Given a set of n random variables, a covariance matrix M is an n × n matrix where M(i, j) = cov(Zi, Zj). Recall that cov(Zi, Zj) = titj?ij, where ti is the standard deviation of asset i and ?ij is the correlation coefficient of assets i and j. Consider the set of n assets, with defaults modeled by n random variables with binomial distributions. Recall that for a binomial process with probability P, the variance is P × (1 – P). For example, flipping a fair coin a number of times constitutes a binomial process with P = 50%. Suppose every asset shares the same correlation coefficient ?. Build the covariance matrix. The sum of all elements of this matrix is the variance of the total number of defaults. Derive an expression for, and plot, the standard deviation of the total number of defaults as a function of different correlation coefficients (from 0% to 100%) for different numbers of assets (1, 10, 100, 1000). What conclusions can you draw?
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