We derive sufficient conditions for ranking performance evaluation systems in multi-task agency models (using both optimal and linear contracts) in terms of a second-order stochastic dominance (SSD) condition on the likelihood ratios. The SSD condition can be replaced by a variance-covariance matrix of likelihood ratios (VCM) condition when the utility function is square-root, the performance measures are normally distributed, and for LEN models. We identify existing results derived under the LEN assumptions that rely on the VCM condition and, thus, also hold for optimal contracts.

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We characterize the necessary and sufficient conditions for communication of private, unverifiable managerial information to be valuable for stewardship purposes. First, the hard, verifiable information must be able to confirm to some extent the truthfulness of the manager's soft, unverifiable report. Second, neither the hard, verifiable information nor the soft, unverifiable managerial report should be too informative for providing effort incentives because that makes it harder to motivate truthful reports from the manager. In particular, this conflict can be sufficiently severe that eliciting reports from the manager has no value.

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The optimization of investment portfolio is the key to financial risk investment. In this study, the investment portfolio was optimized by removing the noise of covariance matrix in the mean-variance model. Firstly, the mean-variance model and noise in covariance matrix were briefly introduced. Then, the correlation matrix was denoised by KR method (Sharifi S, Grane M, Shamaie A) from random matrix theory (RMT). Then, an example was given to analyze the application of the method in financial stock investment portfolio. It was found that the stability of the matrix was improved and the minimum risk was reduced after denoising. The study of minimum risk under different M values and stock number suggested that calculating the optimal value of M and stock number based on RMT could achieve optimal financial risk investment portfolio result. It shows that RMT has a good effect on portfolio optimization and is worth promoting widely.