This one required more care than I initially gave it.
Start by stating the two fundamental constraints for probabilities: each probability must be non-negative and their sum must equal 1. Then solve the resulting inequalities for q, considering the quadratic nature of the expressions and the need for all three probabilities to be valid simultaneously.
Pro tip: After finding the range, verify it by testing boundary values and a midpoint to ensure all probabilities stay within [0,1]. This demonstrates rigor and catches potential mistakes.
Recall that for any probability distribution, each probability must be between 0 and 1 inclusive, and the sum of all probabilities must equal 1.
Write the three inequalities: q ≥ 0, q² ≥ 0, and 1 - q - q² ≥ 0. Also ensure each probability ≤ 1, but note that q ≤ 1 and q² ≤ 1 are automatically satisfied if q is in [0,1] and the third inequality holds.
Solve 1 - q - q² ≥ 0, which is equivalent to q² + q - 1 ≤ 0. Find the roots of q² + q - 1 = 0 using the quadratic formula: q = (-1 ± √5)/2. The inequality holds between the roots.
Intersect the solution of the quadratic inequality with q ≥ 0. The roots are approximately -1.618 and 0.618, so the valid range is 0 ≤ q ≤ (√5 - 1)/2.
Check that at the boundaries and within the range, all three probabilities are between 0 and 1. Conclude that the valid range is [0, (√5 - 1)/2].
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
They framed this as a follow-up but it felt like its own question.
Start by defining the portfolio weight vector w and the payoff matrix A, then express the portfolio payoff as Aw. The non-negativity constraint is simply Aw ≥ 0, which can be written in matrix form as -Aw ≤ 0 or with an identity matrix as -I(Aw) ≤ 0. Finally, connect this to optimization problems like linear programming or quadratic programming, and mention how it ensures no arbitrage or meets risk limits.
Pro tip: In practice, you often need to handle numerical issues and scalability; mentioning that you'd use a solver like scipy.optimize or cvxpy and that the constraint is linear (hence convex) shows you understand both theory and implementation.
Clearly state that A is a 3x3 matrix where rows are states and columns are assets, and w is a 3x1 vector of portfolio weights. The portfolio payoff in each state is given by the vector Aw.
The requirement of non-negative payoff in every state means Aw ≥ 0 componentwise. In matrix form, this is -Aw ≤ 0, or equivalently -I(Aw) ≤ 0 where I is the identity matrix.
Explain that this constraint is linear and can be incorporated into a linear program or quadratic program, e.g., maximizing expected return subject to Aw ≥ 0 and budget constraints.
Mention that such constraints prevent arbitrage or ensure minimum payoffs, and can be extended to include transaction costs, leverage limits, or robust constraints under uncertainty.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.