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
adding risk-free saving and borrowing to any portfolio
the optimum is at tangency
The Frontier in Pictures Kerry Back