Download Return Predictability and Dynamic Asset Allocation: How Often

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M
A
FO
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Return Predictability and
Dynamic Asset Allocation:
How Often Should Investors
Rebalance?
IN
A
N
Y
HIMANSHU ALMADI, DAVID E. RAPACH, AND ANIL SURI
IS
TH
DAVID E. R APACH
EP
R
O
D
U
C
E
is a professor of economics
at the John Cook School
of Business at Saint
Louis University in
St. Louis, MO, and a consultant to Merrill Lynch
Wealth Management in
New York, NY.
[email protected]
IT
IS
IL
LE
TO
L
G
A
is head of portfolio construction and investment
analytics at Merrill Lynch
Wealth Management in
New York, NY.
[email protected]
R
A NIL SURI
16
LE
To shed light on this issue, we analyze the
performance of DAA portfolios constructed
from monthly, quarterly, semi-annual, and
annual out-of-sample forecasts of U.S. stock,
bond, and bill returns. We first evaluate the
forecasts over an out-of-sample period from
January 1965 to December 2012 and show
that stock and bond return forecasts based
on principal components extracted from a
variety of fundamental, macroeconomic, and
technical variables significantly outperform
the constant expected return (i.e., random
walk with drift) baseline forecast. The principal components improve return forecasts
by incorporating information from multiple
potential predictors, while filtering out much
of the noise from individual predictors. In
line with the literature, we also show that
the degree of stock and bond return predictability typically increases markedly with the
forecast horizon.
Treating the stock, bond, and bill return
forecasts as an investor’s active views, we generate posterior expected returns using a modified version of the Black–Litterman model
(Black and Litterman [1991, 1992]) proposed
by Da Silva et al. [2009]. These posterior
expected returns and a set of benchmark
portfolio weights subsequently serve as inputs
for an active portfolio optimization problem
that determines the DAA portfolio weights.
We compute posterior expected returns
R
TI
nvestors require reliable out-of-sample
return forecasts to successfully pursue
dynamic asset allocation (DAA). A spate
of recent studies indicates that dependable out-of-sample return forecasts are indeed
available using improved forecasting strategies (e.g., Ludvigson and Ng [2007], Rapach
et al. [2010], and Neely et al. [forthcoming]).
These recent studies offer investors the
opportunity to substantially improve portfolio performance via dynamic investment
strategies.
To best exploit out-of-sample return
predictability, investors must decide how
frequently to rebalance their portfolios. On
the one hand, out-of-sample return predictability for stocks and bonds is statistically
and economically significant at horizons as
short as one month, suggesting that investors can rebalance more frequently to take
greater advantage of the dynamic investment opportunities afforded by changes in
expected returns. On the other hand, the
degree of stock and bond return predictability generally appears stronger at longer
horizons, suggesting that investors rebalance less frequently in response to the more
reliable signals provided by longer-horizon
forecasts.1
How often investors should rebalance
their portfolios in the presence of return predictability is ultimately an empirical issue.
A
is the director of portfolio
construction and investment analytics at Merrill
Lynch Wealth Management in New York, NY.
[email protected]
C
I
H IMANSHU A LMADI
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JPM-ALMADI.indd 16
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and DAA portfolio weights for each forecast horizon,
assuming that the investor rebalances the portfolio at the
same frequency as the forecast horizon. We use a balanced benchmark portfolio, which allocates 60%, 35%,
and 5% to stocks, bonds, and bills, respectively.2
In the absence of transaction costs, we show that
the DAA portfolios outperform the balanced benchmark at all rebalancing frequencies. Moreover, outperformance is decidedly maximized at the monthly
horizon, and this outperformance is magnified during
the most severe postwar U.S. recessions, including the
recent Great Recession, indicating that DAA is particularly valuable to investors during periods of extreme
economic stress.
As anticipated, the DAA portfolios based on
monthly rebalancing entail higher turnover than do
DAA portfolios based on less frequent rebalancing. We
thus measure outperformance for a range of unit transaction costs. According to the transaction-cost/rebalancing
frontier, monthly rebalancing produces the greatest
outperformance among the DAA portfolios when the
unit transaction cost is less than approximately 50 basis
points; for unit transaction costs above this level, annual
rebalancing nearly always delivers the greatest gains.
