You manage a multi$-$stage VC fund conducting due diligence on an pharmaceutical start$-$up$.$ The required rate of return is $33\%.$ You will participate in an A round initially, a B round in four years, and a C round in seven years.

Required:

In the A round, what equity position do you need to justify a $$1,000,000$ investment if the pre$-$round valuation of the firm is $$2,500,000?$

In the B round, an additional $$4,500,000$ will be raised from a syndication of VCs including your firm. To maintain your equity position, how much must you contribute to the B round of financing? Assume the firm is valued at cost$.$

In the C round, an additional $$10,000,000$ will be raised from the same syndication of VCs$.$ If you wish to continue to maintain the same equity position, how much will you need to invest in this round? Assume the firm continues to be valued at cost$.$

If the firm is acquired for $$90,000,000$ in the $9$th year, is your required rate of return satisfied?