QBUS2310 Chap.10 Optimisation Under Uncertainty and Risk
Optimisation Under Uncertainty and Risk
Deterministic models treat uncertain quantities as known, which can produce fragile decisions. Uncertainty can be handled through expected-value models, worst-case protection, staged recourse, or explicit risk measures. Each choice expresses a decision attitude. This chapter focuses on principles supported by the acquired risk material and avoids claiming a complete course-specific robust-counterpart or software treatment.
These terms let a reader tell a quantile threshold apart from a tail-average loss. Expected Value is a probability-weighted average outcome. Worst-Case Model is a model that protects against the most adverse outcome in a stated uncertainty description. Recourse Decision is an action chosen after some uncertainty has been observed. Value at Risk is a loss quantile at a stated confidence level.
Conditional Value at Risk is the average loss in the specified worst tail of a distribution. Risk Measure is a numerical summary used to compare uncertain losses. The risk figures above are verified against their definitions: VaR at confidence 0.75 is 6 and CVaR is 12 for this equal-probability distribution. VaR identifies a quantile threshold; CVaR measures the mean loss in the specified upper tail.
What this chapter covers
- 01
Sources of model uncertainty
- 02
Expected-value decisions
- 03
Worst-case protection
- 04
Scenario and recourse framing
- 05
Loss distributions
- 06
Value at Risk
- 07
Conditional Value at Risk
- 08
Risk-model limitations
Optimisation Under Uncertainty and Risk worked example
- +1Sort the loss outcomes and form cumulative probabilities: 0.25, 0.50, 0.75, and 1.00.
- +1At confidence 0.75, the smallest loss whose cumulative probability reaches 0.75 is six, so VaR is six.
- +1The worst tail has probability 0.25 and contains loss twelve only, so CVaR is twelve.
- +1State the convention and atom issue. Averaging every observation at or above VaR would incorrectly include the threshold mass in some discrete cases.
Key terms
- Expected Value
- A probability-weighted average outcome.
- Worst-Case Model
- A model that protects against the most adverse outcome in a stated uncertainty description.
- Recourse Decision
- An action chosen after some uncertainty has been observed.
- Value at Risk
- A loss quantile at a stated confidence level.
- Conditional Value at Risk
- The average loss in the specified worst tail of a distribution.
- Risk Measure
- A numerical summary used to compare uncertain losses.
Optimisation Under Uncertainty and Risk FAQ
Why does sources of model uncertainty matter?
Expected Value: A probability-weighted average outcome. Worst-Case Model: A model that protects against the most adverse outcome in a stated uncertainty description. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How can I check scenario and recourse framing?
Worst-Case Model: A model that protects against the most adverse outcome in a stated uncertainty description. Recourse Decision: An action chosen after some uncertainty has been observed. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
What separates expected value from worst-case model?
Recourse Decision: An action chosen after some uncertainty has been observed. Value at Risk: A loss quantile at a stated confidence level. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Which error is most likely around value at risk?
Value at Risk: A loss quantile at a stated confidence level. Conditional Value at Risk: The average loss in the specified worst tail of a distribution. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How should I practise risk-model limitations?
Conditional Value at Risk: The average loss in the specified worst tail of a distribution. Risk Measure: A numerical summary used to compare uncertain losses. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Exam move
Write losses in increasing order with cumulative probability before computing a quantile. For CVaR, identify the exact tail probability and handle any mass at the VaR threshold explicitly. Rehearse the chapter method in this order: Sort the loss outcomes and form cumulative probabilities: 0.25, 0.50, 0.75, and 1.00. At confidence 0.75, the smallest loss whose cumulative probability reaches 0.75 is six, so VaR is six.
The worst tail has probability 0.25 and contains loss twelve only, so CVaR is twelve. State the convention and atom issue. Averaging every observation at or above VaR would incorrectly include the threshold mass in some discrete cases.
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