DAA portfolios based on annual rebalancing continue
to outperform the benchmarks for unit transaction costs
well in excess of 400 basis points. Overall, as long as unit
transaction costs are less than 50 basis points, which is
often the case in practice, the degree of short-horizon
return predictability appears strong enough to warrant a
more aggressive strategy of monthly rebalancing to take
greater advantage of changes in expected returns.
RETURN FORECASTS
Stock Returns
Suppose that we want to forecast rt:stock
t + h , the continuously compounded return on the S&P 500 Index from
the end of month t to the end of month t + h, using N
plausible predictor variables. The most obvious method
for incorporating information from all N predictors is
the multivariate predictive regression model,
rt:stock
t +h
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α h + β1,h x1,t + … + β N ,h x N ,t + εt:t +h
(1)
where xi,t is a month-t predictor variable (i = 1,…,N)
and εt,t+h is a zero-mean disturbance term. Because stock
returns inherently contain such a large unpredictable
component, however, multivariate predictive regressions with only a moderately large number of predictor
variables are highly susceptible to in-sample overfitting. Principal components provide a more promising
approach for incorporating information from multiple
predictors. Consider the predictive regression model,
rt:stock
t +h
h
+ β1,h Fˆ1,1,t + … + β K ,h FˆK ,t + εt:t +h
(2)
where Fˆ t = (Fˆ 1,t, … , Fˆ K ,t )′ is the vector containing the first
K principal components extracted from xt = (x1,t, …, xN,t )′
for K << N. Intuitively, the first few principal components
capture the key comovements among the entire set of
predictors, thereby filtering out much of the noise in the
individual predictors and avoiding in-sample overfitting.
The forecast corresponding to Equation (2) is given by
( )
( )
ˆ (h ) + βˆ 1,h Fˆ11,( t ) + + βˆ K ,h FˆK( ,)t
rˆt:stock
: +h = α
(3)
where Fˆ t (Fˆ 1,t ,…, Fˆ K ,t )′ is the vector containing the first
( )
( )
K principal components extracted from xt ; αˆ h and βˆ j h
(j = 1, …, K) are the ordinary least squares (OLS) estimates
of αh and βj,h ( j = 1, …, K), respectively, in Equation (2);
and K is selected using the adjusted R 2 statistic, all based
on data from the start of the available sample through
month t. Ludvigson and Ng [2007], Rapach and Zhou
[2013], and Neely et al. [forthcoming] show that principal
component forecasts of U.S. stock returns deliver significant out-of-sample forecasting gains.
We generate stock return forecasts using Equation (3) and principal components extracted from the
following set of nine predictor variables:
( )
( )
( )
1. log(D/P): log of a 12-month moving sum of
dividends paid on the S&P 500 index minus
the logg of the S&P 500 price index (i.e.,
12
log ∑ s 1 t ( 1) log( t ) , where Dt and Pt
are the month-t dividend per share and stock
price, respectively).
2. Inflation: calculated from the U.S. Consumer
Price Index (CPI) for all urban consumers.3
3. Term spread: 10-year Treasury bond yield minus
the three-month Treasury bill yield.
(
)
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4. Default spread: difference between Moody’s BAA
and AAA rated corporate bond yields.
5. Output gap: deviation of the log of industrial production from a quadratic trend.4
6. MA(1,12): dummy variable equal to one if the
S&P 500 price index is greater than its 12-month
moving average and zero otherwise.
7. MA(2,12): dummy variable equal to one if the
S&P 500 price index’s 2-month moving average
is greater than its 12-month moving average and
zero otherwise.
8. MOM(9): dummy variable equal to one if the
difference between the S&P 500 price index and
its value nine months ago is positive and zero
otherwise.
9. MOM(12): dummy variable equal to one if the
difference between the S&P 500 price index
and its value 12 months ago is positive and zero
otherwise.
This group of fundamental, macroeconomic, and technical variables constitutes a diverse set of plausible stock
return predictors.
Based on data availability, our sample spans January
1927 to December 2012. The data are from Global Financial Data, Ibbotson Associates, and the Federal Reserve
Economic Database (FRED) at the Federal Reserve
Bank of St. Louis. We use January 1927 to December
1964 as the initial in-sample estimation period, so that
we compute out-of-sample stock return forecasts for
January 1965 to December 2012. The initial estimation
period provides a reasonably long period for reliably estimating the predictive regression model parameters when
computing the initial forecasts. The forecast evaluation
period covers nine recessions identified by the National
Bureau of Economic Research (NBER), including the
three severe postwar recessions surrounding the oil price
shock of the mid 1970s, Volcker disinf lation of the early
1980s, and recent global financial crisis.
