Prof. Thomas J. Chemmanur
MF891: Doctoral Seminar in Corporate Finance
Problem Set-V
1. Dividend Policy and Stock Issues Under Perfect Capital Markets
Assume perfect capital markets. Consider a company with the following market value balance sheet:
Assets Liabilities
Cash 1000 Debt 0
Fixed
Assets 9000 Equity 12,000
New Project
NPV 2000
------
12,000 12,000
------
Assume that the firm has currently 120 shares outstanding. Implementing the new project requires an investment outlay of $1,000. Compute the wealth of a shareholder who owns 1 share in the firm under each of the following alternatives:
(a) The firm implements the project using the $1,000 available to it, and pays no dividends.
(b) It pays the $1,000 as a dividend to its equity holders, and finances the project by issuing new equity.
(c) It implements the project using the $1,000, but pays a 10% stock dividend (i.e., it gives free an extra share for every ten shares).
(d) It uses the $1,000 to buy back equity, but then issues new equity to raise the $1,000 required to implement the project.
(e) It does not implement the project, but pays out the $1,000 cash to equity holders.
2. Dividend Policy Under Asymmetric Information
Consider a scenario where firms differ in their time 0 cash flows. Firms are of two types: Type H firms will have a cash flow of XH, while type L firms will have a cash flow XL,XL<XH. Insiders, who announce the firm’s dividend policy at time 0, know the realization of their firm’s cash flows at that time, while outside investors do not. Outsiders only know that a fraction a of firms are type H firms, while the remaining (1-a) are type L. After announcing their dividend policy, firm managers choose their firm’s investment policy. The investment technology available to either type of firm is the same, and is given below:
f(I)=k1I for I£XL
f(I)= k1XL + k2(I- XL) for XL£I£ If
k2< k1, XL< XH< If
Here f(I) gives the cash flow yielded at time 1 by the firm which invests an amount I at time 0 (all investors will observe this cash flow at time 1). Assume for simplicity that the risk-free rate of return is zero, and all investors are risk-neutral. (The above technology implies that in the absence of asymmetric information, all firms will choose to invest up to the “full-investment” level If, depending on available cash flow, but not above that, since any amount excess of this will not yield incremental cash flows).
The objective of firm insiders in choosing their firm’s dividend and investment policy is to maximize the weighted average of time 0 and 1 firm values (same as cum-dividend equity values, since the firm is and all-equity firm), given by:
Where Vo and V1 are determined by (among other things) the dividend and investment policy chosen by insiders at time 0. Assume for simplicity that the firm can raise no outside financing of any kind, so any amount invested at time 0 by either type of firm has to come from the cash flows realized at time0. Assume also that XH<If, so that neither type of firm has enough cash available to invest up to the full-investment level If. Notice therefore that the trade-off facing each kind of firm is between signaling value (by paying a high dividend) and investing as close to the full investment level as possible. Assume also that only the dividend paid by any firm at time 0 is observable by outsiders; the investment level chosen by the firm is not observable. Assume also that there are no taxes on capital gains or other income.
a. As a benchmark, compute the full information value of equity of each type of firm at time 0? What is the full information level of dividends paid?
b. Now, in the asymmetric information setting specified above, write down the incentive compatibility conditions that have to be satisfied for a separating equilibrium (involving signaling firm value through dividends) to exist. You can use the notion of “efficient Perfect Bayesian Equilibrium,” as the equilibrium concept here (so that the resources incurred by either firm type to signal is the minimum necessary). Write down the market rationality condition as well in this setting.
c. For this section, assign the following numerical values: If=100million, k1=2.2, k2=1.2, XH=80 million, XL=50 million, g=0.5. Does a separating equilibrium of the type specified in (b) exist for these values? Show that it exists or it doesn’t (in other words, demonstrate your answer either way). What is DH* and DL* in this case?
d. Given the equilibrium in (c) above, specify the beliefs of an investor when faced with a firm announcing a dividend of $4 million.
Hint: The above problem can be thought as a variation on the Miller and Rock (1985) model that we discussed in class. However when doing this problem, you will probably be better off simply following any of the standard signaling models we discussed in class (e.g., Spence (1977), or Ross(1977)), keeping in mind the costs and benefits of signaling in this particular context.
