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Optimal Rebalancing Frequency for Bond-Stock Portfolios
David M. Smith, PhD, CFA
University at Albany, SUNY
School of Business
Department of Finance
Albany, NY 12222
ds693@albany.edu
(518) 442-4245
Fax: (518) 442-3045
William H. Desormeau, Jr., CFP®
Strategic Benefit Services
One Empire Drive
Rensselaer, NY 12144
wdesorme@hanys.org
(518) 431-7749
Fax: (518) 431-7601
September 10, 2006
David M. Smith, PhD, CFA, is Associate Professor of Finance and a research associate at the
Center for Institutional Investment Management, University at Albany, SUNY. William H.
Desormeau, Jr., CFP®, is Manager of Strategic Benefit Services, an affiliate of the Healthcare
Association of New York State and provider of employee benefit programs.
The authors are grateful to Bruce Geller for data acquisition, and to Peter Brucato, Indushobha
Chengalur-Smith, and Hany Shawky for helpful comments.
2
Optimal Rebalancing Frequency for Bond-Stock Portfolios
EXECUTIVE SUMMARY
Purpose
Examine the effect of investors’ portfolio rebalancing frequency and out-of-balance
threshold on scaled return (return/risk ratios).
Method
Using 1926-2003 data, examine the return/risk ratio for 19 different bond-stock
portfolios with policy weights of 5%-95%, 10%-90%, and so on through 95% bond-
5% stock. For each of these 19 portfolios, calculate scaled return (mean return
standard deviation of return) under 60 different time-based policies of rebalancing
every 1 month, 2 months, and so on through 60 months. Next, examine 20
threshold-based policies involving rebalancing when portfolio weights deviate from
policy weights by 0.5%, 1%, and so on through 10%. Finally, conduct both
analyses again under two distinct Federal Reserve monetary policies:
expansionary and contractionary.
Conclusions
Rebalancing frequency and threshold have a significant effect on scaled return.
Specific results:
o For many model portfolios examined, deferring rebalancing to even as long
as four years was superior to a monthly or quarterly rebalancing policy.
o For percentage threshold-based policies, the superior (inferior) outcomes
were associated with rebalancing only when portfolio weights were 5% or
more (less than 5%) out of balance.
The best rebalancing policy is dependent on, and can be planned around, the
Fed’s prevailing monetary policy.
3
Optimal Rebalancing Frequency for Bond-Stock Portfolios
1. Purpose
Changes in bond and stock prices necessitate periodic restoration of portfolios’ asset
allocations to their policy weights. This process helps to ensure that portfolios remain
compatible with the objectives and constraints of beneficiaries. Work by Leibowitz and
Hammond (2004) shows that professional investment managers tend to rebalance their
portfolios more frequently than do individual investors. The optimal rebalancing frequency and
threshold remain open questions for both types of investors. The purpose of this paper is to
examine empirically the return-and-risk characteristics of investment portfolios under various
rebalancing frequencies and out-of-balance thresholds.
Our study takes the following approach. Using large-capitalization U.S. stocks and U.S.
government bonds as representative asset classes, we examine the scaled returns for 19 model
portfolios between 1926-2003, for rebalancing frequencies ranging from one month to 60
months and thresholds ranging from 0.5% to 10%. We then re-examine the question for two
subintervals associated with tight versus expansionary monetary policy. In brief, the results
show that the common policy of frequent portfolio rebalancing is inferior to a policy
characterized by greater patience, and that the Fed’s prevailing monetary policy stance should
be taken into account in the rebalancing decision.
The next section summarizes the relevant literature. Section 3 discusses the data and
the methods used. We present our results in Section 4, and Section 5 contains the conclusion.
2. Literature Review
Many extant studies examine rebalancing frequency for relatively long periods, an
extensive range of bond-stock weightings, or taking into account the prevailing stock market or
4
monetary policy environment. To date, none has combined all of these features and also
examined both time-based and threshold-based rebalancing policies.
Ferguson [1986] demonstrates theoretically and empirically that investors can manage
risk effectively by aggressively altering the exposure for two asset classes – one risk-free and
the other risky. Ferguson proposes a Dynamic Asset Allocation (DAA) program, which involves
manipulating portfolios consisting of one risk-free and one risky investment type: U.S. Treasury
Bills and securities tracking the S&P 500 Index.
The initial allocation to the two asset classes is determined by the investor’s minimum
acceptable expected return in any given year. Ferguson tests his hypothesis of using only two
asset classes to manage risk by replacing the S&P 500 Index security with a security leveraged
to 150% of the underlying Index. The DAA program functions as follows. Over time, DAA will
systematically increase exposure to the asset class with superior performance. Using data from
1928 through 1983, Ferguson concludes that the DAA strategy results in a beneficial blend of
market upside capture and market downside protection. Ferguson’s work focuses on tactical
asset allocation; that is, his policy requires frequent portfolio adjustments in response to market
trends.
