Sep 19, 2017

(HEDGING OR GAMBLING ON NET WORTH?). FROOT ET AL. (1993) ANALYZE AN ENTREPRENEUR’S RISK…

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(hedging or gambling on net worth?). Froot et al. (1993) analyze an entrepreneur’s risk preferences with respect to net worth. In the notation of this book, the situation they consider is summarized in Figure 3.12. The entrepreneur is risk neutral and protected by limited liability. The investors are risk neutral and demand a rate of return equal to 0. At date 0, the entrepreneur decides whether to insure against a date-1 income risk

For simplicity, we allow only a choice between full hedging and no hedging (the theory extends straightforwardly to arbitrary degrees of hedging). Hedging (which wipes out the noise and thereby guarantees that the entrepreneur has cash on hand A0 at date 1) is costless. After receiving income, the entrepreneur uses her cash to finance investment I and must borrow I − A from investors, with A = A0 in the case of hedging and A = A0 + ε in the absence of hedging (provided that A I; otherwise there is no need to borrow).

 

Note that there is no overall liquidity management as there is no contract at date 0 with the financiers as to the future investment. This exercise investigates a variety of situations under which the entrepreneur may prefer either hedging or “gambling” (here defined as “no hedging”). (i) Fixed investment, binary effort. Suppose that the investment size is fixed (as in Section 3.2), and that the entrepreneur at date 1, provided that she receives funding, either behaves (probability of success pH, no private benefit) or misbehaves (probability of success pL, private benefit B). As usual, the project is not viable if it induces misbehavior and has a positive NPV (pHR>I>pLR + B, where R is the profit in the case of success). Let A be defined (as in Section 3.2) by

Suppose that ε has a wide support. Show that the entrepreneur

(ii) Fixed investment, continuous effort. Suppose, as in Exercise 3.20, that succeeding with probability p involves an unverifiable private cost 1 2p2 for the entrepreneur (so, effort in this subquestion involves a cost rather than the loss of a private bene- fit). (Assume R and that the support of ε is sufficiently small that the entrepreneur always receives funding when she does not hedge (and a fortiori when she hedges). This assumption eliminates the concerns about financing of investment that were crucial in question (i). Show that the entrepreneur hedges. (iii) Variable investment. Return to the binary effort case (p = pH or pL), but assume that the investment I is variable (as in Section 3.4). The income is RI in the case of success and 0 in the case of failure. The private benefit of misbehaving is B(I) with B > 0. Assume that the size of investment is always constrained by the pledgeable income and that the optimal contract induces good behavior. Show that the entrepreneur

• hedges if B(·) is convex;

• is indifferent between hedging and gambling if B(·) is linear;

• gambles if B(·) is concave.

(iv) Variable investment and unobservable income. Suppose that the investment size is variable and that the income from investment R(I) is unobservable by investors (fully appropriated by the entrepreneur) and is concave. Suppose that it is always optimal for the entrepreneur to invest her cash on hand. Show that the entrepreneur hedges. (v) Liquidity and risk management. Suppose, in contrast with Froot et al.’s analysis, that the entrepreneur can sign a contract with investors at date 0. Show that the entrepreneur’s utility can be maximized by insulating the date-1 volume of investment from the realization of ε, i.e., with full hedging, even in situations where gambling was optimal when funding was secured only at date 1.



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