14.1 Portfolio optimization

Portfolio optimization is a crucial aspect of modern finance, blending mathematics and investment theory. It focuses on maximizing returns while minimizing risk, using tools like the Markowitz model and efficient frontier to guide investment decisions.

This topic explores key concepts like diversification, asset allocation, and risk-return trade-offs. It also delves into mathematical techniques such as mean-variance optimization and quadratic programming, essential for constructing optimal portfolios in real-world scenarios.

Portfolio Optimization Fundamentals

Markowitz Model and Efficient Frontier

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• Markowitz model forms foundation of modern portfolio theory
• Developed by Harry Markowitz in 1952 revolutionized investment management
• Assumes investors are risk-averse and seek to maximize returns for a given level of risk
• Efficient frontier represents set of optimal portfolios offering highest expected return for a given level of risk
• Plotted on a graph with expected return on y-axis and risk (standard deviation) on x-axis
• Portfolios on efficient frontier offer best risk-return trade-off
• Investors choose portfolio along efficient frontier based on individual risk tolerance

Risk-Return Trade-off and Diversification

• Risk-return trade-off fundamental concept in finance describes relationship between risk and potential reward
• Higher risk generally associated with higher potential returns
• Lower risk typically yields lower potential returns
• Investors must balance desire for returns with their risk tolerance
• Diversification key strategy to manage risk in portfolio
• Involves spreading investments across various asset classes (stocks, bonds, real estate)
• Reduces impact of poor performance in single investment or sector
• Aims to achieve smoother overall portfolio performance over time
• Correlation between assets crucial factor in effective diversification
• Negatively correlated assets tend to move in opposite directions, enhancing diversification benefits

Asset Allocation Strategies

• Asset allocation process of dividing investments among different asset classes
• Determines portfolio's overall risk and return characteristics
• Strategic asset allocation involves setting long-term target allocations for each asset class
• Tactical asset allocation allows short-term deviations from strategic allocation to capitalize on market opportunities
• Factors influencing asset allocation include:
  • Investor's risk tolerance
  • Time horizon
  • Financial goals
  • Market conditions
• Common asset classes in allocation:
  • Equities (stocks)
  • Fixed income (bonds)
  • Cash and cash equivalents
  • Real estate
  • Commodities
• Rebalancing periodic adjustment of portfolio to maintain desired asset allocation

Mathematical Techniques

Mean-Variance Optimization

• Mean-variance optimization mathematical approach to construct optimal portfolios
• Balances expected returns (mean) against risk (variance)
• Inputs required for mean-variance optimization:
  • Expected returns for each asset
  • Standard deviation (risk) for each asset
  • Correlation coefficients between assets
• Process aims to maximize portfolio expected return for a given level of risk
• Alternatively, can minimize risk for a given level of expected return
• Utilizes covariance matrix to capture relationships between asset returns
• Results in optimal portfolio weights for each asset

Quadratic Programming in Portfolio Optimization

• Quadratic programming mathematical optimization technique used in portfolio construction
• Solves optimization problems with quadratic objective function and linear constraints
• In portfolio context, objective function typically maximizes expected return or minimizes risk
• Constraints may include:
  • Budget constraint (sum of weights equals 1)
  • Long-only constraint (no short selling)
  • Sector or asset class limits
• Quadratic nature arises from variance term in objective function
• Efficient algorithms available to solve quadratic programming problems (interior point methods)
• Can incorporate additional constraints or objectives (transaction costs, taxes)
• Allows for more complex portfolio optimization scenarios than simple mean-variance approach

Performance Measurement

Sharpe Ratio and Risk-Adjusted Returns

• Sharpe ratio measures risk-adjusted performance of an investment or portfolio
• Developed by William Sharpe in 1966 widely used in finance industry
• Calculated as: $Sharpe Ratio = \frac{R_p - R_f}{\sigma_p}$
  • Where $R_p$ = portfolio return, $R_f$ = risk-free rate, $\sigma_p$ = portfolio standard deviation
• Higher Sharpe ratio indicates better risk-adjusted performance
• Allows comparison of investments with different risk levels
• Limitations of Sharpe ratio:
  • Assumes returns are normally distributed
  • May not capture tail risks effectively
  • Uses standard deviation as risk measure, which penalizes upside volatility
• Other risk-adjusted performance measures:
  • Sortino ratio (focuses on downside risk)
  • Treynor ratio (uses beta instead of standard deviation)

Capital Asset Pricing Model (CAPM) and Beta

• CAPM fundamental model in finance describes relationship between systematic risk and expected return
• Developed by William Sharpe, John Lintner, and Jan Mossin in the 1960s
• Key components of CAPM:
  • Risk-free rate ($R_f$)
  • Market risk premium ($R_m - R_f$)
  • Beta ($\beta$)
• CAPM equation: $E(R_i) = R_f + \beta_i(E(R_m) - R_f)$
  • Where $E(R_i)$ = expected return on asset i, $E(R_m)$ = expected market return
• Beta measures asset's sensitivity to market movements
  • $\beta > 1$ indicates higher volatility than market
  • $\beta < 1$ indicates lower volatility than market
  • $\beta = 1$ indicates same volatility as market
• Uses of CAPM in portfolio management:
  • Estimating required return for individual stocks
  • Evaluating portfolio performance
  • Pricing of risky securities
• Limitations of CAPM:
  • Assumes perfect markets and rational investors
  • Relies on market portfolio, which is theoretical and unobservable
  • Empirical evidence suggests additional factors may explain returns (leading to multi-factor models)