15.451 - FINANCIAL ENGINEERING (FALL 2004) 1
Optimal Rebalancing Strategy for Pension
PlansMarius Albota1, Li-Wei Chen2, Ayres Fan2, Ed Freyfogle3, Josh Grover3, Tom Schouwenaars2,
Walter Sun2
Abstract
Existing approaches to portfolio rebalancing are suboptimal. In particular, pension plans generally
rebalance on a calendar basis or use tolerance bands to trigger rebalancing. In this document, we propose
a different approach to rebalancing portfolios which, over long periods of simulations, outperforms
rebalancing strategies of monthly, quarterly, annual, and 5% tolerance rebalancing using different utility
functions. Specifically, the utility functions we examine are quadratic, log wealth, and power utilities.
We first derive the efficient frontier and construct the optimal portfolios for each of these utilities.
For each portfolio, we determine a certainty equivalence and use this information with transaction costs
in a dynamic programming framework to determine whether or not it is optimal to trade at each time step
(end of each month). We test our method against the traditional methods using Monte Carlo simulations
on simulated data. Finally, we provide sensitivity tests and discuss potential extensions to our work.
We thank Sebastien Page (State Street Associates) and Mark Kritzman (Windham Capital Management Boston, LLC) for
introducing the project and providing valuable guidance throughout.1PhD, Research Laboratory of Electronics2PhD, Laboratory for Information and Decision Systems3MBA, Sloan School of Management
15.451 - FINANCIAL ENGINEERING (FALL 2004) 2
CONTENTS
I Introduction 3
II Background 3
II-A Existing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
II-B Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
III Optimal Rebalancing Using Dynamic Programming 6
III-A Solution Methodology: Dynamic Programming . . . . . . . . . . . . . . . . . . . . 7
III-B Modelling Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
III-B.1 Certainty Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
III-B.2 Variance Penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
III-C Modelling Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
III-C.1 Linear Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . 9
III-C.2 Affine Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . 9
IV Efficient Frontier and Portfolio Weights using Mean-Variance Optimization 9
V Two Asset Model 13
V-A Cost Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
VI Multi-Asset Model 17
VII Sensitivity Analysis 21
VII-A Sensitivity to Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
VII-B Sensitivity to Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
VII-C Sensitivity to Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
VIII Areas for Future Investigation 24
IX Conclusion 26
Appendix I: Problem Statement 27
Appendix II: Derivation of Optimal Portfolio for Two Risky Assets 27
References 29
15.451 - FINANCIAL ENGINEERING (FALL 2004) 3
I. I NTRODUCTION
Pension fund managers develop risk models and optimal portfolios to match their future liabilities
to their expected future returns. One way to model these risk preferences is through the use of utility
functions. This utility is then reflected in the target portfolio, a set of weights for different asset classes
(that the manager must not stray far from) mandated by the trustees or directors. Given the fact that
different asset classes can exhibit different rates of return, a manager cannot maintain this target of
weights over time without active rebalancing. Furthermore, managers also must rebalance if and when
the weights in the target portfolio are altered. This may be done to reflect the directors changing their
market views (e.g., changing the mean returns in their model) or their risk tolerance (as expressed by
their utility function).
Most academic theory ignores trading costs and assumes that a portfolio manager can simply readjust
their holdings dynamically without any problems. However, in practice, trading costs are non-zero and
affect the decision to rebalance. In this paper, we examine a different approach to portfolio rebalancing
through the process of dynamic programming. We show that our method performs better than traditional
methods of rebalancing.
In Section II, we discuss the different utility functions that we consider. We discuss the method of
dynamic programming and how we model tracking error and transaction costs in Section III. In Section IV,
we construct optimal portfolios for each of the utility functions we consider. We then demonstrate the
rebalancing problem on a simple two-asset example in Section V to illustrate our algorithm. Section VI
examines the more general case of multiple assets over long periods of time. We report results from
sensitivity analysis in Section VII, examine areas of future investigation in Section VIII, and conclude
the paper in Section IX.
II. BACKGROUND
A. Existing Methods
Conventional approaches to portfolio rebalancing include periodic and tolerance band rebalancing [1],
[2]. Periodic rebalancing is the most direct and simple to implement. A portfolio manager reviews the
current weights against the target weights periodically (every week, month, quarter, or year) and makes
adjustments to realign the portfolio. Tolerance band rebalancing, slightly more sophisticated, requires
managers to rebalance whenever any asset class deviates outside of some pre-determined tolerance band.
Whenever this event occurs, the manager fully rebalances back to the target portfolio.
The main reason why a portfolio manager might not want to rebalance is transaction costs. Not only
does it cost money to trade these various securities, but it also requires manpower and technology
15.451 - FINANCIAL ENGINEERING (FALL 2004) 4
Utility function Expected utility
Quadratic fq(x) = x− α2(x− x0)
2 Uq(µ, σ) = µ− α2σ2
Log wealth fl(x) = log(1 + x) Ul(µ, σ) = log(1 + µ)− σ2
2(1+µ)2
Power fp(x) = 1− 1/(1 + x) Up(µ, σ) = 1− 1(1+µ)
− σ2
(1+µ)3
TABLE I
UTILITY FUNCTIONS AND THEIR CORRESPONDING APPROXIMATE EXPECTED UTILITIES USED IN THIS PAPER. THE UTILITY
FUNCTIONSf ARE EXPRESSED IN TERMS OF THE RETURNx. THE EXPECTED UTILITY FUNCTIONS ARE SPECIFIED IN TERMS
OF THE MEAN RETURNµ AND THE STANDARD DEVIATION OF RETURNSσ.
resources. In theory, if the transaction costs exceed the expected benefit from rebalancing, then no
adjustment should be made. However, without any quantitative measure for this benefit, there is no
way to accurately determine whether or not to trade.
B. Utility Functions
Evaluating individual preferences to risk and return and making corresponding portfolio allocation
decisions is a difficult task. Investment professionals need analyze various factors to develop portfolios
that allow clients to reach their investment goals while taking into account risks associated with bear
markets and singular events such as crashes.
No single portfolio can meet the needs of every investor. One way to specify an investor’s risk
preference is through the use of utility functions[3]. A utility function tells us how much satisfaction
(utils) we get for a given level of returnx. Clearly, most people prefer higher levels of return to lower
levels of return, so hopefully the utility function is monotonically increasing withx. If the marginal
utility decreases withx (i.e., utility grows sublinearly), then an individual is said to berisk averse.
