use domestic stocks and domestic fixed income for the two-asset case. 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. Table 4 shows the assumed optimal portfolio weights, which as stated before are optimal under the standard mean-variance
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utility function. The choice of the initial portfolio weights is arbitrary. In our example, we use optimal portfolios based on a set of reasonable expectations and a mean-variance utility function. We could just as well substitute optimal weights based on other descriptions of expected utility. Long-only investment managers, for example, would rely on a mean-tracking error utility function, while behavioral investors might use an s- shaped value function. Investors mostly concerned with large loses would use a kinked utility function (see Cremers, Kritzman, and Page, 2005). Also, the assumption that returns are normally distributed is convenient but not necessary for optimal rebalancing – as long as the distributions can be generated via Monte Carlo simulations.
[Insert Table 4 here]
We assume that we have a two-year investment horizon over which we wish to minimize the aggregate total cost; that is, the cumulative sum of trading costs and sub- optimality costs. For the calendar heuristics, we fully rebalance the portfolio at pre- determined time intervals. For the tolerance band heuristics, we fully rebalance the portfolio when asset weights breach pre-determined thresholds. Although we cannot extend the dynamic programming algorithm beyond five assets, we test the MvD heuristic and the other heuristics for portfolios of 10, 25, 50, and 100 assets using individual stocks, which are listed in the appendix.
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As indicated in section 3, for each portfolio we sample several hundred values for d until we find the d which yields the lowest average figure of merit (AFOM)5, which we define as the average total cost over the 200 Monte Carlo runs:
(12)
where CEC MV i,p
1 200 24
AFOM = ∑∑CECMV +TC
200 i=1 p=1
is the certainty equivalent cost for the ith portfolio path in the pth period
under mean-variance utility and TCi , p are the transaction costs (3) for the ith portfolio
path in the pth period.
Table 5 summarizes the results. It shows that the MvD heuristic performs quite
well compared to the dynamic programming solution for the two asset case and substantially better than other heuristics6. As we increase the number of assets we find that the advantage of dynamic programming over the MvD heuristic shrinks and is reversed at five assets. We are not able to apply dynamic programming beyond five assets, but we are able to extend the MvD heuristic up to 100 assets. We find that the MvD heuristic reduces total costs relative to all of the other heuristics by substantial amounts. In the appendix we present a more detailed cost analysis that partitions costs into trading and sub-optimality components.
[Insert Table 5 here]
Although the performance of the MvD heuristic improves relative to the dynamic programming solution as more assets are added, this improvement reflects a growing reliance on approximation for the dynamic programming approach. For the two-asset
i,p i,p
11
case the dynamic programming solution searches within an interval of plus or minus 5% around the optimal portfolio, and divides this range into 5,000 units. For greater than two assets, the search is confined to plus or minus 3% around the optimal portfolio, and this space is divided into increasingly coarser units, as shown in Table 6.
[Insert Table 6 here]
We have no way of knowing how well the MvD heuristic would track the ideal but unobtainable dynamic programming solution, but we are encouraged that its advantage over the next best heuristic increases as we add more assets. Moreover, we would not know ex ante which heuristic is the next best; hence a fairer assessment of the relative efficacy of the MvD heuristic might be to compare it to the average result of the other heuristics.
Part VI. 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.
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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 extent of our knowledge, however, the MvD heuristic is the best alternative by far for rebalancing portfolios with more than just a few assets.
The scalability of the MvD heuristic opens to the door to several new applications of portfolio rebalancing. Passive managers could use the MvD heuristic to optimize the tradeoff between tracking error and transaction costs. Quantitative asset managers could use it to minimize alpha decay between rebalancing dates.
Plan sponsors in particular could benefit from the MvD heuristic, as they are continually confronted with asset mix rebalancing decisions. Moreover, plan sponsors could customize the optimal rebalancing process to existing tolerance bands, tracking error targets, cash inflows, and benefit payments.
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Table 1. The Curse of Dimensionality
Number of Assets
Number of Portfolios
Number of Calculations to Perform
2 3 4 5 6 7 8 9 10
101 5,151 176,851 4,598,126 96,560,646 1,705,904,746 26,075,972,546 352,025,629,371 4,263,421,511,271
5,620,751 14,619,573,351 17,233,228,186,751 11,649,662,254,243,700 5,137,501,054,121,460,000 1,603,471,162,336,350,000,000 374,655,945,665,079,000,000,000 68,281,046,097,460,800,000,000,000 10,015,396,403,505,300,000,000,000,000
Notes. This table shows the number of portfolios as a function of the number of assets, assuming 1% granularity. It also shows the number of calculations one would need to perform in order to solve the dynamic programming problem for a one-year horizon with12 time steps.
