I’ve read through a few related posts, but still don’t understand when to use tangency portfolio vs corner portfolio. Does it all hinge on whether the invester is allowed to buy risky assets on margin and can borrow at risk free rate?

Yes.

if they cannot short, you will use corners.

if you can short, you choose the portfolio with the highest sharpe and borrow at the risk free

Condition for using tangency:

1.highest sharpe ratio, and

- (any combination of) asset allocation using tangency portfolio and risk free will achieve the required rate of return.

if any of the above condition is not met, find your corners

Thanks to both of you for your replies. I agree that if they cannot short, you would have to use corners. So what about Example 10 in CFAI p267. In Solution 1, they use a single Portfolio #4 as the SAA (and reference as a tangency port) instead of a blend of 4 and 5. The Example also states that it is sign-constrained (no shorting)

note that number 4 has highest sharpe.

and it satisfies 6.5% required rate of return without leveraging.

now, let’s say this portfoilio’s sharpe remains the same, but the expected return is 6% instead of 7.25%.

without leveraging, 100% of asset invested in “4” will not meet the required rate of return even if it has the highest sharpe.

if this was the case, you have to find your corners. you will see similar questions in mocks and practice exams.