Common probability distributions — CFA L1 Quant

Distributions describe how probabilities are spread across possible outcomes. The CFA tests a small set of distributions intensively because they show up in valuation models, risk management, and statistical inference. The normal distribution alone shows up in 50%+ of L1 Quant questions. Lognormal handles asset prices. Binomial handles option pricing trees. Master these and you understand the probabilistic backbone of finance.

Foundation

Discrete distributions (countable outcomes): • Binomial: n trials, each with probability p of success. P(k successes) = C(n,k) × p^k × (1−p)^(n−k). Mean = np, variance = np(1−p). • Poisson: rare events per unit time, average rate λ. P(k events) = λ^k × e^(−λ) / k!. Mean = variance = λ. Continuous distributions (any value in a range): • Uniform [a, b]: equal probability density. Mean = (a+b)/2. • Normal: bell curve, defined by mean μ and SD σ. ~68% within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ. • Lognormal: applies when underlying variable can be negative but observed is bounded by zero (stock prices, asset values). If X ~ lognormal, then ln(X) is normal. Which applies when: • Returns: typically modelled as normal • Prices: typically modelled as lognormal (can't be negative) • Default events: Bernoulli/binomial • Trade arrivals: Poisson

Deep Dive

Z-score (standardisation): Z = (X − μ) / σ Transforms any normal variable to standard normal (mean 0, SD 1). Then use Z-table to find probabilities. Key Z-values: Z = 1.00 → 84% (one-tail) Z = 1.65 → 95% (one-tail) Z = 1.96 → 97.5% (one-tail), 95% two-tail Z = 2.33 → 99% (one-tail) Z = 2.58 → 99.5% (one-tail), 99% two-tail Confidence intervals (CI) for known σ: CI = X̄ ± Z × (σ / √n) For unknown σ (use sample SD), use t-distribution with degrees of freedom = n − 1. Lognormal properties: If X ~ lognormal with parameters μ and σ: E[X] = e^(μ + σ²/2) Median[X] = e^μ Mode[X] = e^(μ − σ²) For stock prices: ln(S_T/S_0) ~ Normal(μT, σ²T) → S_T ~ Lognormal. This is the foundation of Black-Scholes option pricing.

Worked example

Worked example — equity return inference NIFTY 50 historical: mean monthly return 1.2%, SD 5%, ~20 years (240 observations). Question 1: What's the probability NIFTY drops more than 10% in a single month? Z = (−10 − 1.2) / 5 = −2.24 P(Z < −2.24) ≈ 1.3% So a single month worse than −10% should occur about once every 77 months (~6 years) under normal assumption. In reality, it has happened ~6 times in 20 years (March 2020, October 2008, etc.) — about once every 40 months. Reality is more frequent than normal predicts — fat tails confirmed. Question 2: What's the 95% confidence interval for the true mean monthly return? Sample mean = 1.2%, sample SD = 5%, n = 240 Standard error = 5 / √240 = 0.323% 95% CI = 1.2 ± 1.96 × 0.323 = (0.567%, 1.833%) per month Or 6.8% to 22% annualised. That's a wide range — even with 20 years of data, the true mean is uncertain. This is why retirement projections quoting "12% per year" should be honest about uncertainty.

Real-world scenario

Real-world scenario — risk-budget for an Indian pension fund A pension fund has ₹500 cr AUM. Current allocation: 60% equity (annual SD 16%), 40% debt (annual SD 4%), correlation 0.2. Portfolio variance: σ²_p = (0.6)²(0.16)² + (0.4)²(0.04)² + 2(0.6)(0.4)(0.16)(0.04)(0.2) = 0.009216 + 0.000256 + 0.000614 = 0.010086 σ_p = 10.04% annualised Daily SD ≈ 10.04 / √252 = 0.633% VaR_99% (one-day): 500 × 0.00633 × 2.33 = ₹7.37 cr The pension fund could lose up to ₹7.37 cr in a single day with 99% confidence. Beyond that level, regulatory inquiry and possible coverage issues. If the fund's internal risk limit is ₹6 cr daily VaR, current allocation breaches. Action: reduce equity to ~50%, raise debt to 50%, recompute. Trade-off: lower expected return for compliance.

