How do you determine whether to combine the corner portfolio with the risk free asset versus another corner portfolio?
I always thought it had to do with whether or not there were constraints against short selling, but I just took the Wiley AM mock and there was a question that stated no short selling was allowed but they still combined the corner portfolio with the RF asset.
Thanks everyone
If short selling is constrained, you cannot borrow to achieve a higher expected return, but you can combine with a higher CP instead.
If you want a lower expected return, you can lend at the risk free rate, which would be better than combining with a lower CP.
Adding to it, if the market portfolio return rate is higher than your expected rate go for the market portfolio and invest excess balance in rfr asset.
+1, this is important.
correct me if I’m wrong, but another issue is there is no risk free asset in the long run. That’s why we need to use CPs, isn’t it?
MVO is single period, so it considers rf asset. Monte Carlo is multiperiod where there is no rf assets.
(at least thats how I remember it)
thanks. Looked it up in CFAI book and it makes sense. Can’t believe I have a doubt like this a week(not even) before the exam. 
So if they give you 5 corner portfolios lets say with the follow e®
1 - 10
2 - 9
3 - 8
4 - 7
5 - 6
and your required return is 7.5 and risk free is 3%, wouldnt we just use corner 3 & 4?
in what situation would we use the risk free rate and use corner portfolio 3 and the risk free rate?
In a situation where the market portfolio is known.
Here, you would just combine them.
So if they say the market portfolio is earning 5.5%, what would change? im still confused as to what the market portfolio has to do with it
Then nothing would change if you’re constrained to shorting the rfr.
If however, the market portfolio was #3, then you would lend at the rfr and get your 7.5% expected return with a better sharpe ratio.
Don’t worry about any of this. 99.9% this won’t come in the exam, it’s not part of the LOS.
Got it! thanks Mr Smart
can we assume that portfolio with highest sharpe ratio is the tangent portfolio? Or they have to specify the market portfolio?
They are both, and it’s also a corner portfolio.
But this is not what the material is teaching (I know it’s contradictory), and I haven’t seen a CP question so far that makes use of that fact. So just combine CPs.
If you borrow at the risk free to invest in the portfolio with the highest sharpe ratio. How do,you solve algebrically? And is it the same formula x = Xw + Rf?1-w) and why are you adding the risk free if ur borrowing??
as I’m getting confused with the return obtained with leverage in fixed income return + d/e (return - Rf) as you can solve for your return required this way. Not sure when to use which formula
I’m not going to open my books to look up to see if this is right but I would use the below formula to determine how much to borrow:
E®portfolio = E®invested + (Borrowed/Equity)*(E®invested-Cost of borrowing)
Set whatever you want whereever you want and solve
Obviously someone correct me if i’m wrong because i refuse to open up my books to check
This may be algebraically the same
But say your portfolio is at 10%, and you need 11%. The RFR is 1%. You would essentially have to borrow more than 10%, since 10% (1.1) - (.1)x1% i.e. the rfr will get you slightly less than 11%. Too lazy to figure out the actual %
It is the same exact formula as with a typical corner portfolio except that when you solve it one of the numbers will be negative and the other positive. Given the right information you can come to the same conclusion with the formula that verse gave away as well.
Per CFAI, if you’re permitted to use leverage/borrow at the risk-free rate then you use the risk-free asset and the tangency portfolio, i.e., the portfolio with the highest Sharpe ratio. You can do the normal interpolation to calculate the weights and then you can check your calc with the levered return calculation (levered return = unlevered return + [D/E * (unlevered return - risk-free rate aka cost of capital)].
looked it up in Kaplan Book 3 page 122, and its explains to just take the weight of the risk free and solve with the tangency portfolio. So basically you get a free loan and get to receive the risk free rate.
Yet this is then contracticted later on with calculation you mentioned before: levered return = unlevered return + (D/E * (unlevered return - risk free rate).