13.1 Asset pricing and risk-return tradeoffs

Asset pricing and risk-return tradeoffs are crucial concepts in financial markets. They help us understand how investors make decisions and how assets are valued based on their risk levels. This knowledge is essential for making informed investment choices and managing portfolios effectively.

The relationship between risk and return is fundamental to financial decision-making. By exploring concepts like systematic risk, diversification, and the Capital Asset Pricing Model, we gain insights into how investors balance potential rewards with the uncertainty of financial markets.

Risk and Expected Return

Measuring and Characterizing Risk

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• Risk in financial markets measured by volatility or standard deviation of returns represents uncertainty of future outcomes
• Systematic risk (market risk) affects all securities and cannot be diversified away
  • Examples include economic recessions, interest rate changes, or geopolitical events
• Unsystematic risk (firm-specific risk) can be reduced through diversification
  • Examples include management changes, product recalls, or labor strikes
• Beta coefficient measures an asset's sensitivity to market movements
  • Quantifies systematic risk relative to overall market
  • Beta of 1 indicates asset moves in line with market
  • Beta greater than 1 indicates higher volatility than market (technology stocks)
  • Beta less than 1 indicates lower volatility than market (utility stocks)

Risk-Return Tradeoff and Investor Behavior

• Risk-return tradeoff principle states higher expected returns generally associated with higher levels of risk
  • Investors demand compensation for taking on additional risk
  • Low-risk assets (government bonds) typically offer lower returns
  • High-risk assets (small-cap stocks) typically offer higher potential returns
• Risk premium represents additional return investors demand for bearing risk
  • Calculated as difference between expected return on risky asset and risk-free rate
  • Example: If stock has expected return of 10% and risk-free rate is 2%, risk premium is 8%
• Risk aversion describes investors' preference for lower risk given same level of expected return
  • Influences asset pricing and market equilibrium
  • Explains why riskier assets must offer higher expected returns to attract investors
  • Degree of risk aversion varies among individuals and impacts portfolio allocation decisions

Graphical Representations of Risk-Return Relationships

• Security Market Line (SML) graphically represents relationship between systematic risk (beta) and expected return for individual securities
  • X-axis represents beta, Y-axis represents expected return
  • Upward sloping line indicates positive relationship between risk and return
  • Intercept of SML represents risk-free rate
  • Slope of SML represents market risk premium
• Capital Allocation Line (CAL) shows risk-return tradeoffs for portfolios combining risky assets with risk-free asset
  • Represents efficient combinations of risky portfolio and risk-free asset
  • Slope of CAL indicates Sharpe ratio of portfolio
• Efficient Frontier depicts set of optimal portfolios offering highest expected return for given level of risk
  • Curved line representing best possible risk-return combinations
  • Portfolios below frontier considered inefficient

CAPM for Required Return

CAPM Framework and Components

• Capital Asset Pricing Model (CAPM) describes relationship between systematic risk and expected return for assets
• CAPM formula: $E(R_i) = R_f + \beta_i(E(R_m) - R_f)$
  • E(Ri) represents expected return on asset i
  • Rf represents risk-free rate (typically short-term government securities)
  • βi represents beta of asset i
  • E(Rm) represents expected return of market
• Beta (β) in CAPM represents sensitivity of asset's returns to market movements
  • Calculated as covariance of asset returns with market returns divided by variance of market returns
  • $\beta_i = \frac{Cov(R_i, R_m)}{Var(R_m)}$
• Market risk premium, E(Rm) - Rf, represents additional return investors expect for bearing systematic risk of market portfolio
  • Historical average market risk premium typically ranges from 4% to 8%

CAPM Applications and Interpretations

• CAPM used to price individual securities
  • Determine if asset is overvalued or undervalued relative to its expected return
  • Example: If CAPM suggests required return of 12% but asset only offers 10%, it may be overvalued
• Evaluate investment opportunities
  • Compare expected returns of different assets or projects with their required returns based on risk
  • Useful for capital budgeting decisions in corporate finance
• Estimate cost of capital for firms
  • Determine required return on equity for company based on its beta
  • Essential for valuation and financial decision-making

