Practical Portfolio Construction: Minimum Variance, Risk Parity and Rebalancing

The previous lesson in this course showed that expected returns are the least reliable input to a portfolio, and a raw covariance matrix is not far behind. Practical portfolio rules can be ranked by how much they need to estimate.

Moving down the table uses more information and exposes the result to more estimation error. None of these rules is best in every market; the choice is about which errors you are prepared to carry. This lesson works through the middle of the table on the same three-asset example used in the first lesson: A (10% volatility), B (15%), C (20%), with correlations 0.3, 0.2 and 0.5.

Weights tell you where the money is. They do not tell you where the risk is. The risk contribution of asset $i$ is its weight times its marginal effect on portfolio variance:

The contributions sum to exactly 100% of the variance (a consequence of Euler's theorem for functions that scale with the weights), so they can be read as a budget.

Apply it to the equally weighted portfolio of A, B and C:

Key idea: Equal money is not equal risk. The "diversified" 1/N portfolio gets almost half its risk from one asset, C, because C is both the most volatile and the most correlated with the others. On a real multi-asset portfolio split between equities and bonds, the same arithmetic leaves the equities dominating the risk.

The equal risk contribution (ERC) portfolio, the formal version of risk parity, chooses weights so that every asset has the same risk contribution. Maillard, Roncalli and Teiletche studied its properties in "The Properties of Equally Weighted Risk Contribution Portfolios" (Journal of Portfolio Management, 2010). Two of their results matter most in practice:

• The ERC portfolio's volatility always lies between that of the minimum-variance portfolio and that of the equally weighted portfolio.
• When all pairwise correlations are equal, ERC weights reduce exactly to inverse volatility, $w_i \propto 1/\sigma_i$.

Solved on the three-asset example (computed with scipy, as in the code step below):

Inverse volatility gets close to equal risk here, because the correlations differ only moderately. The gap grows when one asset is much more correlated with the rest than the others are.

Here are the four risk-based rules on the same three assets, scored with the true inputs.

The ordering matches the Maillard, Roncalli and Teiletche result: ERC at 10.17% sits between minimum variance at 9.25% and equal weight at 11.52%.

ERC has no closed form for general correlations, but it is a small, well-behaved optimisation: minimise the spread between the risk contributions, subject to the weights being positive and summing to one.

import numpy as np
from scipy.optimize import minimize

def risk_contributions(w, Sigma):
    marginal = Sigma @ w
    return w * marginal / (w @ marginal)

def equal_risk_contribution(Sigma):
    n = len(Sigma)
    def objective(w):
        rc = w * (Sigma @ w)
        return np.sum((rc[:, None] - rc[None, :]) ** 2)   # all pairwise gaps
    res = minimize(objective, np.ones(n) / n, method="SLSQP",
                   bounds=[(0, 1)] * n,
                   constraints={"type": "eq", "fun": lambda w: w.sum() - 1},
                   options={"ftol": 1e-14})
    return res.x

vol = np.array([0.10, 0.15, 0.20])
corr = np.array([[1, .3, .2], [.3, 1, .5], [.2, .5, 1]])
Sigma = np.outer(vol, vol) * corr
w = equal_risk_contribution(Sigma)
print(w.round(3), risk_contributions(w, Sigma).round(3))
# [0.488 0.287 0.226] [0.333 0.333 0.333]

The tight ftol matters: the objective is tiny near the solution, and the default tolerance can stop early with visibly unequal contributions. In production, feed it a shrunk covariance matrix, for the reasons the previous lesson measured.

Risk-based portfolios lean toward low-volatility assets, so they tend to have lower volatility and lower expected return than 1/N. On the three-asset example:

Scaling any portfolio by a leverage factor scales its excess return and volatility together, leaving the Sharpe ratio unchanged. So if the ERC portfolio's Sharpe ratio is higher, borrowing to bring it up to 1/N's volatility produces more expected return at the same risk: 7.34% against 7.00% in this example. This is the logic behind risk parity funds that apply leverage to a risk-balanced mix.

Gotcha: The table assumes borrowing at the risk-free rate with no margin calls. Real leverage costs a spread over that rate, needs collateral, and can force selling at the worst moment. The trading risk course on AnyLearn covers how margin and leverage spirals work. A higher Sharpe ratio on paper only becomes a higher return after those costs.

Every weighting rule is a snapshot. The day after you set the weights, prices move and the portfolio drifts. Start the three-asset example at 1/N and suppose that over a year A returns 2%, B 10% and C 25%:

Restoring 1/N means selling 3.8 points of C and buying A and B: trades totalling 7.5% of the portfolio. Drift also moves the risk: C, already the biggest risk contributor, now carries even more of the portfolio's variance.

Definition: Turnover is the fraction of the portfolio traded in a period, often reported one-way (half the sum of absolute weight changes, 3.8% here). The annual cost drag is roughly turnover times the cost per unit traded, including spreads, commissions and market impact, so a strategy's gross advantage has to be judged net of it. DeMiguel, Garlappi and Uppal included turnover as one of their three comparison measures for exactly this reason.

Calendar rules are simple and predictable. Band rules trade only when drift is large enough to matter.

Rebalancing policy is where a weighting rule meets its costs. The main levers:

The last row connects back to estimation error. A minimum-variance or ERC portfolio recomputed from a short, unshrunk covariance window will change its weights every period in response to noise, generating turnover that buys nothing. Longer windows, shrinkage and bands all damp this.

In practice: Estimate turnover before choosing a rule, not after. A strategy whose paper advantage over 1/N is smaller than its extra trading cost is not an improvement, however sophisticated the weights.

Put the three lessons together and a defensible process looks like this:

1. Define the universe and the benchmark. 1/N over the same assets is the minimum bar any rule must clear out of sample.
2. Decide what you are willing to estimate. If you have no return forecast you would bet on, use a risk-based rule and ignore expected returns entirely.
3. Estimate risk robustly. Shrink the covariance matrix, and use enough history for the number of assets.
4. Choose the rule for its risk profile. Minimum variance for the lowest volatility, accepting concentration; ERC for balanced risk; inverse volatility when you want most of ERC's balance with only volatility estimates.
5. Add constraints deliberately. Long-only and position caps limit the damage any single estimate can do.
6. Decide on leverage explicitly, with its real borrowing cost and margin terms.
7. Set the rebalancing policy and measure turnover.
8. Evaluate out of sample, net of costs, against the benchmark from step 1.

Key idea: Portfolio construction is mostly the management of estimation error. The mathematics of diversification is exact; the inputs are not, and the rules that survive are the ones that use the fewest unreliable numbers.