Cointegration and Long-Run Asset Allocation

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Cointegration and Long-Run Asset Allocation
Ravi BANSAL
Fuqua School of Business, Duke University, Durham, NC 27708 and NBER (ravi.bansal@duke.edu)
Dana KIKU
Wharton School, University of Pennsylvania, Philadelphia, PA 19104 (kiku@wharton.upenn.edu)
We show that economic restrictions of cointegration between asset cash flows and aggregate consumption
have important implications for return dynamics and optimal portfolio rules, particularly at long invest-
ment horizons. Whencashflows andconsumption shareacommon stochastic trend(i.e.,arecointegrated),
temporary deviations between their levels forecast long-horizon dividend growth rates and returns, and
consequently, alter the term profile of risks and expected returns. We show that the optimal asset allo-
cation based on the error-correction vector autoregression (EC-VAR) specification can be quite different
relative to a traditional VAR that ignores the cointegrating relation. Unlike the EC-VAR, the commonly
used VAR approach to model expected returns focuses on short-run forecasts and can considerably miss
on long-horizon return dynamics, and hence, the optimal portfolio mix in the presence of cointegration.
We develop and implement methods to account for parameter uncertainty in the EC-VAR setup and high-
light the importance of the error-correction channel for optimal portfolio decisions at various investment
horizons.
KEY WORDS:Asset allocation; Cointegration; Long-run risks.
1.INTRODUCTION
Risks facing a short-run and long-run investor can be quite
different. While at very short horizons, the contribution of cash-
flow news to the variance of return may be small, as the in-
vestment horizon increases, cash-flow fluctuations become the
dominantsource of return variability.Hence,understandingand
modeling the behavior of asset returns, especially at long hori-
zons, depend critically on understanding and modeling the dy-
namics of their cash flows. In this article we argue that de-
viations between cash-flow levels and aggregate consumption
(the error-correction term) contain important information about
means and variances of future cash-flow growth rates, and con-
sequently, returns. Incorporating this cointegration restriction
in return dynamics yields interesting implications for the term-
structure of expected returns and risks, and hence, asset allo-
cations at various investment horizons. In particular, we show
that the error-correction mechanism significantly alters the risk-
return tradeoff and the shape of optimal portfolio rules implied
by models where the long-run adjustment of cash flows is ig-
nored.
Our motivation for including the error-correction mechanism
is based on the ideas of long-run risks developed in Bansal and
Yaron (2004), Hansen, Heaton, and Li (2005), Bansal, Dittmar,
and Lundblad (2005), and Bansal, Dittmar, and Kiku (2009).
These articles, both theoretically and empirically, highlight the
importance of the long-run relation between cash flows and ag-
gregate consumption for understanding the magnitude of the
risk premium and its cross-sectional variation. Built on this ev-
idence, our article aims to explore the effect of long-run prop-
erties of asset cash flows on the optimal portfolio mix at var-
ious investment horizons. Intuitively, if the long-run dynamics
of asset dividends are described by a cointegrating relation with
aggregate consumption, then current deviations between their
levels should forecast future dividend growth rates (see Engle
and Granger 1987). Further, as risks in long-horizon returns are
dominated by cash-flow news, the predictability of asset divi-
dends emanating from the error-correction mechanism may sig-
nificantly alter the future dynamics of multihorizon returns and
their volatilities. This suggests that the error-correction chan-
nel may be very important for determining the optimal asset al-
location at intermediate and long investment horizons. Earlier
portfolio choice literature, including Kandel and Stambaugh
(1996), Barberis (2000), Chan, Campbell, and Viceira (2003),
and Jurek and Viceira (2005), model asset returns via a standard
vector-autoregression, and hence, ignore the consequences of
the long-run dividend dynamics for the risk-return tradeoff and
allocation decisions.
We measure the long-run relation between asset dividends
and aggregate consumption via a stochastic cointegration.