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Table 2. Volatilities and Transaction Costs
Rebalancing Asset Class
Index
Standard Deviation
Transaction Cost
Domestic Equities Domestic Fixed Income Foreign Developed Equity Foreign Bonds
Foreign Emerging Equity
S&P 500
Lehman US Agg MSCI EAFE + Canada CGBI World ex US MSCI EM
12.74% 3.96% 13.41% 8.20% 18.51%
0.40% 0.45% 0.50% 0.75% 0.75%
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Table 3. Correlations
Domestic Domestic Foreign Foreign Equities Fixed income Dev. Equities Fixed income
Domestic Fixed Income Foreign Developed Equity Foreign Bonds
Foreign Emerging Equity
-0.31 0.84 -0.14 0.77
-0.19
0.53 0.16
-0.17 0.83 -0.05
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Table 4. Optimal Portfolios
Two Assets
Three Assets
Four Assets
Five Assets
Domestic Equities Domestic Fixed Income Foreign Developed Equity Foreign Bonds
Foreign Emerging Equity
60.00% 40.00%
40.00% 40.00% 20.00%
40.00% 25.00% 20.00% 15.00%
40.00% 25.00% 15.00% 15.00% 5.00%
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Table 5. Performance Comparison – Total Costs (bps)
Rebalancing Strategy
Two Assets
Three Assets
Four Assets
Five Assets
Ten Assets
Twenty Five Assets
Fifty Assets
Hundred Assets
Dynamic Programming MvD Heuristic
6.31 6.90
6.66 7.03
7.33 7.58
8.76 8.61
NA 25.57
NA 20.38
NA 17.92
NA 12.46
0.25% Bands 0.50% Bands 0.75% bands 1% Bands
2% Bands
3% Bands
4% Bands
5% Bands Monthly Quarterly Semi-annually
15.19 14.11 12.80 11.54 8.73 8.51 9.46 11.20 15.65 11.05 11.13
17.01 15.75 14.09 12.52 9.20 8.66 9.52 11.21 17.25 11.86 11.53
19.81 17.81 15.32 13.15 9.79 10.14 12.08 14.80 20.07 13.51 12.67
21.37 18.92 16.27 14.13 10.73 11.43 13.78 16.77 21.85 14.76 13.95
41.93 41.73 40.05 37.71 41.94 61.29 88.49 120.19 41.92 45.17 69.97
42.96 38.42 32.95 31.95 48.59 73.78 93.23 106.38 42.92 34.32 40.75
41.53 31.15 31.46 36.74 66.96 89.03 98.55 102.38 43.34 33.12 37.33
26.88 21.82 25.02 29.47 39.33 41.54 41.96 42.03 39.75 26.54 24.41
Notes. This table shows results for 5,000 Monte Carlo simulations. For the 10 through 100 asset cases, which employ equally weighted portfolios of stocks drawn from the S&P 500, a dynamic programming solution is unachievable.
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Table 6. Dynamic Programming Discretization Scheme
Notes. Given that large fluctuations from the optimal allocation are improbable we chose to sample with higher density around the optimal allocation.
Number of Assets
Number of Discreetization Points
Number of Portfolios
2 3 4 5
5,000 60 14 7
5,001 3,323 2,174 1,508
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Appendix
Table 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 IBM
CISCO SYSTEMS DELL
ORACLE
EBAY
Y AHOO
FIRST DATA
ADOBE SYSTEMS HOME DEPOT LOWE’SCOMPANIES TARGET
ST ARBUCKS BEST BUY
SEARS HOLDINGS NIKE AMAZON.COM KOHLS
CLEAR CHANNEL COMMUNICAT IONS OMNICOM GROUP
H A R L E Y – D A VI D SO N
YUM! BRANDS
AMERICAN EXPRESS FREDDIE MAC
CAPITAL ONE FINANCIAL
SLM
GOLDEN WEST FINANCIAL PFIZER
JOHNSON & JOHNSON AMGEN
UNITEDHEALTH GROUP MEDT RONIC
ELI LILLY
WYETH
CARDINAL HEALTH GILEAD SCIENCES SCHERING-P LOUGH GUIDANT
CAREMARK RX
STRYKER
VALERO ENERGY BURLINGTON RES
DEVON ENERGY ANADARKO PETROLEUM PROCTER & GAMBLE
WAL MART STORES PEPSICO
WALGREEN ANHEUSER-BUSCH ECOLAB
SIGMA ALDRICH
GENERAL DYNAMICS
DANAHER
CENDANT
GENERAL ELECTRIC
UNITED TECHNOLOGIES
BOEING
3M
TYCO INTL.
UNITED PARCEL SER.
