Lesson 5 · Portfolio ManagementCFA L1

Active Portfolio Management

If the market portfolio is efficient, passive investing suffices — yet active managers bet they can beat the benchmark. This lesson makes that bet measurable: active return and active weights, decomposing value into asset allocation vs security selection, the information ratio and the optimal amount of active risk to take, the Treynor-Black model for turning mispricings into weights, and the fundamental law that ties performance to skill and breadth.

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Active return & weights Allocation vs selection Information ratio Optimal active risk Treynor-Black model Grinold's rule The fundamental law

Active return is the difference between an actively managed portfolio's return and its benchmark: $R_A=R_P-R_B$. It can be viewed ex-ante (expected) or ex-post (realised), and measures the value active management adds.

Learning outcomes

Compute active return from active weights.
Decompose active return into asset allocation and security selection.
Define the information ratio and find the optimal active risk.
Apply the Treynor-Black model to build the optimal active portfolio.
State Grinold's rule and the fundamental law of active management.

Active return and active weights

The active weight on asset $i$ is the deviation from the benchmark, $\Delta w_i=w_i^P-w_i^B$ (positive = overweight, negative = underweight), and active weights sum to zero. Active return is their return-weighted sum:

Active return from active weights

$$ R_A=R_P-R_B=\sum_{i=1}^{N}\big(w_i^P-w_i^B\big)r_i=\sum_{i=1}^{N}\Delta w_i\,r_i, \qquad \sum_i\Delta w_i=0 $$

Worked example

An international equity portfolio

US: benchmark 22%, portfolio 16%, return 20%. UK: 28% / 34% / 23%. Other: 50% / 50% / 16%. Find the active weights and active return.

Show solution

Active weights: US $16-22=-6\%$; UK $34-28=+6\%$; Other $50-50=0$. Active return $=(-0.06)(20\%)+(0.06)(23\%)+0(16\%)=\textbf{0.18%}$ — a small gain from overweighting the higher-returning UK and underweighting the US.

Active weights −6%, +6%, 0; active return 0.18%

Asset allocation vs security selection

When a manager both allocates across asset classes and selects securities within them, the value splits cleanly into two sources (it does not, however, capture their interaction):

Two sources of active value

$$ R_A=\underbrace{\sum_j \Delta w_j\,R_j^B}_{\text{asset allocation}}+\underbrace{\sum_j w_j^P\,R_j^A}_{\text{security selection}} $$

Worked example

Equities and bonds

Equities fund 22% (benchmark 16%); bonds fund −1% (benchmark −2%). Baseline policy 60/40; actual 68/32. Split the value.

Show solution

Active returns: equities $22-16=6\%$; bonds $-1-(-2)=1\%$. Allocation (active class weights +8%, −8%): $0.08(16\%)+(-0.08)(-2\%)=1.44\%$. Selection: $0.68(6\%)+0.32(1\%)=4.4\%$. Total $=1.44\%+4.4\%=\textbf{5.84%}$. Check directly: $R_B=0.6(16\%)+0.4(-2\%)=8.8\%$, $R_P=0.68(22\%)+0.32(-1\%)=14.64\%$, $R_A=5.84\%$. ✓

Allocation 1.44% + selection 4.4% = 5.84% total active return

The information ratio

Good active management is high and consistent. The information ratio rewards both:

Information ratio

$$ \text{IR}=\frac{E(R_A)}{\sqrt{\operatorname{var}(R_A)}}=\frac{E(R_P-R_B)}{\text{tracking error}} $$

The denominator, $\sqrt{\operatorname{var}(R_A)}$, is the active risk (tracking error). The IR is scale-invariant — doubling all active weights doubles $R_A$ and the active risk, so the ratio is unchanged.

Optimal active risk

How much active risk should you take? Mix the benchmark (weight $1-w$) with the active portfolio (weight $w$) and maximise the combined Sharpe ratio. The optimum and the resulting optimal active risk are:

Optimal active weight and risk

$$ w^{*}=\frac{\text{IR}}{\text{SR}_B}\cdot\frac{\sigma_B}{\sigma_A}, \qquad \text{optimal active risk}=w^{*}\sigma_A=\frac{\text{IR}}{\text{SR}_B}\,\sigma_B $$

Worked example

Mixing a fund with its benchmark

Fund: IR 0.2, active risk 9%. Benchmark: Sharpe 0.4, s.d. 12%. Find the optimal weights and active risk.

