$1.$ Carlson Corporation finances its capital needs with $25\%$ long term debt and $75\%$ equity. Carlson$$s outstanding debt carries a coupon rate of $6\mathrm{.}5\%,$ and as of now has a yield to maturity of $6\mathrm{.}8\%.$ By comparison, the yield to maturity on short term U$.$S$.$ Treasury Bills is currently $5\mathrm{.}0\%.$ The beta coefficient for Carlson Corporation$$s equity was recently estimated to be $0\mathrm{.}75.$ Historically, returns on short term treasuries have averaged $3\mathrm{.}5\%$ per year. The expected return on the overall market for upcoming years is $10\mathrm{.}5\%.$

Carlson is considering going into the widget production business, where it has not operated before. Carlson$$s analysts have estimated in light of Carlson$$s funding methods and the competitive nature of the widget business that the beta of the widget cash flows will be $1\mathrm{.}2.$ Carlson$$s marginal tax rate is $21\%.$

$($a$)$ Which of the two beta estimates described above is relevant for computing the NPV of the widget production proposal, and why?

$($b$)$ Provide and interpret an estimate of the cost of capital $($expected return$)$ for Carlson$$s equity investment in the widget business.

$($c$)$ Provide an estimate of the weighted average cost of capital for Carlson$$s overall investment in the widget business. What is the purpose of obtaining this estimate?