A company currently has earnings $(E_0)$ of $$4\mathrm{.}50$ and a dividend $(D_0)$ of $$0\mathrm{.}45.$ The firm's current return on equity $($ROE$)$ is $20\mathrm{.}00\%.$ The firm will maintain the same dividend payout and ROE over the next two periods. Then it will transition in a linear reduction for four steps in years $3,4,$ and $5$ to a growth of $2\mathrm{.}50\%$ in year $6.$ The firm will then grow at $2\mathrm{.}50\%$ to perpetuity. The firm's beta is presently $1\mathrm{.}8,$ but this will transition to $1$ over the same period. The risk$-$free rate is $4\%$ and the market risk premium is $6\%.$ ROE is expected to be $10\%$ in year $5$ to perpetuity. What is the present value of this firm's equity using a three$-$stage model with linear transition in years $3,4,$ and $5?$

$$ $$47\mathrm{.}90$

$$45\mathrm{.}18$

$$52\mathrm{.}75$

$$ $$43\mathrm{.}6$