Your client requests a portfolio with zero Environmental Risk. They are interested in two Exchange Trade Funds and will allow you to short one of the ETFs. What is the weight in the first ETF if you create a zero Environmental Risk with a negative weight in the second ETF. The first ETF's Environmental Score is 2.3 and the second ETFs score is 11. (Answer in % to one decimal place, 40.5% is 40.5)

Question

BlackTom AI · Accepted Answer

To tackle this problem, I’ll restate what’s being asked and identify the key constraints.
We want a zero Environmental Risk for the portfolio, using two ETFs where the second is allowed to have a negative weight (i.e., a short position). The first ETF has an Environmental Score of 2.3, the second has a score of 11, and we need the weight of the first ETF in percent (to one decimal place).
First, set up the two fundamental equations that reflect the constraints of a zero-risk portfolio:
1) The weights must sum to 1 (assuming a fully invested portfolio): w1 + w2 = 1.
2) The environmental risk contributed by the portfolio must be zero: 2.3*w1 + 11*w2 = 0.
Now solve the system. From equation (1), w2 = 1 − w1.
Substitute into equation (2): 2.3*w1 + 11*(1 − w1) = 0.
Expand and collect like terms: 2.3*w1 + 11 − 11*w1 = 0 → (2.3 − 11)*w1 + 11 = 0 → (−8.7)*w1 = −11.
Divide both sides by −8.7: w1 = 11 / 8.7 ≈ 1.264367…
Convert to a percentage by multiplying by 100: about 126.4367%, which rounds to 126.4% when you report to one decimal place.
For completeness, the corresponding w2 is 1 − w1 ≈ 1 − 1.264367 ≈ −0.264367, i.e., about −26.4%, which is consistent with the description of a negative weight (short position) in the second ETF.
Key checks: the math satisfies both constraints (w1 + w2 = 1 and 2.3*w1 + 11*w2 = 0), and the rounding aligns with the requested one-decimal-place format.
In summary, solving the linear system yields a first-ETF weight of approximately 126.4%, with the second ETF carrying about −26.4%.

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