We compute return forecasts at monthly (h = 1),
quarterly (h = 3), semi-annual (h = 6), and annual
(h = 12) horizons. The forecasts at the quarterly, semiannual, and annual horizons do not overlap, because we
ultimately use the forecasts as inputs for portfolios that
rebalance at the forecast horizon. We generate the outof-sample forecasts using recursive (i.e., expanding) estimation windows. For example, the initial monthly stock
return forecast for January 1965 is based on data covering
18
January 1927 to December 1964; the second monthly
return forecast for February 1965 is based on data covering January 1927 to January 1965, and so on.
T he con st a nt ex pected ret u r n model,
rt:stock
=
α h + εt:t +h , serves as a natural baseline for assessing
t +h
stock return forecasts. This model corresponds to the
canonical random walk with drift model for the log of
stock prices, which implies that stock returns are not
predictable (apart from their long-run average return).
We straightforwardly compute the constant expected
return forecast as the historical average of stock returns
from the start of the available sample through the month
of forecast formation.
Exhibit 1 presents out-of-sample results at each
horizon for S&P 500 return forecasts based on Equation (3) and principal components extracted from the
nine predictor variables. We report the Campbell and
2
Thompson [2008] out-of-sample R 2 statistic, ROS
, which
provides a convenient metric for comparing a predictive regression forecast to the historical average. The
2
statistic is analogous to the familiar in-sample R 2
ROS
statistic and measures the proportional reduction in
mean squared forecast error (MSFE) for the predictive regression forecast vis-á-vis the historical average.
2
Along with the ROS
statistic, Exhibit 1 reports the Clark
and West [2007] statistic for testing the null hypothesis
that the historical average MSFE is less than or equal to
the predictive regression MSFE against the alternative
hypothesis that the historical average MSFE is greater
than the predictive regression MSFE (corresponding to
2
2
≤ 0 against H A : ROS
H 0 : ROS
> 0). Finally, Exhibit 1 also
reports correlations between the predictive regression
forecasts and actual stock returns.
2
The ROS
statistics for the principal component
forecasts of S&P 500 returns are positive at all hori2
zons in Exhibit 1. At first glance, the ROS
statistic of
1.10% at the monthly horizon seems small. However,
because stock returns inherently contain a large unpre2
dictable component, even a monthly ROS
statistic near
0.5% can indicate economic significance (e.g., Campbell
and Thompson [2008] and Zhou [2010]). According to
the Clark and West [2007] statistics, the MSFE for the
principal component forecast is significantly below that
of the historical average at three of the four horizons.
2
Observe that the ROS
statistics for the principal component forecasts of S&P 500 returns (as well as the correlations) increase markedly at longer horizons, rising to a
quite sizable 8.03% (0.27) at the annual horizon.
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JPM-ALMADI.indd 19
Notes: The S&P 500 (Ten-year Treasury bond) return forecast is based on a predictive regression model in which the return is regressed on a constant and up to the first three lagged principal components extracted from nine (seven) predictor variables. The 30-day Treasury bill return forecast is based on a bivariate predictive regression model in which the return is regressed on
2
a constant and the lagged three-month Treasury bill yield. ROS
measures the proportional reduction in MSFE for the predictive regression forecast relative to the historical average baseline
forecast. Clark & West stat. is the Clark and West [2007] statistic for testing the null hypothesis that the historical average MSFE is less than or equal to the predictive regression model
2
2
≤ 0 against H A: ROS
> 0 ); ***, **, and *
MSFE against the alternative that the historical average MSFE is greater than the predictive regression model MSFE (corresponding to H 0 : ROS
ρ
indicate significance at the 10%, 5%, and 1% levels, respectively. r̂ ,r is the correlation between the return forecast and actual return.
Return Forecasting Results at Different Horizons, January 1965 to December 2012
EXHIBIT 1
SUMMER 2014
Bond Returns
We also compute bond return forecasts using
bond
Equation (3), with rt:stock
t + h replaced by rt:t + h , the continuously compounded return on a 10-year Treasury bond
from the end of month t to the end of month t + h. We
extract the principal components from the following
seven predictor variables for bond returns:
1.
2.
3.
4.
5.
6.
Bond yield: 10-year Treasury bond yield.
Inflation: as previously defined.
Term spread: as previously defined.
Credit spread: as previously defined.
Output gap: as previously defined.
