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# Bonds Portfolios - Statistical Calculations

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Assume a bond fund manager manages a portfolio of 50 high yield (speculative) bonds which are mostly rated B. Suppose, based on past experience, the probability of default of one of these bonds in the next year is approximately 11.9%. Suppose the default potential of each bond in this bond portfolio is a binary random variable with mean p and variance p(1-p). Assume default is independent across bonds, so that the number of defaulted bonds in the portfolio will follow a binomial distribution with mean 50p and variance 50p(1-p) where p = 0.119.

(a) Over the next year, what is the expected number of defaults in the portfolio assuming a binomial model for defaults?
(b) Estimate the standard deviation of the number of defaults over the next year.
(c) What is the probability that the number of defaults will be 4 or less over the next year?
(d) Find a 95% confidence interval around the mean for the number of defaults we are likely to see in the coming year (Hint: you may use the following formula: PLEASE SEE JPEG ATTACHMENT).
(e) Critique the use of the binomial distribution in this context.
(f) What would happen to your confidence interval in part (d) if the bond probabilities were not independent of each other?

#### Solution Preview

Let n be the number of bonds (50), p be the probability of each bond defaulting in the next year (0.119), and X be the random variable for the number of bonds defaulting in the next year.

a) Expectation = np = 50(0.119) = 5.95.

b) Standard deviation
= Sqrt(Variance)
= Sqrt(50p(1-p))
= Sqrt(50(0.119)(0.881))
= Sqrt(5.24195)
= 2.29

c) P(X ≤ 4)
= P(X = 0) + ...

#### Solution Summary

The solution determines statistical calculations for bonds portfolios.

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