Wealth Strategies Journal - "What Trustees Should Know about Asset Management Approaches and Rebalancing Elections"

If the variability (uncertainty) of return is too great, even spectacular percentage rates of returns will, in the end, produce only modest wealth. Variance drain is easy to demonstrate mathematically. If an investor puts $1 into each of two investments with equal average return but different volatility, the end results will not match. If for, example, investment A earns –10% year one and +10% year two, its average return is 0% and its ending wealth is $1(.9)(1.1) = $0.99. If investment B earns –20% year one and +20% year two, its average return is also 0%. Its ending wealth, however, is $1(.8)(1.2) = $0.96. Absent cash flows, the order in which the returns are achieved is irrelevant. The important point is that the greater the variability in returns the lower the investor’s spendable wealth. However, trustees can distribute only dollar wealth; therefore the mathematics of variance drain has critical significance for trust beneficiaries.

When faced with the obligation of making periodic income distributions to the current beneficiaries, the order in which returns occur matters very much. When a portfolio makes periodic distributions, such distributions act as a multiplier of downside results ⁴² and as a cap on upside results. Following a period of negative return, the act of removing dollars from the portfolio means that fewer dollars remain to assist in the recovery of wealth; following a period of positive return, the act of removing dollars from the portfolio means that fewer dollars remain available to continue compound growth. Variance drain can erode portfolio wealth despite the fact that all investment risk/return parameters operate according to the money manager’s forecasts.

Here is a highly stylized example: we know (because multiplication is commutative) that 3*2*1 = 6; and that 1*2*3 = 6. The order of the ‘returns’ does not matter. This principle holds for any compound return series in which there are no cash flows. Consider, however, what happens when we add cash-flow requirements to the series:

The order in which returns are earned matters when there are cash flows. An average return “target” is no longer sufficient in the presence of portfolio distribution requirements and return variance. Cash flows change the definition of ‘required return,’ which, in turn, may change both the trustee’s asset allocation preferences and rebalance strategy. A key element in skilled asset management is the trustee’s ability to coordinate the portfolio’s asset allocation policies, rebalance strategy and distribution requirements. This suggests that trust investment policy should be ‘n’-dimensional rather than a one-dimensional hunt for attractive rates of investment returns.

If there is a benefit to variance reduction, so also there is a cost for forsaking the opportunity to rebalance. Erb and Harvey term the cost of not rebalancing ‘covariance drag:’ “…on average the portfolio weights of individual assets in an unrebalanced portfolio covary negatively with the returns of the individual assets. This results in a negative impact of not rebalancing.” ⁴³ The authors quantify the cost of not rebalancing for a hypothetical equally-weighted portfolio of 40 uncorrelated securities each of which have a zero expected risk premium and a standard deviation of thirty percent. They simulate 10,000 45-year return histories for each of the 40 securities, for a portfolio that is never rebalanced, and for an equally weighted portfolio that is rebalanced annually. On average, the individual securities generated an average annual return in excess of the risk free rate close to zero. The compound return of the rebalanced portfolio (4.3%) exceeded the compound return of the drifting portfolio (3.8%) by 50 basis points per year. The rebalanced portfolio generated more ending wealth in 71% of the simulations and had a better reward-to-risk tradeoff (Sharpe Ratio) in all 10,000 trials. In this simulation, the cost of not rebalancing (i.e., the covariance drag) equals 50 basis points per year.

The authors derive the following equation to estimate the diversification return of an equally weighted rebalanced portfolio of uncorrelated assets where each asset’s returns are not serially correlated: ⁴⁴

Diversification Return =

Where compound return increases with the number of securities in the portfolio (K); increases with the average standard deviation of the securities (s); and, increases as the average correlation of securities diminishes (r). They indicate, “when asset variances are high and correlations are low, the diversification return can be very high.” However, “…positive autocorrelation of returns might lower… diversification return.” ⁴⁵

Although rebalancing strategies are important tools to preserve the integrity of the trust’s investment policy, the algebra of diversification provides additional justifications for a trustee to consider the design and implementation of portfolio rebalance policy:

Mathematical Studies: Maximizing Expected Utility

Hayne Leland at the University of California, Berkeley, authored a seminal mathematical study of rebalance tradeoffs. ⁴⁶ Drawing upon earlier studies by George Constantindes, and B. Dumas & E. Luciano, ⁴⁷ Leland’s study argues that when rebalancing costs are proportional to the amount rebalanced, ⁴⁸ the optimal strategy involves:

Leland does not specify a particular utility function for the investor, but develops a cost function for deviations from the targeted asset allocation. If the target allocation is well synchronized to the investor’s wealth accumulation and consumption objectives, deviation from the target will result in a measure of “disutility” to the extent that it makes the attainment of future economic goals more uncertain. ⁴⁹

Leland’s model uses two asset classes (stocks = S and bonds = B) that follow log random walks (Brownian Motion). Therefore, according to the fundamental law of the evolution of wealth, the instantaneous rate of change for the stock position equals

and, for the bond position, equals

These equations state that the change in wealth follows a process dependent on three factors: the expected return (m = mean or average return); the expected standard deviation of return (s = standard deviation) and an adjustment to the expected standard deviation (Z = a random process) that is characterized by a zero mean and unit variance (i.e., a random draw from a standard log-normal distribution). The mean sets the general direction for the evolution of the vector of wealth over time, but the evolution of wealth is uncertain because the portfolio’s volatility generates a range of possibly negative and positive values for each vector component (period return).

The ratio of stocks to bonds will determine the future wealth of the investor. Thus, if we define wealth by the letter ‘w,’ the instantaneous rate of change in the investor’s wealth equals:

where
r = the correlation between stocks and bonds, and,
t = the applicable interval of time.

As stated, any difference between the target portfolio asset allocation (presumed to be optimal for the needs, goals and circumstances of the trust) and the actual portfolio will generate a loss of utility. Utility loss (L) is measured by the degree of tracking error [the squared deviations of the actual portfolio (w(t)) from the optimal portfolio (w*)], with the utility cost-per-unit-of-tracking-error measured by the parameter lamda (l):
Utility Loss = L = l(w(t)-w*)2dt.
Over the applicable planning horizon, the investor will want to minimize the discounted (present valued) integral (sum) of the cost of divergence (l) plus any trading costs associated with the rebalancing function. Leland’s model is “self-financing” in that any costs associated with rebalance trades are paid from without rather than from within the portfolio. This makes the mathematics more tractable because any change in the value of the stock allocation (dS) is offset exactly by a change in the value of the bond position with the change having the opposite sign (dS = -dB).

A brief review may be helpful. Leland posits a “no trade” region around the stock and bond positions because the costs of rebalancing small variations from the optimal portfolio may be greater than the cost in “disutility” by not maintaining the strict asset allocation targets of the optimal portfolio. Costs, therefore, involve two terms: a monetary term (k) measured by trading costs and a utility term (l) measured by the increased uncertainties of not adhering to the asset allocation target. The total cost function, therefore, includes k+l with k equal to $0.00 whenever the asset drift remains within the bounds of the no trade region (wmax, wmin). Putting it all together, the investor wishes to achieve the most favorable total cost function (V) over the integral of all periods within the applicable planning horizon:
V(w(t); wmax,wmin) =

This equation says that investor utility is maximized when the present value costs of tracking error plus the present value costs of rebalance transactions are minimized for any given level of wealth. ⁵⁰

After solving a complex differential equation, ⁵¹ Leland’s monograph continues by calculating the optimal rebalance strategy. According to the model, it is optimal to return the portfolio’s stock weighting to the edge of the no trade boundary rather than to the 60% allocation target; and Leland compares the costs and estimated turnover of this strategy with a quarterly calendar-based rebalance formula that maintains the exact allocation target. He estimates a cost savings of approximately 50% when the investor uses the optimal strategy. ⁵² Needless to say, if there are other costs (i.e., taxes), the width of the no-trade region expands: “In many cases, the no-trade interval changes with the cube root of the parametric changes.” ⁵³

⁴² A condition sometimes termed ‘feeding the bear.’ Weiss, Gerald R., “Dynamic Rebalancing,” Journal of Financial Planning (February, 2001). Weiss recommends a rebalance strategy that, during periods in which equity performs poorly, portfolio withdrawals are financed primarily from fixed income portfolio investments (no equity is sold to fund distributions); otherwise, withdrawals and portfolio rebalancing objectives are achieved by annually rebalancing to a target allocation. This dynamic strategy is compared to both a straightforward inflation-adjusted distribution policy and a “keep an emergency reserve” to buffer market volatility policy; and is found to be the superior alternative.

⁴³ Erb and Harvey, Supra, p. 60.

⁴⁴ Serial correlation is present when an asset’s returns are correlated with each other over successive time intervals. Serial correlation is also termed autocorrelation—the correlation of a security with itself.