Perold and Sharpe [1988] examine dynamic strategies for rebalancing portfolios in
response to the tendency of risky assets to increase in value relative to less risky assets, over
time. They define a constant-mix strategy as one that buys risky assets (stocks) as they fall in
price (and in relative value within the portfolio), and sells them as they rise in price. Perold and
Sharpe observe that the popularity of a dynamic strategy tends to eliminate its effectiveness.
Specifically, market forces will remove the advantage of a dynamic strategy practiced by more
than a minority of market participants.
Perold and Sharpe find that a constant-mix strategy would outperform a buy-and-hold
strategy in a volatile market, without sustained moves either up or down. The greater the
volatility, the greater is the advantage for a constant-mix strategy. The constant-mix strategy will
5
underperform the buy-and-hold approach when there are no reversals in market direction. That
is, it will underperform during extended bull or extended bear markets. This is a pioneering work
in showing that certain market patterns and volatility render some dynamic strategies more
effective than others. Nonetheless, some of the strategies cannot easily be applied because
investors often fail to realize until afterward whether they are in a bull or bear market.
Constant-proportion portfolio insurance and option-based portfolio insurance are two
strategies that sell stocks as their prices fall, and buy stocks as their prices rise. In effect, they
take the opposite actions of the constant-mix strategy. These two forms of portfolio insurance
will outperform the buy-and-hold approach during sustained market moves, either up or down.
They underperform the buy-and-hold approach in trendless, volatile markets.
Using market data from 1963 through 1988, Dennis, Perfect, Snow and Wiles [1995]
examine the effects of rebalancing on portfolios that conform to rigid quantitative criteria.
Screening firms using an approach similar to that of Fama and French [1992], they restrict their
portfolios to small capitalization firms exhibiting a high Book-to-Market-Equity (BE/ME) ratio.
They find that the rebalancing interval producing the highest return was two years. They
observe that their optimal portfolios were well diversified as indicated by the large number of
Standard Industrial Classification (SIC) codes represented.
Horvitz [2002] promotes the idea that rebalancing costs are substantial, and fall into two
broad categories: trading costs and tax costs. He argues that even institutional investors tend to
underestimate trading costs. Trading costs are not limited to brokerage commissions, but also
include execution inefficiency (slippage) and opportunity costs. Moreover, the tax effects of
portfolio turnover are complicated by “the 12-month hurdle,” which determines whether a gain is
treated as short-term or long-term for tax purposes. The federal tax rate for long-term gains is
considerably more favorable for investors than the rate for short-term gains.
Horvitz concludes that in light of rebalancing costs, asset classes move materially off
their targets less frequently than many investors realize. Therefore, rebalancing in taxable
6
portfolios can safely be avoided over short time periods. The rebalancing of minor asset classes
(those comprising 10% or less of the portfolio) may not contribute meaningfully to the desired
risk reduction, and hence may not be warranted. Although the present study does not take
trading costs and taxes into account explicitly, Horvitz makes clear that frequent rebalancing
strategies carry a built-in disadvantage, which must be kept in mind as our results are
interpreted.
Lynch and Balduzzi [2000] find that the predictability of returns has an effect on the
rebalancing behavior of investors. When returns are predictable, investors rebalance more often
and bear higher transaction costs. Lynch and Balduzzi show that rebalancing frequency
decreases when transaction costs rise beyond a level deemed acceptable by the investor. They
also find that, under circumstances where the predictive variable is less persistent (reliable), the
investor is less likely to rebalance even when confronted with an extreme value.
Tsai [2001] examines the effect of various rebalancing strategies on five model
portfolios, each representing a range of risk profiles and consisting of between four and seven
asset classes. She compares the return and risk profiles of the portfolios from following a
strategy of never rebalancing versus four active alternatives, all between 1986-2000. The four
alternatives are rebalancing monthly, rebalancing quarterly, and rebalancing monthly or
quarterly only if any asset class drifts by more than 5% from its policy weight. A buy-and-hold
strategy (never rebalancing) produces the lowest risk-adjusted returns, as measured by Sharpe
ratio, for all five portfolios tested. For the other four active rebalancing strategies tested Tsai
found small (insignificant) variations in performance and risk. No one strategy outperformed on
a risk-adjusted basis across all the portfolios. Tsai concludes that portfolios should be
rebalanced because the small additional return associated with not rebalancing fails to
compensate for the significantly increased risk. Our study follows in the spirit of Tsai’s, but it
extends the investment period to 78 years, accounts for 19 portfolio weightings, and considers
alternative monetary policy environments.
7
Weiss [2001] examines the portfolio management needs of the retiree who needs to
earn a reasonable return and manage risk while taking systematic withdrawals for living
expenses. He uses Monte Carlo Simulations to test annual and dynamic rebalancing strategies.