Based on the implications of the cointegrating relation, we
model dividend growth rates, price-dividend ratios, and returns
using an error-correction specification of a vector autoregres-
sion (EC-VAR) model. Our time-series specification allows us
to compute the term profile of conditional and unconditional
means and the variance–covariance structure of asset returns,
which we subsequently use to derive the optimal conditional
and unconditional portfolio rules. To highlight the importance
of the error-correction dynamics in dividends, we compare the
resulting allocations with those impled by a standard VAR
model, which excludes the error-correction variable from in-
vestors’ information set.
We solve the portfolio choice problem for buy-and-hold
mean-variance investors with different investment horizons,
ranging from 1 to 15 years, and different levels of risk aver-
sion. To emphasize the implications of long-run cash-flow dy-
namics for the risk-return tradeoff and optimal portfolio mix,
we focus on equities that are known to display large disper-
sion in average returns and opposite long-run (cointegrating)
© 2011 American Statistical Association
Journal of Business & Economic Statistics
January 2011, Vol. 29, No. 1
DOI: 10.1198/jbes.2010.08062
161

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162Journal of Business & Economic Statistics, January 2011
characteristics. In particular, we consider investors who allocate
their wealth across value and growth (i.e., high and low book-
to-market) stocks and the one-year Treasury bond. We distin-
guish between conditional and unconditional portfolio choice
problems and highlight the differences between the two.
To keep the analysis simple and transparent, we abstract from
any types of dynamic rebalancing and focus on the first-order
effect of the error-correction mechanism captured by the solu-
tion to the mean-variance problem. As shown in Jurek and Vi-
ceira(2005), regardlessofinvestors’riskaversion,theintertem-
poral hedging demand contributes a very small portion (less
than 5%) to the variation of the overall portfolio weights across
time. Thus, the volatility of the optimal portfolio is largely
dominated by its myopic component, which our approach cap-
tures. Considering reasonable alternative preference specifica-
tions, while straightforward, is unlikely to materially alter our
evidence.
We establish several interesting results. Consistent with
Bansal, Dittmar, and Lundblad (2001), Hansen, Heaton, and
Li (2005), and Bansal, Dittmar, and Kiku (2009), we find
that value and growth stocks significantly differ in their ex-
posures to long-run consumption risks. While cash flows of
value firms respond positively to low-frequency consumption
fluctuations, growth firms display a negative response in the
long run. Importantly, we find that current deviations in the
dividend-consumption pair (the cointegrating residual) contain
distinct information about future dynamics of both cash flows
and multihorizon returns, which is missing in the VAR setup.
In particular, if the error-correction dynamics are ignored and
returns are modeled via the standard VAR, one is able to ac-
count for about 11% and 52% of the variation in growth and
value returns at the 10-year horizon. With the cointegration-
based specification, the long-run predictability of growth and
value returns rises to striking 42% and 65%, respectively.
The forecasting ability of the error-correction term signifi-
cantly alters variances (and covariances) of asset returns rel-
ative to the growth rates-based VAR, especially at intermedi-
ate and long horizons. As expected, the EC-VAR model gener-
ates a declining pattern in the term structure of unconditional
volatilities of both value and growth stocks. The standard de-
viation of value returns in the VAR specification, on the other
hand, is slightly increasing with the horizon. Hence, the EC-
VAR model potentially is able to capture much larger benefits
of time-diversification relative to the traditional VAR approach.
Indeed, if the error-correction channel is ignored, the uncondi-
tional allocation to value stocks steadily declines: the VAR in-
vestors reduce their holdings of value stocks from 66% to about
52% as the investment horizon changes from 1 to 10 years. This
pattern is consistent with the VAR-based evidence of Jurek and
Viceira (2005). In contrast, relying on the cointegration-based
specification, investors tend to allocate a much larger fraction
of their wealth to value stocks as the investment horizon length-
ens.Inparticular,theholdingofvaluefirmsincreasesfrom76%
at the one-year horizon to about 96% for the 10-year invest-
ment. Thus, optimal portfolio prescriptions based on the stan-
dard VAR and EC-VAR models can be very different—these
differences are a reflection of the error-correction mechanism
between asset cash flows and aggregate consumption and the
ensuing time-diversification effect. Given a strong economic
appeal of cointegration in dividend-consumption relation, our
evidencesuggeststhatinvestorsshouldrelyontheoptimalport-
folio mix based on the EC-VAR model.