Numerous other risk characteristics can be imparted through the shape of the utility function. A manager
then chooses portfolio weightsw so that expected utility is maximized. Although several utility functions
with plausible characteristics have been proposed, one must keep in mind that utility and risk are highly
subjective in nature, dynamic in time, and highly difficult to model.
To decide on a rebalancing policy (or even to create an optimal portfolio) without knowing the actual
future returns, we need the expected utility. It has been shown by Levy and Markowitz [4] that for
most relevant utility functions this expected utilityU can be approximated using truncated Taylor series
expansions to be a function of mean and standard deviation,U(µ, σ).
15.451 - FINANCIAL ENGINEERING (FALL 2004) 5
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
Returns
Util
s
QUAD (α=.06)QUAD (α=.04)LOGPOWER
Fig. 1. Plots of the quadratic utility (for two differentα’s), log wealth utility, and power utility as a function of returns.
In Table I, we list three utility functions and the corresponding expected utilities that we use in this
paper as shown in Cremerset al. [5]. For each utility,fi(x) for i = q, l, p represents the utility in utils
given a returnx (what we will sometimes refer to as the empirical utility).Ui(µ, σ) for i = q, l, p is
the expected utility (also in utils). Figure 1 plots the three empirical utility functions as a function of
return. The absolute value of the functions is not important (we could arbitrarily scale the utility functions
without affecting the corresponding optimal portfolio). The relative difference in utility for differentx is
what is important.
Quadratic utility is a commonly used function, and using it is akin to doing standard mean-variance
optimization. Regardless of whether or not the assets are Gaussian-distributed, the expected utility only
involves the first two moments, so any higher order moments are ignored. Theα parameter can be
adjusted to indicate risk tolerance. A larger number indicates that an investor is more risk averse.
Even though it is true that the expected utility can be written just in terms of the mean and variance,
the expression forUq(µ, σ) is only an approximation. The true value should be:
Uq(µ, σ) = µ− α
2(σ2 + (µ− x0)2) . (1)
Note then that if we knewµ a priori, then we would just choosex0 = µ. But µ is a function of the
portfolio weightsw, so we cannot fix it ahead of time. But in the regime we are typically operating in,
15.451 - FINANCIAL ENGINEERING (FALL 2004) 6
µ(w) ≈ µ(w∗) because we are rebalancing when the portfolios are too unbalanced. So we can treat this
Uq as a reasonable approximation to the true expected utility as long as we choose an appropriatex0.
We can choosex0 to be µ(w∗) minus a couple of basis points (bps) because we are usually operating
in a slightly suboptimal region. This leaves a much simpler expected utility function (especially in terms
of µ) which is useful for doing the analysis.
For α > 0, quadratic utility does display risk aversion. The main difficulty with quadratic utility is
that it has the odd behavior that for a large enough return, it istoo risk averse and the utility function
actually prefers a smaller return (becauselimx→∞ fq(x) = −∞). This behavior begins atx = x0 + 1/α,
the maximum of the quadratic function.
The derivation of the expected utilities for log wealth and power is non-obvious. Let’s examine log
wealth utility. We can expand the utility function around the pointx = µ using a Taylor series:
log(1 + x) = log(1 + µ) +11!
f ′l (1 + µ)(x− µ) +12!
f ′′l (1 + µ)(x− µ)2 + · · ·
≈ log(1 + µ) +x− µ
1 + µ− (x− µ)2
2(1 + µ)2. (2)
Thus we see that
Ul(µ, σ) = E[log(1 + x)]
≈ E
[log(1 + µ) +
x− µ
1 + µ− (x− µ)2
2(1 + µ)2
]
= log(1 + µ)− σ2
2(1 + µ)2.
Additional terms of the Taylor expansion may be used to improve the approximation. These will then
involve the skewness and the kurtosis and higher-ordered moments. A similar method may be applied to
derive the approximations for power utility as well as other arbitrary utility functions.
III. O PTIMAL REBALANCING USING DYNAMIC PROGRAMMING
In this section, we investigate optimal rebalancing strategies for portfolios with transaction costs. In
general, we consider a multi-asset problem where we are given an optimal portfolio consisting of a set
of portfolio weightsw∗ = {w∗1, . . . , w∗N}, whereN is the total number of assets. The optimal strategy
should be to maintain a portfolio that tracks the optimal portfolio as closely as possible, while minimizing
the transaction costs.
We consider a model where we observe the contents of the portfolio once a month, and at the end
of each month we have the option of rebalancing the contents of the portfolio. In general, the decision
to rebalance should be based on a consideration of three costs: the tracking error associated with any
15.451 - FINANCIAL ENGINEERING (FALL 2004) 7
deviation in our portfolio from the optimal portfolio, the trading costs associated with buying or selling
any assets during rebalancing, and the expected future cost from next month onwards given our actions
in the current month. The optimal strategy dynamically minimizes the total cost, which is the sum of
these three costs.
A. Solution Methodology: Dynamic Programming
One way to optimally solve the minimum cost problem is through Dynamic Programming [7], [8], [9].
Given that our portfolio today is weighted in each of theN assets according towt = {wt,1, . . . , wt,N},we can express the cost (known in optimization literature as thecost-to-go function) mathematically as:
Jt(wt) = T (wt+1, w∗) + C(wt+1, wt) + Jt+1(wt+1) (3)
where T (w, w∗) is the tracking error over a one-month duration associated with holding portfoliow
instead of the optimal portfoliow∗, C(w′, w) is the trading cost associated with going from a portfolio
of weightsw to w′, andJt+1(w) is the expected future cost fromt+1 onwards given all future decisions.
The optimal strategy chooseswt+1 such that the cost is minimized:
J∗t (wt) = minwt+1
T (wt+1, w∗) + C(wt+1, wt) + J∗t+1(wt+1) (4)
In steady-state, if rebalancing is done optimally at each stage, the cost-to-go should converge, such that
J∗t (w) = J∗t+1(w) = J∗(w). The challenge is therefore to determine the cost-to-go valuesJ∗(w); once
these values are known, then the optimal rebalancing decision is to choose the portfoliowt+1 according
to
wt+1 = arg minwt+1
T (wt+1, w∗) + C(wt+1, wt) + J∗(wt+1) (5)
We can determine the cost-to-go values using a technique calledvalue iteration. The idea behind value
iteration is to choose an arbitrary set of cost-to-go valuesJt(w) for some timet that we imagine to be
very far in the future. We then repeatedly apply (4) to obtain cost-to-go values successively closer to the
present. After a sufficient number of iterations, we will approach a steady-state and the cost-to-go values
should converge on the optimal valuesJ∗(w).