⁴⁵ In a separate paper, Bernstein, William J., “The Rebalancing Bonus: Theory and Practice,” Efficient Frontier (1996) http://www.efficientfrontier.com argues: “only when long term return differences among assets exceed 5 percent do nonrebalanced portfolios provide superior returns, and then only at the cost of increased risks.” Bernstein derives the following formula for estimating the rebalancing benefit for a two asset portfolio:
X1X2(Var1/2 + Var2/2 – Covar1,2)
Where ‘X’ is the weight of the asset within the portfolio, ‘Var’ is the variance of returns; and ‘Covar’ is the covariance of returns. The Bernstein formula takes advantage of the well-known statistical identity: Covariance1,2 = (Standard Deviation 1)( Standard Deviation2)(Correlation1,2). Thus, the Bernstein formula has a term for variance, a term for correlation and a term for covariance drag.

⁴⁶Leland, Hayne E., “Optimal Asset Rebalancing In the Presence of Transaction Costs,” Working Paper No. RPF-261 Walter A. Haas School of Business Research Program in Finance Working Paper Series (August, 1996).

⁴⁷ Constantinides, George, “Capital Market Equilibrium with Transactions Costs,” Journal of Political Economy (1986), and Dumas, B., & Luciano, E., “An Exact Solution to a Dynamic Portfolio Choice Problem Under Transactions Costs,” Journal of Finance (1991), pp. 577-596.

⁴⁸Transactions costs can be fixed (i.e. irrespective of the amount bought and sold the investor has only a fixed charge) or proportional. If proportional costs are, for example, one percent, a trade involving $1,000 of assets will cost $10 while a trade involving $1,000,000 of assets will cost $10,000.

⁴⁹ It is important to note that “target” in this context can have several meanings. In terms of a fixed mix approach to asset management, target means the asset allocation established in the trust portfolio’s investment policy; in terms of an insured portfolio approach, target means the application of the multiplier to the portfolio’s equity surplus. In many studies, the term “rebalancing” refers to all portfolio changes required to reoptimize the portfolio to reflect changing trust objectives, economic circumstances, and investment forecasts. Rebalancing, according to this use of the term, is akin to ‘recalibrating’ the portfolio’s design to maximize utility over the forthcoming period. Deviations from the optimal portfolio (i.e., tracking error) create disutility.

⁵⁰ Leland’s argument also assumes that a passive portfolio management strategy (i.e. drifting mix) has a quantifiable cost that can be measured relative to the ideal asset allocation. Thus, rebalancing mitigates “tracking error.” This has proved to be a fruitful concept in much recent research. For example, one study notes: “What drives the benefit of rebalancing is reducing the tracking error from getting far off-target. As it happens, tracking error is quadratic. It’s proportional to the square of the deviation from the target allocation. For example, when a portfolio with a 30% target for U.S. bonds gets to 32% bonds (2% over target) the tracking error is four times as high as being 1% over target. And if bonds climb to 33% of the portfolio, the tracking error risk is nine times the risk associated with a 1% deviation.” Masters, Seth J., “Rules for Rebalancing,” Financial Planning (December, 2002), pp. 89-93.

⁵¹ Maximizing the first two derivatives of the total value function.

⁵² Although trading frequency may increase, Leland’s model assumes that trading costs are strictly proportional and not fixed. Additionally, the amounts traded will be very small. Small trades generate only small costs (unlike the fixed or step-rate trade commission schedule found at most brokerages) and all “costly trades” within the no-trade zone (i.e., trades that produce more trading costs than investor utility increases) are eliminated. The model’s predicted cost savings may not be attainable under most wealth administration platforms in today’s marketplace.

⁵³ Other studies indicate that the nature of the costs (i.e., fixed, proportional, or a combination thereof), determine whether the rebalance action should bring the asset weighting to the closest boundary of the no-trade region or to a point inside the region. Yip, Kenneth & Donohue, Christopher, “Optimal Portfolio Rebalancing with Transaction Costs,” The Journal of Portfolio Management (Summer, 2003), pp. 49-63 provides an insightful extension of the Leland two-asset class model to multi asset class portfolios. In addition to examining the nature of the no-trade-region in the face of changing transaction costs, risk aversion, and correlations, they simulate various rebalance strategies. They recommend using a no-trade-region approach to rebalancing; but conclude “…the complexity of optimal rebalancing quickly makes it intractable, thus limiting its application to simple portfolios….”