FINC3017 · Investments And Portfolio Management
Active vs Passive, Tracking Error & Allocation
Active vs Passive, Tracking Error & Allocation (Week 8) judges active management and sets portfolio weights through time. You extend Fama-MacBeth to multiple factors to separate exposure (betas) from pricing (lambdas), look at the non-performance costs of active management (fees, turnover, liquidity, leverage, tax), and quantify active risk with tracking error — the volatility of return relative to a benchmark. The information ratio, IR = α/σ(ε), measures active return per unit of that risk, and the Fundamental Law decomposes skill into IR = IC × √breadth. Allocation rules include fixed-fraction and volatility-managed weights.
What this chapter covers
- 01Fama-MacBeth in a multifactor setting: stage-1 betas, stage-2 per-period λ_kt
- 02Separating factor exposure (β) from factor pricing (λ)
- 03Non-performance metrics: fees (2-and-20), turnover, liquidity, concentration, leverage, tax
- 04Tracking error = σ(ε) from r_i − rf = α_i + β_i(r_bench − rf) + ε_t
- 05Information ratio IR = α_i/σ(ε)
- 06Fundamental Law of Active Management: IR = IC × √breadth
- 07Fixed-fraction allocation: R_t = rf + w(R_risky − rf)
- 08Volatility-managed weight w_t = σ_target/σ_t (Moreira-Muir)
Information ratio, the Fundamental Law, and a vol-managed weight
- 2 marks(a) Information ratio IR = α/σ(ε) = 3%/5% = 0.60.
- 2 marks(b) Fundamental Law: IR = IC × √breadth, so √breadth = IR/IC = 0.60/0.05 = 12.
- 1 mark(b cont.) Square it: breadth = 12² = 144 independent bets per period.
- 2 marks(c) Volatility-managed weight w_t = σ_target/σ_t = 12%/20% = 0.60, so de-risk to 60% in the risky asset when volatility is high.
Key terms
- Tracking error
- The standard deviation of a portfolio's return relative to its benchmark, σ(ε), estimated as the volatility of the residual from regressing the portfolio's excess return on the benchmark's. It measures how far an active portfolio strays from its benchmark.
- Information ratio (IR)
- Active return per unit of active risk, IR = α/σ(ε). It is the right measure when judging a manager against a benchmark, the active-management analogue of the Sharpe ratio, with higher values indicating more efficient use of tracking error.
- Fundamental Law of Active Management
- IR = IC × √breadth, decomposing the information ratio into the information coefficient (forecasting skill) and breadth (the number of independent bets). It shows that consistent small skill applied across many independent bets beats occasional large bets.
- Information coefficient (IC)
- The correlation between a manager's forecasts and the realised outcomes — a measure of raw forecasting skill. In the Fundamental Law it is the per-bet skill that, scaled by the square root of breadth, produces the overall information ratio.
- Volatility-managed weight
- A dynamic allocation rule w_t = σ_target/σ_t (Moreira-Muir) that lowers risky exposure when recent volatility is high and raises it when calm. Because high-volatility periods tend to carry poor returns, scaling down then has historically improved risk-adjusted performance.
Active vs Passive, Tracking Error & Allocation FAQ
When should I use the information ratio instead of the Sharpe ratio?
Use the information ratio when you are judging an active manager against a specific benchmark, because it measures benchmark-relative alpha per unit of tracking error. Use the Sharpe ratio when you care about total, standalone risk relative to the risk-free rate. The IR is essentially the benchmark-relative version of the Sharpe ratio.
Why does breadth enter the Fundamental Law as a square root?
Independent bets diversify the noise in your forecasts, and like any diversification of independent risks the benefit grows with the square root of the number, not linearly. So adding bets helps, but with diminishing returns — which is why raising your skill (IC) is more powerful than simply placing more bets of the same quality.
How does volatility-managing a portfolio improve performance?
The rule w_t = σ_target/σ_t cuts exposure precisely when markets are turbulent. Empirically high-volatility episodes deliver below-average returns, so reducing risk then avoids losses without sacrificing much upside, which historically raises the Sharpe ratio. It is a timing strategy based on volatility being more forecastable than returns.
Exam move
Practise the IR computation and the Fundamental Law inversion (solving for breadth or IC) until they are automatic, and know the vol-managed weight rule by heart. Be ready to explain why breadth enters as a square root and which performance measure suits a benchmark-relative versus standalone comparison — these distinctions are common conceptual MCQs.
