You are an original owner of a publicly traded food service company named Smith's Foods $($SF$).$ Your company has been in the high$-$end restaurant business $($HE$)$ for the past ten years. You announced today that the company will be issuing equity today to open a new fast food division $($FF$).$

Opening this new division costs $$15M$ today, which you will fund by issuing equity. Revenues from this new division are expected to be $$10M$ next year and are projected to grow by $3$ percent per year. Costs from this new division are expected to be $$9M$ next year and are projected to grow by $2$ percent per year. The life of this project is $20$ years.

Analyses of other fast food businesses suggest that the beta of a typical fast food division equals $0\mathrm{.}80.$ Assume that the revenues and costs have similar risk. Throughout this problem, assume a risk$-$free rate of $4$ percent and a market risk premium of $5$ percent.

a$)$ What is the NPV of this new investment? Is the investment worthwhile? $($Hint: you will first need to use the CAPM to find the correct discount rate. Then you will need to use the growing annuity formula twice: once for revenues and once for costs.$)$

Based on your analysis of other high$-$end restaurant businesses, you determine that the beta of your highend division equals $1\mathrm{.}6.$ Directly before the investment in the fast food division, the assets of SF were worth $$30M.$ Assume that the announcement of the new division does not affect the value or beta of the high$-$end division.

b$)$ What is the beta of the firm after the announcement of the new division? The beta of a firm can be calculated as the value$-$weighted average of the division betas:

${}_{\text{f}\text{i}\text{r}\text{m}}=\frac{V_{D1}}{V_{D1}+V_{D2}}{}_{D1}+\frac{V_{D2}}{V_{D1}+V_{D2}}{}_{D2}$

where D$1$ represents "Division $1"$ and D$2$ represents "Division $2"($in this case, the high$-$end division and the fast food division$).$ The value of the new division equals the present value of its future net cash flows. When calculating the value of the fast food division, do not subtract the initial $$15M$ cost$,$ as this was paid for using newly$-$issued equity.

c$)$ The value of the existing debt on the balance sheet of Smith's Foods is $$18M.$ Assume that the beta of this debt equals $0\mathrm{.}10.$ What is the beta of the firm's equity after the announcement of the new division? $($Hint: use the ${}_E$ equation on slide $25$ in the Week $2$ lecture slides. To calculate $E,$ add the values of the two divisions to get the value of assets, and then subtract the value of debt.$)$

d$)$ The equity beta of the firm decreased as a result of the announcement. Discuss the two ways in which the announcement decreased the equity beta. $($Hint: refer again to slide $25$ in the Week $2$ lecture slides.$)$