State Preference Model and Capital Markets
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Questions and Answers

What are the end-of-period payoffs for securities X and Y in the given example?

The end-of-period payoffs for security X are $2$ in a bust and $10$ in a boom, while for security Y, they are $5$ in a bust and $4$ in a boom.

How much does security X cost in the capital market today?

Security X costs $3 in the capital market today.

What system of equations needs to be solved to find the prices of the two states, p1 and p2?

The equations are $3 = 2p1 + 10p2$ for security X and $4 = 5p1 + 4p2$ for security Y.

In a perfect and complete capital market, how many states of nature are considered in this example?

<p>There are two states of nature considered: bust and boom.</p> Signup and view all the answers

What are the quantities used to replicate the payoff in security X?

<p>The quantities used to replicate the payoff in security X are $QX ,1 = 2$ and $QX ,2 = 10$.</p> Signup and view all the answers

What is the main assumption of the state preference model regarding market conditions?

<p>The main assumption is that it operates in a perfect capital market with uncertainty about future states of nature.</p> Signup and view all the answers

Define a complete capital market in terms of securities and states of nature.

<p>A complete capital market exists when the number of unique linearly independent securities equals the total number of alternative future states of nature.</p> Signup and view all the answers

Explain what pure securities are within the context of the state preference model.

<p>Pure securities are defined as securities that provide an end-of-period payoff of 1 in a specific state and 0 in all other states.</p> Signup and view all the answers

How do state-contingent payoffs relate to the probabilities of different states in the state preference model?

<p>State-contingent payoffs are determined by the probabilities of various states, which reflect the economic uncertainty in the market.</p> Signup and view all the answers

What role does economic uncertainty play in the state preference model?

<p>Economic uncertainty is captured by various states of nature, influencing investor preferences and the pricing of securities.</p> Signup and view all the answers

Why is the concept of market completeness important in financial modeling?

<p>Market completeness is crucial because it ensures that all possible risks can be traded, allowing for accurate pricing of securities.</p> Signup and view all the answers

What is the significance of one-period models in the context of the state preference model?

<p>One-period models simplify the analysis by focusing on a single decision-making timeframe, reducing the complexities of future uncertainties.</p> Signup and view all the answers

How does the state preference model address the pricing of securities in a perfect capital market?

<p>The state preference model allows for the pricing of securities based on the existing market conditions and the probability-weighted outcomes of different states.</p> Signup and view all the answers

What are pure securities and how do they relate to market securities?

<p>Pure securities are financial instruments that provide a payoff of 1 in a specific future state and can be combined to replicate market securities.</p> Signup and view all the answers

How can a market security's payoff be expressed in terms of pure securities?

<p>A market security's payoff can be expressed as a linear combination of the payoffs from pure securities corresponding to different states.</p> Signup and view all the answers

What is the formula used to determine the price of a market security?

<p>The price of a market security is given by $p_j = \sum_{s} p_s Q_{js}$, where $Q_{js}$ is the number of pure securities used.</p> Signup and view all the answers

What does it imply if a market is described as perfect and complete?

<p>A perfect and complete market allows for the construction of pure securities for every state using existing market securities.</p> Signup and view all the answers

Explain the relationship between the payoffs of pure securities and the number of pure securities employed.

<p>The number of pure securities determines the magnitude of their contribution to the overall payoff of a market security in a specific state.</p> Signup and view all the answers

Can you give an example of a payoff structure for a market security using pure securities?

<p>An example structure could be a market security with a payoff of $4$ in state 1, $2$ in state 2, and $7$ in state 3.</p> Signup and view all the answers

What role do pure securities play in forming an optimal portfolio decision?

<p>Pure securities allow investors to construct optimal portfolios that align closely with their risk preferences and expected payoffs.</p> Signup and view all the answers

What system is solved to find the prices of pure securities in a complete market?

<p>A system of equations is solved to determine the prices of pure securities based on the payoffs and their combinations.</p> Signup and view all the answers

What is the objective function of a greedy investor in the state preference model?

<p>To maximize the expected utility of end-of-period wealth.</p> Signup and view all the answers

What does 'E[U(W)]' represent in this context?

