Probability Trees in Investment Problems

Updated for the 2026-2027 CFA® Level I curriculum.

Investment outcomes often unfold in stages. The economic environment may be revealed first, followed by a company result, credit event, or market return. A probability tree arranges those events in sequence so you can see which probabilities belong to each path.

For CFA Level I, you should be able to formulate an investment problem as a probability tree, calculate the probability of each terminal outcome, and use those outcomes to find an overall probability or expected return.

Key Takeaways About Probability Trees

• A probability tree organizes events in the order they occur.

• Branches leaving the same node must represent mutually exclusive and exhaustive outcomes.

• The probabilities leaving each node must sum to .

• Later-stage branch probabilities are conditional on the path that leads to their node.

• Multiply branch probabilities along a single path.

• Add mutually exclusive path probabilities when they lead to the same final event.

• Terminal path probabilities must sum to when the tree covers every possible outcome.

• Returns or payoffs are usually attached to terminal outcomes.

• A probability tree can be used to calculate conditional expectations and an overall expected value.

What You Need to Know for CFA Level I

For CFA Level I, focus on translating a written scenario into the correct sequence of branches.

You should be able to:

• Identify the first and later stages of an investment problem.

• Create mutually exclusive and exhaustive branches.

• Label later branches with the correct conditional probabilities.

• Calculate the joint probability of a complete path.

• Combine several paths to calculate an overall event probability.

• Confirm that all terminal path probabilities sum to .

• Attach a return or payoff to each terminal outcome.

• Calculate the expected return across the complete tree.

• Explain how the tree represents conditional expectations.

• Distinguish a probability tree from a decision tree involving choices.

What Is a Probability Tree?

A probability tree is a diagram that represents uncertain events in sequence. It begins with a starting point and splits into branches for the possible first-stage outcomes. Each of those branches can then split again when a later event depends on what happened earlier.

A probability tree usually contains four main elements:

Consider an investment whose return depends first on the economy and then on the company’s earnings result. The first branches may represent expansion and slowdown. Each economic branch can then split into an earnings beat or miss.

The tree keeps the condition attached to every probability. A probability of an earnings beat following an expansion may differ from the probability of a beat following a slowdown.

Probability Tree vs Decision Tree

A probability tree maps uncertain events. No person chooses which branch occurs.

A decision tree may include choices available to management, an investor, or another decision-maker. Those choice branches can be followed by uncertain outcomes.

The terms are sometimes used loosely in search queries, but they represent different analytical structures. This study note covers probability trees built from uncertain events.

How to Build a Probability Tree

Step 1: Identify the Sequence of Events

Determine which event occurs first and which events follow.

For example:

1. The economy either expands or slows.

2. The company either beats or misses earnings expectations.

3. The asset produces a return based on the complete sequence.

Step 2: Draw the First-Stage Branches

The first-stage branches should represent every possible outcome at that stage.

If expansion and slowdown are the only possible economic states:

These probabilities are unconditional relative to the information available at the beginning of the tree.

Step 3: Add Later-Stage Branches

Add the possible second-stage outcomes beneath every first-stage branch.

The probability on a later branch is conditional on the path that reaches its node. For example:

and

are separate probabilities.

Even when events are independent and the numerical probabilities are the same, the later branch is still interpreted conditional on reaching that node.

Step 4: Check Each Node

The probabilities leaving each node must sum to .

For an earnings result beneath the expansion branch:

Perform the same check at every branching point.

Step 5: Attach Terminal Outcomes

Place the final return, payoff, default result, or other outcome at the end of each complete path.

A terminal outcome should reflect the full sequence that leads to it.

How to Calculate Probabilities From a Probability Tree

Three calculations cover most probability-tree questions.

Path Probability

Multiply the probabilities along a complete path to calculate the joint probability of that sequence.

For a two-stage path:

For example:

The result is the probability that both events occur in the specified order.