CAPM Assumptions and Limitations

• CAPM assumes investors hold well-diversified portfolios, eliminating unsystematic risk
  • In reality, many investors may not hold perfectly diversified portfolios
• Assumes market portfolio is efficient
  • Difficult to define and measure true market portfolio in practice
• Model relies on simplifying assumptions
  • Perfect capital markets, no transaction costs, and homogeneous investor expectations
• Empirical challenges in testing CAPM validity in real-world markets
  • Some studies suggest factors beyond beta may explain asset returns (size effect, value effect)
• Single-factor model may not capture all relevant risks
  • Multi-factor models (Fama-French three-factor model) attempt to address this limitation

Diversification Impact on Portfolios

Principles of Diversification

• Diversification spreads investments across various assets to reduce overall portfolio risk without sacrificing expected returns
• Based on fact different assets often do not move in perfect correlation with each other
• Portfolio risk measured by weighted average of individual asset risks, adjusted for correlations between asset returns
• Diversification effect demonstrates risk of portfolio typically less than weighted average of risks of its individual components
  • Example: Portfolio with 50% in Stock A (20% volatility) and 50% in Stock B (15% volatility) may have overall volatility of 16% if stocks are not perfectly correlated

Implementing Diversification Strategies

• As number of uncorrelated or lowly correlated assets in portfolio increases, unsystematic risk decreases
  • Approaches zero in fully diversified portfolio
  • Diminishing marginal benefits of diversification as more assets added
• International diversification provides additional risk reduction benefits
  • Includes assets less correlated with domestic markets
  • Example: U.S. investor adding European or emerging market stocks to portfolio
• Asset allocation across different classes (stocks, bonds, real estate) enhances diversification
  • Each asset class responds differently to economic factors
• Sector diversification within asset classes further reduces risk
  • Investing in multiple industries (technology, healthcare, finance) rather than concentrating in one sector

Limits and Considerations of Diversification

• Limits of diversification reached when only systematic risk remains
  • Cannot be eliminated through diversification alone
  • Represents "market risk" affecting all securities
• Over-diversification can lead to diminishing returns
  • Transaction costs and complexity may outweigh marginal benefits of adding more assets
• Correlation between assets can change over time
  • May reduce diversification benefits during market stress (financial crises)
• Importance of regular portfolio rebalancing
  • Maintain desired risk-return profile as asset values fluctuate

Portfolio Efficiency and the Efficient Frontier

Concept and Construction of the Efficient Frontier

• Efficient Frontier represents set of optimal portfolios offering highest expected return for given level of risk or lowest risk for given level of expected return
• Derived from Modern Portfolio Theory (MPT) developed by Harry Markowitz
  • Quantifies benefits of diversification
• Shape of Efficient Frontier determined by risk-return characteristics and correlations of available assets in investment universe
• Portfolios lying on Efficient Frontier considered efficient
  • Those below it suboptimal, offering either lower returns for same risk or higher risk for same return
• Construction requires estimates of expected returns, volatilities, and correlations
  • Subject to estimation error and may change over time

Analyzing Portfolio Efficiency

• Tangency point between Capital Market Line (CML) and Efficient Frontier represents optimal risky portfolio when combined with risk-free asset
  • CML represents risk-return tradeoffs for portfolios combining market portfolio with risk-free asset
• Performance measures evaluate efficiency of portfolios relative to Efficient Frontier
  • Sharpe ratio: excess return per unit of total risk
    • $Sharpe Ratio = \frac{R_p - R_f}{\sigma_p}$
    • Rp represents portfolio return, Rf represents risk-free rate, σp represents portfolio standard deviation
  • Treynor ratio: excess return per unit of systematic risk
    • $Treynor Ratio = \frac{R_p - R_f}{\beta_p}$
    • βp represents portfolio beta

Practical Applications and Limitations

• Efficient Frontier used in portfolio construction and optimization
  • Helps investors identify portfolios that maximize return for given risk tolerance
• Mean-variance optimization techniques used to find optimal asset allocations
  • Example: Determining weights of stocks and bonds in portfolio to maximize Sharpe ratio
• Limitations in practice
  • Assumes normal distribution of returns, which may not hold for all assets
  • Sensitive to input parameters, small changes in estimates can lead to significant portfolio shifts
• Dynamic nature of financial markets
  • Efficient Frontier shifts over time as asset characteristics and correlations change
  • Requires periodic reassessment and portfolio rebalancing