<p>It represents the expected utility of wealth based on various probable end-of-period outcomes.</p> Signup and view all the answers

How are investment decisions made according to the state preference model?

<p>Investment decisions are made by comparing the expected utility of end-of-period wealth across different probability distributions.</p> Signup and view all the answers

Explain the role of securities in investment decisions.

<p>Securities allow firms to fund investments in real assets and enable individuals to claim ownership of these assets.</p> Signup and view all the answers

What is the significance of equilibrium in the context of securities?

<p>Equilibrium occurs when demand equals supply, resulting in a consistent set of security prices.</p> Signup and view all the answers

What is the relationship between risk and investments in securities?

<p>Investing in securities inherently involves taking on risk and uncertainty.</p> Signup and view all the answers

What does optimal decision making under uncertainty entail?

<p>It involves making informed investment choices based on the expected utility for individuals and firms given security prices.</p> Signup and view all the answers

Define the term 'expected utility' as used in the context of this model.

<p>Expected utility is the weighted average of utilities across all uncertain alternatives, reflecting both probability and outcome.</p> Signup and view all the answers

What are the three determinants of the price of a pure security?

<ol> <li>Time preferences for consumption and productivity of capital, 2. Expectations of state-probabilities, 3. Risk attitude towards variability in end-of-period wealth.</li> </ol> Signup and view all the answers

How is a riskless portfolio constructed?

<p>A riskless portfolio is constructed as a combination of one pure security for each state.</p> Signup and view all the answers

How does the risk-free rate ($rf$) affect the price of a riskless claim?

<p>The price of a riskless claim is given by $pf = ps / (1 + rf)$, reflecting the time value of money and the productivity of capital.</p> Signup and view all the answers

What is the significance of state probabilities in determining security prices?

<p>State probabilities represent individuals' beliefs about future market occurrences and affect the expected payoffs of securities.</p> Signup and view all the answers

What happens when individuals have homogeneous expectations of state probabilities?

<p>When individuals have homogeneous expectations, expected end-of-period payoffs can be calculated uniformly across investors.</p> Signup and view all the answers

What role do time preferences play in financial decision-making?

<p>Time preferences indicate how much individuals value present consumption over future consumption, influencing their investment choices.</p> Signup and view all the answers

How does an investor's attitude toward risk affect their investment decisions?

<p>An investor's risk attitude determines their willingness to accept variability in potential outcomes, influencing their portfolio choices.</p> Signup and view all the answers

What does the formula $pf = ps / (1 + rf)$ illustrate in terms of asset pricing?

<p>The formula illustrates the concept of discounting future payoffs to present values taking into account the risk-free rate.</p> Signup and view all the answers

What is the initial objective for a greedy and rational investor regarding their wealth?

<p>The objective is to maximize the utility function, which is expressed as $u(C) + \sum_{s} \pi_s U(Q_s)$.</p> Signup and view all the answers

Describe the role of the Lagrange method in the optimal portfolio choice.

<p>The Lagrange method is used to solve the optimization problem subject to the wealth constraint, represented as $\sum_{s} p_s Q_s + C = W_0$.</p> Signup and view all the answers

What does a firm aim to maximize when making investment decisions, according to the content?

<p>A firm aims to maximize the expected utility of its current shareholders.</p> Signup and view all the answers

Explain the Fisher separation principle under certainty in a capital market.

<p>Under certainty, the Fisher separation principle states that actions maximizing firm share prices also maximize shareholders' wealth and utility.</p> Signup and view all the answers

What are the conditions under which the Fisher separation principle holds true under uncertainty?

<p>It holds true in a capital market that is perfectly competitive, frictionless, and complete.</p> Signup and view all the answers

In optimal investment decision-making, how are firms characterized?

<p>Firms are characterized by their production functions, expressed as $Q_{sj} = \phi_j(I_j, s)$, where $I_j$ is the initial investment.</p> Signup and view all the answers

What is the importance of wealth constraint in the optimal portfolio decision for individuals?

