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A stock has 30%30\% volatility, the index 15%15\%, and their correlation is 0.60.6. What is the stock’s beta to the index?

Answer with a number. Fractions, powers and expressions like 23/6 or C(52,5) are read correctly in practice.

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Answer

1.21.2

Beta is a correlation scaled by the ratio of standard deviations: β=ρ σy/σx=0.6×(0.30/0.15)=1.2\beta = \rho\,\sigma_y/\sigma_x = 0.6 \times (0.30/0.15) = 1.2. The two inputs do different jobs. The correlation says how much of the stock’s movement the index explains, and the volatility ratio converts into the stock’s units. A beta above one therefore need not mean tight co-movement: here only 36%36\% of the variance is explained, and the rest is idiosyncratic risk that a beta hedge does nothing about.

Worked solution

  1. Formula
    β=ρ σyσx\beta = \rho\,\frac{\sigma_y}{\sigma_x}
  2. Substitute
    =0.6×0.300.15= 0.6 \times \frac{0.30}{0.15}
  3. Solve
    =0.6×2= 0.6 \times 2
  4. Answer
    β=1.2\beta = 1.2

Takeaway: A beta is a correlation scaled by the ratio of standard deviations.

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A stock has 30% volatility, the index 15%, and their correlation is 0.6. · QuantMax