Free lesson · Portfolio construction

Explain results and marginal diversification

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Start with the idea

Benchmark-relative performance and absolute return answer different questions. Attribution identifies the exposures that produced the result, while marginal-risk calculations help evaluate a proposed addition to the book.

Symbols, units & horizon

aₜ: active return R_p−R_b
R_p,R_b: portfolio and benchmark periodic returns
ā: mean active return
s_a: sample SD of active returns, tracking error
IR: annualised information ratio
A: observations per year under IID annualisation
wᵢ,ₜ₋₁: asset weight entering period t
R_s: proposed sleeve return
ε: small added sleeve allocation funded by zero-return cash in this calculation
σ_s²: sleeve variance
Δσ_p²: change in portfolio variance
Cov: covariance

When and why to use this

Use information ratio for benchmark mandates and marginal covariance for portfolio changes. Reconcile attribution to the actual net P&L ledger.

a_{t} = R_{p, t} - R_{b, t}, I R = \frac{a}{s_{a}} \sqrt{A}

Algebra and arithmetic

Compute active return and information ratio

Subtract benchmark return period by period: aₜ=Rp,ₜ−Rb,ₜ. Compute its mean and sample SD, the tracking error.
Under the same IID additive-return argument used for Sharpe, annual IR is $\frac{\sqrt{A} \overline{a}}{s_{a}}$ . Zero tracking error makes the ratio undefined.

Work it by hand

Mean monthly active return=.2%, tracking error=1%: IR=.2√12=.69282.

Active return is portfolio return less benchmark return. Tracking error is its standard deviation, and the information ratio measures active return per unit of tracking error. A periods per year gives the usual annualisation only under suitable serial dependence assumptions.

R_{p, t} = \sum_{i} w_{i, t - 1} R_{i, t}, Δ σ_{p}^{2} = 2 ϵ Cov (R_{p}, R_{s}) + ϵ^{2} σ_{s}^{2}

Algebra and arithmetic

Expand the risk of adding a sleeve

New return is $R_{n e w} = R_{p} + ϵ R_{s}$ . Expand variance to $σ_{p}^{2} + 2 ϵ Cov (R_{p}, R_{s}) + ϵ^{2} σ_{s}^{2}$ .
Subtract old variance. For a funded reallocation use the difference between the incoming and outgoing sleeve as Rₛ; the simpler formula assumes no other exposure changes.

Work it by hand

ε=.1, covariance=.002, sleeve variance=.04 gives Δvariance=.0004+.0004=.0008. Expected net return must justify that change.

The second expression is for adding an ε-sized sleeve without otherwise changing the existing book; a funded reallocation requires including the sleeve being reduced. Correlation alone cannot determine whether a new strategy is valuable. Expected net return, marginal risk, tail dependence, capital usage, and capacity all matter.

Research sources, review dates and limitations

Extend the research question

Compare feasible portfolio weights under two plausible covariance estimates. Recheck final constraints after rounding and transaction costs.

Continue with the connected research module →

Connect the ideas: Dependence and diversification

Retrieve: Joint behavior matters when combining uncertain outcomes.

Check the change: Correlation, cointegration, covariance and event dependence answer different questions.

Statistics → Linear algebra → Time series → Portfolio management theory → Prediction foundations

Explain it yourself: Why does a highly correlated pair not automatically provide a converging spread?

Self-assessed. Write your explanation before opening this comparison.

Correlation measures co-movement under a chosen sample and horizon. It does not establish a stationary combination or contractual convergence; those require separate definitions, tests and implementation checks.

Connect the ideas: Constraints and survival

Retrieve: A desired position must fit available capital and explicit limits.

Check the change: Portfolio weights, venue collateral, working orders and redemption obligations impose different constraints.

Risk → Optimization → Arbitrage → Crypto derivatives → Execution & microstructure → Fund operations & capstone

Explain it yourself: Can an offsetting terminal payoff remove a margin problem today?

Self-assessed. Write your explanation before opening this comparison.

No. Cash may be required before the hedge pays, or in another account. Check the path, collateral location and feasible transfer times.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from statistics import mean, stdev
from math import sqrt

def information_ratio(portfolio_returns, benchmark_returns, periods=252):
    if len(portfolio_returns) != len(benchmark_returns):
        raise ValueError("Aligned return arrays required")
    active = [p-b for p,b in zip(portfolio_returns,benchmark_returns)]
    return mean(active)/stdev(active)*sqrt(periods)

def weighted_return(weights, returns):
    if len(weights) != len(returns):
        raise ValueError("Aligned assets required")
    return sum(w*r for w,r in zip(weights,returns))

def added_sleeve_variance(allocation, covariance_with_book, sleeve_variance):
    return 2*allocation*covariance_with_book+allocation**2*sleeve_variance

print(added_sleeve_variance(.1,.005,.04))

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