Allocation of Wealth by Individuals. 1. Multiperiod investment-consumption decision. B. Williams (1938), Theory of Investment Value.

Transcription

1 Allocation of Wealth by Individuals I. A Brief History of Financial Economics A. Fisher (1930), The Theory of Interest. 1. Multiperiod investment-consumption decision. 2. Fisher Separation Theorem: Investment decision can be separated from financing decision. B. Williams (1938), Theory of Investment Value. Value additivity principle C. Hicks (1939), Value and Capital. Term Structure of Interest Rates and the role of expectations in pricing of assets. D. von Neuman and Morgenstern (1947), Theory of Game and Economic Behavior, and Savage (1954), Foundation of Statistics. 1. Expected utility hypothesis and decision making under uncertainty. 2. Game Theory E. Markowitz (1952), Portfolio Selection Investment decision making under uncertainty 2. Using mean-variance framework to measure risk and return. 1-1

2 F. Arrow (1953), The Role of Securities in Optimal Allocation of Risk- Bearing, and Debreu (1959), Theory of Value. 1. First Equilibrium model of an economy under uncertainty 2. Role security markets and securities in optimal allocation of resources G. Modigliani and Miller (1958), Cost of Capital and Capital Structure..., (1961), Dividend Policy..., and (1963), Corporate Income Taxes..., 1. Firm s financing decision and its effects on the firm s value. 2. Firm s dividend policy and its effects on the firm s value. H. Arrow (1964), Some Aspects of the Theory of Risk Bearing, and Pratt (1965), Risk Aversion in Small and Large. 1. Studied and quantified risk aversion and behavior toward risk I. Sharpe (1964), Capital Asset Prices. 1. The Capital Asset Pricing Model (CAPM) 2. Measurement of risk and the valuation of risky assets 1-2

3 J. Cootner ed. (1967), The Random Character of Stock Market, and Fama (1970), Efficient Capital Markets. 1. Time series properties of security prices 2. Efficient market hypothesis 3. Role of information in determination of asset prices K. Black and Scholes (1973), 1. Arbitrage pricing approach to pricing of securities. 2. Valuation of corporate securities and contingent claims. L. Alchian and Demsetz (1972), Production, Information, Costs, and Economic Organization, and Jensen and Meckling (1976), The Theory of the Firm. 1. Agency relationships and managerial behavior 2. Firm s financing and investment decisions when done by agents M. Merton (1971), Optimal Investment..., (1973), An Intertemporal Model of Asset Prices, Rubinstein (1976), Valuation of Uncertain Income..., and Lucas (1978), Asset Prices in an Exchange Economy. 1. Multiperiod investment-consumption decisions 2. General equilibrium asset pricing models 1-3

4 N. Grossman (1976), On the Efficiency of Financial Markets..., and Grossman and Stiglitz (1980), On the Impossibility of Informationally Efficient Markets. 1. Aggregation of information by prices. 2. Noise traders and formation of security prices. II. Time Value of Money A. Value of an asset = f(estimates of future income) Risk Time For the time being we will ignore risk B. Present Value Model V 0 C1 = ; Value of an asset is equal to its next period cash flow 1 + r 1 discounted at the one-period rate of interest. Time 1 C 1 Slope = ( + r ) 1 1 V 0 Time 0 1-4

5 The PV implies value additivity. Does this hold in reality? Volume discounts Closed-end funds III. Individual Investment-Consumption Decision A. Utility function: a welfare index, U(C) U(C) Common Properties of utility functions: C U ( C) C > 0 Non Satiation; Positive Marginal Utility 2 U ( C) C 2 < 0 Diminishing marginal Utility 1-5

6 B. Two-period utility: U(C 0,C 1 ) Where: U C U U U U > 0, > 0, < 0, < 0, < C C C C C Indifference curve: Let U(C 0,C 1 ) = U 1. What is the contour plot of U? C 1 U 1 Higher Utility U 0 C 0 U ( C, C ) = U < U

7 C. Simple Consumption Decision when the individual has perishable endowments: X 0 and X 1 X 1 Maximum Possible Utility X 0 The optimization model is: max U ( C, C ) s. t. C X, C X C, C Using the Lagrange method we have max L = U ( C 0, C 1) λ 0( C 0 X 0) λ1( C 1 X 1) C, C 0 1 The First order conditions are: L U = λ 0 = 0 (1) C0 C0 L U = λ1 = 0 (2) C1 C1 L λ 0 = ( C0 X0) λ 0 = 0 (3) λ 0 1-7

