Two assets have identical 20% volatility and a correlation of 0.95. Their expected excess returns are 5.0% and 5.5%. What weight does the unconstrained maximum Sharpe portfolio put on the first asset?

Two assets have identical 20% volatility and a correlation of 0.95. Their expected excess returns are 5.0% and 5.5%. What weight does the unconstrained maximum Sharpe portfolio put on the first asset?

Approach: Tangency weights are proportional to the inverse covariance matrix times the excess return vector. With equal volatilities the inverse correlation matrix is enough.

-3/7. Tangency weights are proportional to Sigma^{-1}*mu, and with equal volatilities the inverse correlation matrix is proportional to [[1, -0.95], [-0.95, 1]], so the unnormalised weights are 0.05 - 0.95*0.055 = -0.00225 and 0.055 - 0.95*0.05 = 0.0075. They sum to 0.00525, giving -3/7 = -42.9% and 10/7 = 142.9%. A half point difference in an expected return that cannot be estimated to better than two points has produced a 143% long against a 43% short, because dividing by (1 - rho^2) = 0.0975 amplifies any difference in mu by a factor of ten. This is the sense in which unconstrained optimisation is error maximisation, and it is why weight bounds or shrinkage of mu are applied before the optimiser sees the inputs.

Follow-up: How large must the standard error on each expected return be before the sign of the first weight is uninformative?

Key concepts: tangency portfolio, error maximisation, correlation, estimation error.