Probability concepts — CFA L1 Quant

Probability quantifies uncertainty. Every investment decision is implicitly a probability statement — you bet that an outcome is likely enough to justify the price. CFA tests probability formally because most retail investors (and many professionals) reason about probability poorly. Master conditional probability and Bayes' theorem here, and you'll spot the cognitive errors that separate sophisticated investors from naive ones.

Foundation

Three building blocks: • Joint probability P(A and B): both events occur. If independent: P(A and B) = P(A) × P(B). • Conditional probability P(A | B): A occurs given B has occurred. Defined as P(A and B) / P(B). • Independent events: P(A | B) = P(A). Knowing B doesn't change the probability of A. Bayes' theorem updates probability based on new information: P(A | B) = P(B | A) × P(A) / P(B) It looks abstract but it's the most useful formula in applied probability — let me show you why.

Deep Dive

Expected value (mean): E[X] = Σ x_i × P(x_i) For a stock: weighted average of possible returns × probabilities. Variance: σ² = Σ (x_i − E[X])² × P(x_i) Standard deviation: σ = √σ² Covariance — how two variables move together: Cov(X, Y) = Σ (x_i − E[X]) × (y_i − E[Y]) × P(x_i, y_i) Correlation: ρ(X, Y) = Cov(X, Y) / (σ_X × σ_Y), bounded between −1 and +1. Portfolio variance with two assets: σ²_p = w_A² σ_A² + w_B² σ_B² + 2 w_A w_B Cov(A, B) = w_A² σ_A² + w_B² σ_B² + 2 w_A w_B σ_A σ_B ρ_{AB} The key insight: when ρ < 1, portfolio variance is less than the weighted average of individual variances. Diversification works. Worked: Asset A has SD 18%, Asset B has SD 4%, correlation 0.1, weights 60%/40%: σ²_p = 0.6² × 0.18² + 0.4² × 0.04² + 2 × 0.6 × 0.4 × 0.18 × 0.04 × 0.1 = 0.0117 + 0.000256 + 0.000346 = 0.0123 σ_p = √0.0123 = 11.1% Compared to weighted average: 0.6 × 18 + 0.4 × 4 = 12.4%. Diversification saved 1.3% of volatility — meaningful.

Worked example

Worked example — Indian banking stress test A credit model assesses Indian PSU banks vs private banks. Base rates over 10 years: P(NPA spike at PSU bank in any year) = 8% P(NPA spike at private bank in any year) = 3% A risk indicator (e.g., gross NPL > 5%) flags banks. Past performance: P(flagged | NPA spike) = 80% (model is reasonably good) P(flagged | no NPA spike) = 10% (false positive rate) Question: A PSU bank is flagged by the indicator. What's the probability it actually has an NPA spike? Apply Bayes: P(spike | flagged) = P(flagged | spike) × P(spike) / P(flagged) = (0.80 × 0.08) / [(0.80 × 0.08) + (0.10 × 0.92)] = 0.064 / (0.064 + 0.092) = 0.064 / 0.156 = 41% So even when the indicator flags a PSU bank, the probability of actual stress is 41% — far from certainty. For private banks (lower base rate 3%), the same flagging gives: = (0.80 × 0.03) / [(0.80 × 0.03) + (0.10 × 0.97)] = 0.024 / 0.121 = 20% Low base rates make positive flags harder to trust. Senior credit analysts internalise this; junior ones over-react to flags.

Real-world scenario

Real-world scenario — Mr. Iyer's "hot tip" on a small-cap Mr. Iyer's broker tells him a small-cap stock is "set to triple" based on insider information. Mr. Iyer is excited. Probabilistic analysis: Base rate — fraction of small-caps that triple in a year: ~5% P(triple) = 5% P(broker calls something a "tripler" | actually triples) = 30% (brokers do recognize some real winners) P(broker calls something a "tripler" | doesn't triple) = 20% (false-positive rate is high — brokers always have hot tips) P(actually triples | broker calls it) = (0.30 × 0.05) / [(0.30 × 0.05) + (0.20 × 0.95)] = 0.015 / (0.015 + 0.19) = 0.015 / 0.205 = 7.3% So despite the confident "tripler" call, only 7% chance it actually triples. 93% chance it doesn't. The base rate is killing the call. Mr. Iyer should treat hot tips as approximately 50/50 information at best — and never the basis for concentrated bets. This is why behavioural finance professionals say: "Listen to the base rate, not the salesman."

Advanced

Bayes' theorem in practice — the most important application: Base rates matter. A 1% base rate of default, with a credit model 90% accurate at flagging defaulters and 5% false positive on non-defaulters: P(default | flagged) = P(flagged | default) × P(default) / P(flagged) = (0.90 × 0.01) / (0.90 × 0.01 + 0.05 × 0.99) = 0.009 / 0.0585 = 15.4% Despite the model flagging the loan, only 15% chance it's actually a default risk. Base rates dominate. This counterintuitive result is why "this fund has beaten the market 7 years in a row" doesn't mean what you think — base rate of luck-based outperformance is high. CFA hammers conditional probability everywhere: prepayment given interest-rate move (fixed income), default given credit rating (credit risk), stock-up given earnings beat (event studies). Master Bayes once and it shows up everywhere.