MOMBY(6): dummy variable equal to −1 (1) if
the bond yield is more than five basis points above
(below) its six-month moving average and zero
otherwise.
7. MOMBY(12): dummy variable equal to −1 (1) if
the bond yield is more than five basis points above
(below) its 12-month moving average and zero
otherwise.
This set of predictors augments the four predictors
used by Ilmanen [1997]. The last two predictors represent technical indicators for the bond market. When
the bond yield exceeds its six-month moving average,
MOMBY(6) identifies a positive yield trend—and thus
a negative price trend—and produces a bearish signal
for future bond returns. MOMBY(12) is a smoothed
version of MOMBY(6).
Exhibit 1 indicates that the principal component
2
forecasts of bond returns perform well: the ROS
statistics
are all positive, and the Clark and West [2007] statistics demonstrate that the principal component MSFE is
significantly less than the historical average MSFE at all
2
horizons. The ROS
statistics (correlations) increase with
the horizon, rising from 2.04% (0.11) at the monthly
horizon to 18.13% (0.38) at the annual horizon.
Bill Returns
Bill returns are substantially easier to forecast than
stock or bond returns, due to the persistence of shortterm interest rates and relatively limited sensitivity of bill
prices to yield f luctuations. We compute bill return forecasts using the bivariate predictive regression model,
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rt:bill
t +h
α h + βh yt + εt:t +h
(4)
uted. We thus transform μt:t+h and ∑t:t+h to correspond to
simple returns for the relevant horizon:
where rt:bill
t + h is the continuously compounded return on
a 30-day Treasury bill from the end of month t to the
end of month t + h and yt is the three-month Treasury
bill yield. The forecast corresponding to Equation (4)
is given by
rˆ
bill
t:t + h
( )
( )
where αˆ h and βˆ h are the OLS estimates of αh and βh ,
respectively, in Equation (4) based on data from the start
of the available sample through month t.
The last row of Exhibit 1 demonstrates that bill
return forecasts based on the Treasury bill yield are very
2
accurate. The ROS
statistics are above 95% at all horizons,
and the corresponding Clark and West [2007] statistics
clearly indicate that the predictive regression MSFE is
significantly less than the historical average MSFE at
all horizons. The correlations between the predictive
regression bill return forecasts and actual bill returns are
also very high at all horizons.
DAA PORTFOLIO OUTPERFORMANCE
Construction of DAA Portfolios
We construct the DAA portfolios in two steps.
First, we compute posterior return moments using the
modified version of the Black–Litterman (BL) model
proposed by Da Silva et al. [2009]. Denote the vector
of horizon-h continuously compounded return forecasts by
bond
bill
rˆt:t +h = ((ˆr tstock
:t + h , rˆt:t + h rˆt:t + h )′
(6)
bond
stock
bill
where rˆt:t +h and rˆt:t +h (rˆt:t +h ) are the return forecasts based
on principal components (the three-month Treasury bill
yield) described in the previous section. Equation (6)
serves as an investor’s vector of active views on expected
returns for the modified BL model. We use these active
views, together with an exponentially weighted moving-average estimate of the return covariance matrix,
to calculate the posterior vector of expected horizon-h
returns, μt:t+h , and posterior return covariance matrix,
∑t:t+h , as detailed in the Appendix. By applying the BL
model to continuously compounded returns, we implicitly assume that simple returns are log-normally distrib-
20
{
{exp
⎡⎣exp (
(5)
0 5diag(
tt h
St:t +h = expp
( )
= αˆ + βˆ h yt
( )
h
mt:t +h = expp [
)] − ι 3
tt h
0 5diag(
tt h
tt h
0 5diag(
g(
tt h
tt:: h
(7)
}
) }′ )
) − ι3 ι′3 ⎤⎦
(8)
where ι3 is a 3-vector of ones and represents elementby-element multiplication.
Second, we use mt:t+h and St:t+h to calculate the DAA
portfolio weights in the context of the following active
management optimization problem:
A
argm
r ax {
w tDAA
:t + h =
t :t + h }
⎡
⎣( w t t
h
⎤
′
w tbench
t h ) mt:t + h ⎦
(9)
subject to
(
t:t + h
bench
t:t + h
) St:t +h (
t:t + h
bench
bench
t:t + h
) (h 12)TE2 (10)
w t:t +h′ ι3 = 1
(11)
w LB ≤ w t:t +h
(12)
where w tbench
:t + h is the vector of benchmark portfolio weights.