B. Modelling Tracking Error
Note that the cost-to-go values, and hence the optimal strategy, will depend on the cost functions
T (w, w∗) andC(w′, w) chosen. In this section, we discuss strategies for modelling tracking error.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 8
1) Certainty Equivalence:In the certainty equivalence approach, we model the investor’s preferences
using a utility function (see Section II-B). For any portfolio weightsw, we can express the expected utility
asU(µT w, wT Λw). We observe that there exists a risk free rate (which we will denote asrCE(w)) that
produces an identical expected utility. We therefore callrCE(w) the certainty equivalent returnfor the
weightsw. The condition for this isU(rCE , 0) = U(µT w, wT Λw). The certainty equivalents for the three
utility functions that we are using are:
1) Quadratic:rCE(w) = Uq(µT w, wT Λw)
2) Log wealth:rCE(w) = exp(Ul(µT w, wT Λw))− 1
3) Power:rCE(w) = 1/(1−Up(µT w,wT Λw))− 1
One interpretation of the certainty equivalent then is as a risk-adjusted rate of return given risk preferences
embedded in the utility function.
If we hold a suboptimal portfoliow, the utility of that portfolioU(w) will be lower thanU(w∗), with a
correspondingly lower certainty equivalent return. We can interpret this as losing a riskless return (equal
to the difference between the two certainty equivalents) over one period, corresponding to the penalty
paid for tracking error. Therefore, under the certainty equivalence approach, the tracking error has the
cost function
T (w,w∗) = rCE(w∗)− rCE(w) . (6)
The reason why we use a certainty equivalent is because in our cost function,T (·, ·) andC(·, ·) must
have commensurate values. We know that the cost will be in terms of dollars or basis points or some
other absolute measure. It is more straightforward to then convert expected utility suboptimality into a
similar absolute measure using certainty equivalents rather than trying to express the trading costs in
terms of diminished expected utility.
2) Variance Penalty:The optimal portfolio describes the best possible tradeoff between risk and
variance given the preferences of the investor. When the actual portfoliow does not match the optimal
portfolio w∗, a tracking error termw∗−w exists. We can assign a cost per unit varianceα to this tracking
error to encourage less deviation from the optimal portfolio. The tracking error cost can then be written
as:
T (w,w∗) = α(w∗T − wT )Λ(w∗ − w) (7)
15.451 - FINANCIAL ENGINEERING (FALL 2004) 9
C. Modelling Transaction Costs
1) Linear Transaction Costs:The simplest model for transaction costs is simply to assume a linear
cost. Under this model, we assume that for asseti we pay a transaction cost ofci per unit to buy or sell
the asset. Under this model,
C(w′, w) = cT |w′ − w| (8)
wherecT = [c1, . . . , cN ] is the vector of transaction cost coefficients.
A variant of the linear cost model allows for different costs to buy (c+) and sell (c−) assets:
C(w′, w) = cT+ max{w′ − w, 0}+ cT
−max{w − w′, 0} (9)
2) Affine Transaction Costs:Building on the linear cost model, we can allow for fixed costs in
rebalancing as well. This model encourages rebalancing to occur less frequently but in larger transactions:
C(w′, w) = cT+ max{w′ − w, 0}+ cT
−max{w − w′, 0}+ cTf I(w − w′) (10)
wherecf is the vector of fixed costs associated with trading each asset, and the indicator functionI(x)
is 1 if x = 0 and 0 otherwise.
IV. EFFICIENT FRONTIER AND PORTFOLIO WEIGHTS USINGMEAN-VARIANCE OPTIMIZATION
Given the assets at our disposal we constructed various optimal portfolios. For the analysis below
we concentrated on generating optimal portfolios with five funds: US Equity, Developed Market Equity,
Emerging Market Equity, Private Equity, and Hedge Funds. Figure 2 shows their historical monthly
returns. The kurtosis and skewness of the corresponding distributions are given in Table II. They indicate
that most of the assets exhibit approximately normal returns (which ideally have a skewness of 0 of and
kurtosis of 3). Figure 3 depicts this for the US equities case.
Previous research [5], [11] has indicated that full-scale portfolio optimization [14] is not required for
normally-distributed asset returns: optimal portfolios are then located on the efficient frontier resulting
from mean-variance optimization [12]. Although the legitimacy of mean-variance optimization is de-
pendent on restrictive assumptions regarding return distributions and investor risk preferences, full-scale
optimization has only been shown to improve the model in case of highly non-normal returns. Moreover,
as it compares all possible portfolios, it is extremely computationally intensive. For these reasons, we
decided to use mean-variance optimization for our portfolio construction.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 10
0 20 40 60 80 100 120 140 160 180−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Mon
thly
Ret
urns
US EquityDev. Market Eq.Emer. Market Eq.Private Eq.Hedge Funds
Fig. 2. Monthly returns for selected funds used in the portfolio selection analysis.
Kurtosis Skewness
(normal = 3) (normal = 0)
US Equity 3.673 -0.572
Developed Market Equity 3.269 -0.195
Emerging Market Equity 4.713 -0.732
Private Equity 3.821 -0.398
Hedge Funds 7.037 -0.827
TABLE II
KURTOSIS ANDSKEWNESS FORSELECTED FUNDS.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 11
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.150
1
2
3
4
5
6
7
8
Monthly Returns
Fre
quen
cy
Fig. 3. Histogram of monthly returns for US equity fund, with Gaussian fit to the data overlaid.
Since the average asset returns do not affect the approximation error of mean-variance optimization,
we were allowed to change each of the mean returns to generate truly diversified portfolios, rather than
portfolios with assets concentrated in one or two securities. Tables III and IV show the means, standard
deviations, and correlation coefficient matrix used in our analysis.
We first computed the mean-variance efficient frontier by solving a series of quadratic programs, each
minimizing the variance for a given expected portfolio returnµp. This task was accomplished using the
Quadprog.m function as part of a larger Matlab routine. Since short sales are not allowed for this
pension fund, the optimization problem has the following form:
minw w′Σw
s.t. w′µ = µp
∑i wi = 1
w ≥ 0,
(11)
15.451 - FINANCIAL ENGINEERING (FALL 2004) 12
Mean Return (%) Std. Dev.
