Consumo musical
OMNÍVOROS
3.1.5 Papel de los medios masivos de comunicación y TIC en el consumo musical
Our empirical results obtained from the estimation of optimal consumption and portfolio rules using only financial assets and financial wealth can be summarized as follows:
1. The intercept parameter η0is non-zero and significant.
2. The wealth-dependence term ηw is positive and significant.
3. The curvature parameter γ is positive, realistic, and lower under WDU.
4. Risk aversion is counter-cyclical.
Because there is a relative paucity of empirical results in goods space for these models, it is difficult to establish whether or not our findings make sense on a comparative basis. Nonetheless, we can use the implied theoretical restrictions in price space to obtain further perspective on our goods-space results.
We consequently consider the implications of our model and of our results for asset returns. For that purpose, we relax the assumption that the investment set is constant. Again, we can resort to the iso-morphism result of Schroder and Skiadas (2002) to map the result from the dual problem to the primal problem. A standard argument establishes that equilibrium state-price deflator in the dual market is given by the marginal utility of consumption:
ˆ
πt= e−ρtUˆˆc,t. (41)
Based on this we can:
1. compute the process for dˆπt/ˆπtusing dual utility (10) and dual consumption (9),
2. compute the dual short rate and risk premia processes ˆrt, µp,tˆ − ˆrt using (14) and (16),
3. map those dual expressions back into their primal counterparts rt, µp,t− rtby inverting (15) and (17).
These calculations reveal that the risk premia is:
µp,t− rt= RtrcCovt
The risk-free rate is given by:
are the consumption and wealth relative risk aversion indices.
Risk premia The premia (42) is a two-factor pricing model, with the C-CAPM consumption beta supple-mented by the CAPM total wealth return beta. In particular, (7) reveals that the quantity of consumption risk, i.e. Covt(dCt/Ct, dPt/Pt), is priced by the Arrow-Pratt risk aversion level, measured with respect to consumption, i.e. Rrct . Similarly, the quantity of the market risk, i.e. Covt(dWt/Wt, dPt/Pt), is priced by the corresponding Arrow-Pratt risk aversion, measured with respect to wealth, i.e. Rrwt . The model can thus be interpreted as a linear combination of a static CAPM (ηc= 0), and a standard C-CAPM (ηw= 0), where the weights depend on the relative contributions of consumption and wealth to the agent’s utility.
Duffie and Epstein (1992) also obtain a two-factor model, although their model is derived under non-expected utility, rather than VNM preferences. In addition, the relative weights depend on the distance between risk aversion, and the inverse of the elasticity of inter-temporal substitution. Hence, an expected-utility maximizer (risk aversion inversely equal to elasticity of inter-temporal substitution) does not price market risk. In contrast, our agent maximizes expected utility but, for ηw6= 0, nonetheless values market risks. Moreover, the relative weights under WDU reflect the importance of consumption versus wealth risk aversion. This is fortunate to the extent that it provides an intuitively appealing interpretation where each risk is being priced by its corresponding risk aversion measure.
Our estimates indicate that optimal consumption is not proportional to wealth (finding 1). This has important consequences for the pricing equations. To see this consider the case where η0 = 0 in (19).
Then, the consumption/wealth ratio is constant, and the growth rates on consumption and wealth are equal:
dCt/Ct = dWt/Wt. Consequently, so are the covariance terms. Substitute in the premium (42) to obtain that:
which is simply the standard C-CAPM with CRRA preferences, in which wealth dependence plays no role.
Hence, our finding 1 that η06= 0 is important to allow for wealth dependence to impact asset returns. In this perspective, our unequivocal rejection of the CRRA model can be interpreted as the dual in goods space of its empirical anomalies in price space.
Second, the presence of a second source of IMRS risk is a welcomed addition in finding a solution for the equity risk premium puzzle. Finding 2 establishes that ηw> 0 such that the price of the market risk, Rrwt , is positive. This result is consistent with the multi-factor empirical literature which finds that market risk is positively valued by the market (Chen et al., 1986; Ferson and Harvey, 1991). If the quantity of market risk is also positive, then a high equity premia need not be explained by consumption risk alone. This is also confirmed in our data set. Table 4 establishes that the total wealth risk of corporate stocks is much larger (by a ratio of 91:1) than consumption risk. A consequence of estimating ηw> 0 is that this larger market risk can justify the high observed premia at a lower level of risk aversion. This is consistent with our finding 3 that the curvature parameter γ, and the risk aversion estimates in general, are lower under WDU. These results parallel those of A¨ıt-Sahalia et al. (2004) who derive the pricing kernels under direct preference for luxury goods (measured essentially by durables, and thus a component of total wealth). Imposing positive marginal utility of luxury goods (equivalent in our case to ηw> 0), they find that luxuries, who are mainly owned by rich people, are more covariant with returns. Consequently, a larger quantity of risk, and a lower risk aversion justify the observed premia.
