CFA Level 1 · Module 02 Quantitative Methods · Chapter 6
The previous reading described a set of returns that had already happened. This one describes returns that have not happened yet. When the future is uncertain, an outcome is a random variable, and the way its possibilities are spread out is a probability distribution. Learning the handful of distributions that finance leans on, and the two summary numbers that define each, is what lets an analyst put odds on the future instead of merely hoping.
The reading has three movements. First, the machinery of a random variable: its expected value, its variance, and the difference between discrete and continuous outcomes. Second, the specific distributions that recur throughout the curriculum, above all the normal distribution for returns and the lognormal distribution for prices. Third, the idea that a probability is not fixed but should shift as new information arrives, formalized in conditional expectations and Bayes’ formula.
As always the arithmetic is light and the interpretation is everything. The exam wants you to compute an expected value, read a probability off the normal distribution by standardizing, and revise a prior belief with a new signal, and to know which distribution belongs to prices and which to returns.
A random variable is a quantity whose value is not known in advance because it depends on chance, such as next year’s return on a share. A probability distribution lists the possible values and how likely each is. The distinction that organizes everything is between discrete and continuous variables. A discrete variable takes separate, countable values, like the number of years out of five that a fund beats its benchmark. A continuous variable can take any value in a range, like a return that could be any number between negative 100% and something large.
Discrete distributions are described by a probability mass function, which gives the probability of each exact value, and those probabilities must sum to one. Continuous distributions are described by a probability density function, whose height shows where outcomes cluster, but for which the probability of any single exact value is zero; only the probability of landing in a range is meaningful, and it is the area under the density over that range. Both kinds also have a cumulative distribution function, which gives the probability of an outcome at or below a given value and always climbs from zero to one as the value increases.
These three functions are just different views of the same information. The density shows where the mass is, the cumulative function accumulates it from the left, and reading a probability off a distribution almost always comes down to finding an area under the density or, equivalently, a difference of two points on the cumulative function. Keeping the picture in mind, mass on the left, cumulative rising to the right, prevents most errors before they start.
The expected value of a random variable is its probability-weighted average, the mean you would approach by repeating the gamble many times. It is the single best forecast of the outcome, and it is computed by multiplying each possible value by its probability and summing. The variance measures how far outcomes are likely to fall from that expected value, again probability-weighted, and its square root, the standard deviation, expresses the risk in the original units.
where xi are the possible outcomes, pi are their probabilities summing to one, and E[X] is the expected value. Compared with the previous reading, the probabilities replace the equal weighting of a historical sample.
Setup. A share of Meridian Alloys has three possible one-year returns: positive 20% with probability 0.3, positive 5% with probability 0.5, and negative 10% with probability 0.2. Find the expected return, the variance, and the standard deviation.
Answer: expected return 6.5%, variance 110.25, standard deviation 10.5%. The expected value is what you forecast; the standard deviation of 10.5% warns that any single year is likely to land well away from that 6.5%. Both numbers are needed to describe the bet.
| Outcome | Probability | Contribution to mean |
|---|---|---|
| +20% | 0.3 | 6.0 |
| +5% | 0.5 | 2.5 |
| −10% | 0.2 | −2.0 |
| Expected value | 1.0 | 6.5 |
Averaging the outcomes instead of weighting them by probability. The expected value is not the simple mean of the possible returns; it is each return multiplied by its own probability, then summed. Treating a 20% outcome with probability 0.3 the same as a 5% outcome with probability 0.5 throws the answer off. Always check that the probabilities sum to one before you begin.
Expectation extends naturally to two assets at once. When outcomes are described by scenarios that specify a return for each asset together, a joint distribution, the covariance is the probability-weighted average of the products of the two assets’ deviations from their own expected values. It is the forward-looking twin of the covariance from the previous reading, built from probabilities of future scenarios rather than a history of realized returns, and it is the input the portfolio risk formulas actually consume.
where each scenario i has probability pi and returns Ai and Bi for the two assets, and E[A] and E[B] are their expected returns.
Setup. Two assets have returns that depend on the economy. In a boom (probability 0.3) A returns 15% and B returns 10%; in a normal year (probability 0.5) A returns 8% and B returns 6%; in a bust (probability 0.2) A returns negative 10% and B returns 2%. Find the covariance.
Answer: a covariance of about positive 23.40. The positive sign says the two assets tend to be strong or weak together, driven by the shared economic scenarios. Dividing by the product of their standard deviations would rescale this into a correlation, exactly as in the previous reading.
