# introducing a risk free asset changes the efficient frontier into a straight line?

Can someone explain this to me?

I was going over a problem and it had this in the explanation -

The introduction of a risk-free asset changes the Markowitz efficient frontier into a straight line

Linear combinations (combinations where the sum of the weights add to 1) of the risk free asset and a risky portfolio have risk-return characteristics such that they fall on a straight line:

The expected return of the combination portfolio is a weighted average of the two assets’ individual returns. And because the covariance of the risk free asset and any risky asset is zero (and the variance of the risk-free asset is zero), the formula for the variance of the portfolio is simply the weight of the risk asset squared times the variance of the risky asset. So, the std deviation is the weight of the risk times the std. deviation of the risky.

These two things mean that that combinations of the risk-free asset and any one risky portfolio will fit on a straight line on the std deviation/expected return graph. So, if you combined the risk-free asset with the portfolio on the risky asset-only efficient frontier that made the resulting line just tangent to the the risky-asset-only efficient frontier, a potfolio on the the resulting line would always give a superior risk-return tradeoff to any other portfolio with the same standard deviation.

In other words, this line would become the “new” efficient frontier.

Here’s a article I wrote comparing CAL, CML, and SML: http://financialexamhelp123.com/cal-vs-cml-vs-sml/. If you look at the CML graph (about 2/3 of the way down), you’ll see the efficient frontier made up of all risky assets, and the CML, which includes the risk-free asset; the CML is the efficient frontier when the risk-free asset is included.