Bayes' Formula and Updated Probabilities

Bayes' formula allows an analyst to revise (update) a prior probability after receiving new information. In an investment context, this is essential for portfolio risk analysis — for example, updating the probability that a market is in a particular regime given observed returns. Key Definitions: $P(A)$ = prior probability of event $A$ $P(B \mid A)$ = likelihood of observing evidence $B$ given $A$ is true $P(B)$ = total (unconditional) probability of evidence $B$ $P(A \mid B)$ = posterior (updated) probability of $A$ given $B$ Bayes' Formula: $P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$ Total Probability Rule (used to compute $P(B)$): $P(B) = P(B \mid A_1) \cdot P(A_1) + P(B \mid A_2) \cdot P(A_2) + \cdots + P(B \mid A_n) \cdot P(A_n)$ where $A_1, A_2, \ldots, A_n$ are mutually exclusive and exhaustive events. Conditional Expectation: $E(X) = \sum_{i} P(A_i) \cdot E(X \mid A_i)$ This formula weights scenario-specific expected values by their (possibly updated) probabilities to produce a composite expected value.

Scenario: A quantitative portfolio manager at an asset management firm monitors a global equity fund. She classifies market conditions into three mutually exclusive and exhaustive states:

• $S_1$: Bull Market — prior probability $P(S_1) = 0.40$
• $S_2$: Neutral Market — prior probability $P(S_2) = 0.35$
$S_3$: Bear Market — prior probability $P(S_3) = 0.25$

The fund's risk model assigns the following conditional expected annual returns and conditional standard deviations for the portfolio under each state:

Market State	Expected Return	Standard Deviation
$S_1$: Bull	$+18\%$	$12\%$
$S_2$: Neutral	$+5\%$	$8\%$
$S_3$: Bear	$-14\%$	$20\%$

The manager's leading economic indicator (LEI) model generates a signal $I$ (indicating "positive momentum") at the start of the year. Based on historical calibration, the conditional probabilities of observing signal $I$ under each state are: $P(I \mid S_1) = 0.70, \quad P(I \mid S_2) = 0.40, \quad P(I \mid S_3) = 0.15$ Using this framework, answer the following:

1. Calculate the unconditional probability of observing signal $I$, i.e., $P(I)$.
2. Using Bayes' formula, calculate the updated (posterior) probabilities $P(S_1 \mid I)$, $P(S_2 \mid I)$, and $P(S_3 \mid I)$.
3. Calculate the unconditional expected return of the portfolio using the prior probabilities.
4. Calculate the updated expected return of the portfolio using the posterior probabilities from part (2).
Calculate the updated total variance of the portfolio using the posterior probabilities, accounting for both within-state variance and across-state variance (i.e., the law of total variance). Express the answer as a variance and as a standard deviation (in \%).

Step 1: Unconditional Probability of Signal $I$

Apply the total probability rule: $P(I) = P(I \mid S_1) \cdot P(S_1) + P(I \mid S_2) \cdot P(S_2) + P(I \mid S_3) \cdot P(S_3)$ $P(I) = (0.70)(0.40) + (0.40)(0.35) + (0.15)(0.25)$ $P(I) = 0.280 + 0.140 + 0.0375$ $\boxed{P(I) = 0.4575}$

Step 2: Posterior Probabilities via Bayes' Formula

$P(S_i \mid I) = \frac{P(I \mid S_i) \cdot P(S_i)}{P(I)}$ For $S_1$ (Bull): $P(S_1 \mid I) = \frac{(0.70)(0.40)}{0.4575} = \frac{0.280}{0.4575} \approx 0.6120$ For $S_2$ (Neutral): $P(S_2 \mid I) = \frac{(0.40)(0.35)}{0.4575} = \frac{0.140}{0.4575} \approx 0.3060$ For $S_3$ (Bear): $P(S_3 \mid I) = \frac{(0.15)(0.25)}{0.4575} = \frac{0.0375}{0.4575} \approx 0.0820$ Verification: $0.6120 + 0.3060 + 0.0820 = 1.0000$ ✓

Step 3: Unconditional Expected Return (Prior Probabilities)

$E(R) = \sum_i P(S_i) \cdot E(R \mid S_i)$ $E(R) = (0.40)(18\%) + (0.35)(5\%) + (0.25)(-14\%)$ $E(R) = 7.20\% + 1.75\% + (-3.50\%)$ $\boxed{E(R) = 5.45\%}$

Step 4: Updated Expected Return (Posterior Probabilities)

$E(R \mid I) = (0.6120)(18\%) + (0.3060)(5\%) + (0.0820)(-14\%)$ $E(R \mid I) = 11.016\% + 1.530\% + (-1.148\%)$ $\boxed{E(R \mid I) \approx 11.40\%}$ The positive momentum signal substantially raises the expected return from $5.45\%$ to $11.40\%$, driven by the large upward revision in the Bull Market probability.

Step 5: Updated Total Variance (Law of Total Variance)

The Law of Total Variance states: $\text{Var}(R \mid I) = \underbrace{\sum_i P(S_i \mid I) \cdot \sigma_i^2}_{\text{Within-state variance}} + \underbrace{\sum_i P(S_i \mid I) \cdot \left[E(R \mid S_i) - E(R \mid I)\right]^2}_{\text{Across-state variance}}$ Let $\mu = E(R \mid I) = 11.40\%$, and note all values below are in \% units. Within-state variance component: $\begin{array}{rcl} W &=& (0.6120)(12^2) + (0.3060)(8^2) + (0.0820)(20^2) \\ &=& (0.6120)(144) + (0.3060)(64) + (0.0820)(400) \\ &=& 88.128 + 19.584 + 32.800 \\ &=& 140.512 \end{array}$ Across-state variance component: $\begin{array}{rcl} B &=& (0.6120)(18 - 11.40)^2 + (0.3060)(5 - 11.40)^2 + (0.0820)(-14 - 11.40)^2 \\ &=& (0.6120)(6.60)^2 + (0.3060)(-6.40)^2 + (0.0820)(-25.40)^2 \\ &=& (0.6120)(43.56) + (0.3060)(40.96) + (0.0820)(645.16) \\ &=& 26.659 + 12.534 + 52.903 \\ &=& 92.096 \end{array}$ Total Variance: $\text{Var}(R \mid I) = 140.512 + 92.096 = 232.608 \; (\%^2)$ Total Standard Deviation: $\sigma(R \mid I) = \sqrt{232.608} \approx \boxed{15.25\%}$

Summary of Results

Metric	Prior (Before Signal)	Posterior (After Signal $I$)
$P(\text{Bull})$	$0.4000$	$0.6120$
$P(\text{Neutral})$	$0.3500$	$0.3060$
$P(\text{Bear})$	$0.2500$	$0.0820$
Expected Return	$5.45\%$	$11.40\%$
Standard Deviation	—	$15.25\%$