Arbitrage

For this question, you want $to$ round $upto$ two digits for the final answer. $If$ the final answer $isin$

percentage form, round $upto$ the nearest basis point.

The following $3$ questions are about arbitrage. They are not related $to$ each other.

$(a)$ Suppose that asset A pays you $$300$ tomorrow for sure, and $it$ costs $$250to$ purchase today.

Suppose that asset $B$ pays you $$500$ tomorrow for sure, and $it$ costs $$410to$ purchase today. $Is$

there $an$ arbitrage opportunity? $Ifso,$ construct $an$ arbitrage. Write down which asset $to$ long,

which asset $to$ short, and which cash flow $to$ match.

$(b)$ Suppose that one$-$factor APT holds, with factor $F.$ Consider two well$-$diversified portfolios, A and

$B,$ with risk premium $E[R^A]=8\mathrm{.}6\%,E[R^B]=4\mathrm{.}1\%,$ and betas ${}_A=0\mathrm{.}8,{}_B=0\mathrm{.}4R^A$ and $R^B{w}_A=1,$ and show that $it$

has nonzero return.

For part $(c)$ and $(d),$ consider the following setup. Suppose excess returns $of$ assets follow a two$-$

factor model, with factors $F_1$ and $F_2.$ There are two well$-$diversified portfolios, $A,B$ and $C,$ with

expected returns $E[R^A]=7\%,E[R^B]=-6\%,E[R^C]=10\%,$ and betas ${}_{A,F_1}=1,{}_{A,F_2}=0,$

${}_{B,F_1}=0,{}_{B,F_2}=2,{}_{C,F_1}=1\mathrm{.}2,{}_{C,F_2}=-0\mathrm{.}4R^A$ and $R^B{F}_1$ and $F_2{R}^C{w}_C=$

$1,$ and show that $it$ has nonzero return.