His hypothesis is that the dynamic rebalancing strategy could either add portfolio lifespan for a
given initial withdrawal rate, or increase the portfolio’s value for a target longevity.
Weiss concludes that the dynamic rebalancing strategy is superior to annual rebalancing
because investors do not need to sell equities during market downturns. Instead, they can meet
their withdrawal requirements from the fixed income or cash portions of their portfolio and “ride
out” the period of low equity valuation.
Jensen and Mercer [2003] show the return-to-risk ratios of Markowitz-efficient portfolios
are improved by basing tactical asset allocation adjustments on turning points in the business
cycle. However, investors rarely detect such turning points as they occur, making such an
investment strategy difficult to apply. Jensen and Mercer propose using changes in the
monetary cycle instead of changes in the business cycle as the trigger point for asset
reallocation because monetary policy changes can be readily identified ex-ante.
They find that an asset allocation approach based on the monetary cycle resulted in
significantly improved risk adjusted returns (after transaction costs) over both the business cycle
and the buy-and-hold approaches. Most of the improvement in returns occurs when the Open
Market Committee of the Federal Reserve Board follows a restrictive monetary policy (40% of
the time), during which time the return-to-risk ratio improves by 82%.
3. Methods
The data for this study are derived from Ibbotson’s SBBI Yearbook, 2003. We use
monthly returns on the S&P 500 with dividends reinvested, and on the U.S. long-term
government bond as reported in the Yearbook.
8
We construct 19 fixed-weight portfolios, from 5% bonds-95% stock to 95% bonds-5%
stock, at 5% intervals, and examine the scaled return for each portfolio. Scaled return is
calculated as the ratio of monthly mean return to the standard deviation of monthly returns.
Given investors’ demonstrated preference for returns and aversion to risk, we assume that all
else equal, they will prefer to maximize scaled return. Our measure is similar to the well-known
Sharpe ratio.
The analysis consists of three main parts. First, we examine scaled returns for the 19
fixed-weight portfolios under varying rebalancing frequencies during the entire 1926-2003 period
and subperiods. In each case, we allow the portfolios’ values to reflect the market’s returns in
unconstrained fashion, until rebalancing takes place on a fixed schedule. At that time, each
portfolio’s weights are returned to the fixed weights appropriate for that portfolio. For example, a
40-60% bond-stock portfolio with a rebalancing frequency of every 60 months is likely to
experience large deviations from its model portfolio weights, particularly if the stock and bond
markets move very differently from one another in the interim. Only at month 60 will the bond
and stock components be adjusted to return the portfolio (temporarily) to its policy weights.
For each of the 19 portfolios, we observe over the 78-year period the scaled return
under 60 separate rebalancing policies (from rebalancing every month to rebalancing every 60
months). We then examine whether an optimum frequency exists, by running the following
regression for each of the 19 portfolios:
p
2
pp2pp1pp ~
Freq
ˆ
Freq
ˆ
ˆ
Return Scaled
, (1)
where Scaled Returnp is the ratio of mean monthly return to monthly return standard deviation
for portfolio p (e.g., 5% bond-95% stock), Freqp is the frequency (in months, ranging from 1-60)
with which portfolio p is rebalanced. If an optimum or optimal range exists, we expect to observe
a positive β1 coefficient estimate and a negative β2 coefficient estimate. In equation 1, by taking
the first derivative of scaled return with respect to Freq and setting the result equal to zero
9
(which yields Freq = 1/22) and substituting in the estimates for 1 and 2, we obtain a point
estimate for the optimal rebalancing frequency.
The second part of our analysis examines the same 19 portfolios, but sets aside the time
factor. Instead, we consider the effects of rebalancing when a portfolio’s market weights deviate
from its policy weights by some threshold percentage. For example, whenever the 40-60%
bond-stock portfolio’s weights deviate by some threshold amount from 40-60, the portfolio
weights are restored to the policy weights. A “5% threshold” would require rebalancing if the
bond weights in the portfolio were less than 35% or greater than 45%. The thresholds we
consider are 0.5% through 10%, at 0.5% intervals. Once again, we are searching for the
rebalancing policy resulting in the highest level of scaled returns.
The third part of our analysis considers the possible effects of Federal Reserve
monetary policy on the optimal rebalancing frequency and threshold levels. It has been well
established that Fed policy is highly influential in the financial markets, and we seek to
determine if that importance extends to the portfolio rebalancing decision. We identify the 32
points, from 1926-2003, when the Fed switched the direction of change in the discount rate. For
example, one of the 16 contractionary monetary policy periods began on August 24, 1999, when
the Fed increased the discount rate from 4.5% to 4.75%. This increase followed an
expansionary period characterized by three discount rate cuts, the first of which was on January
31, 1996, and immediately precedes an expansionary period in which the first cut was on
January 3, 2001. Following in the spirit of the Jensen and Mercer (2003) study, we define the
start of a period of expansionary (contractionary) monetary policy as the beginning of the month
following the first decrease (increase) in the discount rate. We apply the conservative
assumption that investors could rebalance by month’s end in recognition of the new regime, so
the August 24, 1999 rate increase signals the start of a new contractionary period whose returns
measurement begins at the close of September 1999 and ends in January 2001.