Overall Event Probability

An event may be reached through several separate paths. Add the relevant path probabilities to calculate its overall probability.

Using conditional probabilities:

This is the rule of total probability. The paths being added must be mutually exclusive.

Expected Terminal Value

When each terminal outcome has a return or payoff, multiply every terminal value by its path probability and add the products.

Where:

• = expected terminal value

• = probability of terminal path

• = return or payoff at terminal path

• = number of terminal paths

The terminal path probabilities must sum to when the tree is exhaustive.

How Probability Trees Support Conditional Expectations

Each first-stage branch defines a condition. The outcomes beneath that branch form a conditional probability distribution.

Suppose the first branches represent expansion and slowdown. You can calculate:

and

using only the outcomes beneath the relevant branch.

The overall expected return then follows from the law of total expectation:

A probability tree makes this structure visible. The first-stage probabilities weight the state-specific expected returns, while the later branches determine the conditional distribution within each state.

Worked Investment Example

An analyst models an asset return using two stages.

First, the economy can expand or slow:

The company can then beat or miss earnings expectations. The probability of each result depends on the economic state.

The branches leaving each node are complete:

Step 1: Calculate the Terminal Path Probabilities

Expansion and Earnings Beat

Expansion and Earnings Miss

Slowdown and Earnings Beat

Slowdown and Earnings Miss

The terminal probabilities sum to :

The tree accounts for every possible terminal outcome.

Step 2: Calculate the Overall Probability of an Earnings Beat

An earnings beat can occur through two mutually exclusive paths:

• Expansion and beat

• Slowdown and beat

Add their joint probabilities:

The company has an overall 56% probability of beating earnings expectations.

Step 3: Calculate the Expected Return From the Terminal Paths

Convert the returns to decimals and weight each one by its terminal path probability:

The asset’s overall expected return is 6.08%.

Step 4: Check the Result Using Conditional Expectations

During an expansion:

During a slowdown:

Now weight the two conditional expected returns by the economic-state probabilities:

Both methods produce the same overall expected return.

The terminal-path method works directly from the four final outcomes. The conditional-expectation method first summarizes the outcomes within each economic state and then combines the two state-specific expectations.

How to Read a Probability-Tree Question

Use this sequence when working through an exam problem.

1. Mark the Order of Events

Identify which event occurs first. The wording often uses phrases such as “if,” “given,” “following,” or “conditional on.”

2. Separate the Probability Layers

Keep first-stage probabilities separate from later conditional probabilities.

3. Identify What the Question Asks

Look for the operation signaled by the wording:

4. Complete the Tree Before Calculating

List every terminal path. Missing one path can affect both the probability check and the expected-value calculation.

5. Run the Final Checks

Confirm that:

• Branch probabilities leaving each node sum to .

• Terminal path probabilities sum to .

• Every terminal value is paired with the correct path probability.

• Returns use consistent percentage or decimal units.

Common Exam Traps

• Adding probabilities along one path instead of multiplying them.

• Multiplying probabilities from separate mutually exclusive paths instead of adding them.

• Treating a later-stage conditional probability as an unconditional probability.

• Applying a conditional probability from one parent branch to another.

• Forgetting to make the branches leaving each node sum to .

• Omitting a terminal path from the tree.

• Attaching a return to an intermediate branch rather than the complete terminal outcome.

• Calculating an expected return before finding all terminal path probabilities.

• Forgetting to convert percentage returns into consistent units.

• Confusing a probability tree with a decision tree that includes choices.

Practice Question

An analyst models a stock using two stages.

The stock receives an analyst upgrade with probability 0.30 and no upgrade with probability 0.70.

If the stock receives an upgrade, it beats the market with probability 0.80. If it receives no upgrade, it beats the market with probability 0.40.

What is the probability that the stock is both upgraded and beats the market?

1. 0.24

2. 0.52

3. 0.80

FAQs About Probability Trees