L7 State Preference Theory PDF

Introduction State preference model Optimal portfolio decision Principles of Finance Lecture 7: State preference theory (CWS ch. 4) Rikke Sej...

Introduction State preference model Optimal portfolio decision Principles of Finance Lecture 7: State preference theory (CWS ch. 4) Rikke Sejer Nielsen 1 / 34  Introduction State preference model Optimal portfolio decision Expected utility The objective function of a greedy investor (U ′ (W ) > 0) is to maximize expected utility of end-of-period wealth, defined as: S X E[U(W )] = pi U(Wi ), i where S is the entire set of uncertain alternatives, i refers to a specific uncertain alternative, and pi and Wi is the probability and end-of-period wealth of a specific uncertain alternative, respectively. ⇒ Investment decision = choice across N probability distributions of end-of-period payoffs ⇒ Investment decision based on comparison of investments’ expected utility of end-of-period wealth. ▶ Optimal investment = investment with highest expected utility. 3 / 34  Introduction State preference model Optimal portfolio decision Optimal decision making under uncertainty for a given set of security prices. Securities are fundamental for the investment decisions: Firms sells securities to fund investments in real assets (supply of securities) Individuals: Individuals invest in securities in exchange for claims to the firms’ real asset (demand for securities) ⇒ Securities makes it possible to exchange consumption over time through financing of productive activities. ⇒ Securities involve taking a position over time ▶ Involve risk and uncertainty ⇒ use securities to evaluate investment choices. ⇒ In equilibrium (demand = supply) ▶ One set of security prices ⇒ a consistent set of firm and individual investment decisions. Today: Optimal decision making under uncertainty for individuals and firms for a given set of security prices. 4 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium State preference model Setting Perfect capital market One-period model. Wealth today, W0 Uncertainty in the sense of not knowing the future state of nature ⇒ State preference model ▶ S states of nature ▶ P πs = Probability of state, s, where πs = 1 ▶ J market securities are available for a given price today, pj. ▶ Security payoffs are known in the different states (state-contingent payoffs). 6 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium State preference model ⇒ Probability of a state of nature = probability of the associated end-of-period security payoff. ⇒ State of nature captures the economic uncertainty in the economy. 7 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium State preference model 8 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Complete Capital Market Definition of market completeness: The capital market is complete if the number of unique linearly independent securities is equal to the total number of alternative future states of nature Example on the board ⇒ In perfect and complete markets, any security can be prices based on existing securities in the capital market. 10 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Pure Securities Definition of Pure Securities A security with an end-of-period payoff of 1 in a given state, and 0 in any other state. 0 0 payoffs 1 (state s) 0 Notation: ps = Price of pure securities with end-of-period payoff of 1 in state s. Qs = Number of pure securities with end-of-period payoff of 1 in state s. 12 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Pure Securities Every market security can be decomposed to a combination of various pure securities. For three future states, we have three pure securities: 1 0 0 payoffs=1 0 payoffs=2 1 payoffs=3 0 0 0 1 For a market security: 4 payoffsec. 2 7 ⇒ Portfolio (PF) replicating the security payoff: payoffPF = 4payoffs=1 + 2payoffs=2 + 7payoffs=3 ⇒ Market security = a portfolio of the pure securities. 13 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Pure Securities When market is perfect and complete: ⇒ possible to construct pure securities for every state using existing market securities with well-defined prices. Price of a market security j s X pj = ps Qjs , S where Qjs is the number of pure securities used to replicate asset j’s payoff in state s. ⇒ Price of a market security = linear combination of prices of pure securities. Find solution to ps by solving a system of equations. 14 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Pure Securities Example - Derivation of Pure Security Prices We are in a perfect and complete capital market, with Two securities: X and Y. Two states of nature: (1) bust and (2) boom. End-of-period payoffs for the two securities: 2 5 payoffX payoffY 10 4 Each security is sold in the capital market today for pX = 3 and pY = 4. 15 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Pure Securities Example - Derivation of Pure Security Prices To replicate payoff in security X: QX ,1 = 2 and QX ,2 = 10 To replicate payoff in security Y: QY ,1 = 5 and QY ,2 = 4 Price of the securities: 3 = 2p1 + 10p2 4 = 5p1 + 4p2 ⇒ Solve system of equations to find p1 and p2. 15 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Pure Securities Example - Derivation of Pure Security Prices Solve system of equations: 2p1 + 10p2 − 3 =2.5(5p1 + 4p2 − 4) ⇔ 2p1 + 10p2 − 3 =12.5p1 + 10p2 − 10 ⇔ 10.5p1 =7 7 2 ⇔ p1 = = 10.5 3 so p2 must be 3 = 2p1 + 10p2 2 ⇔ 3 =2 ∗ + 10p2 3 4 ⇔ 10p2 =3 − 3 5 1 ⇔ p2 = = 30 6 15 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Pure Securities Derivation of Pure Security Prices Note: 2 a payment of 3 is required for a promise of a payoff of 1 in state 1. 1 a payment of 6 is required for a promise of a payoff of 1 in state 2. The concept of pure security is useful for ⇒ Analytical purposes ⇒ Simple description of uncertainty 16 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Capital market equilibrium Capital market equilibrium: For a given set of market prices, supply = demand for every security. Conditions for market equilibrium in the state preference model: 1 Single-price law of markets or law of one price: Any two securities or portfolios with equal state-contingent payoffs must have the same market price. If this is not the case ▶ buy securities/portfolios priced too low ▶ sell securities/portfolios priced too high ⇒ supply ̸= demand. 