(annual) (annual)
US Equity 6.84 14.99
Developed Market Equity 6.65 16.76
Emerging Market Equity 7.88 23.30
Private Equity 12.76 44.39
Hedge Funds 5.28 10.16
TABLE III
MEAN RETURNS AND STANDARD DEVIATIONS.
US Developed Emerging Private Hedge
Equity Markets Markets Equity Fund
US Equity 1.00 0.46 0.45 0.64 0.29
Developed Markets 0.46 1.00 0.42 0.38 0.09
Emerging Markets 0.45 0.42 1.00 0.40 0.21
Private Equity 0.64 0.38 0.40 1.00 0.36
Hedge Fund 0.29 0.09 0.21 0.36 1.00
TABLE IV
CORRELATION COEFFICIENT MATRIX.
wherew are the unknown portfolio weights,Σ is the covariance matrix of the available assets andµ is
vector of expected asset returns. The efficient frontier computed for our five asset classes is shown in
Figure 4.
Given the discussion above, the efficient frontier can be used to determine optimal portfolios for the
different expected utility functions presented earlier. For the quadratic utility function the optimal weights
can directly be determined by solving the following quadratic program:
maxw w′µ− α2 w′Σw
s.t. w′µ = µp
∑i wi = 1
w ≥ 0
(12)
For the non-quadratic utilities, the portfolio optimization becomes a maximization problem along the
15.451 - FINANCIAL ENGINEERING (FALL 2004) 13
efficient frontier. We do so by evaluating the approximate expected utility of all sample portfolios on
the frontier. The optimal portfolio locations corresponding to the different utility functions are plotted in
Figure 4, the corresponding weights are tabulated in Table V. The latter were used as the target weights
in our portfolio rebalancing simulations, discussed in Section VI.
2 4 6 8 10 12 140.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Monthly Standard Deviation (%)
Mon
thly
Ret
urn
(%)
Efficient Frontier
Efficient FrontierQuadratic UtilityLogarithmic UtilityPower Utility
Fig. 4. Efficient frontier and optimal portfolios for the different utility functions discussed.
Quadratic (α=1.5) Logarithmic Power
US Equity 0.1938497 0.1598783 0.2096114
Developed Market Equity 0.22191063 0.2400950 0.2134736
Emerging Market Equity 0.1846855 0.2751353 0.1427194
Private Equity 0.1564359 0.2916636 0.0936941
Hedge Funds 0.2431183 0.0332279 0.3405014
TABLE V
INITIAL PORTFOLIO WEIGHTS FOR DIFFERENT UTILITY FUNCTIONSU(µ, σ).
V. TWO ASSETMODEL
To introduce the problem of portfolio rebalancing, we first consider an example involving two risky
asset classes (and a third riskless asset). The benefit of the two risky asset model is that the optimal
15.451 - FINANCIAL ENGINEERING (FALL 2004) 14
portfolio can be computed in closed form (see Appendix II for the derivation), and we can visually
examine the changes in portfolio weights since we can plot a single asset’s weight over time and obtain
the full description of our portfolio.
For the purpose of this illustrative model, we assume that we can invest in (1) a public domestic
equity fund, (2) a private equity fund, or (3) short-term US treasuries (approximately a riskless asset).
If we assume that returns are normal, then the mean and covariance statistics sufficiently characterize
the assets. Using the expected returns from the problem statement [13] (also shown in Appendix I) and
covariances from historical data [15], we have
• Expected annual returns
US Equity = 7.06%, Private Equity = 14.13%, Risk-free asset = 2.00%
• Annualized standard deviation
US Equity = 12.8%, Private Equity = 21.0%, Risk-free asset = 0.00% (by definition)
• Correlation coefficient between risky assets = -0.46.
Given this information, we can create the efficient frontier, which graphically indicates the maximum
return possible for a given expected return [10]. Figure 5 shows the efficient frontier on a plot of portfolio
standard deviation versus expected value. For monthly returns less than0.82%, the efficient frontier is
the straight line that is tangent to the efficient portfolios curve (i.e. the curve which plots the minimum
standard deviation for a given return for the risky assets) and goes through the portfolio of the riskless
asset. Since we do not consider short selling, the efficient frontier follows the efficient portfolios curve
for higher expected returns.
To determine where along the efficient fronter we need to be, we have to know the risk-return trade-off
of the investor. This information is captured in a utility function. One simple utility model to consider
is quadratic utility. In this model, the expected utility for a portfolioP is
E(U(P )) = µP − α
2σ2
P . (13)
The α parameter is set based on the risk preference of the investor. For our analysis, consider the
case ofα = 0.4. Using this assumption, the optimal portfolio balance is41.37% in US equities,58.63%
in private equity, and nothing in the risk-free asset (i.e. we are on the curved portion of the efficient
frontier). To provide an example of our rebalancing method, we simulate the returns of the two equities
over a ten year period, assuming normal distribution of returns with the means and variances described
earlier.
Figure 6 shows how the portfolio weight of the US equities asset moves over the 120 month period.
With no rebalancing, the weight drifts from the optimal amount of41.37% down to under20%, resulting
15.451 - FINANCIAL ENGINEERING (FALL 2004) 15
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
0.2
0.4
0.6
0.8
1
1.2
1.4
Monthly Std Deviation (%)
Mon
thly
Ret
urn
(%)
Efficient Frontier for Two Risky & One Riskless Asset
Fig. 5. Efficient frontier of two risky asset case. Because short selling is not considered, the efficient frontier is the straight
red line and then the portion of the efficient portfoliios (blue) curve for returns greater than0.82%.
in large suboptimality costs (the exact costs will be described in the next section). Our optimal rebalancing
strategy rebalances often when necessary. During months40 to 45 and90 to 110, the portfolio rebalances
nearly every month to handle sharp changes in returns, while for months45 to 80, the lack of strong
market movements in either direction allow us to avoid any transaction costs1.
The following subsection quantifies the cost savings using our optimal portfolio method, as compared
with the other common methods of rebalancing.
