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I have a hard time understanding why the sharpe ratio corrresponding to the efficient portfolios is the highest possible.

In my book, it states that the sharpe ratio of the efficient portfolios is the slope of the CML, and so if a portfolio had a higher sharpe ratio, it would lie on a line "steeper" than the CML, and then that would be efficient $\Longrightarrow$ contradiction.

But WHY would it lie on the steeper line? What formula proves this (intuitively reasonable) argument?

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The Sharpe Ratio is a direct measure of reward-to-risk. To see how it helps you in creating a portfolio, please consider the following graph:-

enter image description here

The Sharpe Ratio of X is the slope of the line joining cash with X There are three important things to notice in this graph:

If you take some investment like "x" and combine it with cash, the resulting portfolio will lie somewhere along the straight line joining cash with x. ( it's a straight line, not a curve; cash is riskless, so there's no "damping out" effect between cash and x.)

Since you want the rate of return to be as great as possible, you want to select the x that gives you the line with the greatest possible slope (like we have done in the graph).

The slope of this line is equal to the Sharpe Ratio of x.

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