optimal portfolio with risk free leverage option formula

Anyone know the simple fail safe formula for determining the tangency portfolio percent (and corresponding risk free short) when you are allowed leverage and your required return exceeds the tangent portfolio expected return? I know that if the required return is less than the tangency portfolio the mix determination formula is Rf- 4% Required Return=8% Tangency= 10% So the asset mix would be .08-.04 / .10/.04= .66 and so .66 times the tangent portfolio return of 10 gives you 6.66% and the risk free asset brings you (1-.66) times 4% at 1.36% to meet your required return of 8% and give you the optimal SHarpe ratio… I am just having a brain lock and I am having trouble going the other way with this if the required return exceeds the tangent portfolio and you can use leverage (short the risk free asset). Can anyone help here?

Required return = Wt*(Rt)+(1-Wt)(Rf) Wt= weight tangency Rt= Return tangency Rf= Risk free rate

If W is the weight of the tangency portfolio, then… (W) * (Tangency Return) + (1-W) * (RFR) = (Required Return) Solve for W. If you are leveraging, you should expect W to be > 1 and (1-W) to be negative. I find leveraged portfolios hard to do in my head and have to solve them on paper, although maybe practicing a few times would get it back in there.

Thanks guys! John

I’m looking at a similar question on the 2016 past paper - 4B, and shouldn’t the risk free rate be considered a negative rate because you are borrowing funds to leverage? In the solution to this Q and another I’ve seen the risk free rate used is positive which doesn’t make sense if you are borrowing to leverage??

Any insight much appreciated

The risk free rate is always quoted as a positive value. The weight will determine whether you are investing (w > 0) or borrowing (w < 0) the risk-free asset.