Expected number of bond defaults this year

Question

A small portfolio holds 4 bonds. Over the next year they default with probabilities 0.10, 0.20, 0.05, and 0.15 respectively. On average, how many of the 4 bonds default this year?

Accepted Answer

Put one indicator on each bond: Xi = 1 if bond i defaults, else 0. The expected value of an indicator is just the probability it equals 1, so E[Xi] is that bond's default probability. The total number of defaults is X = X1 + X2 + X3 + X4, and by linearity of expectation the mean of the sum is the sum of the means: E[X] = 0.10 + 0.20 + 0.05 + 0.15 = 0.5. Note the bonds did not need to share the same probability, and they did not even need to be independent, linearity holds regardless. Takeaway. For a sum of yes/no events, the expected count is simply the sum of their individual probabilities. …

Expected number of bond defaults this year

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