Free lesson · Optimization

Derive the fully invested minimum-variance portfolio

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When expected returns are too uncertain to distinguish, minimum variance offers a risk-only allocation benchmark.

Use returns measured over a common horizon and a symmetric positive-definite covariance matrix. The fully invested constraint makes the answer different from an unconstrained zero-risk allocation of no holdings. Short positions remain possible unless prohibited.

Solve a linear system instead of explicitly inverting the covariance matrix. A nearly singular matrix can create huge offsetting weights. Compare against equal weights, inspect gross exposure, and repeat with a shrinkage estimate.

Apply it in a strategy

• Freeze inputs at the stated decision time and record their units.
• Compare sophisticated return-based allocations against this monthly risk-only baseline.
• Recompute the example, then change the material assumption and explain the difference.

Research deliverable

Derive the fully invested minimum-variance portfolio: produce the worked calculation, a timestamped input record and a written decision addressing this limitation: Low estimated variance does not protect against structural breaks, liquidity costs or omitted constraints.

These are synthetic mechanics examples, not historical performance or paper replications. Module evidence and research boundaries record the 12 September 2026 review.

# Python 3.10+; standard library and NumPy only.
# Synthetic teaching inputs; conventions and units are defined in the notation above.
import numpy as np

def minimum_variance(covariance):
    s=np.asarray(covariance,float)
    if s.ndim!=2 or s.shape[0]!=s.shape[1] or not np.allclose(s,s.T): raise ValueError('Symmetric square matrix required')
    np.linalg.cholesky(s)
    z=np.linalg.solve(s,np.ones(len(s)))
    return z/z.sum()

w=minimum_variance([[.04,0],[0,.09]])
assert np.allclose(w,[9/13,4/13])
print(w)