8 L λ 1 λ = ( C X ) λ = (4) We have four equations and four unknowns. What do the first-order conditions mean? If we have non satiation, then U C > 0. What does this say about λ? Since there are no financial markets in this simple case, there are no opportunities for the individual to smooth her consumption. So what will be the role of financial markets in the economy? D. Financial Markets Exist. When financial markets are introduced, the temporal distribution of income is no longer important, rather the person s wealth becomes the crucial factor. What is wealth? Was there such a thing in the previous case? Wealth = PV of future income = PV of future consumption. W X1 C1 = X + = C r 1 + r

9 Graphically it means the following: C 1 X 1 B A Slope: r C 0 X 0 W 0 The individual can now be at point B instead of point A. This investor has too much income in period 0 and not enough income in period 1. Thus she lends X 0 C 0 in period 0 and then gets back C 1 X 1 in period 1. From the geometry of the above graph we can see that ( C X )( 1+ r) = C X Mathematically, we have: max U ( C, C ) s. t. C C1 X = X 0 + C, C 1 + r 1 + r

10 max L = C X U ( C0, C1 ) 1 1 λ C0 + X0, λ 1 r r C C 0 1, L NM The FOC are: L λ C = U 0 C = 0 0 L λ C = U 1 C r = 0 L L C1 X1 = C0 + X0 λ 1 + r 1 + r NM O QP = We have 3 equations and 3 unknowns here. What do the FOC mean? 0 What is the relationship between optimal consumption at time 0 and at time 1? What is the interpretation of λ? O QP The ratio of U C 1 to U C 0 is called the marginal rate of substitution and as you can see it is equal to 1/(1 + r). Suppose the economy is populated with many investors who trade in competitive money market. What are the economic implications of this result? As you can see we do not have any real assets in this economy. That is, no real investments are need to be made by individuals. So how do we introduce real investments? 1-10

11 E. Financial Markets and Real Assets Exist Suppose we have a current endowment of X 0 bushels of corns. We can consume them, lend them, or plant them. The production can be represented by the relationship X1 = f ( X0), where f '( X0) > 0 and f ( X0) < 0. Graphically we can represent this as X 1 or as Investment, X 0 X 1 X 0 Consumption, C

12 Combining the production possibility curve, the borrowing-lend line, and the indifference curve, we get E F G I H J O C K X W This is a complicated but very rich graph in terms of ideas and concepts that it conveys. Investor has an endowment of X. This can be invested, consumed or lent. In the above graph, the investor chooses to invest (X K) in real assets. This will result in a harvest of H next period. At point J the investor notices that the marginal productivity of her investment is becoming less than the 1-12

13 market rate of interest. Therefore, if she is interested in consuming less than K, she will be better off to lend at the rate r rather than planting the corn. So this investor lends (K C), and consumes C. Next period, the investor receives H from harvest and (F H) from lending. Thus, a total amount of F is available for consumption. The point W represents the investor s wealth at time 0. But the investor started with X as her endowment. So what does (W X) represent? Notice that the temporal distribution of endowment is not relevant. What matters is the investor s wealth. Regardless of the distribution of endowment, the investor will make the same consumption decision if the level of wealth is W. This is very important because it says that some one else could have made the investment decision for this investor. As her agent, you will get the endowment and the real asset. You will invest in the real asset as long as its marginal productivity is greater than (1 + r), and once you run out of good plots of lands, you will return whatever is left to her in form of a dividend payment. This is known as the Fisher Separation Theorem. It basically says that when you have perfect financial markets, the investment decision 1-13

14 can be separated from the financing and consumption decision. Notice that the objective of the agent is to maximize W. Once this is done, the investor is as well off as possible. Mathematically the above problem can be stated as: C1 X1 max L = U ( C0, C1) λ C0 + X 0 φ X1 f ( X0) X λ φ H 1 + r 1 + rk C, C,,, What are the FOC? What do they represent? The above maximization problem also can be stated as: F X1 max W0 = X0 φ X1 f ( X0) ( A) X, φ 1 + r C 0 a F H C1 max L = U ( C0, C1) λ C0 + W0 ( B) 0, C1, λ 1 + r f First, problem A is solved by the agent. Then he tells the investor what the value of W 0 is. The investor then solves problem B using the information given to her by the agent. The above represents a mathematical representation of the Separation Theorem. What are the implications of this theorem? I K I a f 1-14

15 What if there are market imperfections (e.g., the investor cannot borrow and lend at the same rate of interest)? How do you think uncertainty affects this result? 1-15