Lognormal Distribution

This reading describes random variables through their probability distributions and summarizes them with the expected value, variance and covariance. It then matches the common discrete and continuous distributions to the financial quantities they model, and shows how conditional expectations and Bayes' formula update a forecast when new information arrives.

LOS 6.a — Unconditional mean, variance and covariance of a random variable

Random variables: discrete vs. continuous

A random variable is a quantity whose value is not known until an outcome is observed. It is also called a stochastic variable.

A probability distribution lists the probabilities of all possible outcomes of a random variable, and these probabilities sum to 1. A probability function $p(x)=P(X=x)$ gives the probability that X equals a particular value. For a discrete variable, $p(x)=0$ for a value that cannot occur and $p(x)>0$ for one that can.

Whether a variable is discrete does not depend on its outcomes being equally likely (the uniform case) or positive. Prices quoted in cents are technically discrete. Because there are so many possible values, analysts usually treat them as continuous and speak of the probability of a range of price changes.

What makes a valid probability function

A function $p(x)$ qualifies as a probability function only if both conditions hold:

A list such as 0.50, 0.40, 0.20, −0.10 sums to one but fails condition 1. Five values of 0.25 each pass condition 1 but sum to 1.25, so they fail condition 2.

For a continuous variable, a single point carries no probability, so including or excluding the endpoints does not change the probability of an interval: $P(a\le X\le b)=P(a<X<b)$. The height of the PDF at a point is a density. A probability is the area under the PDF over an interval.

Cumulative distribution function (CDF)

The cumulative distribution function (CDF), also called the distribution function, gives the probability that X takes a value less than or equal to x:

For a discrete variable, $F(x)$ is the sum of the probabilities of all outcomes up to and including x. For the standard normal distribution (Module 6.2), $F(-1)\approx 0.1587$: the area under the PDF to the left of −1 equals the height of the CDF at −1.

Unconditional expected value (mean)

The unconditional mean (also the unconditional expectation or expected value) weights each outcome by its probability:

The weights are the probabilities, and they must sum to 1, so a probability that is not stated can be found as 1 minus the sum of the others. A simple average of the outcomes weights each by $1/n$, so it is the expected value when every outcome is equally likely; with unequal probabilities it generally gives a different answer. The sample mean of Reading 5 summarizes returns already observed (an ex post measure). An expected value weights possible future outcomes by their probabilities, so it is a forecast (an ex ante measure).

Unconditional variance and standard deviation

The unconditional variance is the probability-weighted squared spread around the expected value, and the unconditional standard deviation is its square root:

Variance is in squared units (e.g., %²), so the standard deviation (back in % units) is the figure compared with returns. An equivalent shortcut is ${\sigma }^2=E(X^2)-[E(X){]}^2$.

Unconditional covariance

Unconditional covariance shows the direction of the linear relationship between two random variables, that is, whether they tend to be above or below their own means at the same time. Its strength is read from the correlation (last point below). The covariance is computed as:

The shortcut form is $\text{Cov}(X,Y)=E(XY)-E(X)E(Y)$.

• Positive covariance: the variables tend to move in the same direction; negative: opposite directions.
• For scenario (state) tables, $P(x_i,y_i)$ is simply the probability of the state. In a joint probability table, cells with zero probability contribute nothing.
• Units are the product of the two variables' units (e.g., %² or decimal²), so the size is hard to interpret. Dividing by the two standard deviations gives the unit-free correlation: $\rho =\text{Cov}(X,Y)/({\sigma }_X{\sigma }_Y)$, which always lies between −1 and +1.

LOS 6.b — Moments and uses of the key distributions in finance

The four principal moments

The most useful features of each distribution are what it models and its shape: bounded or unbounded, symmetric or skewed, thin or fat tails. The moment formulas below are for reference.

Discrete distributions

Discrete uniform. A finite set of $n$ outcomes, each with probability $1/n$ (e.g., the integers 1 to 8, each with probability 0.125); the probability of $k$ outcomes in a range is $k/n$, and the CDF at the $k$th outcome is $k/n$. For a discrete uniform variable on the consecutive integers $a,a+1,\dots ,b$: mean $=(a+b)/2$ and variance $=[(b-a+1{)}^2-1]/12$. For a general equally likely set (e.g., $\{1,3,5\}$), compute the arithmetic mean and the population variance directly: here the mean is 3 and the variance is $(4+0+4)/3=8/3$. The consecutive-integer formula would wrongly give 2.

