I’ve come across this situation a few times now in the Schweser and Finquiz Mocks. Basically, we are told to find the optimal asset allocation for a person with required return x and 5 corner portfolios. However, it does not specificaly address whether leverage (or risk-free asset) are allowed to be used.

So, options are to either 1) Combine Portfolios A+B to meet required return or 2) Choose just the highest Sharpe Ratio Portfolio and adjust leverage/risk-free asset accordingly to meet required return.

My question is: without specifying whether leverage is allowed or not, do people assume it is or isn’t? If it’s not we must do option 1), if it is allowed, often times option 2) becomes preferable. Hopefully CFA makes it more clear than these mocks and it won’t be a problem. Thanks.

Been some time since I looked at Corner portfolios but I thought the inability of being able to borrow (shares or leverage) is one of the principal behind using corner portfolios?

Not necessarily Galli. If the problem asks to find the optimal allocation and leverage is allowed, then it will most likely be an allocation to the Corner Portfolio (highest Sharpe ratio) + borrow/leverage. My default has been, if there is no mention of leverage use, to just combine the two needed Corner Portfolios. However, Finquiz Mock Exam 4 AM had a question like that, and the solutions posted showed combining Corner B with risk-free asset.

I just find it frustrating that they can’t write clear questions. I will hold out hope CFAI can do so.

I thought if tangent portfolio’s return > required return, and borrowing/lending is allowed, then you can combine tangent + risk free…now I’m confused.

Giving this some more thought because it’s important and I can’t seem to remember the specific details, i’ll look later tonight.

To borrow (leverage) I was under the impression have to sell / short something. With Corner portfolios, you are sign constrained which limits you to investing in Corner portfolios, likely a weighting of two that produces the desired required return.

The Rf rate can come into play, as jmsp described, by investing in the tangent portfolio + a weighting in the RF rate that brings the portfolio return down to the require return

I was under the impression from my distant review of this topic long ago was:

If target return was the primary objective, combine the two corner portfolios that surround the target return.

If minimizing risk with a target return was the primary objective, use the corner portfolio that most represents the tangency portfolio and combine it with the risk-free asset.

To the OP, I can’t find an example or even an EOC question that asks a question about leveraging position and investing in a corner portfolio - Actually I can’t find any question that asks about leverage. Only page 222 has in paragraph 2 mentions using leverage to acheive a return higher than the tangency (to the CAL) portfolio. Can you share an example you have for the Kaplan practice exam? if you just want to cite the question number I can look at it.

Regarding corner portfolios, see page 213 in the CFAI text, corner portfolios are defined as being sign-constrained where one asset weight is set to zero. There are limited number of these and they’re used to allocate capital when shorting or borrowing on margin is not possible, as the efficient frontier is a plot of all potential investable portfolios - most of which have a negative weight in 1 or more asset classes. Therefore Corner portfolios are a special case of portfolios along the efficient frontier…

Would a leveraged position have a corner portfolio tangent to the CAL? Doesn’t make sense to me… Thoughts? (again sorry to derail)

EDIT: Found this gem, last comment solidifies it for me:

Galli, referring to the link you provided, my question is with regards to a comment the final poster made: “If the tangency portfolio has a lower return than required: 1) if you aren’t restricted from shorting, you can borrow at the rfr (weight rfr < 0) to achieve the required return using the portf w. the highest Sharpe 2) if you ARE restricted from shorting: you have to go to the portfolio with the highest Sharpe that does meet your required return. Here, you may be able to linearly combine adjacent corner portfolios (adjacent to your req’d return, that is). This linear combination will produce the highest possible Sharpe that meets your return requirements.” My question in the OP is basically if the vignette omits any information on whether borrowing/leverage is allowed, do you default to doing 1) or 2). My default has been to assume it’s not allowed unless specifically noted… hence, combine two Corner Portfolios. However, I have completed mocks where even though it’s not mentioned, the answer shows the risk-free asset being used.

I think if they give you a corner portfolio you assume no leverage is appied

if they give you a portfolio that is tangent to the CAL then you can allocate to the Rf, if a higher return is required than what the tangent portfolio is providing, leverage the Rf and allocate more to the tangent portfolio.

I was given corner portfolios on Finquiz Mock 4 AM and assumed no leverage was allowed as you suggested.

However, the answer included was an allocation combination of the corner portfolio (highest sharpe) with the risk-free asset. So I was left going, “well damnit, when should I know if we are allowed to use leverage or not”. And it’s still not clear. Thanks anyways…

^Just because there’s an allocation to the risk-free asset does not mean you borrowed at the risk-free rate and leveraged your portfolio. It sounds like the corner portfolio with the highest sharpe ratio exceeded the return requirement and thus you were required to lower your risk. The most efficient way to do that would be to allocate w to the corner portfolio and (1-w) to the risk free asset. If you were to allocate funds to another corner portfolio, it would be suboptimal to the allocation i described above because it would have a lower sharpe ratio

Thanks Would You Look and Galli. I think it’s becoming clear to me. So, a brief summary would be: If Corner Portfolio A (highest Sharpe) has a return greater than required return, combine Corner A with risk-free-asset to reduce risk to meet required return. If Corner Portfolio A(highest Sharpe) has a return less than required return, two options: 1) If leverage is allowed, lever Corner Portfolio A up till it meets required return. 2) If leverage is not allowed, combine Corner Portfolio A with an adjacent Corner Portfolio with a higher return so that the required return is met.

Quick follow up to iron everything out. If we get to situation 2 above, as the following: Required Return 9% Corner Portfolio 1 (ER = 12%, Sharpe = 0.70) Corner Portfolio 2 (ER = 11%, Sharpe = 0.60) Corner Portfolio 3 (ER = 8%, Sharpe = 0.90) Corner 4&5 not relevant. Leverage not allowed. So we are going to work with Corner Portfolio 3 (highest Sharpe). But it’s return is short of the required return. Do we combine it with Corner Portfolio 2 since it’s adjacent or Corner Portfolio 1 since it’s Sharpe is higher than 2. Thanks!