擔保債權憑證(CDO)是一種相關性的商品。這個商品的各切層價格反映多資產間聯合違約機率的相關性，因此投資人面臨相關性的風險。投資人必須去衡量這些風險，才能正確的去決定各層的公平價值。在CDO的產品中，去分解相關性矩陣來計算相關性，最常見的技巧是使用Cholesky分解。然而，Cholesky分解只能在矩陣為正定時使用。在本篇論文中，我們認為Spectral分解將可以克服上述的缺點。使用Spectral分解將一定可以獲得多資產蒙地卡羅模擬所需要的矩陣。在Cholesky分解和Spectral分解均可執行時，他們所獲的的模擬結果也將是一致的。而當Cholesky 分解不能執行時(矩陣有負的eigenvalue), Spectral分解可以獲得模擬所需的矩陣，也可以清楚的衡量出模擬結果的好壞。

CDO is a correlation product. The investors of this product involve correlation risks since the prices of respective tranches depend on joint default correlations. To determine a fair return for bearing the correlation risks, the investors must be able to measure these risks. The most common skill used to decompose the correlation
matrix for CDO products is the Cholesky decomposition. However, the Cholesky decomposition can only work for the case of a positive matrix. In this paper, we propose a Spectral decomposition which can overcome the shortcomings of the Cholesky decomposition. Spectral decomposition can always obtain matrix that Monte Carlo simulations need. Spectral decomposition will have consistent results if Cholesky decomposition can work. If Cholesky decomposition can not work (matrix has negative
eigenvalues), Spectral decomposition can still obtain a matrix that is able to measures the simulation results not matter good or bad.

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