Capital Market Theory and Pure Securities
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Questions and Answers

What is the derived price of pure security p1 from the given equations?

The derived price of pure security p1 is $\frac{7}{10.5} = \frac{2}{3}$.

What is the derived price of pure security p2 based on the calculations provided?

The derived price of pure security p2 is $\frac{3}{30} = \frac{1}{10}$.

Explain what is meant by 'capital market completeness' in the context of pure securities.

Capital market completeness refers to a market where all risks can be fully hedged and all future states can be traded.

What payment is required for a promise of a payoff of 1 in state 1?

<p>A payment of $\frac{3}{2}$ is required for a promise of a payoff of 1 in state 1.</p> Signup and view all the answers

What does the term 'pure securities' refer to in capital market theory?

<p>Pure securities are financial instruments that provide a payoff in a specific state of the world without any default risk.</p> Signup and view all the answers

What are the two primary uses of the concept of pure security?

<p>Analytical purposes and simple description of uncertainty.</p> Signup and view all the answers

What must hold true for market equilibrium in the state preference model?

<p>Supply must equal demand for every security.</p> Signup and view all the answers

Explain the single-price law of markets.

<p>It states that any two securities with equal state-contingent payoffs must have the same market price.</p> Signup and view all the answers

What is the implication if the single-price law is not upheld?

<p>It leads to buying low-priced securities and selling high-priced ones, disrupting supply and demand.</p> Signup and view all the answers

Define the no-arbitrage principle.

<p>It suggests that two portfolios with identical state-contingent payoff vectors should have equal prices.</p> Signup and view all the answers

What strategy should an investor adopt when faced with portfolio prices that violate the no-arbitrage principle?

<p>Sell the expensive portfolio and buy the cheap one.</p> Signup and view all the answers

How are the prices of market securities determined?

<p>They are determined by the combination of various pure securities.</p> Signup and view all the answers

What is the relationship between market security prices and pure securities?

<p>The price of a market security reflects the determinants of its underlying pure securities.</p> Signup and view all the answers

What is the primary characteristic of a complete capital market?

<p>A complete capital market has an equal number of unique linearly independent securities and future states of nature.</p> Signup and view all the answers

Define pure securities in the context of the state preference model.

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

How do state probabilities relate to the payoffs of securities according to the state preference model?

<p>State probabilities are equivalent to the probabilities of the associated end-of-period security payoffs.</p> Signup and view all the answers

What assumption is made about the capital market in the state preference model?

<p>The state preference model assumes a perfect capital market.</p> Signup and view all the answers

What does the term 'economic uncertainty' refer to in the state preference model?

<p>Economic uncertainty denotes the unknown future states of nature that affect investment outcomes.</p> Signup and view all the answers

Why can any security be priced in a perfect and complete market?

<p>In a perfect and complete market, any security can be priced using existing securities due to their linear independence.</p> Signup and view all the answers

In the context of the state preference model, what does 'one-period model' signify?

<p>A one-period model signifies that all investments and their outcomes are evaluated over a single time frame.</p> Signup and view all the answers

What role does the probability of a state of nature play in investment decision-making?

<p>The probability of a state of nature aids investors in assessing potential outcomes and making informed choices.</p> Signup and view all the answers

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

<p>Time preferences for consumption, expectations of state-probabilities, and risk attitude towards wealth variability.</p> Signup and view all the answers

How is the price of a riskless portfolio calculated using the prices of pure securities?

<p>The price of a riskless portfolio, pf, is calculated as $pf = \frac{\sum ps}{1 + r_f}$ where ps represents the price of pure securities.</p> Signup and view all the answers

In relation to pure securities, what does the term 'riskless security' refer to?

<p>A riskless security refers to a portfolio of one pure security for each state that guarantees a certain payoff.</p> Signup and view all the answers

Why might individuals have different expectations about state probabilities, πs?

<p>Individuals may have different beliefs, experiences, and information that shape their expectations about state probabilities.</p> Signup and view all the answers

What role does the risk-free rate, r_f, play in pricing securities?

<p>The risk-free rate reflects the time value of money and productivity of capital, influencing the pricing of securities.</p> Signup and view all the answers

Explain how time preferences for consumption affect the price of pure securities.

<p>Time preferences for consumption determine how much value is placed on future payoffs versus current consumption, impacting security prices.</p> Signup and view all the answers

Describe the formula used to express the expected end-of-period payoff for a pure security.

<p>The expected end-of-period payoff for a pure security is expressed as $\sum \pi_s \times payoff_s$ for all states s.</p> Signup and view all the answers

What does the notation $\pi_1, \pi_2$ represent in the context of state probabilities?

<p>$\pi_1, \pi_2$ represent the probabilities associated with different states that influence the expected payoff from a pure security.</p> Signup and view all the answers

What formula represents the price of a pure security related to state s?

<p>The price of a pure security is given by $p_s = heta_s \pi_s$, where $ heta_s$ is the price of an expected dollar payoff if state s occurs.</p> Signup and view all the answers

How does risk attitude affect the attractiveness of pure securities across different states?