A. Cost Comparison
As discussed in the background, different strategies exist to trade on a portfolio. In this section we
provide a numerical analysis of the costs using each technique. Using a quadratic utility with anα
parameter of0.4, the optimal portfolio was41.37% in US equity,58.63% in private equity, and nothing
in cash. While we considered the rebalancing option of selling one risky asset and not buying the other
(i.e. leaving it in the risk-free), whenever our algorithm decided to trade, it found it optimal to rebalance
only by investing all proceeds gained from selling one asset into the other asset. So, the cash position
was always zero. This can be attributed to the fact that the certainty equivalent was sufficiently higher
1To see the market movements during the times cited, examine the change in portfolio weights in the no rebalancing graph.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 16
0 20 40 60 80 100 120
0.2
0.3
0.4
0.5(a) Portfolio Weights with No Rebalancing
0 20 40 60 80 100 1200.3
0.35
0.4
0.45
0.5(b) Optimal Rebalancing
0 20 40 60 80 100 1200.3
0.35
0.4
0.45
0.5(c) 5% Tolerance Band Rebalancing
0 20 40 60 80 100 1200.3
0.35
0.4
0.45
0.5(d) Monthly Rebalancing
0 20 40 60 80 100 1200.3
0.35
0.4
0.45
0.5(e) Quarterly Rebalancing
0 20 40 60 80 100 1200.3
0.35
0.4
0.45
0.5(f) Annual Rebalancing
Fig. 6. Plots of US equities weighting in the two asset example using different trading models. The vertical lines indicate
months where rebalancing was done (for monthly rebalancing, this is omitted since trading occurs in every month).
than the risk-free rate. So it was never optimal to hold any cash given our particular assumptions in this
three asset example.
Table VI shows the costs of trading using different strategies. The costs of trading are assumed to be
20 bps for buying or selling public equity, and 40 bps for buying or selling private equity. The non-
optimal utility cost was determined using the idea of certainty equivalents. For each portfolio, a certainty
equivalent can be computed (in terms of monthly returns). The difference between the certainty equivalent
of a non-optimal portfolio and that of the optimal portfolio is defined as the cost of not being optimal.
From the table, we observe that the aggregate monthly cost is minimized by our method. Over a 10
year period, the cost of our portfolio, assuming $100 million invested, is $281,300. The next best method,
that of yearly rebalancing, costs $337,100. The results for each rebalancing method make intuitive sense.
Monthly rebalancing leads to no deviation from optimality, but at the cost of high trading fees. Infrequent
trading yields smaller trading costs, but higher non-optimality certainty equivalent costs. Our method of
rebalancing whenever the cost of non-optimality exceeds the trading costs allows us to adequately trade-
off costs of non-optimality with that of trading.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 17
Trading Non-Optimal Aggregate
(bps) Utility Cost (bps) Cost (bps)
Optimal 18.14 9.99 28.13
No Trading 0 1509.74 1509.74
Yearly 20.29 13.42 33.71
5% Tolerance 17.59 16.83 34.43
Quarterly 37.78 1.74 39.52
Monthly 62.61 0 62.61
TABLE VI
TRADING COSTS, NON-OPTIMAL UTILITY COSTS, AND AGGREGATE COST USING SIX DIFFERENT TRADING STRATEGIES ON
TWO RISKY ASSETS OVER A TEN YEAR PERIOD.
Asset Expected monthly
Class return (%)
US Equity 0.5883
Developed Market Equity 0.5717
Emerging Market Equity 0.7592
Real Estate 0.4250
Private Equity 1.1775
Hedge Funds 0.3500
Fixed Income 0.2083
Cash 0.1667
TABLE VII
EXPECTED MONTHLY RETURNS AS GIVEN IN THE PROBLEM STATEMENT.
VI. M ULTI -ASSETMODEL
With the expected returns given in Table VII, and the sample covariance of the given data, the optimal
portfolio with α = 0.06 is 3.59% in developed markets, 0.76% in real estate, 86.1% in the hedge fund,
and 9.55% in fixed income (with the other portfolio weights at zero).
Unfortunately, the full optimization using eight assets requires excessive computation time, even if we
allow only a few possible weights for each asset. Imagine that 15 possible weights are allowed for each
asset. Then we have an observation space of approximately 2.6 billion points (and we must develop the
15.451 - FINANCIAL ENGINEERING (FALL 2004) 18
Expected Monthly Expected Monthly
Return (%) Std Dev (%)
US Equity 0.57 4.33
Developed Markets 0.55 4.84
Emerging Markets 0.66 6.73
Private Equity 1.06 12.81
Hedge Fund 0.44 2.93
TABLE VIII
EXPECTED RETURNS AND EXPECTED STANDARD DEVIATIONS ASSUMED OVER A ONE MONTH PERIOD FOR OUR FIVE ASSET
MODEL.
optimal policy for each point). Our current implementation can process around 600,000 points per hour
for the five-asset case. Ignoring the fact that computation time per point increases with number of assets,
this results in a run-time estimate of 178 days with memory storage requirements around 416 GB. As a
result, we examine a similar case with five different assets: US equity, developed market equity, emerging
market equity, private equity, and hedge funds.
The expected values and variances we used are shown in Table VIII, and the the correlation coefficients
are shown in Table IV. It should be noted that it is difficult to find a balanced portfolio for the same group
of assets for all of the utility functions we consider given the drastic variation in risk tolerance among
them. The values in those tables were modified from the original numbers in the problem statement and
in the sample data to get more balanced portfolios.
Tables IX - XI show the results of our algorithm and some existing rebalancing methods on Monte
Carlo simulations. We generated 10,000 sample paths, each for 10 years of monthly return data. For each
sample path, we simulate the various rebalancing methods to generate a return value for each month (net
of transaction costs). One way to evaluate performance is with expected deviation from the idealized
portfolio: monthly rebalancing (so we always begin a month atw∗) with no transaction costs. We can
measure this using the actual trading costs that we incur (the first column) and the decrease in certainty
equivalent (the second column). These two numbers are added together in the third column to obtain an
aggregate expected cost. Another method to evaluate the results would be based on the actual sample
returns. We compute the sample means and standard deviations of these net returns in the fourth and
fifth columns. Using that net return stream, we can also evaluate the empirical utility for each month.