It is well recognized in the literature that asset allocation de-
cisions may be quite sensitive to parameter uncertainty. To en-
sure that our results are robust to estimation errors, we sup-
plement our evidence by deriving optimal allocations of a
Bayesian-type investor who recognizes and accounts for uncer-
tainty about model parameters. The impact of parameter un-
certainty in a standard VAR framework was earlier analyzed in
Kandel and Stambaugh (1996) and Barberis (2000). We extend
their approach and develop a method that allows us to handle
parameter uncertainty in the cointegration setup. We find the
Bayesian-based evidence to be qualitatively similar to the no-
uncertainty case. To be specific, even after accounting for un-
certainty in model parameters, the EC-VAR and VAR specifi-
cations deliver quite different portfolio rules, particularly in the
intermediate and long run. As the horizon increases, the allo-
cation to value continues to rise within the EC-VAR specifica-
tion (from about 47% to 64% at the horizon extends from 1
to 10 years) and keeps on falling in the growth rates VAR
framework (from 43% to 24%, respectively). Further, similar to
Barberis (2000), we find that investors that doubt reliability of
the estimated model parameters tend to shift their wealth away
from equities toward safer securities. Depending on the hori-
zon, the allocation to the Treasury bond increases by 20% to
40% compared to the no-uncertainty case. Taken together, our
evidence suggests that parameter uncertainty affects the scale
but not the shape of optimal asset allocations.
The rest of the article is structured as follows. Section 2 de-
scribes the evolution of risks across different investment hori-
zons and points toward the importance of long-run dynamics
of dividend growth rates for optimal decision rules. Section 3
outlines the portfolio choice problem, highlights implications
of cointegration, and describes the dynamic model for asset re-
turns. Our empirical results and their discussion are presented
in Section 4. Finally, Section 5 concludes.
2.SOURCES OF RISKS AT DIFFERENT HORIZONS
Our motivation for incorporating long-run (cointegration)
restrictions is the changing nature of risks across investment
horizons. Although short and long-horizon investors are con-
fronted by both risks in dividend growth rates and risks in price-
dividend ratios, their concerns about the two are likely to be
quite different since the relative contribution of dividend and
price news to the overall return variation changes considerably
with the horizon. While at short horizons, price risks are very
important, their impact gradually diminishes due to stationar-
ity of the price-dividend ratio. Consequently, at long investment
horizons,variationinreturnsisdominatedbyrisksindividends.
To formalize this intuition, we perform a variance decom-
position of returns using a first-order VAR model for dividend
growth rates and log price-dividend ratios. Specifically, we
project dividend growth of an asset on its own lag and regress
the price-dividend ratio on one lag of the dividend growth, as
well as its own lag. To provide a clean interpretation of the role
of price shocks versus dividend shocks we orthogonalize the
VAR innovations by assuming that dividend news leads to price

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Bansal and Kiku: Cointegration and Long-Run Asset Allocation 163
Table 1. Data summary
ReturnsGrowth rateslog(P/D)
Mean StdDevMeanStdDevMeanStdDev
Growth portfolio
Value portfolio
1-yr Treasury bond
Consumption
8.45
14.09
2.25
19.56
22.13
2.04
1.31
3.23
9.07
16.41
3.91
3.23
0.40
0.43
2.161.13
NOTE:
bond and consumption growth. Value firms represent companies in the highest book-to-market quintile of all NYSE, AMEX, and NASDAQ firms. Growth firms correspond to the lowest
book-to-market quintile. Portfolios are constructed as in Fama and French (1993). Returns are value-weighted, price/dividend ratios are constructed by dividing the end-of-year price
by the annual per-share dividend, growth rates are constructed by taking the first difference of the logarithm of per-share dividend series. Time-series for the Treasury bond are taken
from the CRSP Fama-Bliss Discount Bonds files. Data on the per-capita consumption of nondurables and services come from the NIPA tables available from the Bureau of Economic
Analysis. All data are sampled on at the annual frequency, converted to real using personal consumption deflator, and cover the period from 1954 to 2003.
