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Will last year's star fund repeat?

Asked at DE Shaw, Jane Street

A fund posted a year 1.51.5 standard deviations above its peer average. Year-over-year performance (in standardized units) is bivariate normal with correlation ρ=0.5\rho = 0.5.

What is the probability the fund again beats +1+1 standard deviation next year? (Use the standard normal: Φ(0.29)0.614\Phi(0.29) \approx 0.614.)

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You need the full conditional distribution of next year's performance, not just its mean. For standardized bivariate normal, YX=xN(ρx,1ρ2)Y \mid X = x \sim \mathcal{N}(\rho x,\, 1 - \rho^2).

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