Question at position 15 Without imposing no-shorting constraints, you found the tangency portfolio of 3 assets A, B, and C to have weights [0.6, 0.6, -0.2], respectively. What is the most precise thing you can say about the weights of the tangency portfolio once you impose no-shorting constraints?The weights will be [0.5, 0.5, 0.0].The weight on asset C will increase.The weights will remain the same: [0.5, 0.7, -0.2].None of the statements above are necessarily true.The weights on both assets A and B will decrease.

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BlackTom AI · Accepted Answer

We start by restating the question and options to ensure clarity about what we are evaluating.
Question: When the unconstrained tangency portfolio for assets A, B, and C has weights [0.6, 0.6, -0.2], what is the most precise statement about the weights once no-short constraints (weights ≥ 0) are imposed?
Answer options:
1) The weights will be [0.5, 0.5, 0.0].
2) The weight on asset C will increase.
3) The weights will remain the same: [0.5, 0.7, -0.2].
4) None of the statements above are necessarily true.
5) The weights on both assets A and B will decrease.

Now, let’s analyze each choice carefully.
Option 1: The weights will be [0.5, 0.5, 0.0].
Why this could be tempting: removing the short position on C (setting wC to 0) while keeping total weight 1 might seem like a simple adjustment to satisfy nonnegativity.
Why it is not necessarily true: projecting the unconstrained vector [0.6, 0.6, -0.2] onto the nonnegative simplex (w ≥ 0, sum w = 1) does not in general yield [0.5, 0.5, 0]. The optimal non-short tangency portfolio depends on the underlying return/covariance structure; simply halving the A and B weights to sum to 1 while setting C to 0 ignores the optimization criterion that defined the tangency portfolio and how the covariance with the benchmark interacts with each asset. Therefore, this specific coordinate-wise adjustment is not guaranteed.

Option 2: The weight on asset C will increase.
Why this might be proposed: removing a negative weight could, in some contexts, reduce leverage and potentially shift more weight to higher-return/lower-risk assets, but
Why it is not necessarily true: under no-short constraints, the feasible set is w ≥ 0, sum w = 1. Since wC was negative in the unconstrained solution, projecting onto the feasible set does not automatically increase wC; in fact wC must end up ≥ 0, but the amount by which it increases (if at all) is not determined without solving the constrained optimization. In many cases, wC could remain at 0 after projection, or could become a small positive value, but this is not universally guaranteed from the given information.

Option 3: The weights will remain the same: [0.5, 0.7, -0.2].
Why this is unlikely: under no-short constraints, negative weights are forbidden. Keeping wC = -0.2 violates the constraint wC ≥ 0, so this option cannot be correct as a statement about the constrained solution. It also contradicts the requirement that all weights be nonnegative.

Option 4: None of the statements above are necessarily true.
Why this is the precise, cautious conclusion: Given only the unconstrained weights and the constraint w ≥ 0, the exact constrained tangency portfolio is not determined without performing the constrained optimization (which depends on asset returns and the covariance matrix). The three specific claims in options 1–3 are each not guaranteed to hold in general, so the most precise blanket statement is that none of the above statements are necessarily true.

Option 5: The weights on both assets A and B will decrease.
Why this is not a guaranteed outcome: There is no inherent reason to expect both A and B to decrease when enforcing nonnegativity; the constrained optimum could feasibly allocate more to A or B (or even to C if it gains nonzero weight) depending on the relative expected performance and risk contributions. Without solving the constrained optimization, we cannot assert a decrease for both A and B.

Synthesis: Each of the first three options asserts a specific qualitative change or exact weight vector that would only hold under particular data (returns, covariances) and a particular solution path. Since we do not have those specifics and the constrained optimum may take various forms depending on the inputs, those statements are not necessarily true. The safe, general conclusion given the information is that none of the listed specific claims must hold when imposing no-shorting constraints.

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