Events, conditional probability, and expectation
What you will be able to do
Section titled “What you will be able to do”After this lesson, you should be able to:
- identify events that form a disjoint and exhaustive partition;
- calculate a conditional probability;
- calculate an expectation when a random value is constant on each event;
- use conditional event means when the value varies inside an event.
Outcomes, events, and probabilities
Section titled “Outcomes, events, and probabilities”An outcome is one possible result of a modeled experiment. An event is a specified set of outcomes. Use the local probability-event for one such set and the local probability-measure for the rule that assigns probabilities to events. In this lesson, is a general probability measure. A later finance lesson uses only for real-world probabilities and introduces a different symbol for pricing probabilities. This lesson does not choose between the two.
In this lesson, a finite partition is a finite list of events that are:
- disjoint: no outcome belongs to two different partition events; and
- exhaustive: every possible outcome belongs to one of them.
Use the local probability-event-index to select an event and the local probability-partition-size to count the events. For a disjoint and exhaustive list , additivity of the probability measure gives the sum of the event probabilities:
If the events overlap, the sum of their probabilities can count some outcomes twice. If the events are not exhaustive, the sum omits some outcomes. These event and additivity definitions follow Shreve’s probability-space treatment[1].
Conditional probability restricts the outcomes
Section titled “Conditional probability restricts the outcomes”Suppose the question asks for the probability of event given another event. Write the conditioning event as . When , the conditional probability of given is defined as:
The denominator is the probability of the outcomes that the condition “given ” keeps. For example, suppose 30% of all outcomes are in , and 12% are in both and . Then the conditional probability is:
In a default model, consider the probability of “default during this period given survival to its start.” Its denominator is the probability of survival to the start of the period. This conditional probability is different from the unconditional probability of default by a horizon. Shreve develops conditioning as averaging within the information or event retained by the condition[2].
Expectation from disjoint events
Section titled “Expectation from disjoint events”Use the local probability-random-value for a numerical value determined by the outcome and the expectation operator for its probability-weighted average. In this lesson every random value is constant on each event of a finite partition, so the general integral definition of reduces to the weighted sum below. The expectation has the same unit as ; a random USD payment has an expected value in USD, not in percent.
This section gives two formulas. The first formula requires to be constant on each event. The second formula does not.
Exact value on every event
Section titled “Exact value on every event”First suppose is constant on every partition event. Use the local probability-event-value for its value on . Then the expectation is the sum of the event values, each multiplied by the probability of its event:
Shreve states the expectation of a finite-valued random variable in the form of (1.2.4)[3].
Value varies inside an event
Section titled “Value varies inside an event”If is not constant inside , one representative value is in general not exact. For each event with positive probability, replace the value with the conditional mean of inside that event. The expectation is then:
An event with zero probability contributes zero to the sum and can be omitted. Its conditional mean need not be defined.
The partition identity (1.2.5) follows by splitting an expectation across disjoint events and using the partial-averaging property of conditional expectation. Shreve gives the disjoint-event integral split and the unconditional-average property of conditional expectation[4].
Constant value on each event
Suppose is a payment in USD. It equals USD 0, USD 50, or USD 100 on three disjoint exhaustive events with probabilities 0.10, 0.20, and 0.70.
| Event | Constant payment, USD | Probability | Contribution, USD |
|---|---|---|---|
| 0 | 0.10 | 0.00 | |
| 50 | 0.20 | 10.00 | |
| 100 | 0.70 | 70.00 |
The expectation is the sum of the contributions:
The expectation of USD 80 is a probability-weighted average. An expectation need not be one of the possible payments, and in this example it is not.
Conditional means inside events
Suppose three disjoint exhaustive events have probabilities 0.01, 0.02, and 0.97, and the payment varies inside each event. The conditional means of the payment are USD 40, USD 30, and USD 0. The exact expectation from these conditional means is:
USD 40 is the average payment inside the first event. The example does not claim that every outcome in the first event pays USD 40.
Default inside a period
For a CDS period, one event can be “default occurs in this period.” A protection payment discounted at the actual default time varies within that event. Accrued premium also varies within the event, because a default near the end of the period accrues more premium than a default near its start.
So one midpoint value for the whole period is an approximation. The exact contribution of the event is the interval default probability multiplied by the conditional mean of the discounted payment inside the period. If the period is divided into smaller disjoint default-time intervals, the sum has more terms. In a continuous-time model, the limit of this sum is an integral over the default time. The later CDS lesson gives the hazard-rate probabilities and the cash-flow formulas for that integral.
Check your understanding
Section titled “Check your understanding”These items separately test events, conditioning, and both expectation forms. Each question stays collapsed until you open it; answers and explanations appear once you check.
Knowledge check 1.2.1 Events, conditioning, and expectation
Link to Knowledge check 1.2.1: Events, conditioning, and expectationWhich description is an event in a probability model?
Check your answer to reveal the explanation.
A one-year credit model allows at most one default. Which list is a disjoint and exhaustive partition of outcomes?
Check your answer to reveal the explanation.
Events A and B satisfy P(A and B) = 0.12 and P(B) = 0.30. What is P(A given B)? Enter a decimal probability.
Check your answer to reveal the explanation.
In a frequency example, 80 otherwise comparable names are alive at the start of a period and 2 of those names default during it. What conditional default fraction does the sample imply for that period? Enter a decimal.
Check your answer to reveal the explanation.
A random payoff is constant at 0, 50, and 100 units on three disjoint exhaustive events with probabilities 0.10, 0.20, and 0.70. What is its expected payoff in units?
Check your answer to reveal the explanation.
Three disjoint exhaustive events have probabilities 0.01, 0.02, and 0.97. A payment varies inside the events, but its conditional mean in each event is 40, 30, and 0 USD respectively. What is the unconditional expected payment in USD?
Check your answer to reveal the explanation.
Model boundary and review note
Section titled “Model boundary and review note”This lesson teaches finite partitions and the idea of the limit of a refinement. It does not construct probability spaces, prove measure-theoretic integration, estimate probabilities from data, calibrate risk-neutral probabilities, or evaluate the default-time integrals used by a CDS model. A later lesson distinguishes real-world and risk-neutral measures.
The lesson paraphrases the cited textbook; it remains draft pending human
verification of the printed locators, formulas, examples, and notation
bindings.
References
Section titled “References”- Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 1 §1.1, printed pp. 1-3, Def. 1.1.2 and eqs. 1.1.2-1.1.5. draft ↩
- Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 2 §2.3, printed pp. 66-69, eqs. 2.3.4-2.3.17 and Def. 2.3.1. draft ↩
- Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 1 §1.3, printed pp. 13-18, Thms. 1.3.1 and 1.3.4. draft ↩
- Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 1 §1.3, printed p. 17, Remark 1.3.2; Ch. 2 §2.3, printed pp. 68-72, eqs. 2.3.16-2.3.17 and 2.3.25. draft ↩
Credit default swap, the credit derivative the CDS lessons define and value.
The name of a family of standard credit default swap indices, each a standard portfolio of single-name contracts.
Duration times spread, a spread-risk measure for bonds.
Financial Industry Regulatory Authority.
International Money Market. In the CDS lessons, IMM dates are the standard maturity dates on the twentieth of March, June, September, and December.
International Swaps and Derivatives Association.
International Organization for Standardization.
Jump to default, the loss on an immediate default of the reference entity.
Coordinated Universal Time, the time standard the date arithmetic counts calendar days in.