The tracking error (TE) is set to a constant annualized
value of 2% in Equation (10). Equation (11) restricts the
portfolio to be fully invested, and we set w LB = (0.05,
0.05, 0.05)′ in Equation (12).
The benchmark por tfolio al locates 60 %
to stocks, 35% to bonds, and 5% to bills, so that
w tbench
:t + h = (0.60, 0.35, 0.05)′ in Equation (9). This balanced benchmark comports with conventional advice
for many individual and institutional investors. The
DAA and benchmark portfolios are rebalanced at the
same frequency as the forecast horizon. Matching the
forecast evaluation period, we compute monthly returns
for each DAA portfolio for January 1965 to December
2012 (576 observations).
Exhibit 2 presents DAA and benchmark portfolio
weights for stocks, bonds, and bills. The stock weights for
the DAA portfolios are strongly linked to business-cycle
fluctuations in Exhibit 2. This link is particularly evident
for monthly rebalancing, where the stock weights for the
DAA portfolio typically move well below (above) the
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tt h
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EXHIBIT 2
DAA and Balanced Benchmark Portfolio Weights for Different Rebalancing Frequencies,
January 1965 to December 2012
Notes: The black line in each panel delineates the monthly DAA portfolio weight (in percent) for the asset and rebalancing frequency given in the panel
heading. The gray line in each panel delineates the corresponding monthly balanced benchmark portfolio weight. Vertical bars depict NBER-dated
recessions.
benchmark weights near business-cycle peaks (troughs).
As expected, the adjustments in the DAA stock portfolio
weights generally become less abrupt with less frequent
rebalancing. Similarly to the stock weights, the DAA
bond portfolio weights generally fall below (rise above)
the benchmark weights near cyclical peaks (troughs),
especially for monthly rebalancing. The stock (bond)
weights for the DAA portfolios are usually somewhat
above (below) the benchmark weights throughout economic expansions. The bill weights for the DAA portfolios are frequently small and close to the benchmark
weights, although they tend to spike up near cyclical
peaks, as the stock and bond weights fall.5
Baseline Scenario
The return forecasts from the previous section
frequently produce adjustments in the DAA portfolio
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JPM-ALMADI.indd 21
weights in Exhibit 2 that are closely connected to business-cycle f luctuations. Investors need to know whether
these adjustments improve portfolio performance relative to the benchmark and whether more frequent rebalancing in response to changing economic conditions is
beneficial. We analyze these issues in Exhibit 3, which
compares the performances of the DAA portfolios to
the benchmark for January 1965 to December 2012. As
a baseline scenario, we assume zero transaction costs in
Exhibit 3; we examine the effects of transaction costs
subsequently. The second through sixth columns of the
exhibit report the (geometric) average return, standard
deviation, maximum drawdown, Calmar ratio, and
turnover, respectively, for the DAA and benchmark
portfolios at each rebalancing frequency.
The last four columns of Exhibit 2 present the
average excess return, tracking error, information ratio,
and certainty equivalent return (CER) gain for the DAA
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EXHIBIT 3
DAA Portfolio Performance Statistics Relative to the Balanced Benchmark for Different Rebalancing
Frequencies, January 1965 to December 2012
Notes: Average return is the annualized geometric average monthly return for the DAA portfolio rebalanced at the frequency given in the first column. Standard deviation is the annualized standard deviation of the DAA portfolio return. Maximum drawdown is the maximum percentage reduction in cumulative
wealth for the DAA portfolio. The Calmar ratio is the annualized geometric average return divided by the maximum drawdown. Turnover is the DAA
portfolio’s annualized turnover. The statistics in brackets in the second through sixth columns are for the balanced benchmark portfolio. Average excess return
is the annualized geometric average of the monthly DAA portfolio return in excess of the balanced benchmark portfolio return. Tracking error is the annualized standard deviation of the excess return. The information ratio is the annualized arithmetic average excess return divided by the tracking error. CER
gain is the annualized increase in certainty equivalent return that an investor with power utility and a relative risk aversion coefficient of two would realize
by having access to the DAA portfolio in the place of the balanced benchmark portfolio.
portfolios vis-á-vis the balanced benchmark. The CER
gain is the increase in CER that an investor with power
utility and a relative risk aversion coefficient of two
would realize by having access to the DAA portfolio
instead of the benchmark portfolio. Power utility is given
by U(1 + Rp ), where Rp is the portfolio return, U(x) = [1/
(1 − RRA)]x1−RRA, and RRA is the relative risk aver(U bench ) denote the average
sion coefficient. Let U DAA
A
utility realized by an investor who has access to the DAA
(benchmark) portfolio. Inverting the utility function, the
1 (1
( − RAA
R )
− 1 for
CER is given by CER j = ⎣⎡(1 − RRA)U j ⎦⎤
j = DAA, bench; the CER gain is CER DAA − CERbench.