<p>Individuals with risk aversion will find a state with lower end-of-period wealth (LW) more attractive than a state with higher wealth (HW), leading to $p_{LW} &gt; p_{HW}$.</p> Signup and view all the answers

What does the expected utility of end-of-period wealth formula incorporate?

<p>The formula $E[U(W)] = \sum_{s} \pi_s U(Q_s)$ incorporates the probabilities of each state and the utility derived from the end-of-period wealth associated with pure securities.</p> Signup and view all the answers

What does $ heta_s$ represent in the context of security pricing?

<p>$\theta_s$ represents the price of an expected dollar payoff if a particular state s occurs.</p> Signup and view all the answers

What results from individuals investing according to their end-of-period wealth?

<p>Individuals tend to make investment decisions based on the variability of their end-of-period wealth, exemplifying their risk aversion.</p> Signup and view all the answers

Why is the price of pure security LW greater than that of HW?

<p>Due to risk aversion, the expected payoff in state LW is valued higher, resulting in $p_{LW} = \pi^* \theta_{LW} &gt; p_{HW} = \pi^* \theta_{HW}$.</p> Signup and view all the answers

What does the state preference model aim to explain in financial markets?

<p>The state preference model explains how individuals make optimal portfolio decisions by considering their preferences and aversion to risk associated with different states.</p> Signup and view all the answers

How do market prices respond to risk attitudes in pure securities?

<p>Market prices adjust based on the perceived risk associated with state-variation in end-of-period wealth, favoring securities aligned with lower risk states.</p> Signup and view all the answers

What is the objective function of a rational investor with initial wealth W0?

<p>The objective function is to maximize $u(C) + \sum_{s} \pi_s U(Q_s)$ subject to the wealth constraint.</p> Signup and view all the answers

How is the wealth constraint represented in the optimal investment decision model for individuals?

<p>The wealth constraint is represented as $\sum_{s} p_s Q_s + C = W_0$.</p> Signup and view all the answers

What principle explains the relationship between maximizing share prices and shareholder utility?

<p>The Fisher separation principle states that in a perfect capital market, actions maximizing share price also maximize shareholder utility.</p> Signup and view all the answers

What are the two primary components of the firm's objective function regarding investments?

<p>The components are maximizing the expected utility of current shareholders and increasing the firm's share price.</p> Signup and view all the answers

In the context of firms, what does the production function Qsj = φj(Ij, s) represent?

<p>It represents the output produced by firm j as a function of its initial investment Ij and the state of nature s.</p> Signup and view all the answers

What does the Lagrange method help solve in the context of optimal portfolio choice?

<p>The Lagrange method helps solve for the optimal allocation of consumption and investment given the wealth constraint.</p> Signup and view all the answers

What must be true about the capital market for the Fisher separation principle to hold under uncertainty?

<p>The capital market must be perfectly competitive, frictionless, and complete.</p> Signup and view all the answers

Why is it important for firms to issue securities in financing investments?

<p>Issuing securities allows firms to gather the necessary capital required for investments while also sharing ownership with investors.</p> Signup and view all the answers

Flashcards

Pure Security

A financial instrument that promises a specific payoff in only one particular state of the world.

Derivation of Pure Security Prices

The process of determining the price of a pure security by solving a system of equations.

System of Equations

The mathematical representation of the relationship between the prices of pure securities and the corresponding state-dependent payoffs.

Capital Market Equilibrium

A set of security prices that are consistent with the expected payoffs in each state of the world under the no-arbitrage condition.

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Capital Market Completeness

The condition where, for any combination of payoffs across states, there exists a portfolio of existing securities that can replicate that combination.

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State Preference Model

A financial model representing a future uncertain environment with different possible outcomes or "states of nature." It assumes a perfect capital market, a single period, and wealth available at the beginning of the period.

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Complete Capital Market

A market where the number of tradable securities matches the number of possible future outcomes (states of nature). You can create any desired payoff by combining existing securities.

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State Price

The set of all prices of pure securities in an economy. It shows the value investors place on each potential future outcome.

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Perfect Capital Market

A market where investors can easily buy or sell securities at current market prices, with no limitations on trading activities.

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Optimal Portfolio Decision

The process of allocating wealth across different assets to maximize expected return while considering risk.

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Perfect Capital Market

A market with no information asymmetries, transaction costs, or restrictions on trading. Everyone has equal access to information and can buy or sell anything.

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Riskless Security Pricing

The price of a riskless security is determined by the time value of money and the productivity of capital.

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Riskless Portfolio

A portfolio that combines one pure security for each state of the world, guaranteeing a specific return in every possible outcome.

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Heterogeneous State Probabilities

Individuals may have different views about the likelihood of each state occurring.

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Homogeneous Expectations

A situation where everyone agrees on the probability of each possible outcome.

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Expected End-of-Period Payoff

The average payoff of a security, weighted by the probabilities of each state occurring.

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Expected Payoff of Pure Security

The expected payoff of a pure security depends on the probability of the state in which it pays out.

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Risk Attitude

The preferences of investors regarding the trade-off between risk and return.

