CFA Level I · CFA Level I Exam

Statistical Distributions for Financial Asset Prices and Returns: formula sheet

Full chapter guide

Key formulas

PMF conditions (discrete): 0 ≤ p(x) ≤ 1 for every x; Σ p(x) = 1; p(x) = P(X = x). Use this to find a missing probability in a table.
PDF conditions (continuous): f(x) ≥ 0; total area under f(x) = 1; f(x) is a density, not a probability. It can be greater than 1.
CDF definition: F(x) = P(X ≤ x); Non-decreasing, between 0 and 1. Discrete: F(x) = Σ p(xi) for xi ≤ x.
Interval probability (continuous): P(a ≤ X ≤ b) = F(b) − F(a); For a continuous variable P(X = a) = 0, so endpoints do not matter.
Interval probability (discrete): P(a < X ≤ b) = F(b) − F(a); For discrete variables, check whether the lower endpoint is included. If it is, use F(b) − F(a) + p(a).
Upper tail: P(X > x) = 1 − F(x); Fast way to get 'greater than' probabilities.
Discrete uniform probability: P(X = xi) = 1 ÷ n; n is the number of equally likely outcomes. Cumulative probability for k outcomes up to and including a value is k ÷ n.
Discrete uniform mean (consecutive integers 1 to n): E(X) = (n + 1) ÷ 2; Applies only when outcomes are the integers 1, 2, ..., n. For other lists, compute the simple average of the outcomes.
Discrete uniform variance (consecutive integers 1 to n): Var(X) = (n² − 1) ÷ 12; Same condition as above. Standard deviation is the square root.
Continuous uniform density: f(x) = 1 ÷ (b − a) for a ≤ x ≤ b, and 0 otherwise; The height is constant across the interval.
Continuous uniform probability: P(x1 ≤ X ≤ x2) = (x2 − x1) ÷ (b − a); Requires a ≤ x1 ≤ x2 ≤ b. If an interval extends outside [a, b], cut it back to the range first.
Continuous uniform CDF: F(x) = (x − a) ÷ (b − a) for a ≤ x ≤ b; F(x) = 0 below a and 1 above b.
Continuous uniform mean: E(X) = (a + b) ÷ 2; The midpoint of the range.
Continuous uniform variance: Var(X) = (b − a)² ÷ 12; Standard deviation = (b − a) ÷ √12.
Bernoulli mean and variance: E(X) = p; Var(X) = p(1 − p); X = 1 for success, 0 for failure. One trial only.
Binomial probability: p(x) = n! ÷ [(n − x)! x!] × p^x × (1 − p)^(n − x); Probability of exactly x successes in n independent trials with constant p.
Binomial mean and variance: E(X) = np; Var(X) = np(1 − p); standard deviation = √[np(1 − p)]; Valid under the same conditions: independent trials, constant p.
Number of arrangements: nCx = n! ÷ [(n − x)! x!]; On the BA II Plus: n [2nd] [nCr] x [=].
One-period binomial tree: S(up) = S0 × u; S(down) = S0 × d; d = 1/u; u > 1 and d < 1. Recombining when d = 1/u.
Expected price after one period: E(S1) = p × S0 × u + (1 − p) × S0 × d; A probability-weighted average of the two possible prices. For n periods, use the one-period expected multiplier raised to the power n: E(Sn) = S0 × [p·u + (1 − p)·d]^n.
Cumulative probability: P(X ≤ k) = p(0) + p(1) + ... + p(k); For 'at least' questions, use P(X ≥ k) = 1 − P(X ≤ k − 1).
Z-score (standardization): z = (x − μ) ÷ σ; Counts standard deviations from the mean. Use σ, not σ².
Reverse standardization: x = μ + zσ; Use when a probability is given and you need the cutoff value.
Symmetry of cumulative probabilities: N(−z) = 1 − N(z); Lets you use positive-z tables for negative z.
Probability above a value: P(X > x) = 1 − N(z); The table gives the area to the left only.
Probability between two values: P(a < X < b) = N(zb) − N(za); Standardize both values first.
Approximate confidence intervals: 68%: μ ± 1σ; 95%: μ ± 1.96σ; 99%: μ ± 2.58σ; The rule of thumb rounds 1.96σ to ±2σ. Keep 99% (±2.58σ) separate from ±3σ, which covers about 99.7%. Use 1.96 and 2.58 when exact.
Common one-sided z values: N(1.28) ≈ 0.90; N(1.645) ≈ 0.95; N(2.33) ≈ 0.99; Useful for 5% and 1% tail cutoffs.
Linear transformation of a normal variable: If X ~ N(μ, σ²), then aX + b ~ N(aμ + b, a²σ²); Normal stays normal under linear combinations.
Roy's safety-first ratio: SFRatio = (E(Rp) − RL) ÷ σp; E(Rp) is expected portfolio return, RL is the threshold return, σp is the portfolio standard deviation. Pick the portfolio with the highest ratio.
Shortfall probability: P(Rp < RL) = N(−SFRatio); Valid when returns are normally distributed. Higher SFRatio means lower shortfall probability.
Sharpe ratio (for comparison): Sharpe = (E(Rp) − Rf) ÷ σp; Same form as SFRatio but uses the risk-free rate Rf in place of RL.
Continuously compounded return: r(0,T) = ln(S_T ÷ S_0) = ln(1 + HPR); Uses the natural log. Works for price relatives and for holding period returns.
Price from a continuously compounded return: S_T = S_0 × e^r; The reverse of the log return. Always gives a positive price.
Simple return from a log return: HPR = e^r − 1; Use this to move back to a holding period return.
Additivity over periods: r(0,T) = r(0,1) + r(1,2) + ... + r(T−1,T); Log returns add. Simple returns compound by multiplying (1 + HPR) terms.
Scaling mean and variance: Mean over T = T × μ; Variance over T = T × σ²; Std dev over T = σ × √T; Holds when periodic log returns are independent and identically distributed.
Definition of lognormal: X is lognormal if ln(X) is normal; Lognormal values are positive, positively skewed and right-tailed.
Annualizing from a holding period: Annual cc return = (1 ÷ T years) × ln(S_T ÷ S_0); Divide the total log return by the number of years.
t-statistic for a mean: t = (x̄ − μ₀) ÷ (s ÷ √n); Degrees of freedom = n − 1. Use when the population variance is unknown.
Chi-square statistic for a variance: χ² = (n − 1) s² ÷ σ₀²; Degrees of freedom = n − 1. Assumes a normally distributed population. Right-skewed, never negative.
F-statistic for two variances: F = s₁² ÷ s₂²; Degrees of freedom: n₁ − 1 (numerator) and n₂ − 1 (denominator). For a two-sided test of equal variances, put the larger sample variance in the numerator and use the upper-tail critical value for α/2. For a one-sided test, the alternative hypothesis sets which variance is the numerator.
Degrees of freedom for a sample mean: df = n − 1; One parameter (the mean) is estimated from the sample.
Shape summary: t: symmetric, fat tails; χ² and F: right-skewed, ≥ 0; t approaches the standard normal as df rises.
Standard error of a simulated mean: Standard error = s ÷ √N; s is the standard deviation of the simulated outcomes and N is the number of trials. Quadrupling the trials halves the standard error.
Converting a standard normal draw: X = μ + σ × Z; Z is a random draw from the standard normal distribution. X is a draw from a normal variable with mean μ and standard deviation σ.
Lognormal price draw (one step): S(t+Δt) = S(t) × exp(r), where r is the simulated continuously compounded return; Simulating the return as normal keeps the price positive.
Estimate from simulation: Estimated value = average of the N simulated outcomes; For a derivative, average the discounted payoffs. Percentile estimates read from the sorted outcomes give measures such as VaR.

