Questions / Quant research, Portfolio & factor risk
A second forecast can add no independent precision
Two unbiased forecasts estimate the same alpha. Their error variances are 1 and 4, with error covariance 1 in common squared units. Combine them as w times forecast 1 plus (1−w) times forecast 2. What w minimizes forecast-error variance?
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Answer: 1 weight on forecast 1
- Error variance simplifies to 3w²−6w+4. Its derivative 6w−6 vanishes at w=1, so the noisier second forecast receives zero weight; its covariance makes its extra variation unhelpful.
Interview shortcut. Optimal weight = (σ₂² − c)/(σ₁² + σ₂² − 2c) = 3/3.
In the app this has a follow-up where one thing about the setup changes. Try the follow-up.