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Single-Price Law of Markets

The principle that two securities or portfolios with identical state-contingent payoffs must have the same market price. This ensures no arbitrage opportunities.

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No-Arbitrage Condition

When short-selling is allowed, there are no opportunities to make riskless profits by buying and selling securities with different prices but identical payoffs.

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Why are Pure Securities Useful?

The concept of pure security is helpful for understanding and describing uncertainty and for analyzing the relationship between risk and return.

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How do Pure Securities Determine Market Security Prices?

The concept of pure securities helps explain the price of market securities by decomposing them into a combination of pure securities.

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What Determines the Price of a Pure Security?

By analyzing the determinants of individual pure securities, we can understand what drives the prices of market securities.

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Price of a Pure Security

The price of a pure security is calculated by multiplying the probability of the state occurring by the price of an expected dollar payoff in that state.

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Price of an Expected Dollar Payoff (θs)

The price of an expected dollar payoff in a specific state of the world. It reflects how valuable a dollar is in that state.

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Risk Aversion

A measure of an individual's attitude towards risk. Risk aversion means individuals prefer a certain outcome to a risky one with the same expected value.

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Expected Utility of End-of-Period Wealth

The expected utility of an individual's end-of-period wealth is calculated by summing the utilities in each state of the world, weighted by the probability of each state occurring.

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Investor's Objective Function

The investor aims to maximize the expected utility of their consumption in different states of the world, represented by the sum of utilities across all possible states.

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Wealth Constraint

Investors allocate their initial wealth (W0) between current consumption (C) and investments in securities (Qs) across different states (s).

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Firm's Production Function

Firms use securities to fund their investments and are characterized by their production functions, which determine output (Qsj) based on initial investment (Ij) in various states (s).

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Fisher Separation Principle

In a perfect capital market, actions that maximize the firm's share price also maximize both the wealth and utility of its current shareholders. This holds true even under uncertainty if the market is perfectly competitive, frictionless, and complete.

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Maximizing Share Price = Expected Utility

By maximizing the price of current shares, firms maximize the current shareholders' expected utility in a perfect and complete market. This is because maximizing share price benefits all stakeholders.

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Study Notes

State Preference Theory

  • This lecture covers state preference theory, a model for optimal portfolio decisions under uncertainty.
  • The objective of a greedy investor is to maximize the expected utility of end-of-period wealth.
  • This is calculated using the formula: E[U(W)] = ∑ piU(Wi), where:
    • S is the set of all uncertain alternatives
    • i represents a specific alternative
    • pi is the probability of alternative i
    • Wi is the end-of-period wealth associated with alternative i

Optimal Portfolio Decision

  • Investment decisions are made by choosing across probability distributions of end-of-period payoffs.
  • The choice is based on comparing expected utilities of different investments.
  • The optimal investment is the one with the highest expected utility.

Optimal Decision Making Under Uncertainty

  • Securities are crucial for investment decisions.
  • Firms sell securities to fund real asset investments (supply).
  • Individuals invest in securities to claim real assets of firms (demand).
  • Securities enable the exchange of consumption over time.

State Preference Model

  • The model assumes a perfect capital market, a one-period model, and uncertainty about the future state of nature.
  • There are different states of nature, each with a probability.
  • The probabilities sum up to one.
  • The model considers different market securities available for a given price today with their corresponding state-contingent payoff. The security payoffs are known in different states.

Complete Capital Market

  • The capital market is complete if the number of unique linearly independent securities is equal to the total number of possible future states of nature.
  • In a complete market, any security can be priced based on existing securities in the market.

Pure Securities

  • A pure security has an end-of-period payoff of 1 in a given state and 0 in all other states.
  • The price (ps) of a pure security with a payoff of 1 in state s and 0 in all other states depends on the state.
  • The number of pure securities (Qs) in state s can be used to replicate a market security.

Capital Market Equilibrium

  • A capital market is in equilibrium if the supply of each security equals its demand.
  • The single-price law of markets states that any two securities with identical state-contingent payoffs must have the same price.
  • No-arbitrage opportunities exist if short selling is allowed, where selling expensive securities and buying cheaper ones makes money without risk.

Determinants of Security Prices

  • The price of a market security depends on various factors, including time preferences, expectations about state probabilities and risk attitudes.
  • Factors like the time value of money and productivity of capital affect the price of riskless securities while individual risk aversion drives the prices of pure securities.

Optimal Investment Decision (Individuals)

  • Individuals aim to maximize their expected utility of end-of-period wealth, taking into account the weights of pure securities (Qs) and probabilities (πs) associated with each state.

Optimal Investment Decision (Firms)

  • Firms maximize the expected utility of their current shareholders.
  • Firms face a trade-off between investment and current share values.
  • Firms' investment decisions are also driven by the NPV (Net Present Value) of any project, with positive NPV corresponding to better investment.

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L7 State Preference Theory PDF

Description

This quiz explores essential concepts in capital market theory, focusing on the implications of pure securities, market equilibrium, and the no-arbitrage principle. Test your knowledge on derived prices, capital market completeness, and the single-price law within the context of financial markets.

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