Quick revision

A discrete variable has countable outcomes; a continuous variable has probability only over intervals, and any single point has probability zero.
Discrete uniform: each of n outcomes has probability 1 ÷ n.
Continuous uniform on [a, b]: P(x1 ≤ X ≤ x2) = (x2 − x1) ÷ (b − a).
Binomial: mean = np and variance = np(1 − p).
Z-score: z = (x − μ) ÷ σ, and it states how many standard deviations x is from the mean.
The normal distribution is symmetric and fully described by its mean and variance.
Safety-first ratio = (E(R_p) − R_L) ÷ σ_p; choose the higher ratio.
Continuously compounded return = ln(P1 ÷ P0), and it can be added across periods.
The lognormal distribution is bounded below by zero and skewed to the right, so it suits asset prices.
The t distribution has fatter tails than the normal and approaches it as degrees of freedom increase.
Chi-square is used for tests about a variance and is not symmetric; F is the ratio of two independent chi-square variables, each divided by its own degrees of freedom, and is used to compare variances.
Monte Carlo simulation draws random inputs from assumed distributions; its results are only as good as those assumptions.

Common mistakes

Treating the PDF height f(x) as a probability. Fix: Remember that for a continuous variable only area is probability. The height can exceed 1.
Saying P(X = x) is positive for a continuous variable. Fix: There are infinitely many values, so each single point has probability 0. Only intervals have positive probability.
Forgetting to divide by 12 in the variance, or using (b − a)² ÷ 2. Fix: Memorise Var = (b − a)² ÷ 12 as one unit. Test it on [0, 1]: variance should be 1/12 ≈ 0.0833.
Reporting variance when the question asks for standard deviation. Fix: Underline the word asked. Standard deviation = (b − a) ÷ √12.
Leaving out the nCx term and giving only p^x × (1 − p)^(n − x). Fix: Always write three factors: nCx, p^x and (1 − p)^(n − x). Check that n is on the first factor.
Confusing Bernoulli and binomial distributions. Fix: Bernoulli is one trial with mean p and variance p(1 − p). Binomial counts successes in n trials with mean np and variance np(1 − p).
Using variance instead of standard deviation in the z-score. Fix: Always check the symbol. If you see σ² or 'variance', take the square root first.
Reading N(z) as the probability above z. Fix: For 'greater than', calculate 1 − N(z). Sketch and shade the area before computing.
Choosing the portfolio with the highest expected return. Fix: Always compute the SFRatio for every option. A higher return with much higher σ can have a lower ratio.
Subtracting the risk-free rate instead of the threshold return. Fix: Read the stem for the minimum acceptable or threshold return. Use that as RL. Use Rf only if the question says the threshold is the risk-free rate.

Exam tips

Expect conceptual items that test whether a PDF value can be a probability. It cannot.
With a CDF table, subtract rows for intervals and use 1 − F(x) for 'greater than'.
In a PMF table, solve for the missing probability first, then answer.
Watch the wording 'at most', 'less than' and 'at least'. For discrete variables they change which values count.
Do not spend calculator time here. Most items need only addition or subtraction.
Numerical options are listed from smallest to largest, so once you have computed a value you can locate it quickly. The order does not tell you whether an option is correct, so always rely on your own calculation.
A common wrong option is the variance when the standard deviation is asked, or the width squared without the 12. Check which one you computed.
For discrete questions with a fair die or numbered items, count favourable outcomes and divide by n. This takes under 30 seconds.