Builders Fund (55 points) Suppose you are working for a venture capita1 fund that bets on futuristic technologies: investment payoffs are high but un1ike1y. Your job involves talking to wealthy people and convincing them to put money into your fund. One such individua1, Mr. Zorin, has about w=$100mln that he cou1d potentia11y invest. Mr. Zorin's preferences are given by u(w)=log(w) where w is his wea1th measured in mil1ions of US do11ars. For simp1icity, assume that investments pay off or fai1 instant1y. a) ( 10 points) For every $x invested in your fund, Mr. Zorin receives back $9x with probability 0.2 and nothing with probability 0.8. Write down his expected utility from investing $xm1n. b) ( 10 points) If you offered him an opportunity to invest exact1y $50m1n, wou1d he do so or not - and why? c) (20 points) If you 1et Mr. Zorin choose investment size on his own, how much wou1d he put in your fund? d) (15 points) Suppose now that you can charge Mr. Zorin for the opportunity to make the investment. The charge is made right away, so, for examp1e, if you were charging $10m1n your c1ient wou1d have on1y up to $90m1n1eft for investment. What is the maximum fee F you cou1d charge? (You shou1d write down the equation that determines F, but you do not have to so1ve it). Builders Fund (55 points) Suppose you are working for a venture capita1 fund that bets on futuristic technologies: investment payoffs are high but un1ike1y. Your job involves talking to wealthy people and convincing them to put money into your fund. One such individua1, Mr. Zorin, has about w=$100mln that he cou1d potentia11y invest. Mr. Zorin's preferences are given by u(w)=log(w) where w is his wea1th measured in mil1ions of US do11ars. For simp1icity, assume that investments pay off or fai1 instant1y. a) ( 10 points) For every $x invested in your fund, Mr. Zorin receives back $9x with probability 0.2 and nothing with probability 0.8. Write down his expected utility from investing $xm1n. b) ( 10 points) If you offered him an opportunity to invest exact1y $50m1n, wou1d he do so or not - and why? c) (20 points) If you 1et Mr. Zorin choose investment size on his own, how much wou1d he put in your fund? d) (15 points) Suppose now that you can charge Mr. Zorin for the opportunity to make the investment. The charge is made right away, so, for examp1e, if you were charging $10m1n your c1ient wou1d have on1y up to $90m1n1eft for investment. What is the maximum fee F you cou1d charge? (You shou1d write down the equation that determines F, but you do not have to so1ve it)