2 No-arbitrage, if short selling is allowed. 18 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Capital market equilibrium No arbitrage principle Two portfolios with identical state-contingent payoff vectors: Portfolio H with a high price of pH Portfolio L with a low price of pL ⇒ pH > pL ⇒ Strategy: Sell expensive / buy cheap Time CF today CF at end of period Buy Portfolio L −pL payoff (S = s) Sell Portfolio H pH −payoff (S = s) Total pH − pL > 0 0 ⇒ For market equilibrium: no arbitrage opportunities exists. 19 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Determinants of Security Prices What determines the price of a market security? Recall, every market security can be decomposed to a combination of various pure securities in a perfect and complete market. ⇒ Determinants of a market security price = Determinants of individual pure securities ⇒ We explore determinants of individual pure securities! 20 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Determinants of Security Prices Pure securities Determinants of the price of a pure security: 1 Time preferences for consumption and the productivity of capital 2 Expectations of state-probabilities 3 Risk attitude towards the variability across states of end-of-period wealth. 21 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Determinants of Security Prices Pure securities - 1. Time preferences for consumption and the productivity of capital Riskless security = portfolio of one pure security for each state Price of riskless portfolio: S X pf = ps s 1 Recall, the price of a riskless claim: 1+rf , so S X 1 pf = ps = s 1 + rf where risk-free rate, rf , reflect time value of money productivity of capital as seen in the simple world of certainty. 22 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Determinants of Security Prices Pure securities - 2. Expectations of state-probabilities Individuals’ may have different beliefs about state probabilities (πs ) Assuming homogeneous expectations, the expected end-of-period payoff for a pure security with a payoff of 1 in state s is π1 ⊤ 0      π2  0     .  . .  . .  . πs−1  0        = πs ∗ 1 for all states s ∈ S.  πs  1   s+1  0 π    .  .     ..  ..  πS 0 So the price of a pure security ps = πs θs , where θs is the price of an expected dollar payoff if state s occurs. 23 / 34  Introduction State preference model Optimal portfolio decision Capital market completeness Pure Securities Capital market equilibrium Determinants of Security Prices Pure securities - 3. Risk attitude towards the variability across states of end-of-period wealth. Recall, Individuals invest according to their end-of-period wealth Individuals are typically risk aversion For two equally likely states (π ∗ = πLW = πHW ) State with low end-of-period wealth (LW ) State with high end-of-period wealth (HW ) ⇒ An expected dollar payoff is worth more in state LW than in HW : θLW > θHW ⇒ Price of the two pure securities: pLW = π ∗ θLW > pHW = π ∗ θHW ⇒ Pure security LW relatively more attractive than Pure security HW ⇒ Market prices adjust according to the risk attitude towards the state-variation in the end-of-period wealth 24 / 34  Introduction State preference model Optimal portfolio decision For individuals For firms Optimal investment decision (for individuals) Individual’s expected utility of end-of-period wealth: S X E[U(W )] = πs U(Qs ), s where Qs in this context is the number of state s pure securities (replicating the market securities that the individual buys) as well as his or her end-of-period wealth if state s occurs. 27 / 34  Introduction State preference model Optimal portfolio decision For individuals For firms Optimal investment decision (for individuals) Optimal portfolio choice in a perfect and complete market Objective function of a greedy and rational investor with initial wealth W0 is " S # X max u(C) + πs U(Qs ) C,Q s subject to the wealth constraint S X ps Qs + C = W0 s ⇒ Solve using Lagrange method On the board! 28 / 34  Introduction State preference model Optimal portfolio decision For individuals For firms Optimal investment decision For firms Recall Firms fund investments by issuing securities. Firms are characterized by their production functions Qsj = φj (Ij , s) where Ij is the initial investment. The objective function of a firm: maximize the expected utility of its current shareholders. 30 / 34  Introduction State preference model Optimal portfolio decision For individuals For firms Optimal investment decision For firms Recall, the Fisher separation principle under certainty: In a perfect capital market (a frictionless and perfectly competitive market), actions that maximize the price of the firm’s shares maximize both the wealth and the utility of each current shareholder. Under uncertainty, Fisher Separation Principle still holds if the capital market is: perfectly competitive and frictionless, and complete ⇒ By maximizing the price of current shares = maximizing current shareholders’ expected utility. ⇒ The objective function of a firm j: maximizing the price of current shares 31 / 34  Introduction State preference model Optimal portfolio decision For individuals For firms Optimal investment decision For firms The price of firm j’s current shares (Market Value of firm j today) S X pj = Qjs ps , s where Qjs is the state s-contingent end-of-period payoff on firm j’s existing securities. Price depends on ▶ Production functions: Qsj = φ(Ij , s) ▶ Initial investment, Ij 32 / 34  Introduction State preference model Optimal portfolio decision For individuals For firms Optimal investment decision For firms Firm j’s market value prior to making the investment Ij : S X Yj = Qjs ps − Ij , s ⇒ NPV of project’s state-contingent net cash flow Investment decision rule: Invest if NPV > 0 ⇔ Invest if Yj > 0. Example on the board 33 / 34  Introduction State preference model Optimal portfolio decision For individuals For firms References CWS, ch. 4 34 / 34