Here is a portoflio optimalization.
Two sub-projects:
- Markowitz theory
- Shapre theory
Investment portfolio management is based on two key theories: the Markowitz theory and the Sharpe theory. Although the two concepts are related, they differ in their assumptions, objectives and methodology.
Markowitz theory, developed by Harry Markowitz in 1952, focuses on optimising a portfolio by balancing risk and return. Its key elements are:
Objective: Minimising risk for a given return or maximising return for a given level of risk. Risk: Defined as the standard deviation of portfolio returns. Diversification: The use of correlations between assets to reduce risk. Effective frontier: The set of portfolios with the best risk-return ratio. Markowitz theory assumes that investors are rational and risk averse.
- the expected rate of return of a portfolio
Where: E(Rp): The expected rate of return of the portfolio, wi: Weight of asset i in the portfolio, E(Ri): Expected rate of return of asset i, n: The number of assets in the portfolio.
- the risk (variance) of the portfolio σp² = ΣΣ(wi * wj * σij)
Where: σp²: The variance of the portfolio, wi, wj: Weights of assets i and j, σij: Covariance between assets i and j.
- Effective frontier The efficient frontier is the set of portfolios that maximise return at a given level of risk or minimise risk at a given level of return. The formula for the optimal asset weights for a portfolio:
W = (Σ-¹ * (R - Rf * 1)) / (1' * Σ-¹ * (R - Rf * 1))
Where: W: Vector of optimal asset weights, Σ-¹: Inverse covariance matrix of assets, R: Vector of expected returns, Rf: Rate of return of the risk-free asset, 1: Unit vector, 1': Transpose of unit vector.
- the standard deviation of the portfolio σp = √(σp²)
Where: σp: Standard deviation of the portfolio, σp²: The variance of the portfolio.
Sharpe theory, developed by William Sharpe in 1964, is based on the Capital Asset Pricing Model (CAPM). The main assumptions are:
Objective: Maximise portfolio efficiency, i.e. risk premium relative to volatility. Systematic risk: Focus on market risk that cannot be eliminated by diversification. Risk-free assets: The introduction of a market portfolio and risk-free investments as a benchmark. Sharpe ratio: A measure of portfolio efficiency:
Description of formulas in Sharpe theory. The Sharpe ratio: Measure of portfolio efficiency:
S = (Rp - Rf) / σp
Where:
- S - Sharpe ratio,
- Rp - rate of return of the portfolio,
- Rf - rate of return of the risk-free asset,
- σp - standard deviation of portfolio return (total risk).
** CAPM (Capital Asset Pricing Model)**: Determines the expected rate of return of an asset based on systematic risk:
E(Ri) = Rf + βi * (E(Rm) - Rf)
Where:
- E(Ri) - expected rate of return of the asset,
- Rf - the risk-free rate of return of the asset,
- βi - beta coefficient of the asset (a measure of systematic risk),
- E(Rm) - expected rate of return of the market,
- (E(Rm) - Rf) - market risk premium.
Objective: Markowitz: Minimisation of risk or maximisation of return. Sharpe: Maximising the ratio of risk premium to volatility (Sharpe ratio).
Risk: Markowitz: Captures the total risk of the portfolio, measured by standard deviation. Sharpe: Focuses on systematic risk (risk associated with the market as a whole).
Diversification: Markowitz: Key element of risk reduction through correlation analysis between assets. Sharpe: Assumes that unsystematic risk has already been eliminated through diversification.
Reference point: Markowitz: Effective portfolio frontier, i.e. the set of portfolios with the best risk-return ratio. Sharpe: Market portfolio (optimal portfolio) and risk-free asset.
Tools: Markowitz: Analysis of the covariance matrix of asset returns. Sharpe: Capital Asset Pricing Model (CAPM) and Sharpe ratio.
Narzędzia: Markowitz: Analiza macierzy kowariancji stóp zwrotu aktywów. Sharpe: Model CAPM (Capital Asset Pricing Model) i wskaźnik Sharpe’a.