PROBABILITY MODELS FOR PORTFOLIO RETURN AND RISK :
- Expected Return of Portfolio: The expected return of a portfolio with n assets is the weighted average of the individual asset returns, where weights are determined by the market value of each asset.
- Covariance: Measures how two assets move together. It can range from negative to positive values. A positive covariance indicates assets tend to move in the same direction, while negative covariance means they move in opposite directions.
- Covariance Matrix: A covariance matrix shows the covariances between asset returns, where diagonal terms represent variances. For n assets, there are n variances and n(n-1)/2 unique covariance terms.
- Portfolio Variance: Portfolio variance is calculated using asset weights, variances, and covariances. Lower covariance reduces portfolio variance and standard deviation.
- Correlation: The correlation coefficient, derived from covariance, can be used in place of covariances to calculate portfolio variance using a correlation matrix.
- Joint Probability Function: A joint probability function can calculate covariance between asset returns using given probabilities for various economic states.
- Shortfall Risk: Shortfall risk refers to the probability that a portfolio return will fall below a target value. It is crucial in understanding the risk of portfolio underperformance.
- Roy’s Safety-First Criterion: This criterion aims to minimize the probability that portfolio returns will fall below a minimum acceptable level. It is based on calculating the safety-first ratio, which helps identify the optimal portfolio.
- Safety-First Ratio: This ratio determines the number of standard deviations a portfolio’s return is below the expected return. The portfolio with the higher safety-first ratio is preferred.
- Portfolio Selection with Roy’s Criterion: For portfolios with normally distributed returns, the optimal choice is the one with the highest safety-first ratio, minimizing the probability of returns falling below the threshold level.
KEY CONCEPTS :
- Expected Portfolio Return: Calculated as the weighted average of the individual asset returns.
- Portfolio Variance (Two Assets):
- Expressed using covariance: Var(P)=wA2σA2+wB2σB2+2wAwBCovAB\text{Var}(P) = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_Aw_B \text{Cov}_{AB}Var(P)=wA2 σA2 +wB2 σB2 +2wA wB CovAB
- Or using correlation: Var(P)=wA2σA2+wB2σB2+2wAwBρABσAσB\text{Var}(P) = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_Aw_B \rho_{AB} \sigma_A \sigma_BVar(P)=wA2 σA2 +wB2 σB2 +2wA wB ρAB σA σB
- Covariance Calculation: Derived from joint probabilities of asset returns.
- Shortfall Risk: Probability portfolio return falls below a specified threshold.
- Safety-First Ratio: E(RP)−RTσP\frac{E(R_P) – R_T}{\sigma_P}σP E(RP )−RT ; higher values mean lower shortfall risk.
- Roy’s Safety-First Criterion: Selects portfolio with minimal shortfall risk.