Why the Maximum Sharpe Portfolio Defines the Capital Market Line

Summary

The document explains the geometric link between a portfolio’s Sharpe ratio and the capital market line. For a risky portfolio, the ratio corresponds to the slope from the risk-free asset, or cash, to that portfolio in a risk-return diagram. Combining cash with a risky investment traces a straight line between their points because cash has no return variability to offset through covariance with the risky asset.

An investor seeking the greatest expected return for a given level of risk therefore favors the risky portfolio whose line from cash is steepest. That portfolio has the highest Sharpe ratio and determines the capital market line; another portfolio with a steeper line would offer a better reward-to-risk tradeoff. The answer gives an intuitive geometric argument rather than deriving it algebraically. It assumes the standard setup in which cash is riskless and portfolios formed from cash and a risky asset lie on straight lines in mean-standard-deviation space. The discussion does not address estimation error, borrowing constraints, or departures from those assumptions.

Key ideas

• A portfolio’s Sharpe ratio is the slope connecting its risk-return point to the risk-free asset.
• Mixing cash with a risky portfolio traces a straight line in mean-standard-deviation space.
• The steepest feasible line identifies the risky portfolio with the highest Sharpe ratio.
• That maximum-slope portfolio determines the capital market line under the stated assumptions.

Tags

# Sharpe ratio highest amongst efficient portfolios?


# Sharpe ratio highest amongst efficient portfolios?












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?

## Answer by user18663 (score 2)

https://quant.stackexchange.com/a/27604

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:-

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.