Optimal Asset Allocation Between Two Risky Stocks With Perfect Correlation

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.

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_B

The 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_B

Since \rho = 1, the standard deviation becomes:

\sigma_p = w_A \sigma_A + w_B \sigma_B

Optimal 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.