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Benchmark Basics

One-line definition

A benchmark is the reference portfolio a manager's holdings and performance are measured against.

Vocabulary

  • Benchmark weight w_b: the weight of an asset in the benchmark index.
  • Portfolio weight w_p: the weight of an asset actually held.
  • Active weight a = w_p - w_b: over/underweight versus the benchmark. Sums to zero when both are fully invested in the same universe.
  • Active return: r_p - r_b, the portfolio's return in excess of the benchmark's return.
  • Tracking error: the volatility of active return — see tracking error.

Why PMs care

Almost no institutional mandate is judged on absolute return alone. Sizing, risk budgets, and compensation are usually defined relative to a benchmark, so "am I long or short this name/sector versus the benchmark" is a more common question than "how much of this do I own."

Worked example

Portfolio holds 60% US equity / 40% intl equity. Benchmark is 50% / 50%.

Active weights: +10% US equity, -10% intl equity.

If US equity returns 10% and intl equity returns 4% this period: r_p = 0.6*0.10 + 0.4*0.04 = 0.076 r_b = 0.5*0.10 + 0.5*0.04 = 0.070 active_return = 0.076 - 0.070 = 0.006 (60 bp)

Code recipe

def active_weights(portfolio_weights, benchmark_weights):
    return np.asarray(portfolio_weights, float) - np.asarray(benchmark_weights, float)

def active_return(portfolio_return, benchmark_return):
    return float(portfolio_return - benchmark_return)

Common mistakes

  • comparing a portfolio to a benchmark it isn't actually mandated against
  • forgetting that active weights only sum to zero when portfolio and benchmark share the same universe and are both fully invested
  • treating a positive active return in one period as skill rather than noise — see information ratio

Limitations / approximations

This page covers vocabulary only. The full quantitative treatment lives in src/pm/active.py: tracking error, information ratio, information coefficient, and the Fundamental Law (which covers breadth and the transfer coefficient).

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