[ Insert Table 4 here ]
Third, note that the prices of both risks will in general be time-varying. Our results indicate that relative risk aversion with respect to consumption (wealth) was counter- (pro-) cyclical, with the overall indirect utility risk aversion being counter-cyclical (finding 4). This result would be consistent with the predictability puzzle whereby the conditional premia are observed to fall during booms, and pick up during recessions (Cochrane, 1997; Guvenen, 2003a). In the absence of strong conditional heteroscedasticity effects in the quantities of consumption or market risks, predictability would be explained in our model by cyclical movements in risk aversion. Similar counter-cyclical properties are obtained by Yogo (2003). Resorting to direct preferences for durables, he explains the predictability of expected stock returns through cross effects in marginal utility over durables and non-durables. During a recession, durables consumption typically falls. For sufficiently high elasticity of substitution between durables and non-durables, the marginal utility of non-durables increases. Consequently, the required premia for holding the risky asset also increases during troughs. In our case, as discussed earlier, ηw> 0 yields a similar increase in the marginal utility of consumption when wealth falls during recessions.
Risk-free rate As is well known, the risk-free rate puzzle is a by-product of the equity premium puzzle (Weil, 1989; Kocherlakota, 1996). A high risk aversion implies a low elasticity of inter-temporal substitution, and a high risk-free rate to induce savings. We have already mentioned that wealth dependence result in lower curvature indices (finding 3), thereby potentially addressing the risk-free rate puzzle.
Nonetheless, it is interesting to study the impact of WDU for the predicted risk-free rate. As in the standard case, the risk-free rate (43) captures a first-order and a second-order effect reflecting the mean and the variance of the IMRS. In our model however, marginal utility depends on movements in both consumption and in wealth.
Our empirical findings would be consistent with a low risk-free for a number of reasons. First, the effective discount rate in (43) is now ρ − ηw/ηc; a positive ηw consistent with blas´e behavior (finding 2) reduces it and consequently helps in reproducing the low observed rt. Second, a low rtis achieved if the second-order effect on IMRS is stronger than the first-order one. More precisely, allowing for wealth dependence affects
both the mean (through the conditional mean terms for consumption and wealth growth) and the variance of the IMRS (through their conditional variance and covariance terms).
In particular, regardless of the sign of ηw, the variance of innovations in wealth enters negatively and reduces the risk-free rate. Table 4 shows that the volatility of wealth growth is more than 60 times larger than that of consumption growth (Lettau and Ludvigson, 2003, p. 2, also find that measured wealth growth is much more volatile than consumption growth over short horizons). This effect should therefore be important towards reducing the predicted rates. Moreover, the sample moments indicate that consumption growth is positively correlated with wealth growth. Since ηw was estimated to be positive this covariance in (43) tends to reduce further the predicted rate. Note however that ηw> 0 implies that the mean growth rate of wealth affects positively the predicted risk-free rate. Since the empirical moments in Table 4 indicate that mean consumption and wealth growth rates are roughly equal, this first-order effect could be important in increasing the predicted rate.
Internal vs external WDU Our WDU model has been derived under an internal wealth-dependence effect, whereby the agent’s own wealth affects his utility. In contrast, Campbell and Cochrane (1999a) consider an external habit where bliss is determined by the other agents’ consumption levels. It seems relevant therefore to ask how our results are modified if we substitute the aggregate wealth level, say ¯Wt, instead of the personal wealth Wtin the preferences (3).
It can be shown that the premium (42) is unaffected when wealth preferences are external. Since aggregate wealth is beyond the agent’s control in the latter case, external wealth is simply an exogenous state variable that conditions preferences. Following Cox et al. (1985), if this variable is valued by the investor, it is priced, if in addition it covaries with other assets, then it warrants a premium.13 However, under external WDU, the risk-free rate (43) is modified compared to internal wealth preferences. In particular, the use of Envelope theorem under internal wealth preferences implies that the effective discount rate is ρ − ηw/ηc; under external wealth preferences, this rate is simply ρ. The agent internalizes the fact that he can (partially) control future wealth, and therefore future bliss. Consequently, at the optimum, the agent’s subjective rate
of time preference is affected. A ratchet investor (ηw < 0) is more impatient since higher future wealth raises bliss and its associated marginal utility risk; a blas´e investor (ηw> 0) is more patient for the opposite reasons.
To conclude, our wealth-dependent framework has the theoretical potential to successfully address the three main pricing anomalies of the C-CAPM. Our empirical findings in goods space are consistent with a WDU explanation of empirical asset returns puzzles. Whether or not similar estimation results in price space are obtained will require further analysis which we leave on the research agenda.
