The minimum-variance portfolio

Of all the ways to split £1bn between equities and government bonds, exactly one has the lowest risk. For RiskAtoms Group it holds 14.65% in equities and has a volatility of 5.41%. With a different pair, the lowest-risk answer tells the group to sell one of its assets short. Why?

L3.1.03: Two correlated assets drew every mix of two assets as a curve on the risk-return chart, bulging to the left. Its leftmost point, the nose, is the minimum-variance portfolioThe portfolio with the lowest possible variance among all mixes whose weights add up to 1.. It matters for three reasons. It is where the efficient part of the curve begins (L3.1.05: The efficient frontier). It is what an investor should hold who cares only about risk, such as a treasurer parking a reserve. And its maths is exactly the maths of hedging: a hedge is a two-asset portfolio built to have as little variance as possible.

L3.1.02: Two independent assets found the nose when the correlation was zero: the calmer asset gets more weight. With a correlation the answer has one more ingredient, the covariance $\sigma_{12}$:

Read the top line as a tug of war. The $\sigma_2^2$ pulls money towards asset 1 when asset 2 is volatile. The $-\rho\,\sigma_1\sigma_2$ pulls it away when the two move together, because then asset 1 offers little offset. The bottom line is the variance of the difference $R_1 - R_2$: always positive, so it only scales the answer.

For RiskAtoms Group's equities and bonds (ρ = −0.10) the top line is $0.0036 + 0.00096 = 0.00456$ and the bottom $0.03112$, so $w_1^* = 14.65\%$. Its volatility, 5.41%, is below the bonds' 6%, and its expected return, 4.01%, is above their 3.5%.

When the answer is a short sale

The top line turns negative when $\rho\,\sigma_1\sigma_2 > \sigma_2^2$, that is when $\rho > \sigma_2/\sigma_1$. Then the lowest risk comes from selling asset 1 short and holding more than 100% of asset 2. Why would that ever reduce risk? Because when the two are strongly correlated, the volatile asset moves like an amplified copy of the calm one. Shorting a little of it cancels part of the calm asset's own swings. Only the more volatile asset can be shorted: for the calmer one the condition would need a correlation above 1.

The emerging-market case is a warning as much as a result. The short position is small, and it lowers the volatility from 16.00% (global equities alone) to just 15.99%. When two assets are highly correlated, the minimum-variance answer is fragile: a small change in the estimated correlation swings the weights a lot, for very little gain. L3.1.15: Estimation error, shrinkage and Black-Litterman returns to this.

The same maths in a hedge

L2.6.02: Hedging with forwards and futures hedged a position $S$ by selling $h$ futures $F$. The hedged position's variance is $\sigma_S^2 - 2h\,\rho\,\sigma_S\sigma_F + h^2\sigma_F^2$, a parabola in $h$. Setting its derivative to zero gives $h^* = \rho\,\sigma_S/\sigma_F$, the minimum-variance hedge ratioThe amount of a hedging instrument per unit of exposure that minimises the variance of the hedged position.. It is the two-asset problem with the futures weight free and the position's weight fixed at 1.

Two derivations of the same weight. The first uses the constraint $w_1 + w_2 = 1$ to remove one unknown and then sets a single derivative to zero.

It is a minimum because the second derivative is positive: $\frac{d^2\sigma_p^2}{dw^2} = 2\big(\sigma_1^2 + \sigma_2^2 - 2\sigma_{12}\big) = 2\operatorname{Var}(R_1 - R_2) > 0$.

Substituting is easy with two weights but hopeless with fifty. The general tool keeps every weight and handles the constraintA condition the answer must meet, here that the weights add up to 1. with a Lagrange multiplierA method for optimising with constraints: add each constraint, times a new unknown λ, to the objective and set all partial derivatives to zero. (M-B04: Optimisation and Lagrange multipliers). It is the method L3.1.07: Mean-variance optimisation uses for any number of assets.

What the multiplier tells you. Multiply the first condition by $w_1$, the second by $w_2$, and add:

And each condition, divided by 2, reads $w_1\sigma_1^2 + w_2\sigma_{12} = \operatorname{Cov}(R_1, R_p)$ on the left. So at the minimum:

Every asset has the same covariance with the minimum-variance portfolio. If one had a higher covariance, moving a little money out of it into the other would lower the variance, so you were not at the minimum. The multiplierThe new unknown λ in the Lagrange method. It measures how much the optimum would change if the constraint were relaxed a little. λ is the price of the constraint: the extra variance per unit of extra money you are forced to invest.

The minimum variance itself. Use $\sigma_*^2 = \operatorname{Cov}(R_1, R_p) = w_1^*\sigma_1^2 + w_2^*\sigma_{12}$ with $w_1^* = (\sigma_2^2 - \sigma_{12})/K$ and $w_2^* = (\sigma_1^2 - \sigma_{12})/K$, where $K = \sigma_1^2 + \sigma_2^2 - 2\sigma_{12}$:

At $\rho = 0$ it is the result of L3.1.02: Two independent assets; at $\rho = -1$ the top is zero: the perfect hedge.

When it needs a short sale. Since $K > 0$:

A correlation cannot exceed 1, so only the asset with the larger volatility can ever have a negative weight.

The orange line is the tangent at your weight; its slope is the derivative. Drag the weight until the tangent is flat and the slope reads close to zero, then press “Go to the bottom” to check. Then press “Try EM and global equities”: the bottom moves into the shaded short-sale region. Drag ρ down past σ₂/σ₁ and watch it come back.

A minimum-variance answer with a negative weight is only useful if the investor may sell short. Many cannot. UCITS funds may not make uncovered short sales (Article 89 of Directive 2009/65/EC). Where short selling is allowed it is supervised: under the EU Short Selling Regulation (Regulation (EU) No 236/2012), a net short position in a company's shares must be reported to the regulator from 0.1% of its issued share capital and published from 0.5%. Insurers and pension funds usually run long-only mandates. That is why L3.1.07: Mean-variance optimisation solves the problem again with a no-short-sales constraint.

• Minimum-variance weight: w₁* = (σ₂² − ρσ₁σ₂) / (σ₁² + σ₂² − 2ρσ₁σ₂); its variance is σ₁²σ₂²(1 − ρ²) / (σ₁² + σ₂² − 2ρσ₁σ₂).
• Calculus: substitute w₂ = 1 − w₁ and set the derivative to zero. Lagrange: keep both weights, add −λ(w₁ + w₂ − 1), set every partial derivative to zero.
• At the minimum every asset has the same covariance with the portfolio, σ*², and λ = 2σ*².
• The more volatile asset is sold short when ρ > σ₂/σ₁. The less volatile one never is.
• For the group's equities and bonds: 14.65% in equities, 5.41% volatility, 4.01% expected return.
• A minimum-variance hedge ratio, h* = ρσ_S/σ_F, is the same calculation.

Minimum-variance portfolio

The portfolio with the lowest possible variance among all mixes whose weights add up to 1.

Constraint

A condition the answer must meet, here that the weights add up to 1.

Lagrange method

A method for optimising with constraints: add each constraint, times a new unknown λ, to the objective and set all partial derivatives to zero.

Lagrange multiplier

The new unknown λ in the Lagrange method. It measures how much the optimum would change if the constraint were relaxed a little.

Gradient

The vector of partial derivatives of a function: it points in the direction the function rises fastest.

Minimum-variance hedge ratio

The amount of a hedging instrument per unit of exposure that minimises the variance of the hedged position.