# 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.
CML
$$E\left(r_p\right)=r_f+\frac{\sigma_p\left[E\left(r_m\right)-r_f\right]}{\sigma_m}$$

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.