06 — Benchmark-Relative Portfolio Management¶

Learning objectives¶

Calculate active weights and tracking error from a covariance matrix; calculate the realized information ratio from a return series; connect information coefficient, breadth, and the transfer coefficient through the Fundamental Law of Active Management.

Free learning pack¶

1. CFA Institute, Analysis of Active Portfolio Management https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/analysis-active-portfolio-management
2. Information ratio — Wikipedia https://en.wikipedia.org/wiki/Information_ratio
3. Fundamental Law of Active Management — Financial Edge Training https://www.fe.training/free-resources/portfolio-management/fundamental-law-of-active-management/
4. reference/concepts/tracking_error.md, information_ratio.md, information_coefficient.md, fundamental_law.md

Do not search for more material until these are insufficient.

PREDICT¶

A fund holds four names, two overweight and two underweight versus its benchmark by the same amount. Without calculating: does total portfolio volatility have to be higher than the benchmark's just because tracking error is nonzero?

Formula (active weights, tracking error)¶

a = w_p - w_b

TE = sqrt(a^T Sigma a)

In [ ]:

import numpy as np

portfolio = np.array([.30, .25, .25, .20])
benchmark = np.array([.25, .25, .30, .20])
cov = np.array([
    [.040, .010, .005, .002],
    [.010, .020, .004, .001],
    [.005, .004, .010, .003],
    [.002, .001, .003, .030],
])

# MANUAL FIRST:
# compute active weights, then tracking error, by hand.
active = None
tracking_error_value = None
print(active, tracking_error_value)

import numpy as np portfolio = np.array([.30, .25, .25, .20]) benchmark = np.array([.25, .25, .30, .20]) cov = np.array([ [.040, .010, .005, .002], [.010, .020, .004, .001], [.005, .004, .010, .003], [.002, .001, .003, .030], ]) # MANUAL FIRST: # compute active weights, then tracking error, by hand. active = None tracking_error_value = None print(active, tracking_error_value)

PREDICT (information ratio)¶

Two managers both beat their benchmark by 2% on average per year. Manager A's active return bounces between +8% and -4% month to month; Manager B's stays between +3% and +1%. Without calculating, which manager has the higher information ratio?

Formula (realized information ratio)¶

active_returns_t = portfolio_return_t - benchmark_return_t

IR = mean(active_returns) / std(active_returns) * sqrt(periods_per_year)

In [ ]:

portfolio_returns = [0.03, 0.05, 0.02]
benchmark_returns = [0.01, 0.02, 0.01]

# MANUAL FIRST:
# compute the active return each period, then the (unannualized,
# periods_per_year=1) information ratio by hand.
information_ratio_value = None
print(information_ratio_value)

portfolio_returns = [0.03, 0.05, 0.02] benchmark_returns = [0.01, 0.02, 0.01] # MANUAL FIRST: # compute the active return each period, then the (unannualized, # periods_per_year=1) information ratio by hand. information_ratio_value = None print(information_ratio_value)

PREDICT (Fundamental Law)¶

A manager has modest skill — an information coefficient of just 0.05 — but applies it across 400 independent bets a year (a broad, diversified stock-picking process). Another manager has excellent skill (IC = 0.15) but makes only 4 big, concentrated bets a year. Without calculating, whose expected information ratio is higher?

Formula (Fundamental Law of Active Management)¶

IR ~= IC * sqrt(BR) * TC

IC: information coefficient (skill) · BR: breadth (independent bets) · TC: transfer coefficient (1.0 = unconstrained)

In [ ]:

# MANUAL FIRST:
# compute the expected IR for each manager above (assume TC=1.0 for both)
# and confirm your prediction.
manager_a_ic, manager_a_breadth = 0.05, 400
manager_b_ic, manager_b_breadth = 0.15, 4

ir_a = None
ir_b = None
print(ir_a, ir_b)

# MANUAL FIRST: # compute the expected IR for each manager above (assume TC=1.0 for both) # and confirm your prediction. manager_a_ic, manager_a_breadth = 0.05, 400 manager_b_ic, manager_b_breadth = 0.15, 4 ir_a = None ir_b = None print(ir_a, ir_b)

Experiment¶

Manager A's 400 "independent" bets are actually just 20 real sector calls, each expressed through 20 highly correlated single-name positions (average pairwise correlation within a sector call ~0.6). Use effective_breadth(400, average_correlation=0.6) to find Manager A's true breadth, then recompute their expected IR. Does Manager A still beat Manager B?

Reference¶

reference/concepts/tracking_error.md reference/concepts/information_ratio.md reference/concepts/information_coefficient.md reference/concepts/fundamental_law.md

Promote¶

Use src/pm/active.py (active_weights, tracking_error, information_ratio, information_coefficient, effective_breadth, transfer_coefficient, fundamental_law_ir) only after your own implementation.

Test¶

pytest tests/test_active.py

ORAL CHECK¶

Explain to a PM why two managers with the same average active return can have very different information ratios, and why a manager's effective breadth can be far smaller than their raw bet count if those bets are correlated.

Try /tutor information ratio or /tutor fundamental law for an adaptive walkthrough.