^ This makes sense .

yup:) BUT as we are giving the CFAI exam, we need to follow CFAI ways, though it goes against our thinking, if any

Suppose your req 8.7%

Tangency portfolio (TP) offers - 9.1%…so you can combine tangency portfolio & risk free…since you require less than offered by highest sharpe portfolio you could lower risk by combinging it with Rfr

X * 9.1 + (1 - X) *4.5 = 8.7%

=> X = 0.913 —>weight in #4

( 1 - 0.913) = 0.087 —> weight in Rfr

Check—> 0.913 X 9.1 + 0.087 X 4.5 = 8.70%

Risk (linear approximation ) ----- 0.913 X 9.1 = 8.31%

**rules**

- Always choose highest sharpe ratio portfolio (TP)-----> if you require return more than the available on TP, you short at Rfr & invest more than 100% to meet your return objective (Provided shorting is allowed)

**or**

Choose corner portfolios which envelope required return (if shorting is disallowed)

- If return requirement is less than TP----> you invest portion into TP & Rfr----effect lower Std Dev. (again provided lending is allowed)

**or**

Choose corner portfolios which envelope required return (if lending is disallowed)

Hope this helps!

^ thanks makes sense…

So to make it short, in case lev is allowed, we just solve the equation for W of TP and Rf and we can find out in case we forgot the rule if we’re short or long in rf (based on the weights).

In case lev is not allowed, corner portfolios. usually corner portfolio / req return / corner portfolio.