Discuss whether each of following statements is true or false and WHY.

a$.$ It is due to the noise trader risk that the stock price of a company goes up when it drops from Russell $1000$ index to Russell $2000$ index. $(10$ points$)$

b$.$ Suppose there is no arbitrage in the financial market, we can show that a particular stock$$s return ri and its $\backslash$beta i satisfies

E$($ri$)$rf $=\backslash$beta i$($E$($rm$)$rf$),$

where rm is the market portfolio$$s return and rf is the risk$-$free rate.

c$.$ Consider the equity carve$-$outs example $($i$.$e$.,3$Com and Palm case$),$ one of the reasons why the law of one price fails is the difficulty of short selling Palm$$s shares.

d$.$ The Siamese Twin Companies of Royal Dutch and Shell is an example of risk$-$ free arbitrage.