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CFA Level I · CFA Level I Exam

Statistical Distributions for Financial Asset Prices and Returns

This chapter covers the probability distributions used to model asset prices and returns: uniform, binomial, normal, lognormal, t, chi-square and F, plus Monte Carlo simulation. To solve questions, identify the distribution, standardize or apply its formula, then read the probability or interpret the result.

What this chapter covers

This chapter gives you the standard probability models that finance uses. It starts with random variables, then moves through simple distributions (uniform, binomial) to the normal distribution, which sits at the centre of the chapter. From the normal you build the lognormal distribution for asset prices and the sampling distributions (t, chi-square, F) used in testing.

The chapter also covers two applied tools. The safety-first ratio helps you choose a portfolio when you care about avoiding a minimum return. Monte Carlo simulation lets you estimate outcomes when no formula is available.

These ideas feed directly into later work. Sampling and estimation, hypothesis testing and regression all rely on the normal, t, chi-square and F distributions. Portfolio Construction uses means, variances and normality assumptions. Derivatives uses binomial trees for option pricing, and risk measures such as value at risk use simulation and normal assumptions.

Quantitative Methods carries a meaningful share of the Level I exam, and this chapter supplies the base for several later chapters. Questions are usually short and calculation-based, so they suit the 90-second pace if you know the setup. Z-score, binomial and continuously compounded return questions are quick marks once practised. Weak knowledge here also hurts you in hypothesis testing, regression and derivatives, so the effort pays off more than once. With no penalty for wrong answers, you can also eliminate options by checking whether an answer is reasonable, for example a probability above 1 or a negative lognormal price.

Statistical Distributions for Financial Asset Prices and Returns: topics in the order to study them

  1. 1Discrete and Continuous Random VariablesStart here because every other distribution is built on the ideas of probability functions, cumulative distribution functions and discrete versus continuous outcomes.
  2. 2Discrete and Continuous Uniform DistributionsThese are the simplest cases, so you can practise reading a probability function and a cumulative probability before moving to harder shapes.
  3. 3Binomial Distribution and Binomial Tree ModelsIt extends discrete thinking to repeated two-outcome trials and prepares you for the tree models that return in derivatives.
  4. 4Normal Distribution and Standardization (Z-scores)This is the core topic of the chapter, and the next three topics depend on it.
  5. 5Safety-First Ratio and Shortfall RiskIt applies the normal distribution and z-score logic directly, so it is easiest right after you master standardization.
  6. 6Lognormal Distribution and Continuously Compounded ReturnsIt builds on the normal: if continuously compounded returns are normal, prices are lognormal.
  7. 7Student's t, Chi-Square and F DistributionsThese are derived from the normal and are needed for confidence intervals and hypothesis tests later, so learn their shapes and uses now.
  8. 8Monte Carlo SimulationStudy it last because it uses the distributions above as inputs and is mostly conceptual, which suits a final review.

How to prepare Statistical Distributions for Financial Asset Prices and Returns

Spend most of your time on the normal distribution and its links to other topics. Keep the other topics short but accurate.

  1. Read the random variable and uniform topics once and write the formulas for expected value and the uniform probability in your own words.
  2. Practise binomial calculations: expected value np, variance np(1 − p), and a one-period tree with up and down moves and their probabilities.
  3. Drill z-scores until automatic: z = (x − μ) ÷ σ. Memorise the common figures for a normal distribution: about 68% of outcomes fall within 1 standard deviation of the mean. For the other intervals, 1.65 is the two-tailed 90% value (one-tailed 95%), 1.96 is the two-tailed 95% value, and 2.58 is the two-tailed 99% value.
  4. Solve safety-first questions by choosing the portfolio with the highest SFRatio = (E(R) − R_L) ÷ σ, where R_L is the threshold return.
  5. Practise the lognormal and continuous return link: continuously compounded return = ln(P1 ÷ P0). Use the ln and e^x keys on your TI BA II Plus or HP 12C, and check the order of key presses on your calculator.
  6. Make a one-page table of t, chi-square and F: shape, degrees of freedom, and the test each is used for.
  7. Finish with mixed three-option MCQs under time pressure, and review every wrong answer by naming the distribution and the trap.

Common mistakes in Statistical Distributions for Financial Asset Prices and Returns

  • Using variance instead of standard deviation in the z-score or safety-first ratio.

    Fix: Check the units first. Take √variance before dividing, and read the question for which one is given.

  • Treating a lognormal variable as if it can be negative or symmetric.

    Fix: Remember that the log return is normal and the price is lognormal, so the price is never below zero and is right-skewed.

  • Adding simple returns across periods when continuously compounded returns are required.

    Fix: Use ln(P1 ÷ P0) for continuous returns and add them across periods. Simple returns must be compounded, not added.

  • Choosing the portfolio with the highest return in a safety-first question.

    Fix: Compute the ratio for each portfolio and select the highest, which is the one with the lowest probability of falling below the threshold under normality.

  • Mixing up when to use the t, chi-square and F distributions.

    Fix: Link each to its purpose: t for means with unknown variance, chi-square for one variance, F for comparing two variances.

  • Forgetting that a continuous distribution gives zero probability at a single value.

    Fix: Always calculate probabilities over a range, and use the cumulative distribution function.

Last-day revision: Statistical Distributions for Financial Asset Prices and Returns

  • 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.

Statistical Distributions for Financial Asset Prices and Returns practice questions

Statistical Distributions for Financial Asset Prices and Returns in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Statistical Distributions for Financial Asset Prices and Returns: frequently asked questions

Which topic in this chapter is the most important for the exam?

The normal distribution and z-scores are the most important. Safety-first, the lognormal distribution and later hypothesis testing all rely on them. If time is short, master standardization first.

Do I need to memorise normal distribution tables?

You should know the common values: 1.65 for a two-tailed 90% interval (one-tailed 95%), 1.96 for two-tailed 95% and 2.58 for two-tailed 99%. Questions that need other values normally provide the table or the cumulative probability in the question. Practise reading values given in the question.

How is the t distribution different from the normal distribution?

Both are symmetric and bell-shaped. The t distribution has fatter tails, which reflects extra uncertainty when the variance is estimated from a sample. As the degrees of freedom rise, it gets closer to the normal.

Why do we use continuously compounded returns?

They add across time periods, which makes multi-period analysis simple. If they are normally distributed, the price is lognormal, which keeps prices from going below zero.

What should I know about Monte Carlo simulation for Level I?

Know what it is and why it is used: it generates many random outcomes from assumed distributions to estimate a result when no simple formula exists. Know its limit too. It gives estimates, not exact answers, and depends on the quality of the assumptions.