💹Financial Mathematics Unit 10 – Asset Pricing Models

Key Concepts and Definitions

• Asset pricing involves determining the fair value or theoretical price of an asset based on its expected risk and return characteristics
• Assets include financial instruments such as stocks, bonds, derivatives, and real estate
• Risk refers to the uncertainty or potential for loss associated with an investment
  • Systematic risk affects the entire market and cannot be diversified away (market risk, interest rate risk)
  • Unsystematic risk is specific to individual assets and can be reduced through diversification (company-specific risk)
• Return represents the gain or loss on an investment over a specific period, typically expressed as a percentage
• Discount rate is the rate used to determine the present value of future cash flows, reflecting the time value of money and risk
• Arbitrage involves simultaneously buying and selling an asset in different markets to profit from price discrepancies
• Market efficiency suggests that asset prices fully reflect all available information, making it difficult to consistently outperform the market

Fundamental Principles of Asset Pricing

• The law of one price states that identical assets should have the same price in different markets, preventing arbitrage opportunities
• The principle of no arbitrage implies that investors cannot earn risk-free profits by exploiting price discrepancies
• Risk-return tradeoff suggests that higher expected returns are associated with higher levels of risk
• Time value of money recognizes that money available now is worth more than an equal amount in the future due to its earning potential
• Diversification helps reduce unsystematic risk by spreading investments across different assets or asset classes
• Market equilibrium occurs when supply and demand for an asset are balanced, resulting in a stable price
• Rational investors aim to maximize their expected utility by making decisions based on available information and risk preferences

Types of Asset Pricing Models

• Capital Asset Pricing Model (CAPM) describes the relationship between systematic risk and expected return for an asset
  • Assumes investors are risk-averse and hold diversified portfolios
  • Uses beta ($\beta$) to measure an asset's sensitivity to market movements
• Arbitrage Pricing Theory (APT) is a multi-factor model that explains asset returns based on various macroeconomic factors
  • Does not rely on the efficient market hypothesis or utility assumptions
  • Factors may include inflation, GDP growth, and changes in interest rates
• Fama-French Three-Factor Model expands CAPM by adding size and value factors to explain stock returns
  • Size factor captures the outperformance of small-cap stocks over large-cap stocks
  • Value factor represents the excess returns of value stocks (high book-to-market ratio) over growth stocks
• Black-Scholes Model is used for pricing options contracts
  • Considers the underlying asset price, strike price, time to expiration, risk-free rate, and volatility
• Consumption-based models link asset prices to investors' consumption and marginal utility across different states of the economy

Risk and Return Relationship

• Expected return is the anticipated return on an investment based on its probability distribution of possible outcomes
• Standard deviation measures the dispersion of returns around the expected return, indicating the level of risk
• Beta ($\beta$) coefficient represents the sensitivity of an asset's returns to market movements
  • Assets with $\beta > 1$ are more volatile than the market
  • Assets with $0 < \beta < 1$ are less volatile than the market
  • Assets with $\beta < 0$ have returns that move in the opposite direction of the market
• Sharpe ratio measures the risk-adjusted return of an investment by comparing its excess return to its standard deviation
• Treynor ratio evaluates the performance of a portfolio by comparing its excess return to its systematic risk (beta)
• Jensen's alpha measures the excess return of a portfolio above what is predicted by the CAPM, indicating a portfolio manager's skill

Valuation Techniques

• Discounted Cash Flow (DCF) analysis estimates the intrinsic value of an asset by discounting its expected future cash flows to the present
  • Requires forecasting future cash flows and determining an appropriate discount rate
  • Commonly used for valuing stocks and investment projects
• Relative valuation compares an asset's value to similar assets or benchmarks in the market
  • Price-to-Earnings (P/E) ratio compares a stock's price to its earnings per share
  • Price-to-Book (P/B) ratio compares a company's market value to its book value
  • Enterprise Value-to-EBITDA (EV/EBITDA) ratio considers a company's total value relative to its earnings before interest, taxes, depreciation, and amortization
• Option pricing models, such as the Black-Scholes model, determine the theoretical price of an option based on various inputs
• Risk-neutral valuation assumes that investors are indifferent to risk and value assets based on their expected payoffs discounted at the risk-free rate

Market Efficiency and Asset Pricing

• Efficient Market Hypothesis (EMH) suggests that asset prices fully reflect all available information, making it difficult to consistently outperform the market
  • Weak-form efficiency implies that prices reflect all historical information
  • Semi-strong form efficiency suggests that prices quickly adjust to new public information
  • Strong-form efficiency asserts that prices reflect all public and private information
• Anomalies are patterns or events that contradict the EMH, such as calendar effects or value premium
• Behavioral finance examines the psychological factors that influence investor behavior and decision-making
  • Cognitive biases, such as overconfidence and loss aversion, can lead to irrational investment decisions
  • Herd behavior occurs when investors follow the actions of others rather than relying on their own analysis
• Adaptive markets hypothesis combines principles of EMH and behavioral finance, suggesting that market efficiency varies over time and across markets

Applications in Financial Markets

• Portfolio management involves constructing and managing a collection of investments to meet specific goals and risk tolerance
  • Asset allocation determines the proportion of a portfolio invested in different asset classes (stocks, bonds, cash)
  • Security selection involves choosing specific investments within each asset class
• Risk management focuses on identifying, assessing, and mitigating potential losses
  • Value at Risk (VaR) estimates the maximum potential loss over a given time horizon and confidence level
  • Hedging involves taking an offsetting position to reduce the risk of adverse price movements
• Performance evaluation assesses the returns and risk of a portfolio or investment strategy relative to a benchmark
  • Sharpe ratio and Treynor ratio are commonly used risk-adjusted performance measures
• Algorithmic trading uses computer programs to automate trading decisions based on predefined rules and market conditions
• Robo-advisors provide automated, algorithm-driven financial planning and investment management services

Challenges and Limitations

• Model assumptions may not always hold in real-world markets, leading to inaccuracies in asset pricing
  • CAPM assumes that investors have homogeneous expectations and can borrow and lend at the risk-free rate
  • APT relies on the correct identification of relevant macroeconomic factors
• Parameter estimation errors can arise when estimating inputs for asset pricing models, such as expected returns and risk measures
• Market anomalies and inefficiencies can persist, challenging the assumptions of rational investors and perfect information
• Behavioral biases and irrational investor behavior can lead to mispricing and market inefficiencies
• Liquidity risk arises when an asset cannot be easily bought or sold without significantly affecting its price
• Changing market conditions and economic environments can impact the validity and reliability of asset pricing models over time
• Regulatory changes and market disruptions, such as financial crises or technological advancements, can alter the dynamics of asset pricing