All statistics are annualized (with the exception of the
maximum drawdown, which cannot be annualized in
a natural way).
The average return for each DAA portfolio is
higher than that of the balanced benchmark by at least
60 basis points in Exhibit 3. The DAA portfolios have
somewhat higher standard deviations than the benchmark, while the maximum drawdowns for the DAA
portfolios are below (reasonably close to) those of the
benchmark for monthly and semi-annual (quarterly and
annual) rebalancing. The Calmar ratios for the DAA
portfolios with monthly, semi-annual, and annual
rebalancing are greater than their benchmark counterparts; the Calmar ratios are the same with quarterly
22
rebalancing. The information ratios in the penultimate
column of Exhibit 3 are all ample, ranging from 0.34 to
0.48, while the CER gains in the last column range from
44 to 78 basis points. In sum, the DAA portfolios generally outperform their balanced benchmark counterparts
by economically sizable margins in Exhibit 3.6
With respect to the relative performances of the
DAA portfolios in Exhibit 3, the portfolio based on
monthly rebalancing stands out. Among the DAA portfolios, monthly rebalancing produces the highest average
return, Calmar ratio, average excess return, information ratio, and CER gain, as well as the lowest standard
deviation and maximum drawdown. The relative outperformance of the DAA portfolio based on monthly
rebalancing is often substantial; the average excess return
and CER gains are at least 20 and 26 basis points higher,
respectively, for monthly rebalancing vis-á-vis the other
rebalancing frequencies.
While the results in Exhibit 3 demonstrate significant outperformance by the DAA portfolios with
respect to the benchmark over the entire January 1965
to December 2012 evaluation period, they mask pronounced outperformance during particular episodes.
This is evident in Exhibit 4, which illustrates time
variation in outperformance via rolling information
ratios (top row) and CER gains (bottom row) computed
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EXHIBIT 4
Rolling Three-Year DAA Portfolio Information Ratios and CER Gains, January 1965 to December 2012
Notes: Panels A through D show rolling annualized information ratios for the DAA portfolio relative to the balanced benchmark portfolio computed using
a 36-month window for the rebalancing frequency given in the panel heading. Panels E through H show rolling annualized percentage CER gains for the
DAA portfolio relative to the balanced benchmark portfolio computed using a 36-month window for the rebalancing frequency given in the panel heading.
The horizontal axis indicates the end of the 36-month window. Vertical bars depict NBER-dated recessions.
using a 36-month window. Panels A and E of Exhibit 4
indicate that the information ratios and CER gains for
the DAA portfolios based on monthly rebalancing are
substantially higher during the most severe postwar
recessions, namely, the recessions surrounding the oil
price shock of the mid 1970s, Volcker disinf lation of
the early 1980s, and recent global financial crisis. The
information ratios for monthly rebalancing in Panel A
are around 1.0 during the mid 1970s and close to 1.5
during the early 1980s and recent Great Recession.
The CER gains accruing to monthly rebalancing in
Panel E reach 200, 400, and 300 basis points during the
mid 1970s, early 1980s, and Great Recession, respectively. The other panels in Exhibit 4 show that the
information ratios and CER gains are also magnified
during the mid 1970s for quarterly rebalancing; during
the early 1980s for quarterly, semi-annual, and annual
rebalancing; and during the Great Recession for annual
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JPM-ALMADI.indd 23
rebalancing. All of the DAA portfolios also generate
very sizable information ratios and CER gains from
1995 to 1998.
