Abstract

We examine the optimal trading strategy for an investment fund which wishes to maintain assets two assets in fixed proportions, e.g. 60/40 in stocks and bonds. Transactions costs are assumed to be proportional to the amount of each asset traded.

We show that the optimal policy specifies a band about the target stock proportion. As long as the actual stock/bond ratio remains inside this band, no trading should occur. If the ratio goes outside the band, trading should be undertaken to move the ratio to the nearest edge of the band. We compute the optimal band and resulting annual turnover and tracking error of the optimal policy, as a function of transactions costs, asset volatility, the target asset mix, and other parameters. We show how changes in transactions costs and other parameters affect the size of the no-trade band, turnover, and tracking error. Compared to a quarterly rebalancing strategy, an example demonstrates that the optimal strategy can reduce turnover by almost 50%.

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The author gratefully acknowledges support from BARRA. Greg Connor and Ron Kahn have provided valuable comments. Klaus Toft corrected an error in an earlier version. The author bears sole responsibility for errors which may remain.

OPTIMAL ASSET REBALANCING IN THE PRESENCE OF

TRANSACTIONS COSTS

I. Introduction

Many asset funds desire to maintain constant asset proportions, such as a 60/40 ratio of stocks to bonds. As asset values move randomly, asset ratios diverge from the desired target ratio. Funds "rebalance" by trading to restore the asset proportions to their target levels. Rebalancing typically occurs at a regular time interval, e.g. every month or every quarter.

If transactions costs are proportional to the dollar amounts traded, the conventional strategy of periodic rebalancing to the target ratio will not be optimal. Work by Constantinides [1986] and by Dumas and Luciano [1991] (based on a model with power utility functions) shows that the optimal strategy is typically characterized by a "no trade" interval about the target asset ratio. As the ratio of asset values moves randomly within this interval, no trading is needed. When the asset ratio wanders outside the no-trade interval, asset proportions should be adjusted back to the nearest edge of the interval. Subsequently, Dixit [1991] and Dumas [1991] provided further mathematical results for this and related problems.

To illustrate the nature of the optimal policy, we shall later show an example where the target stock/bond ratio is 1.5: a 60/40 stock-to-bond ratio. We show that no readjustments should be made when the ratio remains between 1.44 and 1.56. If the ratio moves (say) to 1.58, then trading should take place to restore the ratio to 1.56. If the ratio moves to 1.42, then the ratio should be restored to 1.44. Thus there may be extended periods with no trading (as the ratio remains inside the no-trade interval), followed by brief periods in which many trades may be required to keep the ratios at the interval boundary. The ratio should never be moved to 1.50!

Our work differs from previous work in several dimensions. First, we take as given the target asset ratios and focus only on how to trade when divergences from the ratios occur. Rather than using a specific utility function over wealth, we postulate a cost function for deviations from the target ratio. Second, we develop a technique for estimating the expected required turnover of the optimal policy. Finally, we consider two risky assets, rather than a risky and a riskless asset.

II. The Optimization Problem

Consider two assets ("stocks" and "bonds"), whose prices S and B follow log random walks

(1) dS(t)/S = :_S dt + F_S dZ_S(t)

(2) dB(t)/B = :_B dt + F_B dZ_B(t),

where dZ_S and dZ_B are the increments of Wiener processes with correlation D.

Using Ito's Lemma, we may easily compute the stochastic process for w(t) = S(t)/B(t):

(3) dw(t)/w = (:_S - :_B + F²_B - DF_B F_S ) dt + F_S dZ_S(t) - F_B dZ_B(t)

which implies

(4) E(dw/w) = (:_S - :_B )dt

(5) E(dw/w)² = (F_S² + F_B² - 2DF_S F_B )dt.

Given whatever utility function s/he may have, the investor wishes to hold stocks to bonds in a target ratio w*. Divergence between the actual ratio, w, and w* cause the investor to lose utility. The absolute loss will depend on the total dollars invested. But we scale the losses relative to wealth, implying that loss depends only on the divergence between w(t) and w*. When this divergence is small, the dollar equivalent cost over a time interval dt may be approximated by

(6) L = 8(w(t) - w*)²dt,

where 8 is a constant representing the cost per unit of tracking error.

Over an infinite horizon, the investor wishes to minimize the discounted integral of this cost of divergence, plus any trading costs associated with adjusting w(t). As is well known, trading costs will be infinite if the investor tries to peg w(t) = w* continuously (see Leland [1985]). Constantinedes [1986] and Dixit [1991] show that, in the presence of proportional trading costs, the optimal policy will specify an interval w_min # w* # w_max . If the current w(t) lies inside this interval, no trading will occur. If w(t) > w_max , stocks will be sold and the proceeds used to purchase bonds in order to reduce the ratio to w_max . If w(t) < w_min, stocks will be bought and bonds sold in order to increase the ratio to w_min .

