A venture capitalist $($VC$)$ is considering providing financing of $$10$ million to a startup firm in return for common stock in the firm. The VC plans to invest in the firm for $5$ years after which the VC expects the firm will sell at an Enterprise Value$-$to$-$EBITDA multiple of $4.$ After $5$ years, the firm is expected to have EBITDA of $100$ million, debt outstanding of $$250$ mln and cash balance of $$50$ mln$.$

$1.$ Assume that the VC will be granted common stock. If the VCs required rate of return is $45\%,$ how much equity ownership must be given to the VC$?$

$2.$ Instead of common stock, the VC is offered convertible debt that pays a $10\%$ annual interest. In Year $5($i$.$e$.,$ at exit$)$ the debt can be converted into equity. Assuming that the VC wants a $38\%$ IRR, what fraction of the firm$$s equity must be given up to the VC at conversion in five years?

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