$1.$ ABC Inc. is an all$-$equity firm that has $29\mathrm{.}625$ million shares outstanding. The firm paid a dividend of $$0\mathrm{.}5$ per share last year and is expected to have a dividend growth rate of $5\%$ for the next $5$ years. After this, the dividend growth rate will decrease to $3\%$ indefinitely. Suppose the market risk premium is $5\mathrm{.}5\%,$ risk free rate is $4\%,$ and ABC$$s beta is $0\mathrm{.}5.($a$)$ Calculate the cost of equity using CAPM model. $($b$).$ Calculate the intrinsic value per share of ABC Inc. using discounted dividend model $($round to the nearest $0\mathrm{.}001).$ State whether the share price is undervalued or overvalued.$($c$)$ ABC Inc. is proposing a rights offering to raise $$79$ million additional funds for a new project. The current market price is $$22$ per share. The subscription price is $$16$ per share.

$($i$)$ Calculate the number of new shares that will need to be issued to raise the additional funds. Also, calculate the number of rights that existing shareholders need to surrender to receive one new share.

$($ii$)$ Calculate the ex$-$right price and the value of a right.

$($iii$)$ Demonstrate that the rights offering does not influence the share value of existing shareholders. Please use an example to illustrate your argument.

$($iv$)$ Interpret the meaning and the process of a rights offering and its advantage to existing shareholders.