KöMaL 2014 B4650

Is there a function of the form $f(x)=\frac{ax+b}{cx+d}$ for which $f(x_1)=x_2$, $f(x_2)=x_3$, $f(x_3)=x_4$, $f(x_4)=x_5$, $f(x_5)=x_1$ is true if $x_1,\ldots,x_5$ are appropriate pairwise different real numbers?