重解股票风险溢价之谜：市场崩盘的影响

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"The equity risk premium 是一个在金融市场中重要的概念，它指的是投资者投资股票市场相对于无风险资产（如国债）所要求的额外回报。在 Mehra 和 Prescott (1985) 的研究中，他们建立了一个基于 Arrow-Debreu 资产定价模型，该模型无法解释为什么实际的股权风险溢价会如此之高。他们因此认为只有非 Arrow-Debreu 模型才能解决这个问题。在《The Equity Risk Premium: A Solution》这篇文章中，作者重新构建了 Mehra 和 Prescott 的模型，引入了可能但不太可能出现的市场崩盘情景。通过这种方式，新模型保留了原模型的吸引力，同时能够解释高股权风险溢价和低无风险回报的现象。关键在于，如果这些市场崩溃事件合理地严重且不是过于不可能发生，那么模型就能在合理的贴现率和风险厌恶程度下运行。文章首先介绍了问题的背景，即 Mehra 和 Prescott 发现的一般均衡模型对于平均股票回报和国债回报的预测与实际数据严重不符。接着，作者提出解决方案，通过考虑市场崩溃的可能性，使得模型能更好地适应实际的市场环境。在 Arrow-Debreu 模型中，资产的价格反映了所有可能未来状态的概率分布，而高股权风险溢价通常归因于市场的不确定性。然而，如果模型只考虑了常规的市场动态，就无法充分解释历史上观察到的高风险溢价。通过引入可能的极端事件，模型可以捕捉到这些罕见但影响巨大的市场崩溃，从而解释为什么投资者需要更高的回报来承担股票投资的风险。此外，作者还讨论了时间偏好和风险厌恶度对模型的影响。在新的设定下，只要市场崩溃的发生概率不是过于微小，且其后果足够严重，即使相对较低的时间偏好和适度的风险厌恶也能导致高股权风险溢价的出现。这篇文章提供了一个修正版的 Arrow-Debreu 模型，该模型通过纳入市场崩溃的潜在影响，成功地解决了原本模型无法解释的股权风险溢价难题。这一方法不仅丰富了金融市场的理论框架，也为理解和预测股票市场的风险提供了新的视角。"

T.A. Rietz, The equity risk pr~:mium 119

The agent displays constant relative risk aversion as given by the utility

function

c :~ -~)- 1

= !.-.' (5)

where a is the parameter of relative risk aversion. When a = 1, the utihty

function is iogarithmie.

The production levels, Yt, and, thus, the consumption levels, c,, evolve

through time according to

Yt+ l "~" Xt+ lYt ~" Ct+l'

(6)

where xt+~¢ {hl,...,hn} ~R + is the gross growth rate, which follows an

ergodie Markov process, i.e.,

Prob(xt+l = ~,j I x, = hi) = ¢,/.

(7)

"Fnis pr~ess allows the apparent non-stationafity we observe in the per capita

consumption stream over the sample period. It also implies that consumption

is autocorrelated and re aliTations of the growth rate affect all later consump-

tion levels°

In this model, the expected utility in eq. (1) exists if, and only if, the matrix

A, whose elements are given by aiy = fl~o-,o, is stable. This also e~tablishes

that a Debreu competitive

equilibrium

exists. 1

The period t asset prices may be expressed as functions of the current state,

(Yt, xt), by

X a

p-(y,,x,) +:(y,+. )},

pf(y,, x,)

Et { fl(Yf+~)(Yt)},

(8)

where Yt and Yt+1 have been substituted for ct and ct+ :, al~d yf~ and Yt+'~

have been substituted for U'(c,) and U'(ct+l), respectively.

Note that Yt and x s are sutiieient for forecasting Yt+l and x~+ 1 and that the

forecast depends only on the levels of Yt and x,, not on the period. Therefore,

l See Mehra and Prescott (1984, 1985). If the t = 0 consumption level is c and its growth rate is

?~i, the discounted expected utility t periods in the future is given by

Eo(Ufc,)) = (A,A'-I,: -~- #9/0 - a},

where A, is the ith row of A and t is a column vector of ones. Thus, the infinite sum in (X) will

converge only when ,4 is stable.

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