This table presents descriptive statistics for returns, dividend growth rates and logarithms of price/dividend ratios of value and growth firms, the return on the one-year Treasury
movements, but price innovations do not lead to contempora-
neous responses in dividends. We implement variance decom-
position for two equity portfolios: growth and value stocks that
we subsequently use in our asset allocation analysis. Growth
and value stocks represent the lowest and the highest quintile
of book-to-market sorted portfolios, respectively. The construc-
tion of portfolios and their dynamics are presented in Table 1.
We find that the contribution of dividends and price-dividend
ratios to return variation changes significantly with the hori-
zon. In the short run, price risks dominate—for both portfo-
lios, about 98% of return variation over the one-year horizon
is attributed to price news. However, as the holding interval in-
creases, risks in returns shift toward risks in dividends. By the
10-year horizon, more than half of return variation is due to div-
idend shocks. As the horizon reaches 20 years, dividend growth
risks account for 75% of variation in growth returns and more
that 90% of risks in value returns. This evidence suggests that
asset allocations at long investment horizons are mostly about
managing dividend risks. Thus, understanding low-frequency
dynamics of asset dividends and integrating them into a model
for the risk-return tradeoff is critical in designing optimal al-
locations for long-horizon investors. In this article, we model
the dynamics of asset dividends via a cointegrating relation
with aggregate consumption and show that the ensuing error-
correction channel has important implications for optimal port-
folio rules at intermediate and long investment horizons. In the
next section, we set out the portfolio choice problem and de-
scribe the dynamics of asset returns that account for long-run
consumption risks in dividends.
3.ASSET ALLOCATION FRAMEWORK
3.1Portfolio Choice Problem
We consider investors with constant relative risk aversion
preferences (CRRA) preferences who follow a buy-and-hold
strategy over different holding horizons. At time t, an investor
chooses an allocation that maximizes her expected end-of-
period utility and is locked into the chosen portfolio till the end
of her investment horizon. Specifically, the s-period investor
solves
max
αs,tEt[Ut+s] = max
αs,tEt
?W1−γ
t+s
1−γ
?
,
(1)
where αs,tis the vector of portfolio weights, Wt+sis the ter-
minal wealth, and γ is the coefficient of risk aversion (RA).
Letting Rp
t+1→t+sdenote the (gross) return on the portfolio held
by the investor,
Rp
t+1→t+s= α?
s,tRt+1→t+s,
(2)
where Rt+1→t+sis the vector of compounded asset returns, the
evolution of wealth is described by
Wt+s= Wt∗Rp
Wedistinguishbetweentheconditionalandunconditionalstock
allocation problems. The conditional problem is stated above
and uses the conditional distribution of future returns. The un-
conditional asset allocation relies on the unconditional distrib-
ution of asset returns to maximize expected utility.
To make the problem tractable, we will assume throughout
that gross asset returns are lognormally distributed. As shown
in Campbelland Viceira(2002), the investor’objectivefunction
in this case can be written as
??
t+1→t+s.
(3)
max
αs,t
Et[rp
t+1→t+s]+1
2Vart(rp
t+1→t+s)
?
−γ
2Vart(rp
t+1→t+s)
?
