The Calmest Blend of a 20% and a 30% Stock Swings 19.64%

Six sevenths in the quieter share beats holding it alone, because what the jumpy share adds at the margin is its correlation times its own swing, fifteen against twenty. The derivative of the variance at a full allocation is exactly +1/50, so the informal test and the first-order condition are one statement. The article carries the exact optimum sqrt(27/700), the convexity making it a minimum, and the correlation threshold of two thirds above which the dip disappears entirely.

Two shares have the same expected return. One swings twenty percent a year, the other thirty, and their correlation is one half. Which blend of the two carries the least risk?

Not the calm one on its own. The best blend holds six sevenths of the money in the twenty percent share, and it swings 19.64 percent, which is under both of its ingredients. A mixture beating both of the things mixed is the part worth understanding.

The floor that is not a floor

The reflex is an averaging argument. A blend of a calm asset and a jumpy one should land somewhere between them, so twenty percent must be the best available and buying any of the thirty percent share can only make things worse.

Averaging is the right instinct for the return, which is genuinely a weighted average of the two expected returns. It is the wrong instinct for the risk, because volatility is not an average of volatilities. It comes out of a variance, and the variance carries a cross term that can be negative relative to what pure averaging would predict.

\sigma_P^2(w) = w^2\sigma_1^2 + (1-w)^2\sigma_2^2 + 2w(1-w)\,\rho\,\sigma_1\sigma_2

(1)

With $w$ the fraction in the calm share, $\sigma_1 = 1/5$ , $\sigma_2 = 3/10$ and $\rho = 1/2$ , so the covariance is $\rho\sigma_1\sigma_2 = 3/100$ .

The test that decides it in one line

Marginal risk

What an asset adds to a portfolio is not its own volatility. It is its volatility multiplied by its correlation with the portfolio you already hold. Written for a single share against the portfolio, that quantity is $\rho\,\sigma_2$ , and it is the derivative of the portfolio's volatility with respect to a small allocation.

Sit at one hundred percent calm share and ask what the jumpy one contributes at the margin. Half of thirty is fifteen. Fifteen is below the twenty you are already carrying, so the first slice of the jumpy share reduces the portfolio's risk rather than raising it. That is the whole argument, and it is not a slogan: differentiating (1) at $w = 1$ gives

\left.\frac{d\sigma_P^2}{dw}\right|_{w=1} = 2\sigma_1\bigl(\sigma_1 - \rho\sigma_2\bigr) = 2 \cdot \tfrac15\left(\tfrac15 - \tfrac{3}{20}\right) = \tfrac{1}{50} > 0

(2)

A positive derivative at $w = 1$ means the variance falls as $w$ comes down off one. The sign of that bracket is exactly the comparison between fifteen and twenty, so the informal test and the first-order condition are the same statement.

Where the curve bottoms out

Setting the derivative of (1) to zero and solving gives the classical two-asset weight:

w^{\star} = \frac{\sigma_2^2 - \rho\sigma_1\sigma_2}{\sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2} = \frac{\tfrac{9}{100} - \tfrac{3}{100}}{\tfrac{4}{100} + \tfrac{9}{100} - \tfrac{6}{100}} = \frac{6}{7}

(3)

So 85.714 percent in the calm share and the remaining 14.286 percent in the jumpy one. Substituting back:

\sigma_P^2\!\left(\tfrac{6}{7}\right) = \tfrac{27}{700}, \qquad \sigma_P = \frac{3\sqrt{21}}{70} = 0.19639610\ldots

(4)

19.64 percent, and it is not a rounding of something above twenty. It is a closed form whose decimal expansion starts 0.196396, thirty-six basis points of pure risk reduction handed over for holding the asset that looks worse.

Fig. 1 — The parabola in the variance, drawn in volatility units. It crosses under the twenty percent line and comes back up to touch it at the right-hand edge.

It is a minimum and not some other stationary point, which is worth one line rather than a gesture. The second derivative of (1) is constant:

\frac{d^2\sigma_P^2}{dw^2} = 2\left(\sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2\right) = \tfrac{7}{50} > 0

(5)

Positive everywhere, so the variance is strictly convex in the weight and the stationary point is the global minimum on the whole line. That convexity is also the reason the bracket in (3) can never vanish for $|\rho| < 1$ , which is what makes the formula safe to use.

The condition that makes an interior answer exist

Six sevenths is inside the interval, so no short selling is needed. That is a fact about these inputs rather than a general feature, and the threshold is easy to read off (2). The derivative at $w = 1$ is positive exactly when:

\rho < \frac{\sigma_1}{\sigma_2} = \frac{1/5}{3/10} = \frac{2}{3}

(6)

With $\rho = 1/2$ the condition holds and the optimum is interior. At exactly $\rho = 2/3$ the derivative vanishes at the boundary, the best available portfolio is the calm share alone, and the answer to the original question really would be twenty percent. Above two thirds the formula in (3) returns a weight greater than one, meaning it wants a short position in the jumpy share, and if shorting is barred the optimum stays pinned at the corner.

Fig. 2 — The same two shares at two correlations. Above ρ = 2/3 the dip disappears and the diversification benefit goes with it.

Reading (6) the other way shows how much of the answer the correlation owns. Hold the two volatilities at twenty and thirty and set $\rho = 0$ : the optimal weight becomes $9/13 = 69.2$ percent and the blend swings $\sqrt{9/325} = 16.64$ percent, a full three and a third points below the calm share on its own. The same two assets, the same volatilities, and more than nine times the risk reduction, purchased entirely by the correlation moving from a half to zero.

What the problem quietly granted

Equal expected returns is doing more work than it looks. It is what turns this into a pure risk question with a single answer. With unequal returns there is no such thing as the best portfolio without saying how much return you want per unit of risk, the minimum-variance point becomes one point on a frontier rather than a target, and six sevenths stops being the answer to anything.

The inputs also have to be the true ones. Volatilities and correlations are estimated, and the weight in (3) is a ratio of differences of estimated quantities, so it inherits their error with amplification. A correlation estimate that is off by a tenth moves the optimal weight materially, which is why minimum-variance portfolios computed from short samples are unstable in practice while the qualitative lesson stays intact: the correlation, not the volatility, decides what an asset does to your risk.

Sources and further reading

The framework equation (1) belongs to — Modern portfolio theory
Where the minimum-variance point sits on the frontier — Efficient frontier
The cross term that makes the dip possible — Covariance
The general principle behind holding the worse-looking asset — Diversification (finance)

The weight in (3) was rebuilt from the first-order condition rather than quoted, and then checked against a brute-force search over 100,001 weights that knows nothing about any closed form: it landed on 0.857140 with a variance matching $27/700$ to $10^{-12}$ . A correlated Monte Carlo built from independent normals reproduced the portfolio swing at three weights, giving 19.641 percent against 19.640 at the optimum. The corner threshold in (6) was checked at $\rho = 2/3$ directly.