Binomial. Counts the number of successes in a fixed number $n$ of independent trials, each with the same success probability $p$ and only two possible results (success/failure). A binomial variable with a single trial is a Bernoulli distribution.

Binomial models are also used for asset prices: at each step the price is multiplied by an up factor $u$ or a down factor $d$, so after $i$ up moves and $j$ down moves the price is $S_0{u}^i{d}^j$. The possible paths are drawn as a probability tree (or decision tree). With $u=1.05$ and $d=1/1.05$, a price of 40 after one up and one down move returns to 40, whatever the order.

Poisson. Counts how many times an event occurs in a fixed interval when events arrive independently at a known, constant average rate $\lambda$. It is the usual model for rare events such as defaults per year, trading halts per month or operational losses per quarter.

Continuous distributions

Continuous uniform on $[a,b]$: every equal-width sub-interval is equally likely, so the CDF rises in a straight line, $F(x)=(x-a)/(b-a)$ for $a\le x\le b$.

Skew 0, kurtosis 1.8 (excess −1.2). Used to generate random draws for Monte Carlo simulation. Example. Uniform on [0, 20]: $P(5\le X\le 9)=4/20=20\%$.

Normal (also called the Gaussian distribution), $N(\mu ,{\sigma }^2)$: symmetric bell curve, fully described by its mean and variance, or equivalently its mean and standard deviation (skew 0, excess kurtosis 0). About 68% of outcomes lie within ±1σ and about 95% within ±2σ. Sums of many independent variables tend toward normality, and a binomial converges to a normal as $n$ grows. It is unbounded in both directions, so it can produce negative values. The normal with mean 0 and variance 1, $N(0,1)$, is the standard normal distribution; its horizontal axis measures standard deviations from the mean.

Lognormal. $Y$ is lognormal when $\ln Y$ is normal; equivalently $Y=e^X$ with $X$ normal.

• Because $e^X>0$ for any $X$, the lognormal is bounded below by zero and never takes a negative value.
• It is positively (right) skewed, with a long right tail.
• That makes it well suited to modeling asset prices. A price following a lognormal cannot fall below zero, so the implied return can never be worse than −100%. A normal price model, by contrast, would allow negative prices (returns below −100%), which is impossible for a limited-liability asset.
• If continuously compounded price returns $\ln (P_1/P_0)$ (no distributions paid) are normally distributed, then prices are lognormal, because $P_T=P_0{e}^r$ with $r$ normal. Over several periods, $r$ is the sum of the one-period log returns (Reading 1), and sums of many independent returns tend toward a normal distribution. A lognormal price model therefore fits well even when single-period returns are not exactly normal.
• Its two parameters are the mean and SD of $\ln Y$, called the location parameter and the scale parameter. They are not the mean and SD of $Y$ itself: the mean of $Y$ is $e^{\mu +{\sigma }^2/2}$.
• Prices can be projected with a geometric Brownian motion model. Here $\mu$ is the expected (arithmetic) growth rate of the price, not the location parameter of $\ln Y$ used above, and $T(\mu -{\sigma }^2/2)$ is the drift term: $\ln S_T=\ln S_0+T(\mu -{\sigma }^2/2)$. Example. $S_0=25$, $\mu =8\%$, $\sigma =20\%$, $T=2$: $\ln S_T=\ln 25+2(0.08-0.02)$, so $S_T=25e^{0.12}=28.19$.
• Exam convention: the price from the drift-term formula is the expected price in $T$ years. Current practice: with $\mu$ as the expected growth rate, that figure is the median of the lognormal price distribution; the mean price is $S_0{e}^{\mu T}$ (here $25e^{0.16}=29.34$), which is higher because of the right skew.

Logistic. Bell-shaped and symmetric like the normal (skew 0), but with fatter tails. Its excess kurtosis is +1.2, so extreme outcomes are more likely than under a normal with the same variance. The variance is ${\pi }^2{s}^2/3$ for scale parameter $s$. The logistic distribution is used in logistic regression, where the dependent variable is binary (0/1), and in machine learning.

