# Corner Portfolios - Blue Box 10 - Page 226

In example ten they combine Corner portfolio with the Rf asset because it doesn’t want to lower the Sharpe Ratio. I would’ve combined Corner 4 with Corner 5. From the passage I have No idea where it came up, that the Sharpe Ratio needs to be maintained.

It says we can NOT meet the Return objective with corner 4 and corner 5 but I dont understant why.

w*(7.24) + (1-w) *5.61 = 6.5 -----> I would just solve this and we would have a comination of corner 4 and 5 that would meet the 6.5% return objectivce. Why is this not right?

Any reason as to why this is the case for this example?

Much appreciated.

I can’t see the example right now, but if there is a risk objective in the question and you are allowed to utilize a risk free asset, an appropriate corner portfolio in combination with the risk free asset will always provide you with a better risk adjusted return. There has to be some clue in there that you can utlize a risk free asset in combination and you would solve for the appropriate weights the same way as you did the corner portfolios.

I’m trying to connect that but I can’t find why they’re utilizing that combo. Using the Return and Risk objective for Corner 4 and Corner 5 seems fine meeting the objectives.

I’m going to take another look at it tonight but I’m not getting it right now. Thanks.

Return objective #1 specifically states that Sharpe ratio should be maximized in different words though. Combination of corner portfolios will not produce the highest Sharpe for 6.5% return. Std dev for combination of corner portfolios will be slightly higher 9.94% vs 9.90% for portfolio based on CP4 and risk-free asset. Whenever they ask to maximize return per unit of risk and allow borrowing at risk-free rate you should use tangent portfolio and risk-free asset for asset allocation.

i think if you can borrow at the Rf rate then you should just use the tangency portfolio, i.e., the portfolio with the highest sharpe, and then you do the interpolation between the Rf rate and that portfolio to get your weightings.

You guys are both right. The main point looking at this Example 10 and Example 9 is HERE --> No Constraint on Short selling like in Example 9. Important little trick here. Example 9 says Thomas does NOT want to borrow but here they say nothing. If no constraint borrow at the Rf rate using leverage to MAXIMIZE returns like jpsi1 said.

Thanks guys.

There is, it is at the bottom of the page if I remember correctly.

anything below Corner Portfolio 4 – viz. 5,6 and 7 cannot be used - because they do not satisfy the minimum return requirement of 6.5%. So they get eliminated automatically. Even if you could combine 7.24 with 5.61 in a proportion to get to 6.5 - here you cannot use 5.61 at all.

How does someone even borrow at the risk free rate in real life? Who would lend a RFR asset? Needs to take borrow credit worthiness into consideration.

Think of the US government borrowing from you via a Treasury note - it is risk free b/c it is backed by the government. This is why government debt instruments are often used as risk free. The loan interest rate you pay is irrelevant…risk free is used to keep the calculation not influenced by other factors when risk is present.

You should also remember that there are problems mentioned in the curriculum with the declaration of anything as 100% risk free.