Show solution

$w^{*}=\frac{0.2}{0.4}\cdot\frac{0.12}{0.09}=0.6667$ in the fund, $0.3333$ in the benchmark. Optimal active risk $=0.6667\times9\%=\textbf{6%}$.

66.7% fund / 33.3% benchmark; optimal active risk 6%

The Treynor-Black model

Treynor-Black turns identified mispricings into a portfolio. In a single-index framework, asset $i$ has a non-zero Jensen's alpha $\alpha_i$, beta $\beta_i$ and idiosyncratic variance $\sigma_{\varepsilon i}^2$. The optimal active portfolio weights each asset by its alpha-to-idiosyncratic-variance ratio, then blends that active portfolio with the benchmark:

Treynor-Black weights

$$ w_i=\frac{\alpha_i/\sigma_{\varepsilon i}^2}{\sum_j \alpha_j/\sigma_{\varepsilon j}^2}, \qquad w_0=\frac{\alpha_A/\sigma_{\varepsilon A}^2}{(E(R_B)-r_f)/\operatorname{var}(R_B)}, \qquad w_A^{*}=\frac{w_0}{1+(1-\beta_A)w_0} $$

Worked example

Four mispriced stocks

Forecasts — A: 20%, β 0.6, σε 60%; B: 17%, 0.6, 50%; C: 14%, 0.5, 45%; D: 8%, 0.2, 40%. Benchmark 10%, s.d. 25%; $r_f=4\%$. Build the optimal active portfolio and the final allocation.

Show solution

Alphas $\alpha_i=E(r_i)-[r_f+\beta_i(E(R_B)-r_f)]$: A $=20\%-[4\%+0.6(6\%)]=12.4\%$; similarly B 9.4%, C 7%, D 2.8%. Weighting by $\alpha_i/\sigma_{\varepsilon i}^2$ and normalising gives $w=(0.28,0.30,0.28,0.14)$. Then $\alpha_A=8.64\%$, $\beta_A=0.516$, $\sigma_{\varepsilon A}^2=0.0697$. So $w_0=\frac{0.0864/0.0697}{(0.06)/(0.25)^2}=1.29$ and $w_A^{*}=\frac{1.29}{1+(1-0.516)(1.29)}=0.8$. Final weights: A 0.22, B 0.24, C 0.22, D 0.11, benchmark 0.20.

Active portfolio (0.28, 0.30, 0.28, 0.14); 80% active / 20% benchmark

Grinold's rule

Treynor-Black takes the alphas as given. Grinold's rule models where they come from:

Grinold's rule

$$ \alpha_i=\text{IC}\times\sigma_i\times S_i $$

The information coefficient (IC) — common to all assets — is the correlation between forecast alphas and realised active returns (high IC = good forecasting skill); $S_i$ is a standardised score for asset $i$ (from fundamentals or analyst forecasts), scaled so the average squared score is 1.

The fundamental law of active management

Substituting Grinold's alpha into the optimal active weights and the expected active return yields the celebrated law:

Fundamental law

$$ \text{IR}=\text{IC}\sqrt{N}=\text{IC}\sqrt{\text{BR}} \qquad\Longrightarrow\qquad \text{(with constraints) } \text{IR}=\text{TC}\cdot\text{IC}\sqrt{N} $$

Skill × breadth

Active quality = skill (the IC) × breadth ($\sqrt{N}$, the number of independent bets / size of the investment universe, BR). Be skilful and apply that skill across many opportunities. Adding portfolio constraints introduces the transfer coefficient (TC) — the correlation between the actual and theoretically-optimal active weights; $\text{TC}=1$ with no friction, and $\text{TC}<1$ measures value lost in implementation.

Limitations: a constant IC across all assets and time is unrealistic (forecasting difficulty varies); "breadth" is ambiguous (many similar stocks are not many independent bets); and the logic is somewhat circular (alpha depends on IC, but IC is defined via alpha) — which is why academia treats the law cautiously.

Key takeaway

Active management is measured by active return $R_A=\sum\Delta w_i r_i$, decomposed into asset allocation and security selection, and judged by the information ratio (active return per unit tracking error). The right amount of active risk is $\frac{\text{IR}}{\text{SR}_B}\sigma_B$; the Treynor-Black model converts alphas into weights proportional to $\alpha_i/\sigma_{\varepsilon i}^2$; and the fundamental law $\text{IR}=\text{IC}\sqrt{\text{BR}}$ says performance is skill × breadth (times a transfer coefficient under constraints) — a clean intuition built on fragile assumptions.

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