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P1.T2.302. Bayes' Theorem (Miller)

Fran

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AIMs: Define and calculate a conditional probability, and distinguish between conditional and unconditional probabilities. Describe Bayes’ Theorem and apply this theorem in the calculation of conditional probabilities.

Questions:

302.1. There is a prior (unconditional) probability of 20.0% that the Fed will initiate Quantitative Easing 4 (QE 4). If the Fed announces QE 4, then Macro Hedge Fund will outperform the market with a 70% probability. If the Fed does not announce QE 4, there is only a 40% probability that Macro will outperform (and a 60% that Acme will under-perform; like the Fed's announcement, there are only two outcomes). If we observe that Macro outperforms the market, which is nearest to the posterior probability that the Fed announced QE 4?

a. 20.0%
b. 27.9%
c. 30.4%
d. 41.6%

302.2. The following probability matrix displays the joint probabilities with respect to two bonds, an investment grade bond and a speculative (junk) bond:



For example, the joint probability that both bonds default is 0.060%; the joint probability that both survive is 96.030%. Consider two posterior probabilities:

I. If we have already observed that the junk bond has defaulted, what is the (posterior) probability that the investment-grade bond defaulted; i.e., Prob [i default | j default]
II. If we have already observed that the investment-grade bond has defaulted, what is the (posterior) probability that the junk bond defaulted; i.e., Prob [j default | i default]

What are these probabilities, respectively?

a. Prob[i default | j default] = 0.06% and Prob[j default | i default] = 2.79%
b. Prob[i default | j default] = 1.98% and Prob[j default | i default] = 6.00%
c. Prob[i default | j default] = 3.27% and Prob[j default | i default] = 4.25%
d. Can't answer, we need unconditional (marginal) probabilities

302.3. Next year the economy will experience one of three states: a downturn, stable state, or growth. The following probability matrix displays joint probabilities of a bond default and the economic state:



For example, the joint probability that the economy is stable and the bond defaults is 1.0%; the unconditional probability that the economy will be stable is 50.0% = 49.0% + 1.0%. If we observe that the bond has defaulted, what is the (posterior) probability that the economy experienced a downturn?

a. 0.60%
b. 19.40%
c. 26.33%
d. 31.58%

Answers:
 
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