,
(4)
where rp
and held up to t+s. The unconditional problem can be restated
analogously by dropping the time subscripts in the expression
above. We will refer to Et[rp
and Et[rp
turn. In the empirical section, the reported mean returns corre-
spond to arithmetic means. To enhance the comparison across
different holding periods, we measure and express all asset re-
turn moments per unit of time, that is, we scale both means and
variances by the investment horizon.
There are three assets available to investors: in addition to
the one-year Treasury bond, they allocate their wealth between
growth and value stocks. We focus on stocks with opposite
book-to-market characteristics that, historically, are known to
display large dispersion in average returns (as shown in Ta-
ble 1). The data employed in our empirical work are sampled
on the annual frequency, thus, a single investment period corre-
sponds to one year. We set risk aversion at five in our bench-
mark case; in addition, we highlight the implications of in-
vestors’ preferences by entertaining a higher risk aversion level
of 10.
t+1→t+sis the log return on a portfolio bought at time t
t+1→t+s] as the expected log return
2Vart(rp
t+1→t+s]+1
t+1→t+s) as the arithmetic mean re-

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164Journal of Business & Economic Statistics, January 2011
3.2Modeling Asset Returns
3.2.1
run dynamics of dividends and consumption via a cointegrating
relation,
Cointegration Specification.
We describe the long-
dt= τ0+τ1t +δct+?d,t,
(5)
where dtis the log level of an asset’s dividend, ctis the log
level of aggregate consumption, and ?d,t∼ I(0) is the cointe-
grating residual or the error-correction term. It follows from
Equation(5)thatdividendgrowthevolvesas ?dt≡ τ1+δ?ct+
??d,t. Hence, a time-series model for ?d,tand ?ctis sufficient
to model the dynamics of cash-flow growth rates.
Our specification implies that dividends and consumption
share a common stochastic trend. The two, however, may ex-
hibit different exposures to the underlying long-run risks as we
do not impose a unit restriction on the cointegration parame-
ter, δ. In addition, by including the time-trend in Equation (5),
we allow for differences in deterministic trends in asset divi-
dends and aggregate consumption. As we argue, imposing re-
strictions on either τ or δ may not be appropriate for the divi-
dend series we rely on.
Following the existing asset pricing literature, we focus on
dividends constructed on the per-share basis. These dividends
correspond to a trading strategy of holding one share of a firm’s
stock at each point in time. An investor following the one-share
strategy will consume all the dividends and reinvests only cap-
ital gains. Consider, alternatively, an investor who plows a por-
tion of the received cash back into the firm. If the amount of
reinvested income matches the net share issuance, such an in-
vestor will hold a claim to the total equity capital of the firm.
Consequently, payout series associated with this alternative in-
vestment, which we refer to as aggregate dividends, are propor-
tional to the firm’s market capitalization. Notice the difference
between the two measures—while the per-share series account
for the growth of the share price, aggregate dividends reflect the
appreciation of the firm’s equity capital.
It may be theoretically appealing to omit the time trend and
restrict the cointegration parameter of aggregate dividends on
the market (or a particular sector of the economy) to one, as it
will yield balanced growth paths of aggregate payouts and ag-
gregate consumption. There is, however, no economic rationale
for such restrictions for dividends per share. In order to illus-
trate and reinforce this important point, Figure 1 plots the log
of the stock market dividend to consumption ratio for the two
dividend measures. While the ratio of aggregate dividends to
consumption appear to be stationary, the ratio of per-share div-
idends to consumption displays a dramatic decline over time.
The reason per-share dividends fail to catch up with the level
of aggregate consumption is due to the fact that per-share se-
ries, by construction, do not account for capital inflow in equity
markets. Bansal and Yaron (2006) provided further discussion
of the difference between the two trading strategies and implied
dividend series.