Let k_S be the transactions cost per dollar of stocks traded, and k_B be the transactions cost per dollar of bonds traded. Since the portfolio must be self-financing, any change in the dollar value of stocks from trading, *S, will be matched by an opposite amount of change in the dollar value of bonds: *B = -*S. Total trading costs *C from a given trade *S in stocks will therefore be

(7) *C = k_S|*S| + k_B|*B| = (k_S + k_B )|*S|.

We may readily ascertain that the change in w due to a trade *S is

(8) *w = (1 + w)*S/B,

implying

(9) *C/|*w| = B(k_S + k_B )/(1 + w).

Clearly costs are proportional to wealth, since B is an absolute number. Normalizing costs by wealth W = B + S gives (for *c = *C/W)

(10) *c/|*w| = (k_S + k_B )/(1 + w)².

Let V(w(t); w_min ,w_max ) be the cost function associated a trading strategy characterized by no trading whenever w(J) + *w(J) , [w_min, w_max ]. That is,

where PV{transactions costs} is the present value of the costs associated with implementing the strategy.

When w(J) + dw(J) , [w_min , w_max ], the only cost over the time interval dJ is the cost of not being at the optimal w*. From the definition of V(w(t); w_min ,w_max ) ,

2(11)

Since the optimal strategy moves w(t) to w_min if w(t) + dw(t) < w_min , and to w_max if w(t) + dw(t) > w_max , it follows that for small excursions of w(t) outside the no-trade interval,

(12) V(w(t); w_min ,w_max ) = V(w_min ; w_min ,w_max ) + *c*/|*w|(w_min - w(t))

(13) V(w(t); w_min ,w_max ) = V(w_max ; w_min ,w_max ) + *c**/|*w|(w(t) - w_max )

when w(t) + dw(t) < w_min and w(t) + dw(t) > w_max , respectively.

From (10), *c*/|*w| = (k_S + k_B )/(1 + w_min )², and *c**/|*w| = (k_S + k_B )/(1 + w_max )². From (12) and (13), smoothness of the function V w.r.t. w at w = w_min and w = w_max implies that

(14) V₁(w_min ; w_min ,w_max ) = -*c*/|*w| = -(k_S + k_B )/(1 + w_min )²

(15) V₁(w_max ; w_min ,w_max ) = *c**/|*w| = (k_S + k_B )/(1 + w_max )²

where V_k(!;!,!) denotes the derivative of V with respect to its kth argument.

Expanding the expectation term of (11) and simplifying gives the differential equation

3(16)

where

a = :_S - :_B + F²_B - DF_B F_S ; b = F_S² + F_B² - 2DF_SF_B .

Equation (16) has the solution

4(17)

where

x = (-2a + b - Sqrt[(2a - b)² + 8br])/2b; y = (-2a + b + Sqrt[(2a - b)² + 8br])/2b,

and C₁ and C₂ are determined by the boundary conditions (14) and (15), and depend upon w_min and w_max . The latter are determined by the "smooth pasting" conditions, which require that V(w; w_min , w_max ) be minimized w.r.t. w_min and w_max . This provides the two final conditions needed for optimization, that at the optimal w_min = w*_min and w_max = w*_max :

(18) V₂(w*_min ; w*_min , w*_max ) = 0;

(19) V₃(w*_max ; w*_min , w*_max ) = 0.

Following Dumas [1991], it can in turn be shown that these conditions imply

(18') V₁₁(w*_min ; w*_min , w*_max ) = 0

(19') V₁₁(w*_max ; w*_min , w*_max ) = 0.

Solving (17) subject to the conditions (14), (15), (18'), and (19') generates solutions for the optimal strategy parameters w*_min and w*_max , and for the function V(w; w*_min , w*_max ). In section IV below, we examine the nature of the optimal solution.

III. Turnover

Following Leland and Connor [1995], we may utilize a further technique to estimate the turnover associated with any particular policy. Consider the special case of V(w; w*_min , w*_max ) when 8 = 0. Call this function T(w; w*_min , w*_max ). Now this is the total discounted value of the transactions costs alone, since the penalty for diverging from w* has been set to zero. The value of T is the net present value of transactions costs forever, which can be viewed as the capitalized value of annual transactions costs. Therefore annual transactions costs are simply rT, and annual (one-way) turnover is rT/(k_S + k_B ). We set w = w*, and define T* = T(w*; w*_min , w*_max ). Our reported turnover results assume T = T*.

