back start next
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [ 113 ] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183]
113 1 + P = Ef (1 + rUi) (7.35> Finally, it is convenient to let gj i denote the growth rate of consumption from t to f b 1, (Ct+i/Ct)  1, and to omit the time subscripts. Thus we ha\e "The original CAPM assumes that investors are concerned with the mean and variance of the return on their portfolio rather than the mean and variance of consumption That version of the model therefore focuses on market betasthat is, the covariances of assets returns with the returns on the market portfolioand predicts that expectedreturn premia are proportional to market betas (Lintner, 1965; Sharpe, 1964). consmnption is zero. Tlius the riskfree rate, Tf+i, satisfies a + p)u{Ct) btlU (Lr+i)J Subtracting (7.32) from (7.31) then gives us FfH 1 r aCovtd + r/Vi.Q+j) ttlU (Lf+i)J Equation (7.33) states that the expectedreturn premium an asset must offer relative to the riskfree rate is proportional to the covariance of its return with consumption. This model of the determination of expected asset retums is known as the consumption capitalasset pricing model, or consumption CAPM. The covariance between an assets return and consumption is known as its consumption beta. Thus the central prediction of the consumption CAPM is that the premiums that assets offer are proportional to their consumption betas (Breeden, 1979; see also Merton, 1973, and Rubinstein, 1976). Empirical Application: The EquityPremium Puzzle One of the most important implications of this analysis of assets expected retums concerns the case where the risky asset is a broad portfolio of stocks. To see the issues involved, it is easiest to return to the Euler equation, (7.28), and to assume that individuals have constantrelativeriskaversion utihty rather than quadratic utility. With this assumption, the Euler equation becomes Cr=j~EAa + ri,)CrAl (7.34) where is the coefficient of relative risk aversion. If we divide both sides by Cf" and multiply both sides by 1  p, this expression becomes
+ ( + 1){(£[0 )2+ (0)}  p. (7.38) When the time period involved is short, the Elr]Elg] and (£[0*]) terms are small relative to the others, Omitting these terms and solving the resulting expression for E[r] yields £[r] p b eElg] + eCov{r,g<=)  { + l)Var(0<). (7.39) Again, it is helpful to consider a riskfree asset. For such an asset, (7.39) simplifies to P + [ ]  e(e + l)Var(0<). (7.40) Finally, subtracting (7.40) from (7.39) yields £[r]r = eCov(r,0<). (7.41) In a famous paper, Mehra and Prescott (1985) show that it is difficult to reconcile observed asset returns with equation (7.41). Mankiw and Zeldes (1991) report a simple calculation that shows the essence of the problem. For the United States during the period 18901979 (which is the sample that Mehra and Prescott consider), the difference between the average return on the stock market and the return on shortterm government debtthe equity premiumis about six percentage points. Thus if we take the average return on shortterm government debt as an approximation to the average riskfree rate, the quantity fir]  r is about 0.06. Over the same period, the standard deviation of the growth of consumption (as measured by real purchases of nondurables and services) is 3.6 percentage points, and the standard deviation of the return on the market is 16.7 percentage points; the Indeed, for the continuoustime case, one can derive equation (7.39) without any approximations. £[(1 + )(1+0" "] = 1+P. (7.36) To see the imphcatlons of (7.36), we take a secondorder Taylor approximation of the lefthand side around r = g = 0. Computing the relevant derivatives yields (1 + r)(l +g)< 1 +   + ( l)g. (7.37) Thus we can rewrite (7.36) as E[r]  e£[0<]  e{£[r]£[0<] + Cov(r,0<)}
Cochrane and Hansen (1992) provide an overview of work on the puzzle and a framework for thinking about proposed explanations. For some proposed explanations, see Mankiw (1986b); Mankiw and Zeldes (1991); Constantinides (1990); Campbell and Cochrane (1995); Weil (1989b); Epstein and Zin (1991); and Problem 7.10. correlation between these two quantities is 0.40. These figures imply that the covariance of consumption growth and the return on the market is 0.40(0.036)(0.167), or 0.0024. Equation (7.41) therefore implies that the coefficient of relative risk aversion needed to account for the equity premium is the solution to 0.06 = e(0.0024), or = 25. This is an extraordinary level of risk aversion; it implies, for example, that individuals prefer a 17% reduction in consumption with certainty to a 1 in2 chance of a 20% reduction. As Mehra and Prescott describe, other evidence suggest that risk aversion is much lower than this. Among other things, such a high degree of aversion to variations in consumption makes it puzzling that the average riskfree rate is close to zero despite the fact that consumption is growing over time. In addition, the problem becomes even more severe if we focus on the postwar period. Mankiw and Zeldes report that for the 19481988 period, the average equity premium is 8 percentage points, the standard deviation of consumption growth is 1.4 percentage points, the standard deviation of the market return is 14.0 percentage points, and the correlation of consumption growth and the market return is 0.45. These numbers imply a value of of 0.08/[0.45(0.014)(0.140)]  91. The large equity premium, particularly when coupled with the low riskfree rate, is thus difficult to reconcile with household optimization. This equitypremium puzzle has stimulated a large amount of research, and mam explanations for it have been proposed. No clear resolution of the puzzle has been provided, however.! 7.6 Alternative Views of Consumption The permanentincome hypothesis provides appeahng explanations of many important features of consumption. For example, it explains wh\ temporary tax cuts appear to have much smaller effects than permanent ones, and it accounts for many features of the relationship between current income and consumption, such as those described in Section 7.1. Yet there are also important features of consumption that appear inconsistent with the permanentincome hypothesis. For example, as described in Section 7.3, both macroeconomic and microeconomic evidence suggest that consumption responds to predictable changes in income. And as we just saw, simple models of consumer optimization cannot account for the equity premium.
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [ 113 ] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183]