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Notes
1Static problems define utility over terminal wealth, but implicitly assume that this wealth is entirely consumed.
2Clearly the two observations are linked to one another under VNM preferences.
3The data is obtained from the Flow of Funds, is made available by the Board of Governors of the Federal Reserve Bank, and is discussed in further details below.
4Our cyclical patterns are robust to the choice of the wealth series in computing consumption and portfolio shares. Replacing ‘Financial wealth’ by the more comprehensive ‘Net worth’ (i.e. including tangible assets, mutual funds, pension plans, . . . net of liabilities) in the denominator in order to derive the consumption and portfolio shares has no qualitative incidence on the patterns identified in Figure 1.
5The de-trended series are measured as deviations from a quadratic deterministic trend for log consump-tion and log wealth respectively.
6See also Falato (2003), or Barberis et al. (2001) for wealth-dependent utility models allowing for rotations in the marginal utility schedules.
7The theoretical restriction (5) stems from the financial problem we are analyzing. Consider the discrete-time analog of maximizing (1) subject to (2). First-order and Envelope conditions yield the following:
Uc,t= exp(−ρ)Et{[Uc,t+1+ Uw,t+1]Ri,t+1},
or,
1 = exp(−ρ)(1 + ηw/ηc)Et
( ηcCt+1+ η0+ ηwWt+1
ηcCt+ η0+ ηwWt
−γ
Ri,t+1
) .
This Euler equation has a familiar representation, with the exception that the subjective discount factor is now modified to allow for wealth dependence. It is reasonable to expect that the effective discount factor, exp(−ρ)(1 + ηw/ηc) ∈ (0, 1). Restriction (5) follows immediately.
8The bliss level is fictitious in the sense of being a subjective, possibly negative, reference level.
9In comparison, typical habit models define the benchmark habit stock in terms of cumulated lagged consumption (e.g. Constantinides, 1990, p. 522):
U = U (Ct, Xt), where Xt≡ Z t
0
exp[−a(t − s)]Csds + exp(−at)X0.
10Following standard practices, the risk-free rate r is calibrated to its mean value.
11For example, pension reserves are typically invested differently by fund managers whether they are defined benefit or defined contribution. Finding a unique pricing index for this series in the absence of detailed information on the funds’ composition is impractical.
12In particular, (20) reveals that, for the risk-free asset:
V0,t= v00+ vw,0Wt (44)
where,
v00= −v10− v20, vw0= 1 − v1w− v2w. (45)
13Clearly, modifying the model so that the agent’s bliss now depend on the difference between own and aggregate wealth, i.e. ηw(Wt− ¯Wt) would eliminate the market risk factor in the premia (42). In equilibrium, a representative agent’s wealth is also the aggregate wealth: Wt= ¯Wt.
Tables
Table 1: Sample moments: Shares of wealth
mean std correlation
Consumption 0.525 0.096 1.000 0.907 -0.233 -0.947 Deposits 0.473 0.123 1.000 -0.519 -0.970
Bonds 0.088 0.031 1.000 0.295
Stocks 0.440 0.110 1.000
Table 2: Sample moments: Real Returns
mean std correlation
Deposits 0.017 0.022 1.000 0.236 0.005
Bonds 0.036 0.130 1.000 0.212
Stocks 0.137 0.342 1.000
Table 3: Parameter Estimates
Param. Estim. T-stat. Estim. T-stat. Estim. T-stat.
A. CRRA B. HARA C. WDU
γ 6.187 4.822 5.799 2.747 5.341 2.937
η0 0 — -17.319 -49.462 -93.815 -3.943
ηw 0 — 0 — 0.019 3.215
µ1 0.007 3.329 0.007 2.395 0.008 4.061
µ2 0.026 7.602 0.028 6.129 0.023 3.846
Q(1, 1) 0.031 19.369 0.031 19.685 0.030 19.894 Q(1, 2) 0.017 2.573 0.017 3.008 0.016 2.935 Q(2, 2) 0.080 19.073 0.079 19.914 0.080 20.204
Note: Estimated model (39). Assets: [Deposits, Bonds, Stocks], Wealth: Wt= V0,t+ V1,t+ V2,t. Sample period: 1952:II–2000:IV. Fixed parameters ρ = (1 + 0.035)1/4− 1, and ηc= 1. µp are the drift parameters, Qpp ≡ Chol(Σpp) is the Cholesky root of the covariance matrix of the returns process.
Table 4: Sample moments: Consumption, wealth growth and stock returns
mean std covariances
Consumption growth 0.02356 0.02079 0.00043 0.00051 0.00051
Wealth growth 0.03658 0.16291 0.02654 0.04701
Stock returns 0.13811 0.34218 0.11709