Regulatory references

CFA Institute Curriculum — Level 1, Quantitative Methods, Reading 3
SEBI Investor Charter — emphasis on base-rate reasoning in advisory
Behavioural finance literature — Kahneman & Tversky on representativeness

Common mistakes & pitfalls

Ignoring base rates — focusing on conditional information without the prior probability.
Confusing P(A | B) with P(B | A) — different probabilities, different answers.
Treating correlated events as independent — overstates joint probability.
Using point estimates of expected value without communicating the variance.
Forgetting that ρ < 1 → diversification benefit, but ρ ≈ 1 → no benefit.

Frequently asked

How do I decide if events are independent?

Logically: does knowing one event change the probability of the other? Stock prices day-to-day are roughly independent. Earnings beats and analyst upgrades are NOT independent. Empirically: check if Cov(X, Y) = 0 in data. CFA exam typically tells you whether to assume independence; in practice, assume dependence unless proven otherwise.

What's the practical use of correlation in portfolio construction?

Lower correlation → bigger diversification benefit. Equity-equity correlation is ~0.6-0.8 (limited diversification). Equity-debt is ~0.0-0.3 (good diversification). Equity-gold is often near zero or negative in crises (great diversification). Build portfolios mixing low-correlation assets to reduce volatility without sacrificing return.

What's "Bayesian" thinking?

Updating prior beliefs based on new evidence in a mathematically consistent way. Frequentist statistics asks "what's the probability of this data given the hypothesis?" Bayesian asks "what's the probability of the hypothesis given this data?" — usually more useful for investment decisions. Most exam Bayes questions are mechanical formula application, but the mindset is what stays with you.

Practice questions

Click each question to reveal the answer and explanation.

Q 1

P(A) = 0.4, P(B) = 0.3, A and B are independent. P(A and B) =

(a)0.10
(b)0.12
(c)0.40
(d)0.70

Correct: (b) 0.12

For independent events: P(A and B) = P(A) × P(B) = 0.4 × 0.3 = 0.12.

Q 2

A coin is flipped 3 times. Probability of exactly 2 heads is:

(a)1/8
(b)3/8
(c)1/2
(d)3/4

Correct: (b) 3/8

C(3,2) × (0.5)^2 × (0.5)^1 = 3 × 0.25 × 0.5 = 0.375 = 3/8.

Q 3

Two assets have SDs 20% and 10%, correlation 0.5. Equal-weighted portfolio variance is closest to:

(a)0.0125
(b)0.0150
(c)0.0175
(d)0.0250

Correct: (a) 0.0125

σ²_p = (0.5)²(0.20)² + (0.5)²(0.10)² + 2(0.5)(0.5)(0.20)(0.10)(0.5) = 0.01 + 0.0025 + 0.005 = 0.0175. Wait... let me recalculate: σ²_p = 0.25(0.04) + 0.25(0.01) + 0.5(0.20)(0.10)(0.5) = 0.01 + 0.0025 + 0.005 = 0.0175. Answer is 0.0175 (option c).

Q 4

P(A | B) = 0.6, P(B) = 0.4. P(A and B) =

(a)0.10
(b)0.16
(c)0.24
(d)0.40

Correct: (c) 0.24

P(A and B) = P(A | B) × P(B) = 0.6 × 0.4 = 0.24.

Q 5

A test for fraud has 95% sensitivity (correctly flags fraud) and 5% false-positive rate. Base rate of fraud is 1%. If a transaction is flagged, probability it's actually fraud is closest to:

(a)16%
(b)50%
(c)90%
(d)95%

Correct: (a) 16%

Bayes: P(fraud|flagged) = (0.95 × 0.01) / [(0.95 × 0.01) + (0.05 × 0.99)] = 0.0095 / 0.059 = 16%. Despite the test's 95% sensitivity, low base rate means most flagged transactions are NOT fraud. Critical insight for AML / KYC operations.

Q 6

For a portfolio, lower correlation between assets:

(a)Always increases expected return
(b)Reduces portfolio volatility for given weights
(c)Eliminates all risk
(d)Has no effect

Correct: (b) Reduces portfolio volatility for given weights

Lower correlation = bigger diversification benefit = lower portfolio volatility for given weights. Doesn't change individual expected returns.

Educational purposes only. The numbers, returns, and examples used in this lesson are illustrative. Past performance does not guarantee future results. Mutual fund and securities investments are subject to market risks. This lesson is not investment advice; for advice tailored to your circumstances, consult a SEBI-registered Investment Adviser. Read our full disclaimer.