15.451 - FINANCIAL ENGINEERING (FALL 2004) 1 Optimal ...ssg.mit.edu/group/waltsun/docs/rebalancing04.pdfConventional approaches to portfolio rebalancing include periodic and tolerance

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


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


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


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).


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,


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


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.


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)


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.


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.


−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)


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


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


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


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.


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.


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


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


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


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


Trading Suboptimal Aggregate Net Standard Utility



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


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


Trading Suboptimal Aggregate Net Standard Utility



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


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


Trading Suboptimality Aggregate Net Standard Utility



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


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.


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

DP

5% tolerance

quarterly

monthly

yearly

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

quarterly

monthly

yearly

Fig. 8. Sensitivity to mean rate for private equity. The dotted line shows the estimated rate used by the dynamic program.


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


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.


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.


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)


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.


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Economic Review, 69:3 (June 1979).

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Paper Series, Financial Economics 272-12, (November 22, 2003).

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[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

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Documents

15.451 - FINANCIAL ENGINEERING (FALL 2004) 1 Optimal ...ssg.mit.edu/group/waltsun/docs/rebalancing04.pdfConventional approaches to portfolio rebalancing include periodic and tolerance