Introduction
As a finance expert, I often analyze how investors should allocate capital between risky assets. One scenario that demands attention is asset allocation between two stocks with perfect correlation. While diversification typically reduces risk, perfect correlation changes the game. Here, I break down the math, logic, and practical implications of this problem.
Table of Contents
Understanding Perfect Correlation
Correlation measures how two assets move relative to each other. A correlation of \rho = +1 means they move in lockstep. If Stock A rises 5%, Stock B also rises 5%. This eliminates diversification benefits, forcing investors to rely solely on expected returns and volatility.
Mathematical Representation
The correlation coefficient \rho_{AB} between two stocks A and B is:
\rho_{AB} = \frac{\text{Cov}(R_A, R_B)}{\sigma_A \sigma_B}For perfect correlation, \rho_{AB} = 1. The portfolio return R_p with weights w_A and w_B (where w_B = 1 - w_A) is:
R_p = w_A R_A + w_B R_BThe portfolio variance \sigma_p^2 simplifies to:
\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_BSince \rho = 1, the standard deviation becomes:
\sigma_p = w_A \sigma_A + w_B \sigma_BOptimal Allocation Strategy
With perfect correlation, the optimal allocation depends on risk and return trade-offs.
Case 1: Equal Expected Returns
If both stocks have the same expected return (\mu_A = \mu_B), the portfolio return is the same regardless of allocation. However, risk varies.
Example:
- Stock A: \mu_A = 10\%, \sigma_A = 20\%
- Stock B: \mu_B = 10\%, \sigma_B = 30\%
The portfolio risk is:
\sigma_p = w_A (20\%) + (1 - w_A)(30\%)To minimize risk, allocate 100% to Stock A since it has lower volatility.
Case 2: Different Expected Returns
If \mu_A \neq \mu_B, investors face a trade-off. Higher returns come with higher risk.
Example:
- Stock A: \mu_A = 8\%, \sigma_A = 15\%
- Stock B: \mu_B = 12\%, \sigma_B = 25\%
The efficient frontier is a straight line connecting the two stocks. The optimal allocation depends on the investor’s risk tolerance.
Sharpe Ratio Analysis
The Sharpe Ratio (S_p) helps assess risk-adjusted returns:
S_p = \frac{\mu_p - r_f}{\sigma_p}Where r_f is the risk-free rate.
Example:
Assume r_f = 2\%.
| Allocation (A/B) | Portfolio Return (\mu_p) | Portfolio Risk (\sigma_p) | Sharpe Ratio |
|---|---|---|---|
| 100% / 0% | 8% | 15% | 0.40 |
| 50% / 50% | 10% | 20% | 0.40 |
| 0% / 100% | 12% | 25% | 0.40 |
Here, the Sharpe Ratio remains constant. Thus, the optimal choice depends purely on risk appetite.
Leverage and Short-Selling
If short-selling is allowed, investors can create leveraged portfolios.
Example:
- Allocate 150% to Stock A, -50% to Stock B.
- Expected return: 1.5(8\%) + (-0.5)(12\%) = 6\%
- Risk: 1.5(15\%) + (-0.5)(25\%) = 10\%
This could be useful for hedging or speculative strategies.
Practical Implications
1. No Diversification Benefit
Since the stocks move together, diversification fails. Investors must rely on individual stock performance.
2. Sector-Specific Risks
Perfect correlation often occurs within the same sector (e.g., two tech stocks). A downturn affects both equally.
3. Portfolio Rebalancing
With perfect correlation, rebalancing doesn’t reduce risk. Instead, tactical shifts based on fundamentals matter more.
Behavioral Considerations
Investors often assume diversification works automatically. But with perfect correlation, this is a fallacy. Overconfidence in correlated assets can lead to concentrated risks.
Final Thoughts
Asset allocation between two perfectly correlated stocks is straightforward mathematically but requires careful judgment.