From an econometric perspective, the distinction between
aggregate and per-share dividends has important implications
for modeling the dynamics of asset returns. As Figure 1 shows,
per-share dividends and aggregate consumption tend to drift
apart over time and the cointegrating relation between the two
Figure 1. Dividend to consumption ratio. This figure plots the
logarithm of per-share dividends to aggregate consumption ratio
(solid line), and the log ratio of aggregate dividends to consumption
(dash-dotted line). Dividend series represent cash flows of the aggre-
gate stock market portfolio. Per-share dividend series are constructed
as Dt+1= Yt+1Pt, where Y and P are the dividend yield and the price
index respectively; the latter evolves according to Pt+1= Ht+1Pt, P0
is normalized to 1, and H is the price gain. Aggregate dividend series
are constructed as Dagg
t+1= Yt+1Kt, where K is the market capitaliza-
tion. The data are real and span the period from 1954 to 2003.
cannot be established under the {δ = 1 & τ = 0} restriction as
confirmed by the augmented Dickey–Fuller test. Hence, omit-
ting the time trend and imposing the {δ = 1 & τ = 0} restric-
tion in the data leads to explosive/nonstationary return dynam-
ics. This issue is also discussed in Bansal, Dittmar, and Kiku
(2009).
The asset pricing literature typically focuses on the per-share
dividends as their present value corresponds to the price of the
asset, which is not true for aggregate dividend series. We follow
this tradition and use series constructed on the per-share basis.
Given the earlier discussion, we do not impose any restrictions
on parameters that govern their long-run dynamics, letting the
data decide on the underlying cointegrating relation between
per-share dividends and aggregate consumption.
3.2.2Return Dynamics.
To describe the distribution of
asset returns at various investment horizons, we model the dy-
namics of single-period returns and state variables jointly via
the following EC-VAR,
⎛
⎜
rt+1
⎜
⎜
⎝
bt+1
?ct+1
?d,t+1
zt+1
⎞
⎟
⎟
⎟
⎠=
⎛
⎜
⎜
⎜
⎝
ab
ac
a?
az
ar
⎞
⎟
⎟
⎟
⎠+
⎛
⎜
⎜
⎜
⎝
•
0
0
0
0
0
•
•
•
•
0
0
•
•
•
0
0
•
•
•
0
0
0
0
0
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎛
⎜
⎜
⎜
⎝
bt
?ct
?d,t
zt
rt
ub,t+1
uc,t+1
u?,t+1
uz,t+1
ur,t+1
⎞
⎟
⎟
⎟
⎠
+
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠.
(6)
That is, we project log bond return, bt, and consumption
growth, ?ct, on their own lags, and regress the cointegrat-

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Bansal and Kiku: Cointegration and Long-Run Asset Allocation 165
ing residual, ?d,t, log price-dividend ratio, zt, and log return,
rt, on their lags (excluding lagged return) and past consump-
tion growth. Denoting X?t= (bt
rewrite the EC-VAR in a compact matrix form,
?ct
?d,t
zt
rt), we can
Xt+1= a+AXt+ut+1,
(7)
where a is the vector of intercepts, the matrix A is defined ear-
lier,and u is a (5×1)-matrixof shocks thatfollowa normaldis-
tribution with zero mean and variance–covariance matrix ?u.
For expositional purposes, we focus of the first-order EC-VAR.
It is easy to allow for higher-order dynamics as they always can
be mapped into the first-order representation.
The error-correction specification is the key dimension that
differentiates our article from the existing portfolio choice lit-
erature. The latter typically models asset returns via a simple
VAR that incorporates information on the price-dividend ratio
(see Kandel and Stambaugh 1996; Barberis 2000; Chan, Camp-
bell, and Viceira 2003; and Jurek and Viceira 2005 among oth-
ers). In contrast to the traditional VAR approach, we describe
the dynamics of asset returns using the error-correction frame-
work that exploits the implications of long-run relation between
dividends and consumption.