We computed the sample average of these empirical utilities and in the sixth column are showing the
15.451 - FINANCIAL ENGINEERING (FALL 2004) 19
Trading Suboptimality Aggregate Net Standard Utility
Cost Cost Cost Returns Deviation Shortfall
(bps) (bps) (bps) (%) (%) (utils x 104)
Ideal 0.00 0.00 0.00 7.45 14.84 0.00
Optimal DP 4.04 1.72 5.75 7.40 14.86 5.55
No Trading 0.00 71.72 71.72 6.77 14.96 71.36
5% Tolerance 7.39 0.70 8.09 7.37 14.83 8.03
Monthly 23.66 0.00 23.66 7.22 14.84 23.72
Quarterly 13.68 0.28 13.96 7.32 14.85 14.28
Annual 6.84 1.55 8.39 7.40 14.94 8.24
TABLE IX
TRADING COSTS, NON-OPTIMAL UTILITY COSTS, AND AGGREGATE COST USING SIX DIFFERENT TRADING STRATEGIES ON
FIVE RISKY ASSETS SIMULATED OVER A10 YEAR PERIOD10,000TIMES USING QUADRATIC UTILITY.
difference between the ideal utility and the utility of each algorithm.
One thing that’s interesting to note is that the units on the utils when multiplied by104 are similar
to basis points (which we use in the first three columns). This is intuitively clear for the quadratic case
where the certainty equivalent was equal to the utility. For the log wealth case, we can look at the Taylor
series expansion aroundx = 0. We can see thatlog(1 + x) = 0 + 11!x − 1
2!x2 . . . ≈ x − 0.5x2. For
power utility, 1− 1/(1 + x) ≈ 1− (1− x + x2) = x− x2. So for both log wealth and power utility, the
utilities are dominated by the linear term for smallx, so we get something similar to basis points when
multiplying by 104. There should also be some relationship between expected aggregate cost (which is
based on mathematical expectation) and average utility shortfall (due to the law of large numbers). Thus,
it is not surprising to see that we get similar percentage improvements in expected aggregate cost and
average utility lost.
For the quadratic utility case shown in Table IX, we do 29% better in terms of expected cost and 31%
better in terms of average utility over the next-best method, 5% tolerance bands.
For the power utility version discussed in Table X, our expected loss is 24% less than the runner-up,
5% tolerance band rebalancing again. The sample-based empirical utility shortfall is reduced by 22%. The
benefits for this method are reduced from the quadratic utility case primarily because less rebalancing is
needed overall because the power utility portfolio has the lowest variance.
Note that even though tolerance bands do better than annual rebalancing in this example (and also for
15.451 - FINANCIAL ENGINEERING (FALL 2004) 20
Trading Suboptimal Aggregate Net Standard Utility
Cost Cost Cost Returns Deviation Shortfall
(bps) (bps) (bps) (%) (%) (utils x 104)
Ideal 0.00 0.00 0.00 6.89 12.38 0.00
Optimal DP 3.47 1.21 4.67 6.87 12.48 4.43
No Trading 0.00 81.70 81.70 6.77 14.95 82.31
5% Tolerance 5.30 0.83 6.13 6.83 12.36 5.75
Monthly 20.05 0.00 20.05 6.69 12.38 19.96
Quarterly 11.59 0.18 11.78 6.77 12.39 11.90
Annual 5.82 1.02 6.84 6.84 12.46 6.64
TABLE X
TRADING COSTS, NON-OPTIMAL UTILITY COSTS, AND AGGREGATE COST USING SIX DIFFERENT TRADING STRATEGIES ON
FIVE RISKY ASSETS SIMULATED OVER A10 YEAR PERIOD10,000TIMES USING POWER UTILITY.
quadratic utility, but not for log wealth), this should not necessarily be taken as an indicator that tolerance
bands a superior method to periodic rebalancing. Better performance can be obtained by tweaking the
threshold parameter or the periodicity of rebalancing. For instance, setting the rebalancing time to two
years for the power utility case results in an expected loss of 6.32 bps per annum. This is achieved by
accruing more than twice as much expected suboptimal risk-adjusted return (2.21 bps versus 1.03 bps),
but also reducing trading costs by 29% (4.11 bps versus 5.81 bps). A more exhaustive search of possible
fixed-interval rebalancing strategies could presumably yield an even better result.
For the log wealth utility case shown in Table XI, our expected loss is 30% less than the best alternative
(annual rebalancing). And the average simulated utility deficit is also 30% less than annual rebalancing.
This is a clear win as we tie for the highest net return while we have the lowest standard deviation (except
for the no rebalance case where in many cases, the high-variance/high-return assets become small quickly,
and without rebalancing, we are stuck in low-variance/low-return assets). You can see the effect of the
higher-variance portfolio in the trading cost numbers for the 5% Tolerance method. In the quadratic case,
the trading costs are only marginally higher than the annual rebalance method. But in the log wealth
case, thop are 49% higher because the tolerance bands are breached more often. It’s possible that better
performance could be achieved by loosening the tolerance band as there is currently very little loss to
portfolio suboptimality.
Before we complete this section, we address the possibility of a different trading cost function. In
15.451 - FINANCIAL ENGINEERING (FALL 2004) 21
Trading Suboptimal Aggregate Net Standard Utility
Cost Cost Cost Returns Deviation Shortfall
(bps) (bps) (bps) (%) (%) (utils x 104)
Ideal 0.00 0.00 0.00 8.65 20.57 0.00
Optimal DP 4.87 2.26 7.13 8.57 20.49 7.09
No Trading 0.00 91.51 91.51 6.77 14.98 87.82
5% Tolerance 11.99 0.44 12.43 8.53 20.60 12.74
Monthly 28.14 0.00 28.14 8.37 20.58 28.18
Quarterly 16.25 0.40 16.65 8.49 20.59 17.13
Annual 8.06 2.17 10.22 8.57 20.67 10.18
TABLE XI
TRADING COSTS, NON-OPTIMAL UTILITY COSTS, AND AGGREGATE COST USING SIX DIFFERENT TRADING STRATEGIES ON
FIVE RISKY ASSETS SIMULATED OVER A10 YEAR PERIOD10,000TIMES USING LOG UTILITY.
particular, while the numbers used are consistent with trading costs cited in other research papers [6],
some may wonder if the results would be different for alternate trading costs. Table XII shows the results
when we reduce the proportional trading costs in half and apply it to the quadratic utility strategy. We do
only 20% better in expected cost, and 21% better in average utility, down from a 30% advantage with the
original costs. Transaction costs for the other methods are cut in half, while suboptimality remains the
same. Because in the original version transaction costs ranged from 82% of the aggregate cost for annual
rebalancing to 100% of the cost for monthly rebalancing while they were only 70% for our method. If
our transaction costs were simply cut in half and we did not alter our trading strategy, we would expect
the aggregate cost to decline by 35%. It actually declines by 39% because we adjust our strategy to trade
more frequently and incur smaller suboptimality penalties.