IV. Results

We first derive the optimal trading strategy for a base case, with target stock/bond mix equal to 60/40. Where appropriate, all base case rates are in annual units:

:_S - :_B = 3.6%

r = 7.5%

F_S = 20.0%

F_B = 10.0%

D = 0.3

w* = 1.5

8 = 0.35

k_S = 1.0% (one way)

k_B = 0.5% (one way)

The optimal policy for this set of parameters is

w_min = 1.421

w_max = 1.573

Therefore, the optimal policy puts a lower bound on the percentage of stock at 1.421/2.421 = 58.69%, and an upper bound on the percentage of stock at 1.573/2.573 = 61.14%. No trading should take place as long as the stock percentage remains between these bounds. If the ratio goes beyond the bounds, trading should occur such as to move the ratio to the

nearest boundary. Expected annual one-way turnover for this strategy is 8.95%, costing approximately 13.4 basis points. The average instantaneous standard deviation of tracking errors about w* = 0.0440. A one standard deviation upward move in w would bring the asset mix to 1.544/2.544 = 60.69%. Therefore the annual standard deviation of the tracking error of the stock percentage about 60% is approximately 0.69%.

We can compare this result with the common practice of rebalancing to w* at a constant time interval *t. Quarterly rebalancing, for example, sets *t = 0.25. Approximating the lognormal distribution with a normal distribution with zero drift gives an expected percent change in w over the period *t:

(20) E[|dw|/w] = (2/B)^0.5F_w(*t)^0.5.

In the case examined above, F_w² = b = 0.038 and w = 1.5. Over a quarter of a year, the expected change E[|*w| from (20) will be 0.1167. From (7), it can be shown that *S/W = *S/(B+S) = *w/(1+w)². Therefore adjusting *w = 0.1167 requires adjusting stock (as a percentage of wealth) by an amount equal to 0.1167/(1+1.5)² = 1.87%. Trading back to w* each quarter will therefore require an expected annual turnover of four times the quarterly amount, or 7.47%.

The average instantaneous variance of the tracking error (w(t) - w* ) over the quarter will be one-half of the end-quarter variance of F_w²w²*t, or 0.0107. Rebalancing to w* implies that each quarter starts anew with zero instantaneous variance. Therefore the average instantaneous variance over the year equals the average instantaneous variance over each quarter. Taking the square root gives an average instantaneous standard deviation of (w(t) - w*) of 0.1034. A one standard deviation upward move of w implies an asset mix of 1.6034/2.6034 = 61.59%, which in turn implies the standard deviation of the asset mix is about 1.59%. This is to be compared with the average instantaneous standard deviation of about 0.69% for the optimal policy.

Accuracy with quarterly rebalancing is less than the optimal policy, but turnover also is less. By setting 8 at a lower level, however, we can find an optimal strategy which has the same turnover as the quarterly rebalancing strategy. Consider decreasing 8 until the standard deviation of the tracking error of the optimal strategy is 0.1034, the same as quarterly rebalancing. By setting 8 = 0.0276, the standard deviation of w

about w* is .1034, and the standard deviation of the stock percentage is 1.59%--the same as with the quarterly rebalancing strategy. The strategy has a "no trade" interval about w* of [1.307, 1.663], implying no trade when the stock proportion remains between 56.7% and 62.5%. But for this strategy, expected turnover is only 3.76%. In sum, the same degree of tracking accuracy can be achieved with an optimal rebalancing strategy as with quarterly rebalancing to w*, with expected turnover only 3.76/7.47 = 50.3% of the quarterly rebalancing's expected turnover. Cost savings on trading of almost 50% are realized by using the optimal strategy.

V. Comparative Statics

(i) Transactions costs.

Doubling transactions costs increases the size of the no-trade interval to [1.400, 1.592]. Expected annual one-way turnover falls to 7.10%. Halving transactions costs reduces the no-trade interval to [1.438, 1.559], with one-way turnover 11.30%. To a close approximation, the size of the no-trade interval (w_max - w_min ) varies directly as the cube root of transactions costs. Turnover varies inversely with the cube root of transactions costs. These relationships hold for any given set of parameters, although the constant of proportionality clearly depends upon the chosen parameters. Since cost per unit trade is proportional to transactions costs, total trading cost will vary with transactions costs raised to the two-thirds (2/3) power.

(ii) Relative volatility.

Halving the relative variance (from .038 to .019) reduces the no-trade interval to [1.436, 1.557], and one-way turnover reduces to 5.66%. Doubling the relative volatility increases the no-trade interval to [1.402, 1.594] and one-way turnover to 14.2%. To a close approximation, optimal turnover varies with the variance b of the dw/w process raised to the two-thirds (2/3) power, or equivalently, with the cube root of standard deviation. The size of the no-trade interval is proportional to the cube root of the variance, or equivalently, is proportional to the standard deviation raised to the 2/3 power.