Transaction Costs
The DAA portfolios based on monthly rebalancing
produce the greatest outperformance under the baseline scenario without transaction costs in Exhibit 3. As
expected, the DAA portfolios based on monthly rebalancing also generate the highest turnover relative to the
benchmark in the sixth column of Exhibit 3. To gauge
the importance of turnover for portfolio performance,
we analyze the effects of transactions costs on the outperformance of the DAA portfolios across rebalancing
frequencies in Exhibit 5. For each DAA portfolio and
rebalancing frequency, we compute information ratios
and CER gains for January 1965 to December 2012
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EXHIBIT 5
Effects of Transaction Costs on Outperformance, January 1965 to December 2012
Notes: Each line in Panel A delineates the relationship between the unit transaction cost in basis points (horizontal axis) and the annualized information
ratio (vertical axis) for the DAA portfolio relative to the balanced benchmark portfolio. Each line in Panel B delineates the relationship between the unit
transaction cost in basis points (horizontal axis) and the annualized percentage CER gain (vertical axis) for the DAA portfolio relative to the balanced
benchmark portfolio.
over a grid of unit transaction costs from zero to 450
basis points. Panels A and B of Exhibit 5 report information ratios and CER gains, respectively, for the DAA
portfolios.
The lines corresponding to monthly rebalancing
are relatively steeply sloped in each panel of Exhibit 5,
indicating that outperformance tails off relatively quickly
as unit transaction costs increase, in line with the much
higher turnover associated with monthly rebalancing
for the DAA portfolio relative to the benchmark. Nevertheless, the information ratios and CER gains remain
positive for the DAA portfolios based on monthly rebalancing for unit transaction costs in excess of 100 basis
points. Given the positive relationship between turnover and rebalancing frequency in Exhibit 3, the lines
f latten as we move from quarterly to semi-annual to
annual rebalancing in Exhibit 5. For quarterly rebalancing, the information ratios and CER gains are positive for unit transaction costs of less than approximately
24
200 basis points. Outperformance remains positive for
semi-annual rebalancing for unit transaction costs of
up to around 200 basis points. For annual rebalancing,
the information ratios (CER gains) remain positive for
unit transaction costs of much more than 400 (300)
basis points.
For each panel in Exhibit 5, we can trace out a
transaction-cost/rebalancing frontier that identifies the
rebalancing frequency with the greatest outperformance
for any given unit transaction cost. A consistent conclusion emerges along these frontiers: for unit transaction
costs of up to approximately 50 basis points, monthly
rebalancing produces the greatest outperformance;
annual rebalancing nearly always provides the greatest
gains beyond this level. When unit transaction costs
are low, investors are best served by rebalancing at the
monthly frequency to take greater advantage of changes
in expected returns detected by the monthly return forecasts, despite the higher portfolio turnover. For higher
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unit transaction costs, the more reliable signals provided
by the annual return forecasts and lower portfolio turnover recommend annual rebalancing.
CONCLUSION
We investigate the effects of rebalancing frequency on portfolio performance when investors have
access to monthly, quarterly, semi-annual, and annual
U.S. stock, bond, and bill return forecasts. Incorporating
recent advances from the literature, we compute predictive regression forecasts of stock and bond returns based
on the first few principal components extracted from
a variety of fundamental, macroeconomic, and technical variables. We compute bill return forecasts from a
bivariate predictive regression based on the three-month
Treasury bill yield. With these return forecasts comprising an investor’s active views, we construct DAA
portfolios that rebalance at the same frequency as the
forecast horizon. Along the transaction-cost/rebalancing
frontier, investors reap the greatest rewards from outof-sample return predictability via monthly rebalancing
when unit transaction costs are less than approximately
50 basis points; for unit transaction costs above this level,
annual rebalancing provides the greatest gains, and these
gains persist for unit transactions costs well above 400
basis points.
It is important to keep in mind that DAA portfolios
require reliable return forecasts to deliver reasonably
consistent outperformance over time. Return forecasts
are more likely to be reliable to the extent that they are
economically plausible. The fact that our return forecasts produce DAA portfolio weights that are strongly
linked to business-cycle f luctuations points to a solid
economic foundation. Indeed, the gains accruing to the
DAA portfolios appear to stem significantly from timevarying risk premiums over the business cycle. From
this perspective, our DAA portfolios are in the spirit
of Li and Sullivan [2011], who recommend a dynamic
approach to active management that takes “full advantage of time-varying risk premiums, driven, in large
part, by investors’ cycling between risk aversion and risk
adoration” (p. 33).7
In the present article, we focus on fixed-frequency
rebalancing corresponding to the return forecast horizon,
as many investors employ fixed-frequency rebalancing.
