Modern Portfolio Theory and Mean-Variance Optimization › The Two-Asset Case

The two-asset case provides intuition before we tackle the general problem. By working through this simplified scenario, we can develop a geometric understanding of how risk and return interact, seeing clearly how correlation shapes the set of achievable portfolios. Consider a portfolio with weights $w$ in asset 1 and $(1-w)$ in asset 2.

The expected return is:

$$\mu_p = w\mu_1 + (1-w)\mu_2$$

where:

• $\mu_p$: expected return of the portfolio
• $w$: weight invested in asset 1
• $\mu_1, \mu_2$: expected returns of assets 1 and 2

This formula shows that expected return varies linearly with the weight $w$. As we shift allocation from asset 2 toward asset 1, expected return moves along a straight line between $\mu_2$ (when $w=0$) and $\mu_1$ (when $w=1$). There are no surprises here: return blending is straightforward.

The variance, however, tells a richer story:

$$\sigma_p^2 = w^2\sigma_1^2 + (1-w)^2\sigma_2^2 + 2w(1-w)\rho_{12}\sigma_1\sigma_2$$

where:

• $\sigma_p^2$: variance of the portfolio
• $w$: weight invested in asset 1
• $\sigma_1, \sigma_2$: standard deviations (volatility) of assets 1 and 2
• $\rho_{12}$: correlation coefficient between returns of assets 1 and 2

The first two terms are the weighted individual-variance contributions. Relative to the same weights at zero correlation, the cross-term raises or lowers portfolio variance according to its sign. Whether a combination has less risk than either asset alone depends on all three terms, the weights, and the covariance.

For fixed long-only weights $0\le w\le1$, lowering correlation cannot increase portfolio variance; it decreases when both weights and volatilities are positive: $\partial\sigma_p^2/\partial\rho_{12}=2w(1-w)\sigma_1\sigma_2\ge0$. When correlation is negative, the cross-term is nonpositive. If one weight is negative, the derivative instead becomes negative, so the long-only correlation intuition reverses.

To understand this intuitively, imagine two assets that tend to move in opposite directions. When one zigs, the other zags. With positive holdings in both, these opposite movements partially cancel each other, smoothing the overall portfolio return and reducing its volatility. The lower the correlation, the more pronounced this cancellation effect.

Let's see this numerically:

The figure reveals several key insights about how correlation shapes investment opportunities:

• Perfect positive correlation ($\rho = 1$): For long-only weights, portfolio standard deviation is the weighted average of individual volatilities and cannot fall below the less volatile asset. With unrestricted weights, it is $|w\sigma_1+(1-w)\sigma_2|$; unequal volatilities permit a zero-variance long-short hedge.
• Imperfect correlation ($\rho < 1$): The displayed frontier can curve leftward. Whether a long-only portfolio has lower risk than either asset alone depends on the weights and covariance, not on $\rho<1$ alone. At fixed long-only weights, lowering correlation cannot increase variance.
• Perfect negative correlation ($\rho = -1$): A risk-free portfolio exists. When $w = \sigma_2 / (\sigma_1 + \sigma_2)$, the portfolio variance is exactly zero. This extreme case illustrates the theoretical maximum of diversification: if two assets move in perfectly opposite directions, we can combine them to eliminate all uncertainty.

For the two-asset case, we can find the minimum variance portfolio analytically. This portfolio represents the allocation that achieves the lowest possible risk, regardless of the expected return it delivers. To find the minimum variance portfolio, we differentiate the variance $\sigma_p^2$ with respect to $w$ and set the result to zero:

$$\begin{aligned} \frac{\partial \sigma_p^2}{\partial w} &= \frac{\partial}{\partial w} \left( w^2\sigma_1^2 + (1-w)^2\sigma_2^2 + 2w(1-w)\rho_{12}\sigma_1\sigma_2 \right) \\ &= 2w\sigma_1^2 + 2(1-w)(-1)\sigma_2^2 + 2(1-2w)\rho_{12}\sigma_1\sigma_2 && \text{(differentiate)} \\ &= 2w\sigma_1^2 - 2(1-w)\sigma_2^2 + 2(1-2w)\rho_{12}\sigma_1\sigma_2 && \text{(simplify)} \\ &= 0 && \text{(set to zero)} \end{aligned}$$

The derivative includes three components corresponding to the three terms in the variance formula. Setting this derivative to zero identifies a stationary point with no first-order change in variance. For $D=\sigma_1^2+\sigma_2^2-2\rho_{12}\sigma_1\sigma_2>0$, the variance has positive second derivative $2D$, so the point is its unique unrestricted minimum; any nonzero finite change $\delta$ in $w$ raises variance by $D\delta^2$. Rearranging the first-order condition to isolate $w$:

$$\begin{aligned} 2w\sigma_1^2 + 2w\sigma_2^2 - 4w\rho_{12}\sigma_1\sigma_2 &= 2\sigma_2^2 - 2\rho_{12}\sigma_1\sigma_2 && \text{(group terms with } w \text{)} \\ w(2\sigma_1^2 + 2\sigma_2^2 - 4\rho_{12}\sigma_1\sigma_2) &= 2(\sigma_2^2 - \rho_{12}\sigma_1\sigma_2) && \text{(factor out } w \text{)} \\ w(\sigma_1^2 + \sigma_2^2 - 2\rho_{12}\sigma_1\sigma_2) &= \sigma_2^2 - \rho_{12}\sigma_1\sigma_2 && \text{(divide by 2)} \end{aligned}$$

For $D=\sigma_1^2+\sigma_2^2-2\rho_{12}\sigma_1\sigma_2>0$, the unrestricted minimizer is:

$$w^* = \frac{\sigma_2^2 - \rho_{12}\sigma_1\sigma_2}{\sigma_1^2 + \sigma_2^2 - 2\rho_{12}\sigma_1\sigma_2}$$

where:

• $w^*$: optimal weight for asset 1 to minimize portfolio variance
• $\sigma_1, \sigma_2$: volatilities of assets 1 and 2
• $\rho_{12}$: correlation between assets 1 and 2

This unrestricted-weight formula requires $D=\sigma_1^2+\sigma_2^2-2\rho_{12}\sigma_1\sigma_2>0$. If $D=0$, the centered asset returns coincide and every budget allocation has the same variance, so there is no unique variance-minimizing weight. For long-only weights and $D>0$, clip the unrestricted minimizer to $[0,1]$. When correlation is zero, the formula allocates more to the lower-volatility asset.