3. The following question is in the spirit of the Rock (1986) model (though the problem structure is somewhat different). As in the Rock setting, the IPO market consists of two kinds of outside investors: sophisticated (institutional) investors and unsophisticated (retail) investors. There are two kinds of firms: Higher value firms (H-type), whose true value is VH = $20 million, and lower value (L - type) firms, whose true value is VL = $16 million. Sophisticated investors know the true value of all firms (i.e., they know whether the firm is a higher valued or a lower valued firm). Unsophisticated investors, as well as entrepreneurs, know only that any firm has a 50% chance of being type H or type L.
Each firm has only equity in its capital structure. The equity is divided into 2 million shares, out of which the firm proposes to sell half the shares (1 million shares) to outsiders in an IPO. The firm prices each of the one million shares at a share price F (so that the total amount sought to be raised in the offering is $F million).
Assume for simplicity that each investor can bid for only one share in the firm, and has only just enough wealth to buy one share in the IPO (bidding for a fraction of a share is not allowed). If an investor chooses not to bid for shares (or does not get an allocation of one share in the IPO) they invest the $F in the risk-free asset. The number of sophisticated investors in the IPO market is 0.8 million, and the number of unsophisticated investors is 1 million (i.e., this is the size of the potential demand from the two categories of investors). Finally, assume that if the number of shares demanded exceeds 1 million, the firm allocates shares to investors in a fair manner (random allocation, with all investors having an equal chance of getting an allocation), with each investor either getting one share allocated to him/her or no shares (i.e., no fractional allocations allowed).
(i) Assume that all investors are risk-neutral. What is the highest value of the share price F at which unsophisticated investors will participate in the IPO? What is the extent of underpricing? What will be the bidding strategy of sophisticated investors at this price?
(ii) Assume now that uninformed investors are risk-averse, expected utility maximizers, with a utility function for wealth given by U (W) = square root of W. (You can continue to assume that sophisticated investors are risk neutral.) Give the highest value of F at which unsophisticated investors will participate in the IPO in this case. How does the extent of underpricing change in this case, and why?
(iii) Assume that the number of bidders for shares in the IPO becomes known before the shares start trading in the secondary market. What will be the opening price per share in the secondary market for (a) truly H- type firms and (b) for truly L-type firms? Give your reasoning in a line or two.
4. The following is a special case of the Gale-Hellwig Model that we discussed in class, where the non-pecuniary cost is incurred by the entrepreneur in the event of bankruptcy is set equal to zero, and the state verification cost incurred by the outside investor is a constant (i.e., independent of the state). Also, the assumption here is that the entrepreneur has zero wealth to begin with, and that his/her firm’s prior indebtedness is zero.
A risk-neutral entrepreneur has a project that requires an investment of $100 and yields the following distribution of returns:
Returns / $70 / $90 / $100 / $120 / $140 / $160Probabilities / 0.05 / 0.10 / 0.10 / 0.25 / 0.30 / 0.20
A risk-neutral potential lender with an opportunity cost of 10 percent interest would be willing to lend the required money but has to employ an auditor at a cost of $10 to monitor the return of the entrepreneur. If the lender were to enter a loan contract with the entrepreneur she would have to rely on the entrepreneur’s statement about the return achieved or have to spend the money to audit the entrepreneur.
Notice that the various possible cash flows that I have specified above can themselves be thought as states of the world (or, equivalently, outcomes influenced by the underlying “states).
a. What must a contract between the entrepreneur and lender specify? What conditions have to hold for the contract to be feasible in each state?
b. Define incentive compatibility. Provide an argument for why any incentive compatible debt contract requires a payment of a constant amount in the solvent (non-verification) states.
c. Consider a “standard debt contract” which requires the entrepreneur to either pay the lender $120, or to declare bankruptcy, in which case the lender will audit him and takes the remaining value of the project available. For this contract, do the following:
(i) Write down the payoff to the entrepreneur and the lender (net of audit costs), in each possible state of the world.
(ii) Write down the ex ante expected return to both the entrepreneur and the lender. Check whether the lender’s expected return (now using the word “return” as in the conventional usage in finance) from investing in the project under this contract is at least equal to the expected return from his alternative investment opportunity.
(iii) Check whether the above contract is incentive compatible.
(iv) Show that the above contract maximizes the expected return of the entrepreneur subject to satisfying the individual rationality constraint of the lender.
Hint: You can do this numerically, by checking whether it is possible to increase the expected return of the entrepreneur by increasing or decreasing the fixed contracted amount of $120 to be paid in the solvent states.