Tracking error measures the volatility of return differences between a portfolio and a reference portfolio. For portfolio weights w, reference weights p, and covariance matrix Σ, it is √((w − p)^TΣ(w − p)). A limit on this quantity keeps an optimized portfolio close to its reference in risk-adjusted return behavior rather than merely constraining its total volatility. section 10.2.2
This makes tracking error a practical trade-off control: a portfolio can pursue another objective while limiting how far it may depart from the reference. ESG-tilted indices use that trade-off differently; some permit larger departures to emphasize social-responsibility characteristics, while others allow only small tilts. section 4.4.2 section 10.3.4
In constrained portfolio optimization, tracking error can be used to:
- Bound the quadratic active-risk expression with a tracking-error tolerance. section 10.2.2
- Compare optimized weights with reference weights; separately constrain total absolute weight deviations with an L1 norm when needed. section 10.2.2
- Trace an ESG frontier by resolving the portfolio at progressively larger permitted tracking-error levels. section 10.3.4
ESG portfolio versus a reference portfolio
Maximize the weighted average ESG score while requiring full investment, nonnegative weights, and tracking error no greater than σb. With a 10% volatility reference portfolio, increasing permitted tracking error from 0.00% to 2.50% raises the reported ESG quality score from 6.02 to 8.13, while reported volatility remains near 12.8%. The result depends on the covariance estimate and available ETF return history. section 10.3.4
Table 10.3 Results from the ESG frontier
| Tracking error | Volatility | ESG quality score |
| 0.00% | 12.87% | 6.02 |
| 0.25% | 12.84% | 6.34 |
| 0.50% | 12.83% | 6.66 |
| 0.75% | 12.80% | 6.97 |
| 1.00% | 12.78% | 7.27 |
| 1.25% | 12.75% | 7.52 |
| 1.50% | 12.74% | 7.68 |
| 1.75% | 12.75% | 7.81 |
| 2.00% | 12.76% | 7.92 |
| 2.25% | 12.77% | 8.03 |
| 2.50% | 12.82% | 8.13 |
Compare the reported ESG-score gain against each permitted tracking-error level and the accompanying portfolio volatility. section 10.3.4
