Equilibrium in Markets for Securities

Microeconomics for Finance

Juan F. Imbet

Master in Finance, 1st Year - Paris Dauphine University - PSL

Introduction to Securities Markets

  • Securities markets involve trading of financial instruments such as stocks, bonds, and derivatives.

  • In equilibrium, supply equals demand for each security.

  • Prices adjust to clear markets.

  • Key concepts: portfolio choice, market completeness, and the role of information.

Portfolio Theory

  • Investors choose portfolios to maximize expected utility given their risk preferences.

  • Mean-variance analysis: portfolios characterized by expected return and variance.

  • Efficient frontier: set of portfolios with maximum expected return for given risk level.

  • Capital Asset Pricing Model (CAPM): relationship between expected return and systematic risk.

The Capital Asset Pricing Model (CAPM)

  • Assumptions:

    • Investors are risk-averse and maximize expected utility
    • Single-period investment horizon
    • Frictionless markets (no transaction costs, taxes)
    • Investors can borrow/lend at risk-free rate
    • Homogeneous expectations

  • Key Result: Expected return on asset i:

  • Key Result: Expected return on asset \(i\): \[\mathbb{E}[R_i] = R_f + \beta_i \left(\mathbb{E}[R_m] - R_f\right)\]

    where \(\beta_i = \dfrac{\mathrm{Cov}(R_i, R_m)}{\mathrm{Var}(R_m)}\)

Market Equilibrium

  • In equilibrium, all investors hold the market portfolio (combination of all risky assets).

  • The market portfolio is mean-variance efficient.

  • Risk-free asset is tangent to the efficient frontier.

  • Separation theorem: portfolio choice separates into two decisions:

    1. Which efficient portfolio to hold
    2. How much to invest in risky vs. risk-free assets

Complete Markets

  • A market is complete if any contingent claim can be replicated by a portfolio of existing securities.

  • In complete markets, all risk can be hedged.

  • Arrow-Debreu securities: pay 1 in one state, 0 otherwise.

  • State prices: prices of Arrow-Debreu securities.

  • Any security’s price is the expected value of its payoffs discounted by state prices.

Incomplete Markets

  • Most real-world markets are incomplete.

  • Cannot hedge all risks.

  • Equilibrium concepts: Radner equilibrium, constrained efficiency.

  • Role of derivatives in completing markets.

Information and Asset Pricing

  • Asymmetric information: some investors have private information.

  • Efficient markets hypothesis: prices reflect all available information.

  • Forms of market efficiency:

    • Weak: prices reflect past price information
    • Semi-strong: prices reflect all public information
    • Strong: prices reflect all information (public and private)

Behavioral Finance

  • Traditional finance assumes rational investors.

  • Behavioral finance incorporates psychological biases:

    • Overconfidence
    • Loss aversion
    • Mental accounting
    • Herding

  • Implications for asset pricing and market anomalies.

Conclusion

  • Securities markets play a crucial role in allocating resources and managing risk.

  • Equilibrium concepts help understand price formation.

  • Extensions to incomplete markets and behavioral considerations provide more realistic models.

  • Applications in portfolio management, risk management, and financial regulation.