< Insert Exhibit 1 about here >
10
4. Findings
Exhibit 1 shows the maximum and minimum scaled returns resulting from various
rebalancing policies, along with the time interval associated with maxima and minima.
Regardless of the policy weights, the maximum scaled return is achieved at a rebalancing
frequency of every 44 months, and the minimum at 1 month frequencies. We generally observe
that longer intervals between rebalancing dominate shorter intervals. Between 1926-2003, the
five best rebalancing intervals are found in the 39-44 month range. The lowest scaled returns
tend to appear for more frequent rebalancing (1-6 months). In the 50-50% through 95-5% bond-
stock portfolios, the results are unambiguous, while portfolios with more than 50% stock perform
relatively less well under a 30-36 month rebalancing policy. Exhibit 1 also shows that the scaled
return range is above 0.145 for portfolios with bond-stock weights between 60-40% and 25-
75%, suggesting that the rebalancing frequency decision matters most for these portfolios.
< Insert Exhibit 2 about here >
Exhibit 2 provides the same information graphically. Each change of shade represents
an increment of 0.04 in the scaled return (mean/standard deviation) ratio. The graphs in Exhibit
3 show the rebalancing frequency/scaled return relationships for six distinct 13-year periods.
Among the notable conclusions from these graphs is that portfolios heavy in bonds
outperformed stock-heavy portfolios in five of the six subperiods. Moreover, for most model
portfolios, a longer-term rebalancing interval outperformed monthly or quarterly rebalancing in
four out of six subperiods. The exceptions were 1965-1977 and 1978-1990.
< Insert Exhibit 3 about here >
Exhibit 4 shows minimum and maximum scaled returns, for 1926-2003, for the nineteen
portfolios under various rebalancing threshold policies. In general, less-restrictive thresholds
(i.e., those allowing higher deviations) perform better than more-restrictive thresholds. Of the 95
situations in which policies are among the five best (19 portfolio weightings × 5 best for each),
only seven involve thresholds below 5%. Of the 95 situations in which policies are among the
11
five worst, only 22 involve thresholds above 5%. It is interesting to note that the ranges between
maximum and minimum scaled returns are lower in Exhibit 4 than in Exhibit 1, suggesting that
returns are ultimately less sensitive to threshold-based rebalancing decisions than to frequency-
based rebalancing decisions. Exhibit 5 contains a graph of results for the entire 1926-2003
period.
< Insert Exhibits 4 and 5 about here >
Exhibit 6 shows the frequency of rebalancing necessitated by following various threshold
policies during the 1926-2003 period. Not surprisingly, low-threshold policies produce in excess
of 600 rebalancing events, and this situation is particularly acute for portfolios nearly equally
divided between bonds and stocks. Less-equally balanced portfolios have only a fraction as
many rebalancings, but the disparity among the portfolios diminishes as the threshold rate rises.
This graph is particularly instructive as investors attempt to keep in mind the costs and fees
associated with trading.
< Insert Exhibit 6 about here >
Exhibit 7 shows the relationship between scaled returns and rebalancing threshold under
expansionary and restrictive monetary policy periods. The exhibit contains maximum and
minimum scaled returns, along with the threshold levels associated with those maxima and
minima. For restrictive monetary periods, all 19 portfolios benefited from a more patient policy
involving a higher rebalancing threshold. The lowest-level threshold of 0.5% almost always
generated the minimum scaled return. In expansionary monetary periods, more patient policies
produced a higher scaled return for 15 out of 19 portfolios. For stock-heavy portfolios during
expansionary times, it is less clear that low-threshold policies are inferior. Indeed, threshold
policy does not appear to matter greatly for stock-heavy portfolios, as the range of scaled
returns is quite small.
< Insert Exhibit 7 about here >
12
Exhibit 8 shows optimal rebalancing frequencies based on regression estimates and
90% confidence limits calculated from equation 1, for periods of contractionary monetary policy.
The regression estimates are statistically significant at the 10% level for all except the 85-15%,
90-10%, and 05-95% bond-stock portfolios. As expected, among the models showing
significance, β1 is positive and β2 is negative. Optimal frequencies are calculated in the range
between 32 and 37 months, with 90% confidence limits ranging between 11-100 months for the
lower-stock portfolios and 21-50 months for the higher-stock portfolios. With such a wide band,
the results do not indicate definitively what the optimal rebalancing strategy is. However, the
Exhibit 8 does provide guidance on strategies that do not work optimally, namely rebalancing
more frequently than every 10 months in the case of lower-stock portfolios and every 20 months
in the case of higher-stock portfolios.