(iii) Relative return and the riskfree rate.

Doubling the expected return premium of stocks over bonds marginally reduces the optimal no-trade interval, although not to the second decimal place. Similarly, turnover is marginally affected. Changes in the riskless interest rate also have negligible effects.

(iv) The target stock/bond ratio w*.

Reducing w* to 1.0 (a 50/50 stock/bond ratio) gives a no-trade interval of [0.929, 1.064], and lowers annual one-way turnover to 7.00%. The standard deviation of the stock fraction is 0.96%. Compared with our base case, turnover is lower and accuracy is less. Increasing 8 will raise turnover and lower accuracy. When 8 = .725, turnover (8.95%) is identical, and both total cost V and turnover T at the new optimum are the same as when w* = 1.5. Thus V - T, the costs of the tracking error or "(in)accuracy costs", also remain unchanged. Therefore, with an appropriate adjustment in 8, a no-trade interval exists for each w* which keeps turnover and (in)accuracy costs invariant to w*. To a close approximation, the adjustment to 8 which leaves turnover and accuracy costs unchanged is proportional to (w*/(1+w*))⁴. Thus if w* moves from 1.5 to (say) 3, 8 must be reduced by a fraction F = [(1.5/2.5)⁴/(3/4)⁴] = (6/7.5)⁴ = .4096. Since the original 8 = 0.35, the adjusted 8 must equal 0.35*F = 0.1434. We conclude that it is always possible to change 8 so that turnover and (in)accuracy costs are invariant to changes in w*.

(v) The cost per unit of tracking error 8.

When parameters other than 8 remain fixed, to a close approximation turnover is proportional to the cube root of 8, and the no-trade interval is inversely proportional to the cube root of 8.

(vi) The tradeoff between turnover, no-trade interval, and tracking error.

We alter 8 and compute turnover and tracking error, keeping other parameters constant. Tracking error is measured by the annualized standard deviation of w. Table I shows this tradeoff, and also shows (w*_max - w*_min ), the size of the no-trade interval around w* = 1.5.

TABLE I

Cost/unit Tracking Error One-way Turnover Std.Dev.[w] No-Trade 8 Interval

______________________________________________________________________________

5.0 21.81% 1.81% .0627

2.5 17.30% 2.28% .0790

1.0 12.73% 3.10% .1072 0.5 10.09% 3.90% .1351

0.35 8.95% 4.40% .1522

0.25 7.99% 4.92% .1703

0.10 5.86% 6.70% .2314

______________________________________________________________________________

It can be verified that, to a close approximation:

(i) The expected turnover is inversely proportional to the width of the no-trade interval;

(ii) The standard deviation of tracking error is inversely proportional to

the expected turnover; ergo

(iii) The standard deviation of tracking error is proportional to the width of the

no-trade interval.

VI. Conclusion

We have shown that the optimal strategy for rebalancing requires the investor to determine a "no trade" interval about the target asset mix, w*. If the actual asset mix w lies within this interval, no trading is required. If the actual asset mix randomly moves outside the interval, rebalancing should take place--not to w*, but to the nearest boundary of the no-trade interval [w_min , w_max ]. We have derived explicit techniques for determining the optimal no-trade interval. It will depend upon the riskfree interest rate, the expected return differential between stocks and bonds, the relative volatility of stocks to bonds, the cost per unit of tracking error, and the transactions costs of stocks and bonds. In many cases, the no-trade interval changes with the cube root of the parametric changes.

We have also determined the expected annual turnover associated with the optimal strategy. Turnover also moves with the cube root of various parametric changes. For typical parameters, turnover following an optimal policy with equivalent accuracy is approximately one-half the turnover of a quarterly rebalancing to w*. Thus the gains to following the optimal strategy are substantial.

As expected, increases in the cost per unit of tracking error will tighten the no-trade interval, implying better accuracy of tracking w* through time, but higher turnover. We have shown that the accuracy of tracking (as measured by the annual standard deviation of w about w*) will be inversely proportional to annual turnover.

There are two important ways which this analysis can be generalized. Following the work of Dixit [1991], we can incorporate a fixed charge per transaction as well as a proportional trading cost. The optimal strategy will be defined in terms of two intervals, one inside the other. No trading will take place as long as asset proportions remain in the larger (outer) interval. Once asset proportions move beyond this outer interval, the asset proportion will be adjusted to the nearest boundary of the inside interval. A more difficult generalization will study the optimal trading strategy when there is a temporary shift in parameters (e.g. trading costs). Since now we will have a time-dependent optimization problem, numerical techniques will almost surely have to be used.

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