Probability trees, conditional expectations and conditional variance

How do you lay a forecast out as a tree?

A probability tree (also called a decision tree) shows an uncertainty that plays out in stages. Each point where the uncertainty resolves is a node . Branches leave the node, each labeled with its probability, and the probabilities leaving one node add to 1. Probabilities in the second stage are conditional probability values: the chance of an outcome given the branch you are on.

Probability treethe first uncertainty, then the outcomes within each branch; also called a decision tree

• Node Know the idea a point where uncertainty resolves
  • Branch probabilities from one node they add to 1
• End-branch (joint) probability Use it multiply along the path
  • All end branches add to 1
• Conditional expected value Memorize the forecast within one first-stage branch
  • Unconditional E[X] weight each by its branch probability and add

$$P(A \text{ and } B) = P(B \mid A)\,P(A)$$

$$P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}, \quad P(B) \neq 0$$

The two formulas are the same relationship read in opposite directions. Going down a tree you multiply; coming back, you divide by the probability of the branch you are on. For example, P(growth 9% | expand) = 0.4875 ÷ 0.65 = 0.75.

What is a conditional expected value?

A conditional expected value is the expected value of a variable if one particular scenario happens. You compute it exactly like an ordinary expected value, but with the probabilities inside that branch.

$$E[X \mid S] = P(x_1 \mid S)\,x_1 + P(x_2 \mid S)\,x_2 + \cdots + P(x_n \mid S)\,x_n$$

How do you combine the branches into one forecast?

Weight each conditional expected value by the probability of its scenario and add. This is the total probability rule for expected value , and the result is the unconditional expected value, the overall forecast before you know which scenario happens. It works when the scenarios are mutually exclusive (only one happens) and exhaustive (together they cover everything).

$$E[X] = E[X \mid S_1]\,P(S_1) + E[X \mid S_2]\,P(S_2) + \cdots + E[X \mid S_n]\,P(S_n)$$

On the Scenarios tab, move the first-stage probability and watch the two conditional expected values stay put while the overall forecast slides between them. The overall figure always lies between the branch forecasts. The Price tree tab belongs to the price-tree section further down.

The practical advantage is that an analyst can revise one branch, say a higher chance of a slowdown, without rebuilding the forecast. Only that probability and the weighted sum change.

Exam tip If your overall forecast lies outside the range of the branch forecasts, you have mis-weighted something.

What about the variance in a tree?

Variance also has a conditional version. A conditional variance measures spread inside one scenario, around that scenario’s expected value. As with the unconditional variance, there is a shortcut: the conditional mean of the squares minus the square of the conditional mean.

$$\mathrm{Var}(X \mid S) = E\big[(X - E[X \mid S])^2 \mid S\big] = E[X^2 \mid S] - \big(E[X \mid S]\big)^2$$

What about the covariance in a tree?

Know the idea

A conditional covariance measures how two variables move together inside one scenario, around that scenario’s expected values.

$$\mathrm{Cov}(X, Y \mid S) = E\big[(X - E[X \mid S])(Y - E[Y \mid S]) \mid S\big]$$

How do forecasts change as prices arrive?

Use it

Conditional expectations matter most in a multi-period price tree. A share starts at 50 and moves up 5 (probability 0.65) or down 5 (probability 0.35) in each of three periods, independently. This is the binomial tree of Lesson 3 with additive moves. Different paths reach the same price, so count them: three paths reach 55 (up-up-down, up-down-up and down-up-up) and three reach 45.

Price at time 3	Paths	Probability
65	1	0.65³ = 0.2746
55	3	3 × 0.65² × 0.35 = 0.4436
45	3	3 × 0.65 × 0.35² = 0.2389
35	1	0.35³ = 0.0429

The same tree is the start of the binomial option model in Derivatives.

Go deeper Why the overall variance exceeds the conditional ones

Not stated in the curriculum; for intuition only. The overall variance splits into two parts: the probability-weighted average of the conditional variances, plus the probability-weighted squared gaps between each conditional expected value and the overall expected value. For the retailer: 0.65 × 3.00 + 0.35 × 16.00 = 7.55, and 0.65 × (8.0 − 4.85)² + 0.35 × (−1.0 − 4.85)² = 6.45 + 11.98 = 18.43. Together they give 25.98. A tree that separates good and bad scenarios clearly has a large second part.

Go deeper Why each price-tree step adds the same amount

Not required for the exam. Each step of the price tree adds 0.65 × 5 − 0.35 × 5 = 1.5 to the expected price and 25 − 1.5² = 22.75 to the variance. After three steps that gives 50 + 4.5 = 54.5 and 3 × 22.75 = 68.25, the same figures as the table. Each observed move removes one step of risk.

Go deeper More than two stages

Trees can run for as many stages as the problem needs: each end node becomes a new node with its own branches, and you fold the tree back from the right. The price-tree section above is a three-stage example, and the binomial option model has the same shape.