Probabilities, conditional risk and updating evidence

Learning goals

• Compute moments under discrete and common continuous distributions.
• Update probabilities and distinguish within-state risk from between-state risk.

Draft · approximately 60 minutes, excluding further revision.

Write a model with probabilities that add to one

For mutually exclusive and exhaustive scenarios, expected return is ΣpᵢRᵢ and variance is Σpᵢ(Rᵢ−μ)². Covariance uses paired outcomes from the same state: Σpᵢ(Aᵢ−μA)(Bᵢ−μB). Probabilities must be nonnegative and total one. A conditional expectation uses probabilities given information; it need not equal the unconditional expectation.

For a Bernoulli event with success probability p, mean is p and variance p(1−p). For a binomial count of n independent identical trials, mean is np and variance np(1−p). A normal variable has mean μ, variance σ², zero skewness and raw kurtosis 3. A lognormal price is positive and right-skewed: if ln P is normal with mean m and variance v, E[P] = exp(m+v/2) and Var[P] = [exp(v)−1]exp(2m+v). Model choice is a hypothesis about the data, not an assurance that losses are normally distributed.

Condition, then combine

Bayes’ rule is P(H|E) = P(E|H)P(H)/P(E). The denominator includes evidence under both the hypothesis and its alternatives. A very accurate signal can still produce many false alarms when the event is rare. Always show the base rate and the likelihood of the observed signal under each state.

Total expectation is E[R] = E[E[R|S]]. Total variance is E[Var(R|S)] + Var(E[R|S]); simply averaging conditional variances omits the uncertainty between states. Total covariance similarly combines average conditional covariance with covariance of conditional means. This decomposition explains why a calm estimate within each regime can hide substantial regime-switching risk.

A base-rate table prevents misleading probability claims

Among 1,000 hypothetical issuers, 100 default and 900 do not. A signal flags 80 of the defaulters and 90 of the nondefaulters. Sensitivity is 80/100=80%; the false-positive rate is 90/900=10%. Of 170 flagged issuers, only 80 default: P(default|flag)=80/170≈47.06%. The 80% detection rate is not the probability of default after a flag.

The posterior depends on both signal performance and the starting default frequency. Applying a model estimated on distressed issuers to a high-quality universe without recalibration can overstate risk. In a probability tree, each branch must use a conditional probability for its own parent node; joint probabilities multiply along branches and mutually exclusive paths add. Check that terminal probabilities sum to one.

Formula and assumptions

E[R] = ΣpR; Var(R) = E[Var(R|S)] + Var(E[R|S]); P(H|E) = P(E|H)P(H)/P(E)

An imperfect credit warning

1. Prior default probability is 5%. A warning appears for 80% of defaults and 10% of nondefaults.
2. P(warning) = 0.80×0.05 + 0.10×0.95 = 0.135.
3. P(default|warning) = 0.04/0.135 = 29.6296%. The signal raises risk materially but does not make default certain.

Calculation lab

Errors to catch

• Using signal accuracy as the posterior probability.
• Averaging conditional variances without adding between-state variance.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org