1

Portfolio Expected Return

The expected return of a portfolio is the weighted average of the expected returns of the individual assets. Each asset's weight is determined by its market value relative to the total portfolio value.

\( E(R_p) = w_1 E(R_1) + w_2 E(R_2) + \cdots + w_n E(R_n) \)

Where:

  • \( E(R_p) \) is the expected return of the portfolio.
  • \( w_i \) is the weight (proportion) of asset \( i \) in the portfolio.
  • \( E(R_i) \) is the expected return of asset \( i \).

2-Asset Portfolio Return Example

You invest 60% in Stock A (expected return: 10%) and 40% in Bond B (expected return: 5%).

Calculation:
\( E(R_p) = (0.60 \times 10\%) + (0.40 \times 5\%) \)
\( = 6\% + 2\% = \textbf{8\%} \)

The portfolio's expected return is 8%.

2

Covariance and Correlation: How Assets Move Together

Understanding how asset returns relate to one another is essential for managing risk. Covariance and correlation quantify the degree to which two assets move in tandem.

Covariance

Covariance measures the directional relationship between the returns of two assets:

  • Positive: Returns tend to move together.
  • Negative: Returns tend to move in opposite directions.
  • Zero: No consistent directional relationship.

Key Properties:

  • The covariance of a return with itself equals its variance: \( \mathrm{Cov}(R_i, R_i) = \sigma^2(R_i) \).
  • Symmetry: \( \mathrm{Cov}(R_i, R_j) = \mathrm{Cov}(R_j, R_i) \).
  • Covariance with a constant is zero.

Correlation

Correlation standardizes covariance to a range between \(-1\) and \(+1\), making it easier to interpret.

\( \rho(R_i, R_j) = \frac{\mathrm{Cov}(R_i, R_j)}{\sigma(R_i) \sigma(R_j)} \)
  • +1: Perfect positive linear relationship.
  • -1: Perfect negative linear relationship.
  • 0: No linear relationship.
  • Note: Only captures linear dependence; nonlinear patterns may exist.
3

Portfolio Variance and Diversification

Portfolio variance measures total risk and depends not only on individual asset risks but also on how they co-move.

\( \sigma^2(R_p) = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \mathrm{Cov}(R_i, R_j) \)

For a two-asset portfolio:

\( \sigma^2(R_p) = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \mathrm{Cov}(R_1, R_2) \)

Diversification Benefits

When assets are not perfectly correlated, combining them reduces overall portfolio risk. This is the power of diversification.

  • If returns are independent, covariance is zero — reducing portfolio variance.
  • For \( n \) assets, you need \( n \) variances and \( \frac{n(n-1)}{2} \) unique covariances.

Properties of Variance

  • Variance of a constant is zero.
  • Adding a constant to a return does not change its variance.
  • Scaling a return by weight \( w \): \( \sigma^2(wR) = w^2 \sigma^2(R) \).
4

Forecasting Correlation and Key Rules

Accurate forecasts of asset relationships are critical for effective portfolio construction.

  • Covariance from Joint Probabilities: Compute as the weighted average of cross-products of deviations from expected returns.
  • Independent Variables: Two returns are independent if \( P(R_i, R_j) = P(R_i) P(R_j) \).
  • Multiplication Rule: If uncorrelated, \( E(R_i R_j) = E(R_i) E(R_j) \).
5

Applications in Portfolio Management

Portfolio mathematics underpins modern investment decision-making, often assuming normally distributed returns.

Optimal Portfolio Selection

  • Mean-Variance Analysis: Constructs portfolios that maximize return for a given level of risk.
  • Roy's Safety-First Rule: Minimizes the chance of falling below a minimum acceptable return \( R_L \). The best portfolio maximizes the Safety-First Ratio:
    \( \text{SFRatio} = \frac{E(R_P) - R_L}{\sigma_P} \)
  • Sharpe Ratio: Measures excess return per unit of total risk (standard deviation). Higher values indicate better risk-adjusted performance:
    \( \text{Sharpe Ratio} = \frac{E(R_P) - R_f}{\sigma_P} \)

Managing Financial Risk

  • Value at Risk (VaR): Estimates the worst expected loss over a time horizon at a given confidence level (e.g., "5% chance of losing $1M or more tomorrow").
  • Stress Testing & Scenario Analysis: Evaluates portfolio resilience under extreme but plausible market conditions, complementing VaR.
Progress:
Chapter 5 of 11