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Work out the asset weights giving the lowest-variance mix of two risky assets

CFA® Level I · Quantitative Methods · The Return and Risk of a Financial Portfolio · QM.17.a

The idea

Combining two risky assets traces out a curve of possible portfolios, and the point with the lowest variance is the minimum-variance portfolio. Its weight in asset 1 is w1 = (var2 - cov12) / (var1 + var2 - 2 x cov12), with w2 = 1 - w1, where cov12 = correlation x sd1 x sd2. With zero correlation the weights are inversely proportional to variance, not to standard deviation. With perfect negative correlation, w1 = sd2 / (sd1 + sd2) and portfolio risk falls to zero. Portfolios on the curve above the minimum-variance point form the efficient frontier; those below it offer lower return for the same risk and are never chosen.

A practice question

From the question bank, with the reason each wrong option is wrong.

Asset 1 has a standard deviation of 20% and Asset 2 of 30%, and their returns are uncorrelated. The weight of Asset 1 in the minimum-variance portfolio is closest to:

  1. 0.60Why not: Weighted by the inverse of standard deviation instead of the inverse of variance.
  2. 0.31Why not: Put Asset 1's own variance in the numerator, which gives Asset 2's weight.
  3. 0.69 — correct

With zero covariance, w1 = var2 / (var1 + var2) = 0.09 / (0.04 + 0.09) = 0.6923, or about 0.69.

A harder one

Asset A has a standard deviation of 15% and Asset B of 25%, with a correlation of 0.3. The weight of Asset A in the minimum-variance portfolio is closest to:

  1. 0.74Why not: Ignored the covariance term and used var(B) / [var(A) + var(B)].
  2. 0.82 — correct
  3. 0.69Why not: Subtracted the covariance only once in the denominator instead of twice.

Cov = 0.3 x 0.15 x 0.25 = 0.01125. wA = (0.0625 - 0.01125) / (0.0225 + 0.0625 - 2 x 0.01125) = 0.82.

The hardest tier

Asset 1 has a standard deviation of 12% and Asset 2 of 18%, with a correlation of 0.2. The standard deviation of their minimum-variance portfolio is closest to:

  1. 10.83% — correct
  2. 13.58%Why not: Took the weighted average of the two standard deviations, ignoring diversification.
  3. 10.03%Why not: Found the right weights but dropped the covariance term from the portfolio variance.

w1 = (0.0324 - 0.00432) / (0.0144 + 0.0324 - 2 x 0.00432) = 0.7358. Portfolio variance = 0.011737, so the standard deviation is 10.83%.

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