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$.$

a$.$ 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$?$

b$.$ 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?