CATERPILLAR
HONEYWELL INTERNATIONAL
EMERSON ELECTRIC
LOCKHEED MARTIN
FEDEX
BURLINGTON NORTHERN SANTA FE CORPORATION ILLINOISTOOL WORKS
UNION PACIFIC
CITIGROUP
BANK OF AMERICA
AMERICAN INTERNATIONAL GROUP
JP MORGAN CHASE & COMPANY
WELLSFARGO & COMPANY
WACHOVIA
MERRILL LYNCH & COMPANY
MORGAN ST ANLEY GOLDMAN SACHS FANNIE MAE
US BANCORP WASHINGTON MUTUAL PRUDENTIAL FINL. LEHMAN BROTHERS MET LIFE
ALLSTATE
SAINT PAUL TRAVELERS SUNTRUST BANKS
BANK OF NEW YORK
FRANK.RES.
HARTFORD FINANCIAL SERVICES INTEL
HEWLETT-PACKARD QUALCOMM
APPLE COMPUTERS
MOTOROLA
T EXAS INST RUMENT S
CORNING
EMC
APPLIED MATERIALS AUTOMATIC DATA PROCESSING ADVANCED MICRO DEVICES
For example, the first 10 securities in column one constitute the 10 asset portfolio, and the securities in the first column constitute the 25 asset portfolio.
We determine the risks and correlations of the securities in Table A1 based on daily 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.
Tables A2 through A9 show the trading cost and sub-optimality cost components for the various rebalancing algorithms.
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Exhibit A2: Performance Comparison – Two Assets (5,000 Monte Carlo Simulations)
Rebalancing Strategy
Costs (bps)
Trading Sub-optimality Total
Dynamic Programming MvD Heuristic
0.25% Bands
0.50% Bands
0.75% Bands 1% Bands
2% Bands
3% Bands
4% Bands
5% Bands Monthly Quarterly Semi-annually
4.87
4.86 15.18 14.06 12.63 11.19 7.18 5.17 3.88 3.00 15.65 9.31 6.70
1.44 2.04 0.01 0.05 0.17 0.34 1.55 3.34 5.58 8.20 0.00 1.74 4.43
6.31
6.90 15.19 14.11 12.80 11.54 8.73 8.51 9.46 11.20 15.65 11.05 11.13
Exhibit A3: Performance Comparison – Three Assets (5,000 Monte Carlo Simulations)
Rebalancing Strategy
Costs (bps)
Trading Sub-optimality Total
Dynamic Programming MvD Heuristic
0.25% Bands
0.50% Bands
0.75% Bands 1% Bands
2% Bands
3% Bands
4% Bands
5% Bands Monthly Quarterly Semi-annually
4.68
4.73 17.00 15.71 13.94 12.20 7.69 5.40 4.03 3.16 17.25 10.24 7.38
1.98 2.30 0.00 0.04 0.15 0.32 1.50 3.26 5.49 8.05 0.00 1.61 4.15
6.66
7.03 17.01 15.75 14.09 12.52 9.20 8.66 9.52 11.21 17.25 11.86 11.53
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Exhibit A4: Performance Comparison – Four Assets (5,000 Monte Carlo Simulations)
Rebalancing Strategy
Costs (bps)
Trading Sub-optimality Total
Dynamic Programming MvD Heuristic
0.25% Bands
0.50% Bands
0.75% bands 1% Bands
2% Bands
3% Bands
4% Bands
5% Bands Monthly Quarterly Semi-annually
5.10
4.94 19.80 17.73 15.05 12.57 7.29 4.82 3.33 2.29 20.07 11.87 8.50
2.23 2.64 0.00 0.08 0.27 0.58 2.50 5.32 8.75 12.51 0.00 1.64 4.17
7.33
7.58 19.81 17.81 15.32 13.15 9.79 10.14 12.08 14.80 20.07 13.51 12.67
Exhibit A5: Performance Comparison – Five Assets (5,000 Monte Carlo Simulations)
Rebalancing Strategy
Costs (bps)
Trading Sub-optimality Total
Dynamic Programming MvD Heuristic
0.25% Bands
0.50% Bands
0.75% bands 1% Bands
2% Bands
3% Bands
4% Bands
5% Bands Monthly Quarterly Semi-annually
6.21
5.30 21.36 18.81 15.92 13.41 7.70 5.09 3.55 2.46 21.85 12.95 9.29
2.55 3.31 0.01 0.11 0.35 0.72 3.02 6.33 10.23 14.31 0.00 1.82 4.66
8.76
8.61 21.37 18.92 16.27 14.13 10.73 11.43 13.78 16.77 21.85 14.76 13.95
22
Exhibit A6: Performance Comparison – Ten Assets (5,000 Monte Carlo Simulations)
Rebalancing Strategy
Costs (bps)
Trading Sub-optimality Total
MvD Heuristic 0.25% Bands 0.50% Bands 0.75% bands 1% Bands
2% Bands
3% Bands
4% Bands
5% Bands Monthly Quarterly Semi-annually
19.59 41.93 41.68 39.21 34.47 20.76 14.11 10.14 7.42 41.92 24.83 17.69
5.98 0.00 0.05 0.83 3.24 21.18 47.19 78.35 112.76 0.00 20.34 52.28