In ongoing research, we are exploring additional rebal-
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ancing strategies that are potentially optimal in a more
global sense. Instead of rebalancing at a fixed frequency
to the updated policy portfolio, these strategies base
the timing and degree of rebalancing on economically
motivated criteria (e.g., Chan and Ramkumar [2011],
Liebowitz and Bova [2011], Michaud et al. [2012], and
Gârleanu and Pedersen [2013]). Such refinements potentially lower transaction costs while preserving the ability
of dynamic portfolios to take extensive advantage of
dynamic investment opportunities, thereby providing
further gains to investors who exploit out-of-sample
return predictability via DAA.
APPENDIX
This appendix describes the computation of the posterior expected return vector, μt:t+h , and posterior return
covariance matrix, ∑t:t+h. We first generate an exponentially
weighted moving-average estimate of the monthly continuously compounded return covariance matrix, Σˆ t: t + 1, using
data from the start of the available sample through month t.
We set the decay parameter to 0.94, ref lecting a half-life of
approximately 12 months and exponential decay to nearly
zero in about seven years. We estimate the horizon-h covariance matrix for continuously compounded returns using
Σˆ t:t + h = h Σˆ t:t +1.
Following Da Silva et al. [2009], we compute μt:t+h and
∑t:t+h using the BL model with an alternative equilibrium
return vector. The vector of return forecasts, r̂ t:t + h in Equation (6) comprises the vector of active views on expected
returns. Adopting the approach of Idzorek [2004], the variance of each active view is given by
⎛ 1− c j ⎞
τω j = τ ⎜
⎟ p j Σˆ t:t + h p ′j
⎝ cj ⎠
for j = stock, bond, or bill
(A-1)
where τ is a scaling parameter (which we set to unity) and pstock,
pbond, and pbill are the first, second, and third rows, respectively,
of P = I3. Ref lecting the beliefs that stock returns have a larger
unpredictable component than bond returns and that bond
returns in turn have a larger unpredictable component than
bill returns, we set c stock = 0.25, c bond = 0.50, and c bill = 0.75
for the vast majority of months, consistent with the ranking
2
indicated by ROS
statistics computed using data available at
the time of forecast formation. For some months, however,
2
the ROS
statistic for bonds is less than that of stocks (particularly during the early 1980s), so that we use c bond = 0.25
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and cstock = 0.50 for these months. Assuming that the active
views are uncorrelated, the covariance matrix for the vector
of active views is given by τ Ω, where Ω is a diagonal matrix
with ωstock, ωbond, and ωbill along the main diagonal. Setting
each element of the equilibrium return vector to the bill
return forecast, we compute the posterior return moments,
μt:t+h and ∑t:t+h , using the familiar BL model formulas (e.g.,
Equations 20 and 21 in Meucci [2010]).
Campbell, J., and S. Thompson. “Predicting the Equity Premium Out of Sample: Can Anything Beat the Historical
Average?” Review of Financial Studies, Vol. 21, No. 4 (2008),
pp. 1509-1531.
ENDNOTES
Clark, T., and K. West. “Approximately Normal Tests for
Equal Predictive Accuracy in Nested Models.” Journal of
Econometrics, Vol. 138, No. 1 (2007), pp. 291-311.
The views expressed here are those of the authors and
not necessarily those of any entity within and including Bank
of America. We are extremely grateful to an anonymous
referee for extensive comments that significantly improved
the article. We also thank Jun Tu and Guofu Zhou for very
helpful comments on earlier drafts.
1
See Rapach and Zhou [2013] for a survey of out-ofsample stock return predictability, including differences in the
degree of return predictability across horizons.
2
We obtain qualitatively similar results when we use
a mean–variance optimal portfolio based on the constant
expected return forecasts as the benchmark. The complete
results are available upon request from the authors.
3
We account for the delay in the release of monthly CPI
data when computing the forecast.
4
We only use data from the start of the available
sample through the month of forecast formation to compute
the output gap, so that there is no look-ahead bias in the
forecast.
5
The cyclical patterns in the DAA portfolios’ stock,
bond, and bill weights ref lect similar patterns in the return
forecasts themselves, which are available upon request from
the authors.
6
Annualized average returns (volatilities) for the S&P
500, 10-year government bonds, and 30-day Treasury bills
are 9.41% (15.24%), 7.54% (8.26%), and 5.26% (0.87%),
respectively, for January 1965 to December 2012.
7
Also see the illuminating survey of time-varying risk
premiums across asset classes by Ilmanen [2011].
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To order reprints of this article, please contact Dewey Palmieri
at dpalmieri@ iijournals.com or 212-224-3675.
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