VII. SENSITIVITY ANALYSIS
In the preceding sections, we have assumed that our model of each asset is accurate. In practice, this is
usually not the case – mean and variance of the returns of each asset as well as the correlation between
assets must be estimated using the historical observations, and there is usually some error associated
with each estimate. Errors in the parameter estimation will cause inaccuracies in the cost-to-go values
obtained from the dynamic program, leading to suboptimal rebalancing. In this section, we investigate
the impact of errors in each of these parameters on the rebalancing strategy.
We investigated a total of 3 parameters – mean, variance, and correlation. Simulations were conducted
15.451 - FINANCIAL ENGINEERING (FALL 2004) 22
Trading Suboptimality Aggregate Net Standard Utility
Cost Cost Cost Returns Deviation Shortfall
(bps) (bps) (bps) (%) (%) (utils x 104)
Ideal 0.00 0.00 0.00 7.45 14.84 0.00
Optimal DP 2.64 0.87 3.51 7.42 14.85 3.42
No Trading 0.00 71.72 71.72 6.77 14.96 71.36
5% Tolerance 3.69 0.70 4.39 7.41 14.84 4.35
Monthly 11.83 0.00 11.83 7.34 14.84 11.86
Quarterly 6.84 0.28 7.12 7.38 14.85 7.44
Annual 3.42 1.55 4.97 7.43 14.94 4.83
TABLE XII
TRADING COSTS, NON-OPTIMAL UTILITY COSTS, AND AGGREGATE COST USING SIX DIFFERENT TRADING STRATEGIES ON
FIVE RISKY ASSETS SIMULATED OVER A10 YEAR PERIOD10,000TIMES USING QUADRATIC UTILITY.
where in each simulation, 2 parameters were held constant while the 3rd was allowed to vary slightly about
the estimated value. The cost-to-go values used for the rebalancing decisions were the ones calculated
using the estimated values, not the actual ones, thus allowing us to characterize the performance of the
strategy in the cases where the estimated parameters differed from the actual parameters. In the case
of the calendar and tolerance band strategies, we assumed that they would also rebalance to an optimal
portfolio calculated from the estimated, not the actual, parameters. Therefore we expect the performance
of all strategies to degrade if parameter estimation is inaccurate; the question is whether some strategies
are relatively more robust to inaccuracies than others.
A. Sensitivity to Mean
In this section, we assume that the variance and correlation are correctly estimated and investigate
estimation errors in the mean. For each point, 100 sequences of 10-year monthly returns were generated,
and the performance of the dynamic rebalancing strategy was averaged over each sequence.
The dynamic rebalancing strategy was reasonably insensitive to errors in estimating the mean: from
Figures 7 and 8, we can observe that the DP approach outperforms all other strategies over a range of
several percentage points of inaccuracies in the estimation. We can conclude that as long as the mean can
be accurately estimated to within a few percentage points, the dynamic programming-based approach is
a good choice.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 23
6 6.5 7 7.5 80
1
2
3
4
5
6
7
8
9
10
mean rate (%/yr)
erro
r co
st (
bps/
yr)
Rate sensitivity − US Equity
DP5% tolerance yearly
monthly
quarterly
Fig. 7. Sensitivity to mean rate for US equity. The dotted line shows the estimated rate used by the dynamic program.
13 13.5 14 14.5 150
1
2
3
4
5
6
7
8
9
10
mean rate (%/yr)
erro
r co
st (
bps/
yr)
Rate sensitivity − Private Equity
DP
5% tolerance
yearly
monthly
quarterly
Fig. 8. Sensitivity to mean rate for private equity. The dotted line shows the estimated rate used by the dynamic program.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 24
10 11 12 13 14 15 160
1
2
3
4
5
6
7
8
9
10
std. dev. (%/yr)
erro
r co
st (
bps)
Variance sensitivity, US Equity
DP5% tolerancequarterly
monthly
yearly
Fig. 9. Sensitivity to variance for US equity. The dotted line shows the estimated variance used by the dynamic program.
B. Sensitivity to Variance
From Figures 9 and 10, we see that the dynamic programming approach again outperforms the other
approaches even if there are large errors in estimating the standard deviation – it remained the best
performer even given inaccuracies in the standard deviation of several percentage points per year.
C. Sensitivity to Correlation
Finally, in Figure 11 we observe that the dynamic programming approach is very insensitive to errors
in estimation of the correlations between assets – the approach outperforms all others over virtually
all possible correlations. This suggests that correlations do not need to be accurately estimated for the
purposes of the DP.
VIII. A REAS FORFUTURE INVESTIGATION
We have intentionally avoided investigating the implications of tax policy on trading. This works for
pension funds since they are generally tax-free accounts. For other types of funds that are taxable (such
as mutual funds), asset managers should factor tax policy into their decision process. Additionally, we
have modeled transaction costs to be a certain percentage of the amount traded. A more complete analysis
might use real world prices, and these could vary over time. We have always assumed that the act of
15.451 - FINANCIAL ENGINEERING (FALL 2004) 25
19.5 20 20.5 21 21.5 22 22.50
1
2
3
4
5
6
7
8
9
10
std. dev. (%/yr)
erro
r co
st (
bps/
yr)
Variance sensitivity, Private Equity
DP5% tolerance
quarterly
monthly
yearly
Fig. 10. Sensitivity to variance for private equity. The dotted line shows the estimated variance used by the dynamic program.
−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.10
1
2
3
4
5
6
7
8
9
10
correlation
erro
r co
st (
bps)
Correlation sensitivity
DP
monthly
quarterly
yearly
5% tolerance
Fig. 11. Sensitivity to correlation between US and private equity. The dotted line shows the estimated correlation used by the
dynamic program.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 26
making a trade would be instantaneous and would not affect asset price. In reality, this would depend on
the asset and the volumes being traded. Finally, we have not considered the possibility of allowing short
sales in our portfolios. This is consistent with most (but not all) pension and mutual funds.
One point of interest is that our methodology seems to be only marginally better than annual rebal-
ancing. One approach that people seem to take in the literature is to take advantage of the empirical
evidence that returns tend to exhibit mean reversion. We do not currently exploit this information which
may allow for further gains.