< Insert Exhibit 8 about here >
The totality of our evidence suggests that during most of the past century of market
history, following a quick-trigger, mechanistic rebalancing approach would have been much less
profitable than a more patient approach. This result applies to a wide array of portfolio
allocations and both expansionary and contractionary monetary policy regimes. Although
financial advisors may believe that clients need to see them as “taking action” on a frequent
basis in order to justify their continued engagement, our study suggests that the frequent activity
should generally involve something other than portfolio rebalancing.
5. Conclusions
Rebalancing frequency and threshold level are associated with significant differences in
portfolio scaled returns. We show that this is true across a wide range of policy weights. From
the perspective of both frequency and threshold levels, patient rebalancing policies tend to
dominate quick-trigger policies, even before trading costs and taxes are considered. If such
13
costs were taken into account, the advantage in favor of patient policies would be even more
dramatic.
We find that Federal Reserve monetary policy has a discernible impact on scaled returns
due to rebalancing, with restrictive monetary periods associated with less ambiguous
conclusions. During restrictive periods, rebalancing more frequently than every 10-20 months is
a suboptimal strategy.
Overall, our findings over a 78-year period are consistent with important conclusions of
Dennis, et al (1995) and Horvitz (2002). Our basic approach is somewhat in conflict with that of
Ferguson (1986), whose DAA strategy would increase exposure to the better performing of two
asset classes, while our rebalancing toward a model portfolio would by definition reduce
exposure to the better performing asset class.
A question arises about why a relatively long time interval (and higher threshold) for
rebalancing outperforms a shorter time period (and lower threshold). One, partial, explanation
comes from the observation by various researchers including Poterba and Summers (1988) and
Fama and French (1988), of positive short-term autocorrelation among stock returns and
negative longer-term autocorrelation. To the extent that returns are positively correlated in the
short run, investors can take advantage of momentum by sitting tight. In contrast, mean
reversion in returns over 3-5 years suggests a policy of rebalancing about that often.
14
References
Dennis, Patrick; Steven B. Perfect, Karl N. Snow, and Kenneth W. Wiles. 1995. “The Effects of
Rebalancing on Size and Book-to-Market Ratio Portfolio Returns.” Financial Analysts
Journal, Vol. 51 No. 3 (May/June), 47-57.
Fama, Eugene F., and Kenneth R. French. 1988. “Permanent and Temporary Components of
Stock Prices.” Journal of Political Economy, Vol. 96, 246-273.
Fama, Eugene F., and Kenneth R. French. 1992. “The Cross-Section of Expected Stock
Returns.” Journal of Finance, Vol. 47 No. 2 (June), 427-465.
Ferguson, Robert. 1986. “How to Beat the S&P 500 (without losing sleep).” Financial Analysts
Journal, Vol. 42 No. 2 (Mar-Apr), 37-46.
Leibowitz, Martin L., and Brett P. Hammond. 2004. “The Changing Mosaic of Investment
Patterns.” Journal of Portfolio Management, Vol. 30 No. 3 (Spring), 10-25.
Horvitz, Jeffrey E. 2002. “The Implications of Rebalancing the Investment Portfolio for the
Taxable Investor.” Journal of Wealth Management, Vol. 5 No. 2, (Fall), 49-53.
Jensen, Gerald R., and Jeffery M. Mercer. 2003. “New Evidence on Optimal Asset Allocation.”
Financial Review, Vol. 38 No. 3 (August), 435-54.
Lynch, Anthony W., and Pierluigi Balduzzi. 2000. “Predictability and Transaction Costs: The
Impact on Rebalancing Rules and Behavior.” Journal of Finance, Vol. 55 No. 5
(October), 2285-2309.
Perold, Andre F., and William F. Sharpe. 1988. “Dynamic Strategies for Asset Allocation.”
Financial Analysts Journal, Vol. 44 No. 1 (Jan-Feb), 16-27.
Poterba. James M., and Lawrence H. Summers. 1988. “Mean Reversion in Stock Prices:
Evidence and Implications.” Journal of Financial Economics, Vol. 22, 27-59.
Tsai, Cindy Sin-Yi. 2001. “Rebalancing Diversified Portfolios of Various Risk Profiles.” Journal of
Financial Planning, Vol. 14 No. 10 (October), 104-110.
Weiss, Gerald R. 2001. “Dynamic Rebalancing.” Journal of Financial Planning, Vol. 14 No. 2
(February), 100-106.
