Portfolio Theory: Covariance, Diversification, and the Efficient Frontier
1. What is Covariance?
Covariance measures how two assets’ returns move together. Positive covariance indicates that assets move in the same direction; negative covariance indicates opposite movements.
Formula:
Cov(Ri,Rj) = Σ((Ri - E[Ri]) * (Rj - E[Rj])) / (n-1)
Importance: In portfolio theory, covariance determines diversification benefits. Assets with low or negative covariance reduce overall portfolio risk.
2. Covariance Between Similar and Different Asset Types
Assets of the same type, e.g., stocks, often have positive covariance due to exposure to similar macroeconomic factors. Different asset types:
- Treasury Bills vs. Stocks: Low or slightly negative covariance
- Stocks vs. Real Estate: Moderate positive covariance
- Treasury Bills vs. Real Estate: Near zero covariance
3. Correlation vs. Covariance
The correlation coefficient standardizes covariance and ranges from -1 to 1:
ρij = Cov(Ri,Rj) / (σi * σj)
This makes it easier to compare co-movement between assets.
4. Why the Efficient Frontier is Curved
Portfolio variance is non-linear because:
σp^2 = wA^2σA^2 + wB^2σB^2 + 2wAwB Cov(A,B)
Lower correlation between assets curves the frontier upward, reflecting diminishing returns for increasing risk.
5. Numerical Example: Two Stocks Portfolio
| Asset | Expected Return | Std. Deviation |
|---|---|---|
| Stock A | 10% | 20% |
| Stock B | 15% | 25% |
| Correlation | 0.2 | |
Portfolio Return (50% A, 50% B): 12.5%
Portfolio Std. Dev: 18.88%
6. Efficient Frontier Graph
Description: The curve shows optimal portfolios (upper curve). Points below are inefficient. Dominant portfolios offer the highest return for a given risk.
7. Including Risk-Free Asset: Capital Allocation Line (CAL)
With a risk-free rate (RFR = 5%), the CAL shows combinations of the risk-free asset and the tangency portfolio (M).
Equation:
EC[R_C] = Rf + (Sharpe Ratio) * σC
8. Diversification Example: Three Stocks
| Pair | Correlation | Portfolio σ |
|---|---|---|
| K & L | 0.8 | 18.97% |
| K & M | 0.2 | 15.49% |
| L & M | 0.4 | 16.73% |
The lowest portfolio risk is achieved with K & M, due to lowest correlation.
9. Key Takeaways
- Diversification reduces unsystematic risk.
- Efficient frontier shows optimal risk-return trade-offs.
- Risk-free assets allow better risk-adjusted returns (CAL/CML).
- Investor preferences (utility) determine the optimal portfolio.
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