Question $3[5$ marks$]$

In $1990,$ John Deere is looking to acquire a company specializing in developing artificial intelligence $($AI$)$ assisted agricultural machinery. A potential target is AEye Farms, Inc., which has two divisions, namely, Software Development and Robotics Technology. AEye Farms is a private start$-$up firm with no debt in its capital structure. John Deere is considering acquiring one of AEye Farms$$ divisions. The analyst of John Deere has collected information on historical returns on the market portfolio and the relevant industries:

Market portfolio Covariance with the market

Estimation period Mean return

E$[$r$\_$M $]$ Variance

Var$($r$\_$M $)$ Software industry

Cov$($r$\_$SW$,$r$\_$M $)$

Robotics industry

Cov$($r$\_$Rob,r$\_$M $)$

$196019890\mathrm{.}100\mathrm{.}0200\mathrm{.}0560\mathrm{.}024$

Answer questions a$)$ and b$)$ below.

$($Lecture notes p$\mathrm{.}12,$ and the basic thought experiment concerning the discount rate and valuation$)$

Assume that the risk$-$free rate is $5\%.$ Using the information provided, calculate the CAPM betas for AEye Farms$$ Software and Robotics divisions, respectively, and determine the required rates of return on each division.

Answer $($show the steps$/$calculation toward your results$)$:

Despite the analyst$$s estimates, an executive director of John Deere has insisted:

$$Use $15\%$ as the discount rate for the valuation of start$-$up tech firms.$$ Had the director$$s rule been applied, for which division of AEye Farms would John Deere end up overpaying?

Answer: