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# P1.T2.717. Bayes' Theorem (Miller, Ch.6)

#### David Harper CFA FRM

##### David Harper CFA FRM
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Learning objectives: Describe Bayes’ theorem and apply this theorem in the calculation of conditional probabilities. Compare the Bayesian approach to the frequentist approach. Apply Bayes’ theorem to scenarios with more than two possible outcomes and calculate posterior probabilities.

Questions:

717.1. Your firm employs a 95.0% value at risk (VaR) model and monitors its performance by comparing the actual daily profit and loss (P&L) to the VaR level. If the actual loss exceeds the VaR, this is called an "exception." There is an unconditional (aka, prior) 80.0% probability that your firm's VaR model is good; in this case of an accurate model, an exception will occur with 5.0% (conditional) probability. However, if the VaR model is bad (inaccurate), then the conditional probability of an exception is 10.0%. These assumptions are illustrated below.

Exceptions occur independently (i.i.d.). If we observe two (2) exceptions in a row, what is the posterior probability that the model is actually bad (bonus: what is the probability of observing an exception tomorrow)?

a. 20.0%
b. 37.5%
c. 50.0%
d. 80.0%

717.2. The bonds in a large pool (aka, population) are rated either A, B or C. Among the pool, 50.0% are A-rated, 40.0% are B-rated, and 10.0% are C-rated. The migration matrix shown below includes the probability that a bond in a given rating category will default. For example, a C-rated bond has a 15.0% probability of default but an A-rated bond has only a 4.0% probability of default.

According to Bayes' Theorem, if a bond defaults, then which is nearest to the probability that it was an A-rated bond?

a. 23.81%
b. 31.75%
c. 44.44%
d. 50.00%

717.3. An analyst at your firm has developed a new trading strategy called AlphaGen. She claims there is a 70.0% probability on any given day that the strategy offers true alpha and will generate a profit. After observing ten (10) days of performance, in fact the strategy was profitable only four days. Assume there are only two possible states of the world: either the analyst is correct and the strategy offers true alpha; or the strategy does not offer true alpha and the profit/loss outcome is equally likely to be a gain or loss on any given day. Your prior assumption was that the two states of the world are equally likely: there is s 50.0% probability the strategy offers true alpha, and a 50%probability it does not. These assumptions are illustrated below; e.g., if the strategy does generate alpha, then P[Profit | Alpha] = 70.0%, but if the strategy does not generate alpha, then P[Profit | No Alpha] = 50.0%.

We can also represent the prior probabilities as follows: P[p = 0.70] = 50.0% and P[p = 0.50] = 50.0%. Additionally, because you are an FRM candidate, you are already able to compute the the following binomial probabilities:
• Prob[Exactly 4 profits out of 10 days | True Alpha] = binomial[4 successes, 10 trials, p = 0.70, false = p.m.f.] = 0.70^4*0.30^6*C(10,4) = 3.68%
• Prob[Exactly 4 profits out of 10 days | No Alpha] = binomial[4 successes, 10 trials, p = 0.50, false = p.m.f.] = 0.50^4*0.50^6*C(10,4) = 20.51%.
Given the evidence that the strategy was profitable on four days out of ten, which is nearest to the (posterior) probability that the analyst is correct and the strategy generates true alpha? (note: this question is inspired by Miller's EOC question 6.4).

a. 15.20%
b. 25.00
c. 31.80%
d. 50.0%

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