The conceptual difference between a standard VAR and the
EC-VAR specifications is summarized by the error-correction
variable, ?d,t. To see its implications for returns consider a Tay-
lor series approximation of log returns (as in Campbell and
Shiller 1988),
rt+1= κ0+?dt+1+κ1zt+1−zt,
where κ’s are constants of linearization. As long-horizon re-
turns can be computed via summing up both sides of Equa-
tion (8), multiperiod returns will depend on the dynamics of
long-horizon dividend growth rates. Thus, long-run predictable
variation in dividend growth via the cointegrating residual
(Engle and Granger 1987) might alter predictability, and hence,
distribution of multiperiod returns. In fact, cointegration be-
tween dividends and consumption has potentially the same eco-
nomicconsequencesforreturnsastheunitcointegrationrestric-
tion between prices and dividends. While price-dividend ratios
are commonly used to forecast long-horizon returns, we argue
that the error-correction residual, ?d,t, may be equally (if not
more) important for predicting future returns; as such, it may
significantly affect volatilities and correlations of multiperiod
returns. Consequently,includingtheerror-correction variablein
the return dynamics may alter our views of the optimal long-run
allocations.
To highlight the importance of the error-correction mech-
anism in cash flows for the risk-return tradeoff and optimal
portfolio decisions, we will compare the implications of the
cointegration-based EC-VAR to those implied by the tradi-
tional VAR specification. In the VAR setup, the error correction
variable in Equation (6), ?d,t, is simply replaced by dividend
growth, ?dt.
Notice that instead of estimating the dynamics of asset re-
turns directly as in Equation (6), one can infer them from the
joint dynamics of price-dividend ratio and dividend growth
according to the log-linear approximation in Equation (8).
As shown in Campbell and Shiller (1988), the approximation
works well at relatively short investment horizons. However,
(8)
once compounded, the approximation error can lead to sizable
distortions in multihorizon return moments, and hence, long-
horizon asset allocations. Quantitatively, we find that at hori-
zons of 10 to 15 years, the volatility of approximate returns
is likely to over or understate the true return volatility by 2%
to 4%, or 15% to 20% in relative terms (see the Appendix for
details). Although our empirical evidence will not materially
change if we rely on log-linearization of asset returns, by mod-
eling the dynamics of asset returns explicitly we are able to
purge the effect of log-linearization and derive allocations that
are not subject to the approximation error.
3.3Term Structure of Expected Returns and Risks
The solution to the portfolio choice problem in Equation (4)
hinges on the distribution of multiperiod returns, in particular,
its first two moments. The required term-profile of expected re-
turns and risks can be easily computed by exploiting the recur-
sive structure of the EC-VAR as we outline in the following.
3.3.1Unconditional Analysis.
ditional problem is derived by fixing expected log returns on
individual assets at their sample means, i.e.,
The solution to the uncon-
E[rt+1→t+s] =1
s
s
?
j=1
¯ r = ¯ r.
(9)
To compute the unconditional variance of asset returns at vari-
ous investment horizons, we exploit the stationarity property of
EC-VAR variables and present the original specification as an
infinite-order moving average,
Xt+1= (I−AL)−1ut+1=
∞
?
j=0
Ajut+1−j.
(10)
It follows, then, that the unconditional variance of Xtis
?∗
0=
∞
?
j=0
Aj?uAj?,
(11)
and the variance of the sum of s consecutive X’s is given by
?∗
s= s?0+
s−1
?
j=1
(s−j)[Vj+V?
j],
(12)
where Vjis the j-order autocovariance of Xtdefined as Vj=
Aj?0. Scaling ?∗by horizon, ?s≡?∗
ance of multiperiod returns (expressed per-unit time) can be ex-
tracted via,
s
s, the unconditional vari-
Var(rt+1→t+s) = ι?
r?sιr,
(13)
where ιr is a (5 × 1)-indicator vector with the last element
(corresponding to return) set equal to 1. As pointed out ear-
lier, while the expected log returns E[rt+1→t+s] are constant
across horizons, the unconditional variances may change with
the horizon. Thus, although the unconditional problem does not
accommodate market timing, it does exploit return predictabil-
ity via horizon-dependent variances and correlations.