Problem Set - IV
Please note that there are FOUR problems in this problem set.
- Dividend Policy and Stock Issues Under Perfect Capital Markets
Assume perfect capital markets. Consider a company with the following market value balance sheet:
AssetsLiabilities
Cash1000Debt0
Fixed
Assets9000Equity12,000
New Project
NPV 2000
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12,00012,000
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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. Capital Structure Choice Under Asymmetric Information
Suppose firms differ in their possible cash flows. High-cash-flow firms have cash flows ,which might take on any value in the interval [0,H] with equal probability, while the cash flows of low-cash-flow firms are uniformly distributed over the interval [0,L], where L<H. Each manager knows what kind of firm he or she has, but investors do not have this information: All they know is that some fraction of the firms have high cash flow and the rest have low. They do, however, observe the amount D of debt that the firm issues.
Suppose each manager chooses a value of D and is paid in such a way that he or she seeks to maximize a weighted average of the current value of the firm Vo, and its expected value in the next period (at which point the cash flow is realized and becomes publicly known) less a bankruptcy penalty, P, that is incurred by the manager (not the firm’s owners) if earnings turn out to be less than D. This occurs with probability D/H for high-cash-flow-firms and D/L for low ones. The expected value of the firm next period is just its expected cash flow, L/2 or H/2. The current value may depend on D through investor’s inferences. Thus a manager of a firm of type t, t=H or L, who selects a debt of D receives a payoff (proportional to)
where is the weight.
Under the assumption that managers in the two types of firms pick different levels of D, investors will infer firm’s values from the choices. Thus, if a firm has debt DL it will be thought to be worth L/2 in the first period, and similarly for DH, no matter what its true type. Thus Vo is determined: Vo =(DH)=H/2, and Vo (DL)=L/2. However, the firm’s value next period will be determined by its actual type.
Show (using the appropriate incentive constraints) that DH must exceed DL: More profitable firms issue more debt. Show that if DL is set at zero and DH is set as low as possible to meet the incentive constraints, then DH decreases as P increases: Higher bankruptcy costs decrease the amount of debt and the probability of bankruptcy.
Hint: Write down the incentive compatibility conditions for each type of firm (H and L type). Combine these and simplify to obtain a single condition. From this you can: (i) Show that DH > DL; and (ii) After setting DL =0, solve analytically for DH. The resulting expression can be shown to be decreasing in P.
3. 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 of firms are type H firms, while the remaining (1-) 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 IXL
f(I)= k1I +k2(I- XL) for XLI 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. Assume that the equilibrium dividend is greater than (XH- XL). (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, =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.
4. Issuing Securities Under Asymmetric Information
At a point in time, say t=0, an all equity financed firm has assets in place and a positive NPV investment opportunity. The current value of the firm’s assets in place is either XH (type H firm) or XL < XH (type L firm), and is known only to firm’s insiders. Outside investors have an ex- ante belief that the firm is of type H with probability p, and of type L with probability 1-p. The investment project must be undertaken in the current period and it is the same for both types of firms. This project requires an initial investment of I and its cash flow will be realized one period from now (i.e., at t=1), at which time it will be observable by all. The cash flow from this project will have a value C>0 with a probability h; it will be 0 with the complementary probability (1-h) (the values C and h are known to both outsiders and insiders). The insiders (management) of the firm makes its investments and financing decisions with the objective of maximizing the t=1 expected value of cash flows to the firm’s original (t=0) stockholders. You may assume throughout that the risk-free rate of return is zero, and that all agents are risk-neutral.
a. Assume first that the firm management is constrained to raise the requisite investment amount I through selling equity alone (i.e. there is no internal financing available, and securities other than equity cannot be issued). Write down the conditions at which the investment project is undertaken and not undertaken. Discuss the kind of equilibria that may emerge on the equity market as a consequence of the planned equity issue (under different parameter values).
b.For this section, assign the following numerical value:
XH =100, XL=0; p=0.5, C=20, h=0.8, I=10.
(i)Maintain for this subsection that only equity can be used to raise financing. Under these conditions, show that, with the above numerical values, a pooling equilibrium where both types of firms issue equity and implement the project does not exist. Further, demonstrate that (using the appropriate incentive compatibility conditions) that a separating equilibrium where the type L firm issues equity and implements the project, while the type H firm does not, exists. Give the equilibrium beliefs consistent with this separating equilibrium.
(ii)Now allow for the possibility that the firm can issue debt. Show that a pooling equilibrium where both types of firms raise the requisite external financing using debt exists. What is the face value of debt to be issued in this equilibrium (assuming a competitive debt market)? Is this debt risk-free or risky?
Hint: After conjecturing a pooling equilibrium, compute the face value of debt to be issued that will leave the debt holder as well off as if he had invested in his alternative investment opportunity. Then show, using this particular face value of debt that the shareholders in both types of firms are better off (in terms of their time 1 expected cash flow) if they implement the project.
c. Go back to the assumption in section b(i) that the firm can only issue equity. Further, assume that you are the CFO of a firm with favorable private information about its assets in place (i.e., a type H firm), with the various parameter values as in section (b). The investment bank you have hired has advised you that you can avoid the problems of asymmetric information in the above situation by spinning-off the new project into a separate firm, and then issuing equity against the new firm, thereby raising the amount I required to implement it. Is this a good idea? If so why (if not, why not)?