<p>The wealth constraint ensures that the total consumption and investment do not exceed initial wealth $W_0$.</p> Signup and view all the answers

What does maximizing the price of a firm's shares imply for its shareholders?

<p>Maximizing the price of shares implies maximizing the expected utility of current shareholders.</p> Signup and view all the answers

Flashcards

Expected Utility

The goal of an investor is to maximize the expected utility of their wealth at the end of the investment period.

Set of Uncertain Alternatives (S)

All possible outcomes of an investment decision, each with its own probability of occurring.

Uncertain Alternative (i)

The specific outcome of an investment decision, with its probability of happening and the corresponding wealth at the end of the investment period.

Probability (pi)

The likelihood of a particular uncertain alternative occurring.

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End-of-Period Wealth (Wi)

The wealth that an investor anticipates having at the end of the investment period, depending on the specific uncertain alternative.

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Optimal Investment

The process of choosing the investment option that provides the highest expected utility, maximizing the potential for a favorable outcome.

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Investment Decision based on Expected Utility

The value of an investment is determined by comparing the expected utility of different investment options and choosing the one with the highest value.

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Securities and Investment Decisions

Firms raise capital by issuing securities, while individuals invest in securities to fund their future consumption and investment plans.

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State Preference Model

A model that considers different possible future states of the world and the payoffs of securities in those states. Investors make decisions based on their preferences for these payoffs in each state.

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Complete Capital Market

A market where investors can trade securities to achieve their desired level of risk and return. It ensures all possible combinations of payoffs are attainable by investors.

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Perfect Capital Market

A market that has no frictions like taxes, transaction costs, or restrictions on borrowing and lending. All information is readily available to investors.

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Pure Security

A security that pays a specific payoff in one state of nature and zero in all other states.

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State of Nature

The state of the economy at the end of the period. This can be high or low economic growth, good or bad market conditions.

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Probability of a State

The probability that a particular state of nature will occur.

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Security Price (pj)

The price at which a security is traded in the market.

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State-Contingent Payoffs

The payoff a security provides in a given state of nature at the end of the period.

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Price of Pure Security

The price of a pure security is the price of a claim to 1 unit of currency in that specific future state.

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Market Security Decomposition

Every market security can be represented as a combination of pure securities.

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Complete Market & Pure Securities

In a complete market, you can construct pure securities for every future state using existing market securities.

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Market Security Price

The price of a market security is the sum of the prices of the pure securities used to replicate its payoff in each state.

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Solving for Prices of Pure Securities

A system of equations is used to determine the prices of pure securities by using the known prices of market securities and the quantities of pure securities needed to replicate their payoffs.

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Replicating Market Security Payoffs

When the payoffs of a market security in each future state are known, the number of each pure security required to perfectly replicate the market security's payoffs can be identified.

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Derivation of Pure Security Prices

The process of finding the prices of pure securities by replicating the payoffs of existing securities in a complete market.

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Capital Market Equilibrium

The equilibrium in a capital market where the prices of securities reflect the expected payoffs of each security in different states of nature, considering the risk aversion of investors.

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Firm's objective function

A firm's goal is to maximize the expected utility of its current shareholders by choosing investments that will lead to the highest possible share price.

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Fisher Separation Principle under uncertainty

The principle that in a perfect and complete market, maximizing the firm's share price will also maximize the wealth and utility of each shareholder.

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Production function of a firm

A function that describes the relationship between a firm's investment and the output it produces in different possible future states.

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Initial wealth (W0)

The initial wealth an investor has before making investment decisions.

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Objective function of a rational investor

The investor's goal to maximize the utility they derive from their consumption and investment choices.

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Price of a riskless portfolio

The price of a riskless portfolio is determined by the present value of its future payoff, calculated using the risk-free rate, which reflects both the time value of money and the productivity of capital.

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Riskless security

A riskless security can be constructed by combining a portfolio of pure securities, one for each possible state of the world.

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Expected payoff of a pure security

The expected end-of-period payoff for a pure security is calculated as the sum of the possible payoffs in each state, weighted by the probability of that state occurring.

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Heterogeneous expectations

Individuals may have different beliefs about the probabilities of future states, leading to disagreements about security prices.

