Exercise 9.2 (start-up and venture capitalist exit strategy). There are three periods, t = 0, 1, 2. The rate of interest in the economy is equal to 0, and ev- eryone is risk neutral. A start-up entrepreneur with initial cash A and protected by limited liability wants to invest in a fixed-size project. The cost of invest- ment, incurred at date 0, is I > A. The project yields, at date 2, R > 0 with probability p and 0 with prob- ability 1 − p. The probability of success is p = pH if the entrepreneur works and p = pL = pH − ∆p (∆p > 0) if the entrepreneur shirks. The entrepre- neur’s effort decision is made at date 0. Left unmon- itored, the entrepreneur obtains private benefit B if she shirks and 0 otherwise. If monitored (at date 0), the private benefit from shirking is reduced to b B. There is a competitive industry of venture capi- talists (monitors). A venture capitalist (general part- ner) has no fund to invest at date 0 and incurs pri- vate cost cA > 0 when monitoring the start-up and 0 otherwise (the subscript “A” refers to “active moni- toring”). The twist is that the venture capitalist wants his money back at date 1, before the final return, which is realized at date 2 (technically, the venture capitalist has preferences c0 +c1, while the entrepre- neur and the uninformed investors have preferences c0 + c1 + c2, where ct is the date-t consumption). Assume that Suppose further that there is a competitive supply of monitors and abundant monitoring capital. At pri- I − pH ( B \ R − ∆p ( > A > I − pH R − b + cA \. ∆p vate cost c, a monitor can reduce the entrepreneur’s private benefit of misbehavior from B to b. Assume that (i) Assume first that the financial market learns (for free) at date 1 whether the project will be suc- cessful or fail at date 2. Note that we are then in andB − b p H ∆p b> c > (∆p)R − pH ∆p two-period model, in which the out- come can be verified at date 1 (one can, for exam- ple, organize an IPO at date 1, at which the shares in (∆p)R > c + B. Show that there exist thresholds A1 A2 A3 such that • if A � A3, the firm issues high-quality public debt (public debt that has a high probability of being repaid); • if A3 > A � A2, the firm borrows from a monitor (and from uninformed investors); • if A2 > A � A1, the firm issues junk bonds (public debt that has a low probability of being repaid); • if A1 > A, the firm does not invest. the venture are sold at a price equal to their date-2 dividend). Show that the entrepreneur cannot be financed without hiring a venture capitalist. Write the two in- centive constraints in the presence of a venture cap- italist and show that financing is feasible. Show that the entrepreneur’s utility is pHR − I − [pHcA/∆p]. (ii) Assume now that at date 1 a speculator (yet un- known at date 0) will be able to learn the (date-2) re- alization of the venture’s profit by incurring private cost cP, where the subscript “P” refers to “passive monitoring.” At date 0, the venture capitalist is given s shares. The date-0 contract with the venture capitalist spec- ifies that these s shares will be put for sale at date 1 in a “nondiscriminatory auction” with reservation By analogy with Diamond’s diversification reasoning (see Chapter 4), argue that the venture capitalist is paid a reward (Rm) only if the two firms succeed. Show that if price P . That is, shares are sold to the highest bidder at a price equal to the highest of the unsuccessful bids, but no lower than P . If left unsold, the venture ( pH R − b + cpH /(pH + pL ) \ ∆p > I − A, capitalist’s shares are handed over for free to the date-0 uninformed investors (the limited partners) in the venture. (a) Find conditions under which it is an equilib- rium for the speculator (provided he has monitored and received good news) to bid R for shares, and for uninformed arbitrageurs to bid 0 (or less than P ). (b) Write the condition on (s, P) under which the speculator is indifferent between monitoring and not monitoring. Writing the venture capitalist’s in- centive constraint, show that P satisfies then financing can be arranged.
Exercise 9.2 (start-up and venture capitalist exit strategy). There are three periods, t = 0, 1, 2. The
rate of interest in the economy is equal to 0, and ev- eryone is risk neutral. A start-up entrepreneur with initial cash A and protected by limited liability wants to invest in a fixed-size project. The cost of invest- ment, incurred at date 0, is I > A. The project yields, at date 2, R > 0 with probability p and 0 with prob-
ability 1 − p. The probability of success is p = pH
if the entrepreneur works and p = pL = pH − ∆p
(∆p > 0) if the entrepreneur shirks. The entrepre- neur’s effort decision is made at date 0. Left unmon- itored, the entrepreneur obtains private benefit B if she shirks and 0 otherwise. If monitored (at date 0), the private benefit from shirking is reduced to b B. There is a competitive industry of venture capi- talists (monitors). A venture capitalist (general part- ner) has no fund to invest at date 0 and incurs pri- vate cost cA > 0 when monitoring the start-up and 0 otherwise (the subscript “A” refers to “active moni- toring”). The twist is that the venture capitalist wants his money back at date 1, before the final return,
which is realized at date 2 (technically, the venture capitalist has preferences c0 +c1, while the entrepre-
neur and the uninformed investors have preferences
c0 + c1 + c2, where ct is the date-t consumption).
Assume that
Suppose further that there is a competitive supply of monitors and abundant monitoring capital. At pri-
I − pH
( B \
R − ∆p
(
> A > I − pH R −
b + cA \.
∆p
vate cost c, a monitor can reduce the entrepreneur’s
private benefit of misbehavior from B to b. Assume that
(i) Assume first that the financial market learns (for free) at date 1 whether the project will be suc- cessful or fail at date 2. Note that we are then in
andB − b
|
∆p b> c > (∆p)R − pH ∆p
two-period model, in which the out- come can be verified at date 1 (one can, for exam- ple, organize an IPO at date 1, at which the shares in
(∆p)R > c + B.
Show that there exist thresholds A1 A2 A3
such that
• if A � A3, the firm issues high-quality public
debt (public debt that has a high probability of being repaid);
• if A3 > A � A2, the firm borrows from a monitor
(and from uninformed investors);
• if A2 > A � A1, the firm issues junk bonds
(public debt that has a low probability of being repaid);
• if A1 > A, the firm does not invest.
the venture are sold at a price equal to their date-2 dividend).
Show that the entrepreneur cannot be financed without hiring a venture capitalist. Write the two in- centive constraints in the presence of a venture cap-
italist and show that financing is feasible. Show that the entrepreneur’s utility is pHR − I − [pHcA/∆p].
(ii) Assume now that at date 1 a speculator (yet un- known at date 0) will be able to learn the (date-2) re- alization of the venture’s profit by incurring private cost cP, where the subscript “P” refers to “passive monitoring.”
At date 0, the venture capitalist is given s shares. The date-0 contract with the venture capitalist spec- ifies that these s shares will be put for sale at date 1 in a “nondiscriminatory auction” with reservation
By analogy with Diamond’s diversification reasoning (see Chapter 4), argue that the venture capitalist is paid a reward (Rm) only if the two firms succeed. Show that if
price P . That is, shares are sold to the highest bidder at a price equal to the highest of the unsuccessful
bids, but no lower than P . If left unsold, the venture
(
pH R −
b + cpH /(pH + pL ) \
∆p
> I − A,
capitalist’s shares are handed over for free to the date-0 uninformed investors (the limited partners) in the venture.
(a) Find conditions under which it is an equilib- rium for the speculator (provided he has monitored and received good news) to bid R for shares, and for uninformed arbitrageurs to bid 0 (or less than P ).
(b) Write the condition on (s, P) under which the
speculator is indifferent between monitoring and not monitoring. Writing the venture capitalist’s in- centive constraint, show that P satisfies
then financing can be arranged.
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