The uniform distribution is the simplest continuous distribution: every value in a range is equally likely, so its density is a flat line. It is not a realistic model of returns, which cluster near a center, but it is the building block for simulation, because a computer’s random number generator produces uniform values that are then transformed into any other distribution needed. Its expected value is simply the midpoint of the range, and it serves mainly as the raw material from which richer distributions are built, a role that matters in the simulation reading that follows.
Because the uniform distribution is flat, it has no peak and thin, abrupt tails, which is the opposite of what real returns look like. Meeting it here is useful precisely as a contrast: once you have seen a distribution with no central tendency and no tails, the bell shape of the normal distribution and the skew of the lognormal stand out as the features that make them fit finance.
The binomial distribution is the workhorse discrete distribution. It models the number of successes in a fixed number of independent trials, each with the same probability of success. Anything that is a repeated yes-or-no event fits: how many of the next five years a fund beats its benchmark, how many bonds in a pool default, how many trading days a strategy is profitable. The probability of exactly a given number of successes comes from a compact formula.
where n is the number of trials, k is the number of successes, p is the probability of success on each trial, and C(n, k) is the number of ways to choose k successes from n trials. The mean is np and the variance is np(1 − p).
Setup. A fund beats its benchmark in any given year with probability 0.6, independently each year. Over five years, what is the probability it beats the benchmark exactly three times, and what is the expected number of wins?
Answer: about a 34.6% chance of exactly three wins, with an expected three wins over five years. Even though three is the single most likely count, its probability is only about a third, a reminder that the expected value is a center of gravity, not a prediction of what will actually occur.
The binomial has three requirements: a fixed number of trials, independence between trials, and the same probability of success on each. If a question describes trials that are not independent, such as defaults that cluster in a downturn, the binomial no longer applies. Watch also for the difference between “exactly” a number of successes, which uses the formula once, and “at least” or “at most”, which sums several terms.
The normal distribution is the famous bell curve and the most important continuous distribution in finance. It is symmetric around its mean, so its skewness is zero, and it is completely described by just two numbers: its mean, which locates the center, and its standard deviation, which sets the width. Returns are often modeled as normal because sums and averages of many small independent influences tend toward normality, a result formalized by the central limit theorem in the next reading.
Its symmetry gives a memorable rule of thumb. About 68% of outcomes fall within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three. These bands let you translate a mean and a standard deviation straight into a rough probability without any table, which is exactly the kind of quick estimate the exam rewards. Two normal distributions can be added, and any linear combination of normals is itself normal, which is why the normal assumption makes portfolio mathematics so tractable.
| Range around the mean | Approximate probability |
|---|---|
| Within one standard deviation | 68% |
| Within two standard deviations | 95% |
| Within three standard deviations | 99.7% |
The drawback is that real returns are not perfectly normal. As the previous reading noted, they show fatter tails than the normal predicts, so extreme moves happen more often than the bell curve allows. The normal distribution remains the default working model because it is simple and tractable, but a careful analyst holds it lightly, aware that it understates the odds of a crisis.
One property makes the normal distribution indispensable for portfolios. Any weighted combination of normally distributed returns is itself normally distributed, so a portfolio of assets whose returns are normal has a return that is also normal, fully described by just its own mean and standard deviation. That closure is what lets the portfolio mathematics of the next readings work with means, variances, and covariances alone, without tracking the whole distribution. No other common distribution behaves so conveniently under addition, which is a large part of why the normal, despite its thin tails, remains the default model of returns.
To find the probability of a specific return under a normal distribution, convert the value into a standard normal score, or z-score, which measures how many standard deviations the value lies from the mean. Once expressed in z, any normal distribution can be read from the same single reference, because the standard normal has a mean of zero and a standard deviation of one.
where x is the value of interest, μ is the mean of the distribution, σ is its standard deviation, and z is the number of standard deviations x lies above (positive) or below (negative) the mean.
Setup. A portfolio’s annual return is modeled as normal with a mean of 8% and a standard deviation of 10%. What is the approximate probability of a loss, that is, a return below 0%?
Answer: about a 21% probability of a loss. Standardizing is the universal move: turn any return question into a z-score, then read the probability from the standard normal. The same method finds the chance of exceeding a target or landing between two values.
Keep the sign of the z-score straight. A value below the mean gives a negative z and a probability below one half; a value above the mean gives a positive z and a probability above one half. Sketching a quick bell curve and shading the region you want prevents the most common error, quoting the complement of the probability actually asked for.
Confusing the variance with the standard deviation when standardizing. The z-score divides by the standard deviation, not the variance. If a question gives a variance, take its square root first. Dividing by the variance produces a z-score that is far too small and a probability that is badly wrong.