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Risk attitude

The price of a pure security is influenced by the risk attitude of investors, with risk-averse investors demanding a higher premium for holding securities that are more volatile.

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Determinants of pure security prices

The price of a pure security is determined by a combination of factors, including time preferences for consumption, productivity of capital, expected state probabilities, and risk attitude.

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Study Notes

State Preference Theory

  • State preference theory is used for optimal portfolio decisions in a given set of security prices.
  • Greedy investors aim to maximize expected utility of end-of-period wealth.
  • Expected utility is calculated as the sum of the probability (𝑝ᵢ) of a specific uncertain alternative (i) multiplied by the utility (U(Wᵢ)) of the end-of-period wealth (Wᵢ).
  • The entire set of uncertain alternatives is represented by S.
  • Each uncertain alternative (i) is associated with a probability (𝑝ᵢ) and an end-of-period wealth (Wᵢ).

Investment Decisions

  • Investment decisions involve choosing across multiple probability distributions of end-of-period payoffs.
  • Decisions are based on comparing the expected utility of investments.
  • The optimal investment is the one with the highest expected utility.

Securities and Investment Decisions

  • Securities are crucial for investment decisions, facilitating the exchange of consumption over time through financing productive activities.
  • Firms issue securities to fund real asset investments (supply of securities).
  • Individuals purchase securities for claims on the firm's real assets (demand for securities).

State Preference Model: Setting

  • The model assumes a perfect capital market.
  • It focuses on a single period.
  • The model considers uncertainty related to future states of nature.
  • There are 'S' possible states of nature (e.g., prosperity, normalcy, recession, depression).
  • The probability of each state is given, and the total probability sums up to 1 (∑πs = 1).
  • The current price of securities, 𝑝ⱼ, and their payoffs in various states are known.

State-Contingent Claims

  • A state-contingent claim has a specific payoff contingent on a particular state of nature and associated probability.
  • Probability of a state of nature is the probability of the associated end-of-period security payoff.
  • States of nature capture economic uncertainty.

Capital Market Completeness

  • A capital market is complete if the number of unique linearly independent securities equals the number of possible future states of nature.
  • In a perfect and complete market, any security can be priced based on existing securities in the capital market.

Pure Securities

  • A pure security has an end-of-period payoff of 1 in a specific state and 0 in all other states.
  • The price (𝑝ₛ) of a pure security is identified.
  • The number (𝑄ₛ) of pure securities is calculated for a given payoff within each state.

Derivation of Pure Security Prices

  • The prices of pure securities (𝑝₁) and (𝑝₂) can be determined by solving a system of equations (e.g. Example given for computing 𝑝₁ and 𝑝₂).

Optimal Investment Decisions: Individuals

  • Individuals maximize their expected utility of end-of-period wealth, which is the sum over each possible state (πₛ U(Qₛ)) .
  • 𝑄ₛ represents the number of pure securities matching a corresponding state s.
  • The equation is solved by the Lagrange method.

Capital Market Equilibrium

  • For equilibrium, supply equals demand for each security.
  • The "single-price law of markets" describes that securities or portfolios with equal state contingent payoffs must have the same price.
  • No-arbitrage opportunities exist if short selling is allowed.

Determinants of Security Prices

  • Time preferences for consumption and the productivity of capital
  • Expectations for state probabilities
  • Risk attitudes towards the variability in end-of-period wealth

Optimal Investment Decisions: Firms

  • Firms maximize the expected utility of their current shareholders.
  • The objective function is to maximize the price of their current shares.
  • Firm's market value (Yj) is calculated prior to an investment, considering the NPV (Net Present Value) of the project's state-contingent net cash flow.
  • Fisher Separation Principle holds under uncertainty if the capital market is perfectly competitive, frictionless, and complete.

References

  • CWS chapter 4

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Description

This quiz explores key concepts of the state preference model and capital market dynamics. Questions cover the pricing of securities, end-of-period payoffs, and the implications of economic uncertainty. Ideal for finance students to test their understanding of market completeness and state-contingent payoffs.

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