Hi

@Eustice_Langham It might help to view the XLS, here is a simple one-sheet version

https://www.dropbox.com/s/q4l43w3n1le5z8h/082520-cml.xlsx?dl=0
This concerns the CML (to my knowledge, it does not require SML/CAPM; for example, Elton who was the prior author in

one of the authoritative texts on MPT established the CML prior to the SML). The

**red dot** is the Market Portfolio; it earns such stats because it has the highest Sharpe ratio on the

**(green****) portfolio possibilities curve (PPC)**. On the single-sheet version I shared, the market portfolio is just a mix of two assets, but in my larger XLS, they are solved-for: they are the mix of risky assets that generates the highest Sharpe ratio (i.e., excess return divided by standard deviation). So the big step before the riskfree asset is introduced is establishing this highest-Sharpe Market portfolio. If we stay on the green PPC, moving in either direction (by definition) implies a decrease in the Sharpe ratio.

They way that I look at this (I say it that way because this is one of the most studied topics in finance, I'm sure there is a more sophisticated way to look at it given the hundreds of finance professors who've analyzed this! ....) is: the riskfree asset gives us (the investor) a way to invest at the same Sharpe ratio over a continuum of risk/reward spectrum as represented by the CML. You can see in the XLS how, if we have a Market portfolio defined, the CML is a dead simple linear function, anchored at the riskfree rate and running through the Market portfolio (aka, tangency to the PPC at the Market Portfolio). Every point on the CML has the same (optimal) Sharpe ratio as the Market portfolio. Moving up/down is not re-mixing risky assets, it is remixing between only two assets: the riskfree rate and the market portfolio. Going "up the CML" is mixing less risk free (by borrowing) and more of the Market Portfolio; going "down the CML" is mixing more riskfree (by lending; aka, investing) and less of the Market Portfolio.

You can see in the XLS how movement up/down the CML is simply shifting the weights:

**E(return) = (%_riskfree) * Rf + (1 - %_riskfree)*E(return_market_portfolio).**
The associated standard deviation, per basic variance properties, is simply

**(1 - %_riskfree)*σ(M) **... because the riskfree asset has no volatility.

If we (as the investor) choose the market portfolio, our CML decision is: 100% market portfolio plus 0% riskfree rate.

If we want less risk, we can "move down the CML" and decide: 50% market portfolio plus 50% riskfree rate. Our expected return and volatility will reduce, but our 50/50 decision will have the same Sharpe ratio (by even visual definition).

So, I think that's what I meant by investing at the riskfree rate: after the Rf has been introduced, the efficient frontier is the CML. Visually, we can see that

*every point on the (green) PPC*, except the Market portfolio, is

*less efficient than any point on the (blue) CML*. Consequently, our best choice is along the CML. And the CML is a mix between x% invested at the riskfree rate (or borrowed at riskfree rate) plus (1 - x%) invested in the same Market Portfolio as the other investors. The CML represents an asset allocation decision between two assets, the riskfree asset and the Market Portfolio (itself already optimal among risky assets).

Hence the way I look at it: the CML (via the Rf assets) "transforms" a single point (Market Portfolio) into a line which avails us of the same efficiency (highest Sharpe!) but at various, desired levels of risk/reward.

I hope that's helpful!

## Stay connected