Matching the distribution to the problem

LOS 6.c — Joint distributions, conditional expectations, variances and covariances

Joint and marginal probabilities

A joint probability distribution gives the probability of every combination of outcomes of two or more variables; for two variables it is bivariate. For discrete variables it is often shown as a contingency table. Row and column totals are the marginal distributions.

For two variables in general, the joint cumulative distribution function gives the probability that both variables are at or below given values: $F_{12}(x_1,x_2)=P(X_1\le x_1,X_2\le x_2)$. A bivariate distribution is described by each variable's mean and variance plus how they move together (their covariance or correlation).

• Marginal: $P(\text{tech})=160/400=0.40$; $P(R>10\%)=120/400=0.30$.
• Joint: $P(\text{tech and }R>10\%)=64/400=0.16$.
• Conditional: $P(R>10\%\mid \text{tech})=64/160=0.40$, using only the technology column.

Conditional mean, variance and covariance

A conditional expectation $E(X\mid S)$ is the expected value of $X$ given that condition $S$ holds, read "the expected value of X given S". A forecast revised for new information is a conditional expectation; for example, "if the merger is approved, my EPS estimate rises to $4.20".

The conditional variance measures dispersion around the conditional mean $E(X\mid S)$ rather than the unconditional mean $E(X)$:

The conditional covariance is defined the same way for two variables, using conditional probabilities and conditional means. It is a different concept from the conditional variance of one variable.

A conditional distribution describes one variable after the other variables in a joint distribution are restricted to certain values. Conditioning changes the relevant probability distribution, so it can change both the expected value and the variance.

Probability trees

A tree shows sequential outcomes; the probability of a full path is the product of the branch probabilities along it. The unconditional expected value weights every end node by its path probability, the product of the branch probabilities along that path. Imposing a condition (for instance, "the price fell in period 1") prunes the tree. Only branches that start from the observed node remain, and their branch probabilities become the conditional probabilities.

• End prices: 60.50 (prob. 0.36), 49.50 (prob. 0.48 via two paths), 40.50 (prob. 0.16). Unconditional $E(P_2)=0.36(60.50)+0.48(49.50)+0.16(40.50)=52.02$.
• Given $P_1=45$ (a fall in period 1): $E(P_2\mid P_1=45)=0.6(49.50)+0.4(40.50)=45.90$; $\text{Var}(P_2\mid P_1=45)=0.6(3.6{)}^2+0.4(-5.4{)}^2=19.44$, so the conditional SD is 4.41.
• Given $P_1=55$: $E(P_2\mid P_1=55)=0.6(60.50)+0.4(49.50)=56.10$. Weighting the two conditional expectations by their branch probabilities gives back the unconditional value, $0.6(56.10)+0.4(45.90)=52.02$.

A common error is to weight the remaining outcomes by their joint (path) probabilities, e.g. $0.4\times 0.6\times 49.50+0.4\times 0.4\times 40.50$, which gives 18.36. That figure is $P(S)E(P_2\mid S)$ rather than the conditional expectation.

LOS 6.d — Updating probabilities with Bayes' formula

Bayes' formula revises a prior probability (also called an a priori probability) of an event into a posterior probability after new information $I$ is observed. It follows from the multiplication rule, $P(I\mid E)P(E)=P(I\text{ and }E)$: writing the joint probability two ways gives $P(E\mid I)P(I)=P(I\mid E)P(E)$, and dividing by $P(I)$ gives

The denominator is the unconditional probability of the information: the sum of its joint probabilities with each state.

The formula can be checked against the 400-firm table in LOS 6.c. Counting directly, 64 of the 120 firms that returned more than 10% are technology firms, so $P(\text{tech}\mid R>10\%)=64/120=0.533$. Bayes' formula reaches the same figure from the reverse conditional probability: $P(R>10\%\mid \text{tech})P(\text{tech})/P(R>10\%)=0.40\times 0.40/0.30=0.533$.

Applying Bayes' formula:

1. Write down the prior $P(E)$ and its complement $P(E^c)=1-P(E)$.
2. Write down the likelihoods $P(I\mid E)$ and $P(I\mid E^c)$; take complements if the information is a "does not happen" event.
3. Multiply to get the joint probabilities of $I$ with each state.
4. Add the joint probabilities to get $P(I)$.
5. Posterior = the joint probability for the state of interest ÷ $P(I)$.

The evidence raises the probability from the 30% prior to about 45%, but because unskilled managers are the majority, a single good year is far from proof of skill.