Decision Theory under Uncertainty and Portfolio Management

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[...] In these models, outcomes are ranked, and probabilities are assigned cumulatively, which helps address some of the aforementioned issues. Ambiguity: The original Prospect Theory does not handle ambiguity (situations where probabilities are not known or not well-defined) well. While it does account for risk (known probabilities) and distortions in the perception of risk, it doesn't provide tools or frameworks for situations of true ambiguity. Rank-Dependent Expected Utility (RDEU) Background: Rank-Dependent Expected Utility (RDEU) evolved from the recognition that individuals tend to rank outcomes and then apply decision weights in a non-linear fashion to these ranked outcomes. [...]

[...] For a given lottery LL with outcome xx and probability pp, the Prospect Theory value is defined as: V(L)=w(p)×u(x)V(L)=w(p)×u(x) Where: - is the decision weight given to outcome xx with probability pp. This function captures the distortion of probabilities, and it is not linear, which means that people often overestimate small probabilities and underestimate large probabilities. - is the utility of the outcome xx. The function is concave for gains and convex for losses, reflecting the diminishing sensitivity to changes in wealth. Using the lotteries provided: 1. L1=(0.11,?1M;0.89,?0)L1=(0.11,?1M;0.89,?0) 2. L3=(0.1,?5M;0.9,?0)L3=(0.1,?5M;0.9,?0) 3. 4. [...]

[...] are ranked in ascending order of desirability or value. - is the decision weight or probability weighting function that captures the distortion of probabilities. This function is applied cumulatively to the ranked outcomes. The cumulative application ensures that the ranking of outcomes influences the decision weights. Key Features: 1. Ranking of Outcomes: Outcomes are ranked by their desirability, and decision weights are applied based on this ranking. 2. Decision Weights: Unlike the linear probabilities in expected utility theory, in RDEU, the decision weights may not equal the objective probabilities. [...]

[...] For the second comparison, the certainty effect kicks in. A sure gain appears more attractive than any gamble, no matter the potential upside. Therefore, L2L2 is preferred over L4L4, even though the expected utility might suggest otherwise. By selecting appropriate parameters for our weighting function (like the right value for the Prospect Theory can explain the preference patterns seen in the Allais Paradox. Critical Problem with Original Prospect Theory The original Prospect Theory was groundbreaking in the way it explained decision-making under uncertainty. [...]

[...] Decision Theory under Uncertainty and Portfolio Management Violation of the Independence Axiom The Independence Axiom: For any three lotteries and NN, and any probability pp in the interval if LL is preferred to MM (i.e., then the lottery which is a pp mixture of LL and NN should also be preferred to the pp mixture of MM and NN. Using the lotteries provided: 1. L1=(0.11,?1M;0.89,?0)L1=(0.11,?1M;0.89,?0) 2. L3=(0.1,?5M;0.9,?0)L3=(0.1,?5M;0.9,?0) 3. 4. L4=(0.10,?5M;0.89,?1M;0.01,?0)L4=(0.10,?5M;0.89,?1M;0.01,?0) Let's examine the Allais Paradox relative to the Independence Axiom: Comparison People prefer L1L1 over L3L3, even though L3L3 offers a chance at a higher amount compared to ?1M). [...]