9. Análisis estadístico de los datos
1.1 Síndromes Falciformes
In the last section, we derived the log excess return on the risky asset for the retirement state, eq. (3.26), as
Etri,t+1− rf + 1 2σ 2 it = κ ψ(1 − δ)σri,t+1,∆ct+1+ κδ ψ σri,t+1,∆ht+1 +(1 − κ)σri,t+1,rp,t+1 (3.30)
where V art[ri,t+1] = σ2it, covt(ri,t+1, ∆ct+1) = σri,t+1,∆ct+1, covt(ri,t+1, ∆ht+1) =
σri,t+1,∆ht+1 and covt(ri,t+1, rp,t+1) = σri,t+1,rp,t+1. As in the retirement state, with
no labour income, our strategy, following Campbell and Viceira (1999), is to characterize the covariance terms as functions of the exogenous risky asset return
15The derivation of log approximation to portfolio return is standard in the literature, see for exampleCampbell and Viceira(2002).
and the stationary consumption-wealth ratio. The covariance between log risky asset return and non-durable consumption growth is written as,
σri,t+1,∆ct+1 = covt(ri,t+1, ∆ct+1)
= covt(ri,t+1, (ct+1− wt+1) − (ct− wt) + ∆wt+1)
= covt(ri,t+1, (ct+1− wt+1)) − covt(ri,t+1, ct− wt) + cov(ri,t+1, ∆wt+1)
= covt(ri,t+1, (ct+1− wt+1)) + covt(ri,t+1, ∆wt+1)
= σri,t+1,(ct+1−wt+1)+ covt(ri,t+1, rp,t+1)
= σri,t+1,(ct+1−wt+1)+ αitvart(ri,t+1)
= σri,t+1,(ct+1−wt+1)+ αitσ
2
it (3.31)
where the second equality is trivial algebra, the third uses properties of the covari- ance operator for random variables, and the rest follows from substituting values for ∆wt+1 and rp,t+1 from the retirement state log linearised equations (3.28) and
(3.29). In addition to these we also use the fact that cov(xt+1, zt) = 0, see Camp-
bell and Viceira(1999). In similar fashion, the covariance between log risky asset
return and durable (housing) consumption growth is derived as σri,t+1,∆ht+1 = σri,t+1,(ht+1−wt+1)+ αitσ
2
it (3.32)
and finally the covariance between log risky asset return is a direct implication of equation (3.29):
σri,t+1,rp,t+1 = covt(ri,t+1, rf + α
0 t(rt+1− rf) + 1 2α 0 tσ 2 t − 1 2α 0 tΣtαt) = αitσ2it (3.33)
Now that we have characterized the three covariance terms, we substitute these terms into (3.30) to get
Etri,t+1− rf + 1 2σ 2 it= κ ψ(1 − δ)(σri,t+1,(ct+1−wt+1)+ αitσ 2 it) +κδ ψ (σri,t+1,(ht+1−wt+1)+ αitσ 2 it) + (1 − κ)αitσ2it (3.34)
which can be rearranged using the fact that the parameter specifying the timing of the resolution of uncertainty is κ = (1 − γ)/(1 − ψ1). We substitute to get the optimal portfolio allocation on the risky assets for the retirement state. The employment state follows the same procedure. These results are described in the following Proposition.
PROPOSITION 1: The optimal portfolio share of risky assets for the retirement state is 16 αrit = 1 γ Etri,t+1− rf +12σit2 σ2 it + 1 1 − ψ 1 − γ γ (1 − δ)σi,(c t+1−wt+1)+ δσi,(ht+1−wt+1) σ2 it ! (3.35) = αM Dr it + α HDr it . (3.36)
where V art(ri,t+1) = σ2it, cov(ri,t+1, ct+1−wt+1) = σi,ct+1−wt+1 and cov(ri,t+1, ht+1−
wt+1) = σi,ht+1−wt+1.
and for the employment state is
αeit = 1 γ Etri,t+1− rf +12σit2 σ2 it + 1 1 − ψ 1 − γ γ X s=e,r πsκh (1 − δ)σ r i,(cs t+1−wt+1)+ δσ r i,(hs t+1−wt+1) σ2 it i ! (3.37) = αM De it + α HDe it . (3.38)
where V art(ri,t+1) = σ2it, cov(ri,t+1, ct+1s −wt+1) = σi,cst+1−wt+1 and cov(ri,t+1, h
s t+1−
wt+1) = σi,hs
t+1−wt+1.
Proof : See Appendix (3.E).
The first equation characterizes the optimal portfolio choice for the risky asset in the retirement state when there is no labour income and the second one with labour income. These equations have two parts. The first part, αM D
it captures
any asset demand induced completely from the current risk premium adjusted for Jensen’s inequality by adding one half the own variance, called the ”myopic demand” of risky asset. The myopic demand corresponds to the single-period
16Equation (3.36) gives us valuable information to the determinants of optimal risky asset allocation, however, it is not a complete solution of the model because the current optimal port- folio allocation is a function of future portfolio and consumption decisions which are endogenous in our model. This dependence on future consumption and portfolio decisions operates through the conditional covariances. The conditional covariances depends on the log non-durable con- sumption to wealth ratio and log durable housing consumption to wealth ratio. These equations can be solved forward and expressed in terms of expectations of future consumption and port- folio returns, see Campbell (1993) equation (3.9).To solve for an exact solution to optimal consumption and portfolio policies, the method ofCampbell and Viceira(1999) can be applied to guess a functional form for these policies and identify the parameters using the technique of undetermined coefficients. This analysis is beyond the scope of this chapter and is left for future work. Instead we characterize the solution as we are only interested in the economic intuition behind these solutions. The quantitative analysis is left for the numerical section.
demand for an asset, when there are no changes in the investment opportunity set, as in the traditional single-period portfolio choice problems. This myopic component is directly proportional to the risk premium, Etri,t+1− rf + 12σ2it, and
inversely proportional to the investor’s risk aversion, γ.