Exhibit 1
Maximum and minimum scaled returns resulting from rebalancing at predetermined time intervals, 1926-2003
This exhibit shows portfolio monthly mean return scaled by its standard deviation for various portfolio policy weights. Starting with the second
column are the maximum scaled return obtained from any rebalancing frequency, followed by minimum values, range (maximum minus
minimum), and the time intervals that yield the five highest and five lowest scaled returns. Results are shown for the 78-year period 1926-2003.
Bond-Stock
Scaled return
Time interval (months)
Max
Min
Range
Max
2nd
best
3rd
best
4th
best
5th
best
Min
2nd
worst
3rd
worst
4th
worst
5th
worst
05-95
0.585
0.555
0.053
44
45
43
42
52
1
2
3
4
6
10-90
0.627
0.574
0.090
44
45
43
42
52
1
2
3
4
6
15-85
0.666
0.594
0.117
44
45
43
42
52
1
2
3
4
6
20-80
0.702
0.615
0.136
44
45
43
42
52
1
2
3
4
6
25-75
0.738
0.637
0.150
44
45
43
42
52
1
2
3
6
4
30-70
0.772
0.661
0.160
44
45
43
42
52
1
2
3
6
4
35-65
0.806
0.686
0.166
44
45
43
42
52
1
2
3
6
4
40-60
0.838
0.711
0.169
44
45
42
43
46
1
2
3
6
4
45-55
0.869
0.738
0.168
44
45
42
43
40
1
2
3
6
4
50-50
0.898
0.765
0.164
44
45
42
43
40
1
2
3
6
4
55-45
0.923
0.792
0.158
44
45
42
43
40
1
2
6
3
38
60-40
0.944
0.817
0.148
44
45
42
43
40
1
2
6
3
38
65-35
0.957
0.839
0.135
44
45
42
43
40
1
2
6
38
3
70-30
0.961
0.855
0.120
44
45
42
43
40
1
2
6
38
4
75-25
0.953
0.862
0.102
44
45
42
43
40
1
2
6
4
36
80-20
0.930
0.856
0.083
44
42
45
43
40
1
2
4
6
36
85-15
0.891
0.836
0.063
44
42
43
45
40
1
2
4
5
8
90-10
0.836
0.801
0.043
44
42
43
40
45
1
2
4
8
5
95-05
0.767
0.750
0.014
44
42
43
39
40
1
8
30
4
5
16
Exhibit 2
Graph of scaled returns for portfolio policy weights and rebalancing frequencies, 1926-2003
Graphs of portfolio monthly mean return scaled by its standard deviation, for portfolios of various bond-stock policy weights under alternative
rebalancing frequencies during 1926-2003.
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 95-05
85-15
75-25
65-35
55-45
45-55
35-65
25-75
15-85
05-95
Rebalancing frequency (months)
Bond-stock
policy weights
Value: 0.54-0.58
Value: 0.94-0.98
Exhibit 3
Scaled returns for portfolio policy weights and rebalancing frequencies
Graphs of portfolio monthly mean return scaled by its standard deviation, for portfolios of various bond-stock policy weights under alternative
rebalancing frequencies during 1926-2003. Panel A shows 1926-1938, while Panels B through F show 1939-1951, 1952-1964, 1965-1977, 1978-
1990, and 1991-2003, respectively.
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 95-05
85-15
75-25
65-35
55-45
45-55
35-65
25-75
15-85
05-95
Rebalancing Frequency (months)
Bond-stock
policy weights
Panel A: 1926-1938
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 95-05
85-15
75-25
65-35
55-45
45-55
35-65
25-75
15-85
05-95
Rebalancing Frequency (months)
Bond-stock
policy weights
Panel D: 1965-1977
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 95-05
85-15
75-25
65-35
55-45
45-55
35-65
25-75
15-85
05-95
Rebalancing Frequency (months)
Bond-stock
policy weights
Panel B: 1939-1951
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 95-05
85-15
75-25
65-35
55-45
45-55
35-65
25-75
15-85
05-95
Rebalancing Frequency (months)
Bond-stock
policy weights
Panel E: 1978-1990
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 95-05
85-15
75-25
65-35
55-45
45-55
35-65
25-75
15-85
05-95
Rebalancing Frequency (months)
Bond-stock
policy weights
Panel C: 1952-1964
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 95-05
85-15
75-25
65-35
55-45
45-55
35-65
25-75
15-85
05-95
Rebalancing Frequency (months)
Bond-stock
policy weights
Panel F: 1991-2003
18
Exhibit 4
Minimum and maximum scaled returns from various rebalancing threshold (%) levels, 1926-2003
When the portfolio’s actual bond-stock weights deviate from the portfolio’s bond-stock policy weights by a threshold number of percentage points,
the portfolio is rebalanced so the actual weights equal the policy weights. This exhibit shows the monthly portfolio mean return scaled by its standard
deviation, for the threshold policies resulting in the highest and lowest scaled returns, along with the range (Maximum minus Minimum) of scaled
returns and the threshold amounts that yield the maximum and minimum scaled returns. Results are shown for the 78-year period 1926-2003.
