The current risk$-$free rate is $4\%$ and the expected rate of return on the market portfolio is $10\%.$ The Brandywine Corporation has two divisions of equal market value, i$.$e$.$ the market value of their assets is the same. The debt to equity ratio $($D$/$E$)$ is $3/7.$ The company$$s debt can be assumed to present no risk of default. For the last few years, the Brandy division has been using a discount rate of $12\%$ in capital budgeting decisions and the Wine division a discount rate of $10\%.$ You have been asked by their managers to report on whether these discount rates are properly adjusted for the risk of the projects in the two divisions.

$($a$)$ What are the betas of typical projects implicit in the discount rates used by the two divisions?

$($b$)$ You estimate that the equity beta of Brandywine is $1\mathrm{.}6.$ Is this consistent with the equity betas implicit in the discount rates used by the two divisions?

$($c$)$ You estimate that the debt$/$equity beta of Kordell Brandy Corp. is $1\mathrm{.}8.$ This company is purely in the brandy business. Its debt to equity ratio is $2/3$ and its debt beta is $0\mathrm{.}2.$ Based on this information and on your estimate of Brandywine$$s equity beta in part $($b$),$ what discount rate would you recommend for projects in the Brandy division of Brandywine?