alphaX Corp. has a target capital structure that calls for $30\%$ debt, $15\%$ preferred stock and $55\%$ common equity. The company has $25$ year maturity, $8\%$ semi$-$annual coupon bonds outstanding currently selling at $$1,257\mathrm{.}30.$ Currently, common and preferred equity in alphaX is trading for $$22\mathrm{.}50$ and $$104\mathrm{.}00,$ respectively. alphaX is a relatively safe investment with a beta of $0\mathrm{.}78$ and it expects to retain $$2\mathrm{.}37$ million in earnings over the coming year. Also, they are a dividend paying company, having just paid a $$2\mathrm{.}10$ dividend and that is expected to grow at $2\%$ per year; their preferred pays a dividend of $8\mathrm{.}25\%$ based on $$100$ par value. Floatation costs of $6\%$ and $18\%$ would have to be paid to issue new shares of preferred and common stock, respectively. The corporation is taxed at a $40\%$ rate. Investors can invest risk free earning $2\%,$ while investing in the market is expected to earn $13\%.$

$($a$)$ What is the firm's cost of debt?

$($b$)$ What is the cost of newly issued preferred stock?

$($c$)$ What is the firm's cost of retained earnings using a dividend discount approach?

$($d$)$ Re$-$answer part $($c$)$ using a CAPM approach.