Bond-Stock
Scaled return
Threshold (%)
Max
Min
Range
Max
2nd
best
3rd
best
4th
best
5th
best
Min
2nd
worst
3rd
worst
4th
worst
5th
worst
05-95*
0.557
0.550
0.007
2.5%
1.5%
0.5%
1.0%
2.0%
3.5%
5.0%
4.5%
3.0%
4.0%
10-90
0.582
0.563
0.019
5.5%
5.0%
3.0%
6.5%
2.5%
8.0%
7.0%
10.0%
7.5%
9.5%
15-85
0.602
0.592
0.010
8.0%
8.5%
4.0%
4.5%
7.5%
9.0%
10.0%
2.5%
6.0%
5.5%
20-80
0.623
0.611
0.012
5.5%
5.0%
10.0%
2.0%
7.0%
6.5%
8.5%
3.0%
3.5%
0.5%
25-75
0.648
0.630
0.019
6.0%
7.5%
9.0%
7.0%
6.5%
10.0%
3.5%
8.5%
0.5%
4.0%
30-70
0.678
0.660
0.018
10.0%
8.5%
7.5%
8.0%
6.5%
9.0%
0.5%
1.0%
4.0%
1.5%
35-65
0.704
0.684
0.020
9.0%
9.5%
7.5%
8.0%
7.0%
10.0%
4.0%
0.5%
1.0%
6.5%
40-60
0.731
0.711
0.020
8.5%
10.0%
8.0%
9.5%
6.0%
0.5%
1.5%
5.5%
1.0%
4.5%
45-55
0.761
0.738
0.024
9.5%
8.5%
10.0%
7.0%
9.0%
0.5%
1.5%
1.0%
5.0%
5.5%
50-50
0.791
0.764
0.027
9.0%
10.0%
6.5%
8.0%
6.0%
0.5%
4.5%
1.5%
1.0%
5.0%
55-45
0.813
0.791
0.022
10.0%
9.0%
6.0%
8.5%
8.0%
0.5%
1.5%
1.0%
4.5%
5.0%
60-40
0.843
0.817
0.026
6.5%
10.0%
9.5%
7.0%
9.0%
0.5%
1.5%
1.0%
2.0%
2.5%
65-35
0.876
0.838
0.037
9.5%
10.0%
8.5%
8.0%
6.0%
0.5%
1.0%
4.0%
1.5%
4.5%
70-30
0.884
0.855
0.029
9.5%
10.0%
6.0%
7.0%
8.0%
0.5%
1.0%
4.5%
4.0%
1.5%
75-25
0.897
0.860
0.037
9.5%
9.0%
8.5%
5.0%
8.0%
4.0%
0.5%
3.5%
1.0%
1.5%
80-20
0.901
0.857
0.044
10.0%
9.5%
8.5%
7.5%
5.5%
3.0%
0.5%
1.0%
1.5%
2.5%
85-15
0.871
0.838
0.033
8.5%
9.5%
10.0%
8.0%
5.5%
0.5%
1.0%
1.5%
2.5%
2.0%
90-10
0.828
0.801
0.026
10.0%
7.5%
8.0%
7.0%
8.5%
0.5%
1.5%
1.0%
3.0%
5.5%
95-05*
0.767
0.752
0.015
0.5%
1.0%
1.5%
2.0%
2.5%
0.5%
1.0%
1.5%
2.5%
3.0%
* Note that for the most extreme allocations (05-95 and 95-05), rebalancing using thresholds above 5% is infeasible.
Exhibit 5
Scaled portfolio returns for various rebalancing thresholds
This exhibit shows a graph of portfolio monthly mean return scaled by its standard deviation, for portfolios
of various bond-stock policy weights under alternative rebalancing thresholds 1926-2003.
0.5% 1.5% 2.5% 3.5% 4.5% 5.5% 6.5% 7.5% 8.5% 9.5% 95-05
90-10
85-15
80-20
75-25
70-30
65-35
60-40
55-45
50-50
45-55
40-60
35-65
30-70
25-75
20-80
15-85
10-90
05-95
Rebalancing Threshold (%)
Bond-Stock
Policy Weights
Value: 0.90-0.92
Value: 0.54-0.56
20
Exhibit 6
Frequency of rebalancing to bond-stock portfolio weights under varying threshold policies, 1926-2003
05-95
25-75
45-55
65-35
85-15
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
4.5%
5.0%
5.5%
6.0%
6.5%
7.0%
7.5%
8.0%
8.5%
9.0%
9.5%
10.0%
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
475
500
525
550
575
600
625
Frequency between
1926-2003
Bond-Stock
Weights
Rebalancing Threshold
21
Exhibit 7
Scaled returns and associated rebalancing threshold points for various portfolio policy weights for periods of expansionary
and restrictive monetary policy, 1926-2003
When the portfolio’s actual bond-stock weights deviate from the portfolio’s bond-stock policy weights by a threshold number of percentage points,
the portfolio is rebalanced so the actual weights equal the policy weights. This exhibit shows the monthly portfolio mean return scaled by its
standard deviation, for the threshold policies resulting in the highest and lowest scaled returns, along with the range (Maximum-Minimum) of scaled
returns and the threshold amounts that yield the maximum and minimum scaled returns. Results are shown for expansionary and contractionary
monetary periods from 1926-2003. Expansionary (contractionary) periods are defined to start the month after the Fed cuts (raises) the discount rate
for the first time following one or more increases (decreases) in the discount rate, and ending in the month of the next change in direction.