Another issue that the sensitivity analysis demonstrated is the dependence on correct assumptions of
the means and variances for the various asset classes. In reality, these model parameters are never known
with absolute certainty. Another extension could be to incorporate robust control techniques to make the
method more resistant to poor model parameters. One issue that comes up when incorporating transaction
costs is that the optimal portfolios will actually change given your rebalancing strategy. Assets that require
less rebalancing should increase in weight.
We acknowledge that our problem can be generalized by changing some of our assumptions and/or
relaxing some constraints. However, given our time frame, we felt that we chose assumptions that best
fit the target audience of pension fund managers.
IX. CONCLUSION
The ad hoc methods of periodic and tolerance band rebalancing provide simple ways of portfolio
rebalancing, but are suboptimal. In this work, we have discussed that through the optimization technique
of dynamic programming, we can reduce the overall costs of portfolio rebalancing. We have found this
to be true for different investor risk preferences. Namely, we have compared the performance of our
technique with the others for three different utility functions: quadratic, log wealth, and power utility.
It is easy to see how transaction costs affect the bottom line. Less obvious are the costs for being
suboptimal. Our use of certainty equivalence to determine the equivalent “risk-free” value of a portfolio
has provided a method for us to reasonably quantify the cost of being suboptimal. Our simulations
confirmed that this optimal method provides slight gains over the best of the traditional techniques of
rebalancing.
ACKNOWLEDGMENTS
The authors would again like to thank Sebastien Page and Mark Kritzman for their guidance in our
project.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 27
Asset Class Benchmark
US Equity Russell 3000 Total Ret
Developed Market Equity MSCI EAFE + Canada ($) Total Ret
Emerging Market Equity MSCI EM Index ($) Total Ret
Real Estate Wilshire Real Estate Securities Index Tot Ret
Private Equity Wilshire LBO Index
Hedge Funds HRF Market Neutral Index
Fixed Income Lehman Agg Tot Ret
Cash JPM Cash 3 mo Tot Ret
TABLE XIII
ASSET CLASSES WITH ASSOCIATED BENCHMARKS.
APPENDIX I
PROBLEM STATEMENT
The objective of this problem is to solve the unsolved problem of optimal rebalancing in the presence
of transaction costs.
Two sub-optimal approaches to portfolio rebalancing currently prevail in the industry. Pension plans
rebalance their allocation either on a calendar basis (weekly, monthly, quarterly, or yearly), or by using
tolerance bands, such as plus or minus 5% around the optimal allocation to each asset class.
The team must propose a model that will determine when and how pension plans should rebalance their
asset allocation. This new approach must reduce transaction costs, and increase expected risk-adjusted
return when compared to the calendar and tolerance band approaches to rebalancing.
Table XIII shows the asset classes to be considered along with the associated benchmarks, while
Table VII shows a given set of expected monthly returns.
APPENDIX II
DERIVATION OF OPTIMAL PORTFOLIO FORTWO RISKY ASSETS
In the special case of two risky assets and quadratic utility, the optimal portfolio weights can be
computed in closed form. Suppose there exist two risky assets A and B having expected returns ofµA
andµB, variances ofσ2A andσ2
B, respectively, and correlation coefficient ofρ.
Suppose a portfolio P is comprised of a fractionx of A, and a fraction(1 − x) of B. The expected
return for such a portfolio is
E(P (x)) = µAx + µB(1− x) (14)
15.451 - FINANCIAL ENGINEERING (FALL 2004) 28
and the associated variance is
σ2P (x) = σ2
Ax2 + σ2B(1− x)2 + 2ρσAσBx(1− x). (15)
Since we consider quadratic utility, we have
E(U(P (x))) = µP (x) −α
2σ2
P (x), (16)
whereα is a risk parameter that is set based on an investor’s risk preferences. In order to find the weight
x that maximizes the expected utility E(U(P)), we take the first derivative of Equation (16) and set it to
zero.dU
dx= µA − µB − α
2[2σ2
Ax− 2(1− x)σ2B + 2ρσAσB(1− 2x)] = 0
This simplifies to
xopt =(µA − µB)/α + σ2
B − ρσAσB
σ2A + σ2
B − 2ρσAσB.
15.451 - FINANCIAL ENGINEERING (FALL 2004) 29
REFERENCES
[1] C. Donohue and K. Yip, “Optimal Portfolio Rebalancing with Transaction Costs,”Journal of Portfolio Management,(Summ
2003) 49-63.
[2] S. J. Masters, “Rebalancing,”Journal of Portfolio Management, (Spring 2003) 52-57.
[3] R. Duncan Luce,Utility of Gains and Losses: Measurement-Theoretical and Experimental Approaches(Lawrence Erlbaum,
2000).
[4] Haim Levy and Harry M. Markowitz, “Aproximating Expected Utility by a Function of Mean and Variance,”American
Economic Review, 69:3 (June 1979).
[5] J. Cremers, M. Kritzman, S. Page, “Portfolio Formation with Higher Moments and Plausible Utility,”Revere Street Working
Paper Series, Financial Economics 272-12, (November 22, 2003).
[6] H. E. Leland, “Optimal Portfolio Management with Transaction Costs and Capital Gains Taxes,”Haas School of Business
Technical Report,, UC Berkeley (Dec 1999).
[7] R. Bellman,Dynamic Programming,Princeton University Press, Princeton, NJ (1957).
[8] R. Bellman and S. Dreyfus,Applied Dynamic Programming,Princeton University Press, Princeton, NJ (1962).
[9] D. P. Bertsekas,Dynamic Programming and Optimal Control,Athena Scientific, Belmont, MA (2000).
[10] R. Brealey and S. Myers,Principles of Corporate Finance,5th Ed., Ch. 8, McGraw-Hill Companies, NY (1996).
[11] J. Cremers, M. Kritzman, S. Page, “Optimal Hedge Fund Allocations: Do Higher Moments Matter?”Revere Street Working
Paper Series, Financial Economics 272-13, (September 3, 2004).
[12] Harry M. Markowitz, “Portfolio Selection,”Journal of Finance, (March 19, 1952).
[13] S. Page, “Optimal Rebalancing Strategy for Pension Plans,”15.451 State Street Problem Statement,(Oct 5, 2004).
[14] P. A. Samuelson, “When and Why Mean-Variance Analysis Gene rically Fails,”American Economic Review, 2003.
[15] K. Terhaar, R. Staub, and B. Singer, “Appropriate Policy Allocation for Alternative Investments,”The Journal of Portfolio
Management(Spring 2003) 101-110.