The second term, αitHD describesMerton (1969,1973)’s ”inter-temporal hedg- ing demand”. The hedge demand corresponds to the additional demand for an asset, when the changes in the investment opportunity set are incorporated in the portfolio choice problem, as in the multi-period portfolio choice problem of
Merton (1973). This component arises when the investor seeks to hedge against
future shocks to the investment opportunity set. As investment opportunities are varying over time, long-term investors care about shocks to investment opportu- nities. In other words, the productivity of wealth also matters and not just the wealth itself.
Samuelson(1969) andMerton(1971) state conditions under which a long term
investor finds it optimal to act myopically, choosing the same portfolio as a short term investor. These include power utility and IID returns. Power utility (also logarithmic utility) implies constant relative risk aversion nullifying our model of recursive preferences. Next, if returns are IID no new information arrives between one period and the next, so there is no reason for the portfolio choice to change inter-temporally. Thus, both conditions imply that there are no changes in time over investment opportunities that might induce changes in consumption (durable and non-durable) relative to wealth. Campbell and Viceira (2001) equate these conditions to a constant consumption-wealth ratio meaning that
σi,ct+1−wt+1 = 0
σi,ht+1−wt+1 = 0.
(3.39)
Thus, we are left with just the myopic part of risky asset demand,
αrit= αM Dr it = 1 γ Etri,t+1− rf +12σit2 σ2 it (3.40)
which is exactly the result ofViceira(2001) for the retirement state. This equation states that optimal portfolio choice is independent of the level of wealth and is only optimized over the mean and variance of the risky return. The presence of a durable good makes no difference to the portfolio rule. In contrast, our model specifies time varying investment opportunity sets, as expected returns are state dependent, and hence the hedging component is non-zero meaning that the presence of the durable good does influence the proportion of wealth invested in
the risky asset. Unfortunately as we do not have a complete analytical solution we cannot exactly pin-point the way in which the durable housing good impacts αt. The only point we make is that housing forms a kind of background risk for
the investor meaning that it is undiversifiable and hence should, all else constant, bring down αt.
Nevertheless, three important results can be derived from the Proposition. Firstly, we find that the relative risk aversion parameter γ is inversely related to αt and the elasticity of intertemporal substitution ψ is directly related to αt.
Both these parameters are thus found to have opposite affects on the optimal risky equity demanded. The fact that increasing risk aversion decreases risky asset demand is universal throughout the literature, see Campbell and Viceira
(1999), Barberis (2000), Campbell (2006) etc. This is true even in the presence
of housing, for example Flavin and Yamashita (2002) find decreasing amount of wealth invested in risky stocks or housing with increasing risk aversion in their quantitative analysis using PSID data. However, existing literature is conflicted regarding the effect of the EIS parameter on αt. For example Vissing-Jorgensen
(2002), Gomes and Michaelidis (2005) and Gˆarleanu and Panageas (2015) find
that higher EIS motivates more consumption smoothing and thereby higher sav- ings and risky asset accumulation. However, Vestman (2012) predicts using a lifecycle portfolio choice model (with housing) that higher EIS lowers risky equity demand. In this chapter consistent with our analytical prediction, Proposition 1, our numerical model also shows the same positive relationship between EIS and αt.
A second result that we can derive from Proposition 1 is that in the absence of any correlation between consumption, house prices, labour income or risky re- turns, the optimal portfolio share of savings in the risky asset simplifies to the myopic demand. In other words there is no hedging component. If we set all the correlations or covariances to zero, we get αit = αM Dit . This proposition becomes
very valuable in our numerical analysis where for the benchmark model we set all correlations to zero. We then impose empirically calibrated covariances or corre- lations of labour income, housing prices etc with returns so that we can quantify the hedging demand. It has to be noted that the absence of hedging motives does not imply that the risky asset demand is fully myopic because investors, specially when returns are mean reverting, can accrue huge wealth by timing the market. That is, optimal strategies contains some planning for the future.
A third and final result that we get from Proposition 1 is that risky asset demand under time varying returns or return predictability can be substantially
different from the IID case. Importantly, if the factor predicting returns is high so will be the expected excess returns and hence a higher risky asset is demanded, refer eq. 3.12 and the subsequent ones. Furthermore, a higher persistence φ of the return predicting factor also results in an increased share of the risky equity demanded. A unit root persistence is suggestive of a market bubble implying under such speculative markets αt goes up. Eraker et al. (2003), Broadie et al.
(2007) and Elkamhi and Stefanova (2015) to name a few finds strong evidence for jumps in returns during the periods of stock market bubbles. A very low realization of the factor can arise due to a market crash, alternatively termed as ”rare disaster”, when there is a huge drop in all macroeconomic variables. Hence, αtis found to be procyclical and as factor processes follow a long run mean growth
rate so will the risky equity demanded. This explains why some recent empirical studies such as Guiso and Sodini (2013) find that the level of wealth invested in risky equities has been steadily increasing.
The proof for these three results are fairly obvious from Proposition 1 and requires no algebraic work, hence omitted from discussion. The analysis in this section abstracted from realistic arguments such as the presence of transaction costs, borrowing constraints, etc. Furthermore, the proposition 1 was more intu- itive, the arguments were not based on an exact solution of either consumption or portfolio choice.