25.57 41.93 41.73 40.05 37.71 41.94 61.29 88.49 120.19 41.92 45.17 69.97
Exhibit A7: Performance Comparison – Twenty-Five Assets (5,000 Monte Carlo Simulations)
Rebalancing Strategy
Costs (bps)
Trading Sub-optimality Total
MvD Heuristic 0.25% Bands 0.50% Bands 0.75% bands 1% Bands
2% Bands
3% Bands
4% Bands
5% Bands Monthly Quarterly Semi-annually
14.16 42.96 37.07 27.60 21.63 10.56 5.91 3.35 1.78 42.92 25.32 17.97
6.22 0.00 1.34 5.35 10.32 38.02 67.87 89.88 104.59 0.00 9.01 22.78
20.38 42.96 38.42 32.95 31.95 48.59 73.78 93.23 106.38 42.92 34.32 40.75
23
Exhibit A8: Performance Comparison – Fifty Assets (5,000 Monte Carlo Simulations)
Rebalancing Strategy
Costs (bps)
Trading Sub-optimality Total
MvD Heuristic 0.25% Bands 0.50% Bands 0.75% bands 1% Bands
2% Bands
3% Bands
4% Bands
5% Bands Monthly Quarterly Semi-annually
12.05 41.22 25.23 17.46 12.93 5.15 1.80 0.59 0.23 43.34 25.57 18.14
5.86 0.31 5.92 14.00 23.73 61.82 87.23 97.95 102.16 0.00 7.55 19.19
17.91 41.53 31.15 31.46 36.66 66.96 89.03 98.55 102.38 43.34 33.12 37.33
Exhibit A9: Performance Comparison – Hundred Assets (5,000 Monte Carlo Simulations)
Rebalancing Strategy
Costs (bps)
Trading Sub-optimality Total
MvD Heuristic 0.25% Bands 0.50% Bands 0.75% bands 1% Bands
2% Bands
3% Bands
4% Bands
5% Bands Monthly Quarterly Semi-annually
7.55 24.75 12.95 8.13 5.39 0.71 0.10 0.02 0.01 39.75 23.46 16.63
4.91 2.13 8.88 16.89 24.08 38.61 41.44 41.94 42.02 0.00 3.08 7.78
12.46 26.88 21.82 25.02 29.47 39.33 41.54 41.96 42.03 39.75 26.54 24.41
24
References
Bellman, R.E. “On the theory of dynamic programming.” Proceedings of the National Academy of Sciences, 38 (1952), pp.716-719
Cremers, Jan-Hein, Kritzman, and Page. “Portfolio Formation with Higher Moments and Plausible Utility.” Revere Street Working Paper Series. Financial Economics 272-12 (2003).
Cremers, Jan-Hein, Kritzman, and Page. “Optimal Hedge Fund Allocations.” Journal of Portfolio Management, Vol. 31 (Spring 2005). No 3.
Kroll, Yoram, Levy, and Markowitz. “Mean Variance Versus direct Utility Maximization.” Journal of Finance, Vol. 39 (1984), No. 1. pp 47-61.
Levy, Haim and Harry M. Markowitz. “Approximating Expected Utility by a Function of Mean and Variance.” American Economic Review, Vol. 69 (1979), No. 3.
Markowitz, Harry M. “Portfolio Selection.” Journal of Finance, (1952), pp.77-91
Markowitz, Harry M and Erik L. van Dijk. Single-Period Mean–Variance Analysis in a Changing World (corrected). Financial Analysts Journal, Vol. 59 (March/April 2003.), 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.” Journal of Portfolio Management, (Winter 2006) pp. 33-43
1 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.
2 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. In our experiments we exactly don’t use g = 1%. We use efficient sampling, which means that g is small around the optimal weights and gets larger as we move further from them.
3 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.
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4 For example, finding the best coefficient d for a 100 asset case would take slightly more than 10 days without grid computing.
5 The term “Figure of Merit” is from in Markowitz and van-Dijk, 2003.
6 Some investors might use more sophisticated heuristics. For example they might use different bands for each asset, or rebalance partially, for example to the edge of the band, rather than back to the optimal weights. Our approach will be useful to these investors, as it will help them optimize these decision rules.
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