Why is the Optimal (Complete) Portfolio different from the Tangency Portfolio?

The optimal (complete) portfolio $w_{CP}$ is

$w_{CP}= y * w_{TP}$ and $w_{rf}= 1 - y$

• $w_{TP}$ collects the portfolio weights of the Tangency Portfolio
• $y$ is a scalar that accounts for the optimal amount of levering up and down
  • For $y<1$: the optimal portfolio of an investor levers down ( investor lends money to the government -> buys bonds)
  • For $y>1$: the investor levers up (borrows money and invests only in risky assets)
• The value of $y$ for concrete investor can be calculated only with knowledge of risk aversion of this investor

The expected return of the investor's optimal (complete) portfolio is

$E[r(w_{CP})] = y * E[r(w_{TP})] + (1 - y) * r_f$

and its variance is

$Var(r(w_{CP})) = y^2 * Var(r(w_{TP}))$

A derivation of these formulas is explained in the Class Material of Week 3