The same standardizing move answers richer questions. To find the probability of a return above a target, standardize and take the upper area; to find the probability of landing between two values, standardize both ends and take the difference of their cumulative probabilities. Every one of these reduces to the same two steps: convert to a z-score, then read the standard normal. Because the distribution is symmetric, the area below negative z equals the area above positive z, a shortcut that saves time when a table gives only one side.
Returns may be modeled as normal, but prices cannot be, and the reason is simple: a price can never fall below zero, while a normal distribution allows any value, however negative. The fix is the lognormal distribution, which is the distribution of a variable whose natural logarithm is normally distributed. A lognormal variable is bounded below by zero and skewed to the right, with a long upper tail, which matches how asset prices actually behave: they cannot go below zero but can, in principle, rise without limit.
The link to returns is elegant. If continuously compounded returns are normally distributed, then prices, which are the exponential of those cumulative returns, are lognormally distributed. This is why the log returns from the first reading matter so much: they are the quantity that can plausibly be normal, and exponentiating them produces the positive, right-skewed prices seen in markets. The two distributions are two ends of the same model, normal for the log return, lognormal for the price it produces.
Model returns as normal and prices as lognormal. The lognormal shape respects the floor of zero and the open-ended upside that real prices have, and it follows automatically once continuously compounded returns are assumed normal. Confusing the two, forcing a normal onto prices, allows the nonsense of a negative price.
This asymmetry also explains a subtle point from the returns readings. Because the lognormal has a long right tail, the arithmetic mean of a set of returns sits above the compound, geometric growth they actually deliver, and the gap widens with volatility. The wealth an investor ends up with follows the compounding, lognormal path, not the simple average, which is why the geometric mean is the honest measure of realized growth. The distribution of prices and the choice of average return are two sides of the same coin, and seeing that connection ties this reading back to the very first one in the module.
Three more distributions appear later in the module, not to model returns but to conduct statistical tests, so it is enough to recognize their roles now. The Student’s t-distribution looks like the normal but with fatter tails, and it is used for inference about a mean when the sample is small or the population standard deviation is unknown, the situation the confidence-interval reading develops. The chi-square distribution is used for tests involving a variance, and the F-distribution for comparing two variances. All three are defined by degrees of freedom, a parameter that reflects the amount of data behind the estimate.
What ties this family together is that each is the reference distribution for a particular kind of test statistic. You do not need their formulas at this level, only the pairing: t for a mean with unknown variance, chi-square for a single variance, and F for the ratio of two variances. Meeting them here means they will be familiar when the hypothesis-testing reading puts them to work.
The degrees of freedom that define these three distributions are worth a word of intuition. They count the number of independent pieces of information behind an estimate, which is roughly the sample size reduced by the quantities already estimated from the same data. As degrees of freedom rise, the t-distribution’s fat tails shrink and it converges to the normal, which is why a large sample lets you treat the two as interchangeable. With a small sample the extra tail thickness of the t-distribution is the price of not knowing the population’s true spread, and it is exactly what keeps small-sample inference honest.
An expected value computed with no extra information is an unconditional expectation. An expected value computed given that some event has occurred is a conditional expectation, and the two can differ sharply. The unconditional expected return blends every scenario weighted by its probability; a conditional expected return fixes a scenario, such as a recession, and asks what to expect within it. Good analysis moves fluidly between the two, using the unconditional figure for a base case and conditional figures to stress-test what happens if a particular state of the world arrives.
The unconditional expectation is the probability-weighted average of the conditional ones. This is why scenario analysis works: estimate what happens in each state of the world, weight each state by its probability, and the pieces roll up into a single base-case forecast, with the conditional figures kept on hand for stress testing.
Setup. Next year the economy will be in expansion with probability 0.7, in which case a portfolio is expected to return 12%, or in recession with probability 0.3, in which case it is expected to return negative 5%. Find the unconditional expected return.
Answer: an unconditional expected return of 6.9%. The unconditional expectation is the probability-weighted average of the conditional expectations, a relationship known as the total expectation, and it is how scenario analysis rolls up into a single base-case forecast.
Probabilities should not stay frozen when new evidence arrives. Bayes’ formula is the rule for revising a prior probability into an updated, or posterior, probability once a signal is observed. It combines the prior belief with how likely the signal would be under each possibility, and it is the mathematical backbone of learning from data. The formula looks compact but its logic is just careful bookkeeping of conditional probabilities.
where P(A) is the prior probability of event A, P(B|A) is the probability of the signal B given A, P(B) is the overall probability of the signal, and P(A|B) is the updated probability of A after observing B.
