Value at risk" and benefits from diversification:-
This exercise looks at the impact of portfolio correlation on capital requirements. An entrepreneur has two identical fixed-investment projects. Each involves investment cost I. A project is successful (yields R) with probability p and fails (yields 0) with probability 1 - p. The probability of success is endogenous.
If the entrepreneur works, the probability of success is
and the entrepreneur receives no private benefit. If the entrepreneur shirks, the probability of success is pL = 0 and the entrepreneur obtains private benefit B. The entrepreneur starts with cash 2A, that is, A per project. We assume that the probability that one project succeeds conditional on the other project succeeding (and the entrepreneur behaving) is
(it is, of course, 0 if the entrepreneur misbehaves on this project). α ∈ [-1, 1] is an index of correlation between the two projects. The entrepreneur (who is protected by limited liability) has the following preferences:
(i) Write the two incentive constraints that will guarantee that the entrepreneur works on both projects.
(ii) How is the entrepreneur optimally rewarded for R¯ large?
(iii) Find the optimal compensation scheme in the general case. Distinguish between the cases of positive and negative correlation. How is the ability to receive outside funding affected by the coefficient of correlation?