Expansionary monetary policy (512 months)
Restrictive monetary policy (424 months)
Bond-Stock
Max
Min
Range
ThresholdMax
ThresholdMin
Max
Min
Range
ThresholdMax
ThresholdMin
05-95
0.709
0.703
0.006
4.0%
2.0%
0.403
0.398
0.005
6.5%
1.5%
10-90
0.733
0.722
0.011
7.0%
1.5%
0.425
0.420
0.008
4.5%
0.5%
15-85
0.758
0.742
0.016
9.5%
5.5%
0.449
0.437
0.012
6.5%
0.5%
20-80
0.779
0.763
0.016
4.5%
7.0%
0.473
0.457
0.016
8.0%
0.5%
25-75
0.800
0.785
0.015
6.0%
8.5%
0.499
0.480
0.019
9.5%
0.5%
30-70
0.825
0.807
0.019
6.5%
3.5%
0.518
0.504
0.014
10.0%
0.5%
35-65
0.853
0.831
0.022
6.0%
4.0%
0.545
0.529
0.016
8.0%
0.5%
40-60
0.881
0.858
0.022
6.5%
0.5%
0.575
0.556
0.020
7.0%
0.5%
45-55
0.906
0.883
0.023
6.5%
0.5%
0.607
0.584
0.023
7.5%
0.5%
50-50
0.932
0.908
0.025
7.5%
0.5%
0.643
0.613
0.029
9.0%
0.5%
55-45
0.961
0.933
0.030
9.0%
0.5%
0.676
0.643
0.033
7.5%
0.5%
60-40
0.988
0.952
0.036
9.0%
0.5%
0.708
0.674
0.035
7.5%
0.5%
65-35
1.014
0.967
0.048
9.5%
0.5%
0.737
0.702
0.035
7.0%
0.5%
70-30
1.017
0.974
0.042
8.5%
0.5%
0.762
0.728
0.035
6.5%
0.5%
75-25
1.028
0.969
0.059
9.5%
0.5%
0.789
0.747
0.042
10.0%
0.5%
80-20
0.998
0.950
0.049
8.0%
0.5%
0.796
0.757
0.039
10.0%
0.5%
85-15
0.951
0.913
0.038
6.5%
8.0%
0.788
0.755
0.033
10.0%
0.5%
90-10
0.888
0.859
0.025
4.5%
0.5%
0.760
0.739
0.021
8.0%
0.5%
95-05
0.796
0.791
0.005
2.0%
2.5%
0.718
0.708
0.009
2.5%
0.5%
22
Exhibit 8
Rebalancing frequency confidence intervals for bond-stock portfolios during restrictive monetary policy periods
Optimal rebalancing frequencies based on regression estimates and 90% confidence limits calculated from the following model:
p
2
pp2pp1pp ~
Freq
ˆ
Freq
ˆ
ˆ
Return Scaled
, where ScaledReturnp is the monthly mean return for portfolio p divided by its standard deviation,
and Freqp is the frequency (in months, ranging from 1-60) with which portfolio p is rebalanced. The estimate of the optimum is calculated as Freq =
-β1/2β2. Data are for the 16 periods between 1926-2003 in which the Federal Reserve raised the discount rate. Regression coefficients for the 85-
15%, 90-10%, and 95-05% bond-stock portfolios were statistically insignificant at the 10% level.
Exhibit 6:
Rebalancing frequency confidence intervals for stock-bond portfolios during restrictive Fed
policy periods
100
100
78
68
65
60
58
56
55
53
53
52
51
51
51
50
37
34
33
32
32
32
32
32
32
32
32
32
32
32
32
32
11
14
15
16
17
18
18
19
19
20
20
20
21
21
21
21
0
10
20
30
40
50
60
70
80
90
100
95-
05
90-
10
85-
15
80-
20
75-
25
70-
30
65-
35
60-
40
55-
45
50-
50
45-
55
40-
60
35-
65
30-
70
25-
75
20-
80
15-
85
90-
10
05-
95
Portfolio weights (bond-stock) after rebalancing
Months
Upper
Expected
Lower
