An important parameter describing the state of capital markets, investment opportunity sets, and asset pricing is the unobservable market risk price. The estimated risk price depends upon the selected asset set, the number of assets, the investment horizon, and the sample period. We document the large theoretical and empirical probability of an unacceptably negative sample estimate for the unbiased estimator of the squared risk price. We address admissibility and dominance under quadratic loss of alternative estimators, concluding that an optimized shrinkage estimator has the smallest expected quadratic loss compared with our alternative estimators. Using market data, we examine estimates for more inclusive investment opportunity sets and provide upper bounded estimates of the risk price and resultant unique Sharpe measure, to complement the existing literature’s Sharpe measure bounds and equity premium estimates. Finally, we examine time series properties, behavior in the business cycle, and relation to some recently advocated economic factors.

描述资本市场，投资机会集和资产定价状态的一个重要参数是无法观察到的市场风险价格。估计的风险价格取决于选定的资产集，资产数量，投资期限和采样期。我们记录了风险价格平方的无偏估计量的不可接受的负样本估计量的大理论和经验概率。我们讨论了替代估计量二次损失下的可容许性和优势，得出结论：与我们的替代估计量相比，优化的收缩率估计量具有最小的预期二次损失。利用市场数据，我们检查了更具包容性的投资机会集的估计值，并提供了风险价格的上限估计以及由此产生的独特的Sharpe度量，补充现有文献的Sharpe度量范围和股票溢价估计。最后，我们研究了时间序列属性，商业周期中的行为以及与一些近期倡导的经济因素的关系。