Excess Return over MTCR’s formula is equivalent to a Sharpe Ratio. The calculation is: (R_{p} R_{f}) / MCTR The Sharpe ratio adjusts for risk, and can help you determine the investment choice that will deliver the highest returns while considering risk.
However, I still cannot understand why an optimal asset allocation involves the fact that ALL assets have MCTR and hence equal Sharpe ratios? Why is not an optimal asset allocation involving assets with highest Sharpe ratios? (but not necessarily equal?)
Thanks!!
It is a mathematical expression. Let us say between a rectangular and a square of equal perimeter, which one will have greater area ? The square.
Similarly, for an optimal portfolio the same Sharpe ratio
The way I understand it is: In an optimal asset allocation, you will have the same Sharpe (slope)
Even if you change the riskreturn combination (higher riskhigher return, lwoer risklower return), with rf as a constant, Sharpe stays the same.
Now the connection to the MCTR is: as you gradually increase the amount of risk taken (relative to overall risk), MCTR will stay the same as Sharpe, following the same idea as above: add a certain amount of risk, you add a certain amount of return as well.
Hence Sharpe and MCTR are the same. Is this correct?
Thank you!
Yes. You are right. Please add the following:

When you extend the CML beyond the optimal portfolio ( the tangency point connecting the Rf with the EF) , there is no incremental benefit ( in absolute terms yes) in a riskreturn spectrum. It is completely proportionate,linearly.Because the Sharpe Ratio is the slope that is constant on the CML.

If you study the graph finer, it would be clearer that there are no portfolios available above the tangency line ( There can be but surely they cannot be termed efficient). And all portfolios available beneath the tangency line and above the EF are definitely inferior to the ones lying on the CML.
If the above are true, then the optimal choice becomes picking up portfolios from the CML. Hence any marginal contribution must be from these portfolios only ( lying on the CML)
I think the above may help you.