Advanced

CFA exam favorite — VaR (Value at Risk) calculation: Value at Risk (VaR): the maximum loss expected at a given confidence level over a given period. With normal distribution assumption: VaR_95% (one-day) = portfolio value × σ_daily × 1.65 VaR_99% (one-day) = portfolio value × σ_daily × 2.33 Example: ₹100 cr equity portfolio, daily volatility 1.5% VaR_95% = 100 × 0.015 × 1.65 = ₹2.475 cr VaR_99% = 100 × 0.015 × 2.33 = ₹3.495 cr Interpretation: 95% confident the loss in one day won't exceed ₹2.475 cr. The critical caveat: real returns have fat tails (excess kurtosis). Normal-distribution-based VaR systematically understates extreme losses. The 2008 GFC exposed this; modern risk management uses CVaR (Conditional VaR) and stress tests, not just normal-VaR. CFA L1 tests recognition of this limitation. L2 introduces conditional value-at-risk; L3 applies in portfolio risk management.

Regulatory references

CFA Institute Curriculum — Level 1, Quantitative Methods, Reading 4
Basel III risk regulations — VaR-based capital requirements
IRDAI Solvency II-equivalent — uses similar distribution assumptions

Common mistakes & pitfalls

Assuming returns are normal when they actually have fat tails — VaR understates true risk.
Using daily VaR × √days for longer horizons — works only if returns are independent (often violated).
Confusing standard deviation of mean (SE = σ/√n) with SD of observations (σ).
Using mean to predict next-period outcome when distribution is skewed.
Forgetting lognormal mean ≠ e^μ; it's e^(μ + σ²/2).

Frequently asked

When should I use binomial vs normal?

Binomial for discrete events (defaults, head/tails). Normal for continuous variables, especially when n is large (CLT). For large n, binomial is approximately normal — but Poisson approximates binomial better for small p, large n.

Why are stock prices modelled lognormal?

Two reasons: (1) prices can't be negative (lognormal is bounded below by zero); (2) returns are roughly normal, and if log-returns are normal, prices are lognormal by definition. This is the foundation of Black-Scholes.

What is "fat tails" and why does it matter?

Fat tails = extreme outcomes more frequent than normal-distribution predicts. In finance: market crashes, lottery-ticket winners, bond defaults. VaR based on normal underestimates these risks. Practitioner risk models often use Student-t distribution or empirical historical simulation to capture fat tails.

Practice questions

Click each question to reveal the answer and explanation.

Q 1

68% of observations from a normal distribution lie within how many standard deviations of the mean?

(a)1
(b)2
(c)2.33
(d)3

Correct: (a) 1

For normal: ~68% within ±1σ, ~95% within ±2σ (more precisely 1.96), ~99.7% within ±3σ.

Q 2

A NIFTY return is normally distributed with mean 1% per month and SD 5%. The probability of a monthly return below −9% is closest to:

(a)1%
(b)2%
(c)5%
(d)10%

Correct: (b) 2%

Z = (−9 − 1) / 5 = −2.0. P(Z < −2.0) ≈ 2.3%. The closest answer is 2%.

Q 3

A 95% one-tailed Z-score is closest to:

(a)1.00
(b)1.65
(c)1.96
(d)2.33

Correct: (b) 1.65

For 95% one-tailed (95% in one direction), Z = 1.65. For 95% two-tailed (95% in middle), Z = 1.96.

Q 4

A binomial random variable with n = 10 trials, p = 0.3 success probability has expected value:

(a)0.3
(b)3
(c)7
(d)10

Correct: (b) 3

E[X] = np = 10 × 0.3 = 3.

Q 5

For a portfolio with daily SD 1%, the 99% one-day VaR (assuming normal distribution) on a ₹100 cr portfolio is closest to:

(a)₹0.5 cr
(b)₹1 cr
(c)₹2.33 cr
(d)₹3.50 cr

Correct: (c) ₹2.33 cr

VaR_99% = portfolio × σ_daily × 2.33 = 100 × 0.01 × 2.33 = ₹2.33 cr. The 2.33 is the 99% one-tailed Z-score.

Q 6

A weakness of normal-distribution-based VaR is:

(a)It's computationally expensive
(b)It assumes returns are independent — usually violated
(c)It understates tail risks because real returns have fat tails
(d)It can't handle multiple assets

Correct: (c) It understates tail risks because real returns have fat tails

Real asset returns have fat tails (kurtosis > 3). Normal-VaR understates the probability of extreme losses, which is exactly when you need risk metrics most. Modern risk management uses CVaR, historical simulation, or Student-t models.

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.