The Frontier in Pictures





Kerry Back

Frontier is a line

  • The frontier portfolios are all scalar multiples of the vector \(C^{-1}(\bar{r}-r_f1_n)\).
  • The scalar determines how much is saved or borrowed.
  • For some value of the scalar, there is no saving or borrowing.
  • For this value of the scalar, the portfolio is called the tangency portfolio.
  • The mean-variance frontier is the line from the risk-free rate through the tangency portfolio.

Tangency

  • Consider the same minimum variance problem but without saving or borrowing.
  • This adds the constraint \(1_n^\top w=1\).
  • This frontier is a hyperbola.
  • The line through the tangency portfolio just touches the hyperbola. It is tangent to the hyperbola at the tangency portfolio.

  • risky assets without saving or borrowing

  • the least-risk hyperbola without saving or borrowing
    HTML tutorial

  • adding risk-free saving and borrowing to any portfolio
    HTML tutorial

  • the optimum is at tangency
    HTML tutorial