The Fisher Model and Financial Markets: the building blocks of a financial market theory

1. The Fisher Model and Financial Markets: the building blocks of a financial market theory By Professor Richard MacMinn Illinois State Universityrichard@macminn.org

2. Outline • Remarks • The classic Fisher model under uncertainty • Securities • Stocks • Bonds • Options • Fisher Separation • The corporate objective function

3. notation

4. Consumption choice • The constrained maximization problem

5. The budget Constraint

6. First order conditions • The first order conditions for the Lagrange function are:

7. Constrained Maximization Problem for the investor • It is also possible to consider the problem from the perspective of an investor. • The investor selects a portfolio of securities to transfer dollars between dates and states. Let xi(x) be the number of state x shares purchased by investor i. • Then the investor’s, problem may be rewritten in terms of the financial assets. Consumption now and then become • and

8. The investment choice • The investor’s constrained maximization problem is

9. The first order conditions • The first order conditions for the maximization problem are of the following form: • This may be equivalently expressed as

10. Basis Stock Demand • Demand • Comparative statics • An increase in risk aversion

11. more notation

12. Other Securities • Equity • Bonds • Options

13. Equity • The consumption pair with equity in corporation f becomes:

14. Equity • The first order condition for corporate shares is

15. Equity value

16. Debt • The consumption pair with debt in the corporation becomes

17. Debt value • The first order condition for debt is • This yields the debt value as

18. Call Option • Consider an option that gives the right to purchase one share of stock in corporation f then at an exercise price ef • The option payoff is • Letting Ef denote the value of the aggregate exercise, the value of the calls is

19. Fisher Separation • Let Pf(I, x) be the random variable representing the investment frontier of the firm. • Suppose that D1Pf(I, x) > 0 • Suppose that Pf(I, x) satisfies the Principle of Increasing Uncertainty, i.e., D2 Pf(I, x) > 0 and D21Pf(I, x) > 0 • Suppose the firm manager makes decisions on personal and corporate account • Manager paid in stock • Manager paid in options

20. yet more notation

21. Share values • If the firm issues stock to finance an investment then the stock value must split into the value of the new shareholder stake and that of the old shareholders.

22. The constrained maximization problem • If the manager is compensated now with mf shares of corporate stock and finances the firm’s investment with stock then the manager’s constrained maximization problem is:

23. The reduced form problem • The finance constraint yields the number of new shares as a function of the investment and so the constrained maximization problem may be expressed as

24. Fisher separation • The first order condition for the Lagrange function with respect to the investment is • This first order condition • demonstrates Fisher separation • shows that the manager makes decisions to maximize current shareholder value since

25. Fisher separation with options • Suppose the manager receives mf call options on corporate shares and, without loss of generality, suppose the manager issue bonds to finance the investment. • The call options give the right to purchase one share of corporate stock then for ef dollars, where ef is a constant.

26. The constrained maximization problem • If the manager has options rather than shares and issues debt rather than equity then the decision problem is

27. The reduced form of the constrained maximization problem • The financing condition implicitly defines the promised payment on debt as a function of the investment. Substituting that into the problem yields • where Wf is the warrant value

28. Fisher separation again • Since the warrant value is • The first order condition for the Lagrange function with respect to the investment is • This first order condition • demonstrates Fisher separation • shows that the manager makes decisions to maximize warrant value • Note that if PIU holds then Iw > Is

29. Concluding Remarks • These results hold for more general settings including some incomplete market settings • The presence of hidden action or hidden knowledge does not imply that separation is not possible. • The corporate objective function is endogenous but acting in the fiduciary interests of shareholders is the exception rather than the rule • The corporate objective functions lead to different operating decisions. • The manager paid in stock options over invests relative to the manager paid in stock if the principle of increasing uncertainty applies. • This model may be used to prove all the theorems in corporate finance