Setup. Before any report, an analyst believes there is a 40% chance a company is financially strong. A strong company issues a good quarterly report with probability 0.8; a weak company issues a good report with probability 0.3. The company issues a good report. Update the probability that it is strong.
Answer: the probability rises from a prior of 40% to a posterior of 64%. The good report was evidence, not proof; it shifted the belief substantially but not to certainty, which is exactly how rational updating should feel. Weaker evidence would move the prior less.
| State | Prior | P(good report) | Joint | Posterior |
|---|---|---|---|---|
| Strong | 0.40 | 0.80 | 0.32 | 0.64 |
| Weak | 0.60 | 0.30 | 0.18 | 0.36 |
| Total | 1.00 | – | 0.50 | 1.00 |
Bayesian updating is proportional reasoning. The posterior for each state is its joint probability, prior times likelihood, divided by the total probability of the signal. Laying the states out in a small table, as above, turns any updating question into three columns and a division.
One caution keeps Bayesian reasoning honest, and the exam likes to test it. When the prior probability of a state is low, even a fairly reliable signal may leave the updated probability modest, because the rare state was unlikely to begin with. Ignoring the prior and reacting only to the strength of the signal is the classic error known as base rate neglect. The formula guards against it automatically, since the prior sits right there in the numerator, but only if you remember to include it rather than jumping straight to the likelihood.
Taken together, these distributions are the vocabulary the rest of the curriculum speaks in. Expected value and variance feed portfolio construction, the normal and its z-score run through confidence intervals and hypothesis tests, the lognormal underlies option pricing, and Bayesian updating reappears wherever new information must revise a view. Learning them here as a connected set, rather than as isolated formulas, is what makes the later readings feel like applications instead of new material.
An investment returns 15% with probability 0.4 and 5% with probability 0.6. What is its expected return?
E[R] = 0.4 × 15 + 0.6 × 5 = 6 + 3 = 9%.
A strategy is profitable on any day with probability 0.5, independently. Over three days, what is the probability of exactly two profitable days?
P(X = 2) = C(3, 2) × (0.5)2 × (0.5)1 = 3 × 0.25 × 0.5 = 0.375, or 37.5%.
Returns are normal with a mean of 10% and a standard deviation of 5%. Using the rule of thumb, within what range do about 95% of yearly returns fall?
Two standard deviations either side of the mean: 10% ± (2 × 5%) = 10% ± 10%, so from 0% to 20%.
Which distribution should be used to model an asset’s price, and which to model its continuously compounded return?
The price is modeled as lognormal, because it cannot fall below zero and is skewed right. The continuously compounded return is modeled as normal, and exponentiating a normal return produces the lognormal price.
A mass function applies to discrete variables and gives the probability of each exact value. A density function applies to continuous variables, where any single value has probability zero, so only the area under the density over a range is a probability. Both are summarized by a cumulative function.
Because it is an average over all outcomes, and the actual result is a single draw. In the binomial example the expected count was three wins, yet the chance of exactly three was only about a third. The expected value is the long-run center, not a forecast of the next observation.
Because it is simple, needs only a mean and a standard deviation, and combines cleanly, since any sum of normals is normal. Those properties make portfolio mathematics tractable. The cost is that it understates extreme moves, so it is used as a convenient default rather than a perfect description.
It rescales any normal variable so it has a mean of zero and a standard deviation of one, expressed as a z-score. That lets every normal probability be read from a single reference distribution instead of a different one for each mean and standard deviation.
Because a price cannot go below zero, but a normal distribution allows any value, including negative ones. The lognormal is bounded at zero and skewed right, which fits prices, and it arises naturally when continuously compounded returns are normal.
When inferring about a mean from a small sample or when the population standard deviation is unknown. The t-distribution has fatter tails than the normal to reflect the extra uncertainty, and it converges to the normal as the sample grows large.
It weighs a prior belief against how well the new evidence fits each possibility. Strong evidence that is much more likely under one state shifts the probability sharply toward that state; weak or ambiguous evidence barely moves it. The posterior is the prior adjusted in proportion to the evidence.
Through calculation of expected values and variances, binomial probabilities, normal probabilities via z-scores, and Bayesian updates, along with recognition questions on which distribution fits prices, returns, or a given test. Know both the mechanics and the pairing of each distribution to its use, and expect at least one item that hinges on reading the sign of a z-score or including the prior in a Bayesian update.
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