In a previous round of financing for a resort spa, an investor contributed $$2$ million in exchange for $1$ million shares of common stock. The entrepreneur retained ownership of $2$ million shares.

Because of massive construction cost overruns and delays, things have not gone well for the spa since the investment. The venture needs another $$1$ million, with which the entrepreneur hopes to complete development. However, the existing investment agreement includes a ratchet provision for the prior investor. Under the terms of the ratchet, the investor will receive enough new shares for free so that the investor's average cost per share is the same as that of any new investor.

A$)$ Suppose that in the absence of the ratchet provision a new investor would be willing to accept $1\mathrm{.}25$ million shares in exchange for the $$1$ million investment. Compute the post$-$money valuation of the venture.

B$)$ Now, based on the valuation in part $($a$),$ giving effect to the ratchet provision, what price per share would the new investor seek, and how many new shares would the existing investor receive? Note: You can do this and part $($c$)$ using Solver, by trial and error, or by finding the solution algebraically.

C$)$ Suppose the ratchet agreement has a floor that limits the average cost of the existing investor to a minimum of $$1$ per share. How would that limitation affect the price per share for the new investor and the number of new shares going to the existing investor?

D$)$ What fraction of the equity would the entrepreneur end up retaining under each of the three scenarios?Refer to the above question. Assume the new investor has renegotiated the terms of the deal and is now willing to invest $$1$ million for $25\%$ of the company. Under a full rachet provision, how many free shares will be issued to the original $($old$)$ investor?