 | Variable Selection for Portfolio Choice
Winner
of the 2001 FAME Research Prize
Authors
Yacine
AÏT-SAHALIA - Princeton University & NBER
Michael W. BRANDT - Wharton School, University
of Pennsylvania & NBER
Date
Feb.
2001
This paper has now been published and is no longer available as
a part of our Research Paper Series. The reference to this paper is:
Aït-Sahalia,
Y., Brandt, M.W. (2001): "Variable Selection for Portfolio Choice". Journal of Finance, 2001,
56, 1297-1351.
Abstract
We
study asset allocation when the conditional moments of returns are partly predictable. Rather than first
model the return distribution and subsequently characterize the portfolio choice, we determine directly
the dependence of the optimal portfolio weights on the predictive variables. We combine the predictors
into a single index that best captures time-variations in investment opportunities. This index helps
investors determine which economic variables they should track and, more importantly, in what combination.
We consider investors with both expected utility (mean-variance and CRRA) and non-expected utility (ambiguity
aversion and prospect theory) objectives and characterize their market-timing, horizon effects, and
hedging demands.
Executive Summary
There
is by now ample evidence in the literature that the means, variances, covariances, and higher order
moments of stock and bond returns are time-varying and predictable. However, it has proven difficult
to translate this evidence of predictability into practical portfolio advice because the different moments
of returns, which in turn determine the optimal portfolio weights, are predicted by different sets of
economic variables. Perhaps because of this difficulty with modeling the conditional return distribution,
most professional investment advice is given solely on the basis of variables that forecast expected
returns, such as the dividend yield or the slope of the term structure.
Looking
beyond expected returns, it is difficult to decide which selection or combination of predictive variables
the investor should focus on. This is true even in the few special cases where we have an explicit asset
allocation formula, such as for mean-variance utility where the optimal allocation is proportional to
the ratio of the conditional mean to the conditional variance of returns.
In
this mean-variance case it is clear that we want to find variables that best predict the ratio of the
first two conditional moments. Choosing variables that best predict the mean and variance separately
is likely to be counter-productive. What should we do if a variable has a positive effect on both means
(which the investor likes) and variances (which are detrimental to the investor)? What should we do
if this variable is highly significant for one of the moments but less so for the other? How do we capture
the relative importance that the investor's preferences place on the different moments? These questions
all suggest that in a portfolio choice context we should select variables to directly predict optimal
portfolio weights, rather than first select variables to predict separate features of the return distribution
and then explore later their implications for asset allocation.
In this
paper, we show how to select and combine variables to best predict an investor's optimal portfolio weights,
both in single-period and multiperiod contexts. Rather than first model the various features of the
conditional return distribution and subsequently characterize the portfolio choice, we focus directly
on the dependence of the portfolio weights on the predictors. We do so by solving sample analogues of
the conditional Euler equations that characterize the portfolio choice, as originally suggested by Brandt
(1999). However, unlike the existing literature, we determine endogenously, for a given set of utility
preferences, which of the candidate predictors are important for the optimal portfolio weights (rather
than important for separate moments of the return distribution).
We form
a linear combination or index of the conditioning variables that best predicts the investor's optimal
portfolio weights and then judge the importance of each individual variable by the role it plays in
this index. We make no further assumptions about the relationship between the optimal portfolio weights
and the predictors for two reasons. First, the dependence of the portfolio choice on the predictors
can be highly nonlinear, even when the conditional moments are approximately linear; and second, the
particular form of the nonlinearities not only varies greatly with the investor's preferences but also
cannot generally be determined explicitly. This leads us to a semiparametric approach, where the optimal
portfolio weights depend nonparametrically on a parametric index of the predictors.
We
study the portfolio choice of investors with both expected utility (mean-variance and CRRA) and non-expected
utility (ambiguity aversion and prospect theory) objectives in order to see how the optimal index composition
depends on the characteristics of the investor's preferences. From a normative perspective, our results
can help investors with any one of these preferences determine which economic variables they should
track and, more importantly, in what single combination. Our index is a parsimonious way to describe
the current state of the investor's investment opportunities, just as in different economic contexts
indices summarize high-dimensional state vectors (the index of leading economic indicators, the business
cycle index, the consumer confidence index, etc.).
Macroeconomic indices
are country-specific, since countries have different characteristics, and for the same reason our investment
opportunities index is investor-specific because investors have different preferences.
For
the purpose of giving portfolio advice, one advantage of our index approach is that it helps investors
understand their conditional asset allocation in a more intuitive manner. For instance, it delivers
simple rules like \if the index increases, the allocation to stocks should increase." By contrast,
it is generally difficult to represent graphically variables in more than two dimensions, let alone
develop economic intuition about their interactions.
At least four stylized
facts emerged from our empirical analysis: - The
slope of the term structure is an ubiquitous variable in our indices, appearing significantly across
all preferences, investment horizons, and rebalancing frequencies. To a lesser extent, but fairly consistently,
a S&P index momentum variable enters our indices at short horizons, while the dividend yield is
the second most important variable at long horizons. The default risk premium generally records the
lowest index loadings.
- All investors, when presented with their
index of investment opportunities, find it optimal to engage in significant market timing.
- Horizon
effects are most pronounced for prospect theory investors, who find stock losses at short horizons to
be prohibitively costly. For investors who are not subject to loss aversion, the relative lack of returns
autocorrelation translates into relatively small horizon effects.
- Hedging
demands are weak and negative because stocks do not provide a good hedge for innovations in the index.
However, the index coefficients vary with the horizons.
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