Risk-free Rate, CML and SML

Capital Market Line (CML) is the best possible capital allocation line (CAL) which is tangent to a broad market portfolio (constructed via Optimal Risky Portfolio with a broad input-list or index) of risky assets or Market Portfolio.

Differentiating w.r.t \(r_f\)

\(\frac{\partial E\left(r_p\right)}{\partial r_f}=\left(1-\frac{\sigma_p}{\sigma_m}\right)\)

We'll consider two cases here -

  1. \(y=\frac{\sigma_p}{\sigma_m}\)< 1
  2. \(y=\frac{\sigma_p}{\sigma_m}\)> 1
Case-1: \(y=\frac{\sigma_p}{\sigma_m}\)< 1
In this case of lending, \(\left(1-\frac{\sigma_p}{\sigma_m}\right)>0\) and thus, if \(r_f\) decreases (as shown in the picture), \(E\left(r_p\right)\) will decrease. So an investor with higher degree of risk-aversion has to allocate more portfolio weightage to the risk-free asset in order to regain previous expected return. Whereas an investor with less degree of risk-aversion will try to allocate more weightage to the risky-asset to regain previous expected return.

Case-2: \(y=\frac{\sigma_p}{\sigma_m}\)> 1
In this case of borrowing, \(\left(1-\frac{\sigma_p}{\sigma_m}\right)<0\) and thus, if \(r_f\) decreases (as shown in the picture), \(E\left(r_p\right)\) will increase. An investor may not change 
his/her leveraged position.

As the Market Portfolio is an Optimal Risky Portfolio constructed through a broad-market index, the mean-variance curve should remain unperturbed.