Suppose that there are only 100 stocks in the stock market.

How do you express the view, in the context of the Black-Litterman asset allocation model with use of vectors, that the expected return of the first stock will outperform an equal-weighted portfolio of all the stocks by 5% ?

Question

Suppose that there are only 100 stocks in the stock market.

How do you express the view, in the context of the Black-Litterman asset allocation model with use of vectors, that the expected return of the first stock will outperform an equal-weighted portfolio of all the stocks by 5% ?

BlackTom AI · Accepted Answer

Question restatement: In the Black-Litterman framework with vector notation for asset allocation, you want to express the view that the expected return of the first stock will outperform an equal-weighted portfolio of all stocks by 5%.

Option 1: P=[1/100, 1/100, ..., 1/100], a 1 by 100 vector.
- This represents an equal-weighting view across all 100 stocks, implying each stock has a +1% (or +1/100) component relative to the benchmark, which does not encode a special view about stock 1 outperforming the equal-weighted portfolio by 5%. It treats all stocks identically, so it cannot express the specific outperformance of stock 1 over the equal-weighted portfolio. So this option fails to capture the stated view.

Option 2: P=[99/100, -1/100, ..., -1/100], a 1 by 100 vector.
- This vector assigns a large positive weight to stock 1 (99/100) and small negative weights to the remaining stocks (-1/100 each). In the Black-Litterman setup, a P vector is used to linearly combine asset returns to express the view: P' * mu_hd = view amount, or in standard form, the view is a linear combination of expected returns. The given pattern effectively assembles a view that stock 1 outperforms the rest in a way that, when compared to the equal-weighted portfolio, yields a 5% outperformance. Intuitively, the 99/100 weight on stock 1 minus a small negative contribution from each other stock aligns with expressing a single-stock outperformance relative to the equal-weighted aggregate. This matches the intended view description.

Option 3: P=[5%, 0, ..., 0], a 1 by 100 vector.
- This suggests a 5 percentage point allocation to stock 1 and zero to all others as the explicit view weights. However, in the Black-Litterman context, P is typically composed of factors applied to the vector of expected returns to form the view; using 5% as a direct weight on a single stock does not align with the standard representation of a relative outperformance over an equal-weighted portfolio. It conveys an absolute adjustment to stock 1 rather than a relative outperformance of stock 1 over the equal-weighted benchmark, making it an incorrect encoding of the described view.

Option 4: P=[5/100, 1/100, ..., 1/100], a 1 by 100 vector.
- This assigns 0.05 to stock 1 and 0.01 to all others, which does not reflect the specific idea that stock 1 should outperform the equal-weighted portfolio by 5%. Instead, it distributes a small positive weight to stock 2 through 100 alongside a minor positive contribution for stock 1, effectively creating a blended view that does not express the stated outperformance relative to the equal-weighted benchmark. Therefore, this encoding misstates the intended relation between stock 1 and the equal-weighted portfolio.

Summary of reasoning for each option:
- Option 1 is incorrect because it imposes equal contributions from all stocks and cannot express outperformance of one stock relative to the equal-weighted portfolio.
- Option 2 is designed to isolate stock 1 with a dominant positive and small negative adjustments to others, aligning with the idea of stock 1 outperforming the equal-weighted mix by a specified margin.
- Option 3 is incorrect because it presents an absolute, not relative, adjustment to stock 1, and does not encode the comparison to the equal-weighted portfolio.
- Option 4 is incorrect because its weight pattern does not capture the precise one-versus-all relative outperformance of stock 1 over the equal-weighted portfolio.

Note: In the Black-Litterman framework, the P matrix is used to define linear views about expected returns by forming linear combinations of the asset returns. The second option’s weight structure most closely matches a view that emphasizes stock 1 relative to the rest, which is consistent with expressing that stock 1 will outperform the equal-weighted portfolio by 5%.

Fall 2025.FIN.5321.02 Final Exam

Suppose that there are only 100 stocks in the stock market. How do you express the view, in the context of the Black-Litterman asset allocation model with use of vectors, that the expected return of the first stock will outperform an equal-weighted portfolio of all the stocks by 5% ?

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