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3_5 Capital Allocation Line

3_5 Capital Allocation Line - 1 w R = 40 ⇒ w RF =(1 w R =...

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The Complete Portfolio and the Capital Allocation Line (CAL) Consider a portfolio on the Markowitz (risky) EF, with E(r R ) = 12% and σ R = 16%. Now we introduce a risk-free asset, where r RF = 4%. We now construct a "new" portfolio (P), referred to as the complete portfolio , that combines the risky portfolio with the risk-free asset. • let w RF = the weight in the risk-free asset • let w R = the weight in the risky portfolio w RF + w R = 1 w RF = 1 - w R • E(r P ) = w RF ( r RF ) + w R E(r R ) E(r P ) = (1 - w R ) r RF + w R E(r R ) E(r P ) = r RF + w R [E(r R ) - r RF ] • by definition, [E(r R ) - r RF ] = risk premium of the risky portfolio E(r P ) - r RF = w R [E(r R ) - r RF ] • by definition, w R [E(r p ) - r RF ] = risk premium of the complete portfolio σ p 2 = w RF 2 σ RF 2 + w R 2 σ R 2 + 2w RF w R σ RF σ R ρ RF,R • By definition, σ RF 2 = 0 σ p 2 = w R 2 σ R 2 and σ p = w R σ R . Now we consider some possible weights.
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Unformatted text preview: 1. w R = 40% ⇒ w RF = (1 - w R ) = 60%. 2. w R = 80% ⇒ w RF = (1 - w R ) = 20%. 3. w R = 100% ⇒ w RF = (1 - w R ) = 0%. 4. w R = 120% ⇒ w RF = (1 - w R ) = -20%. • here the investor is borrowing at r RF Capital Allocation Line E(r P ) lending (investing) at r RF borrowing at r RF CAL r R = 12% w R = 100% r RF = 4% σ P σ R = 16% • CAL slope = ∆ E(r P )/ ∆ σ P • as usual, use E(r P ) and r P interchangeably • CAL slope = (r R - r RF )/ σ R ⇒ (r P - r RF )/ σ P = "reward-to-variability" ratio or, "reward-to-risk" ratio or Sharpe ratio ⇒ on the CAL the reward-to-risk ratio is constant Next step: do the same analysis for other portfolios on the Markowitz Efficient Frontier....
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