Lesson 9 · Portfolio ManagementCFA L1

Revision

A consolidated formula sheet and concept recap of the whole course — from mean-variance optimisation and the CAPM to factor models, value at risk, active management, international investing, the investment policy statement, and trade execution. Use it to pull the threads together; each section links back to the full lesson.

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Risk & return CAPM & performance Factor models & APT Value at risk Active management International IPS & objectives Execution

The unifying logic of the course: summarise assets by mean and variance, build efficient portfolios, price risk in equilibrium (CAPM) or by no-arbitrage (APT), measure risk and performance, then implement the portfolio for a real investor and execute it in the market.

Portfolio risk and return

Core mean-variance results (Lesson 1)

$$ U=\mu-\tfrac{1}{2}A\sigma^2, \qquad \text{CAL: } \mu_c=r_f+\frac{\mu_p-r_f}{\sigma_p}\sigma_c, \qquad y^{*}=\frac{\mu_p-r_f}{A\sigma_p^2} $$

$$ \sigma_p^2=w_A^2\sigma_A^2+2w_A(1-w_A)\sigma_{AB}+(1-w_A)^2\sigma_B^2 $$

The CAL's slope is the Sharpe ratio; the optimal risky portfolio maximises it, after which a risk-preference split completes the allocation (mutual-fund theorem). See Lesson 1.

CAPM and performance evaluation

CAPM, risk decomposition and the four measures (Lesson 2)

$$ E(r_i)=r_f+\beta_i(E(r_M)-r_f), \qquad \operatorname{var}(r_i)=\beta_i^2\operatorname{var}(r_M)+\operatorname{var}(\varepsilon_i) $$

$$ \text{Sharpe}=\tfrac{\mu-r_f}{\sigma}, \ M^2=r_f+\sigma_M\text{Sharpe}, \ \text{Treynor}=\tfrac{\mu-r_f}{\beta}, \ \alpha=\mu-[r_f+\beta(E(r_M)-r_f)] $$

Total risk = systematic + diversifiable; only systematic ($\beta$) is rewarded. Use Sharpe/M² (total risk) for a standalone holding, Treynor/alpha (systematic risk) for part of a diversified portfolio. See Lesson 2.

Factor models and APT

Factor model and the multifactor SML (Lesson 3)

$$ r_i=a_i+\textstyle\sum_k b_{ik}F_k+\varepsilon_i, \qquad E(r_i)=r_f+\textstyle\sum_k b_{ik}\lambda_k $$

$$ \text{Fama-French: } r_i-r_f=\alpha_i+b_1(r_M-r_f)+b_2\text{SMB}+b_3\text{HML}+\varepsilon_i $$

APT derives fair return from no-arbitrage; CAPM is its one-factor case. Smart-beta harvests factor premia (value/size/momentum/quality/low-vol), and much apparent alpha is really factor reward (Buffett). See Lesson 3.

Value at risk

VaR (Lesson 4)

$$ \text{parametric \%-VaR}=-(\mu+z(\alpha)\sigma), \quad N\text{-day}= \text{1-day}\times\sqrt{N}, \quad \text{ES}=-E[r\mid r\le-\text{VaR}] $$

Three methods — historical simulation, parametric (normal, $z(95)=-1.645,\ z(99)=-2.326$), Monte Carlo. The normal understates fat tails; supplement with expected shortfall, backtesting and stress tests. See Lesson 4.

Active management

Active value, IR and the fundamental law (Lesson 5)

$$ R_A=\underbrace{\textstyle\sum_j\Delta w_j R_j^B}_{\text{allocation}}+\underbrace{\textstyle\sum_j w_j^P R_j^A}_{\text{selection}}, \qquad \text{IR}=\frac{E(R_A)}{\sqrt{\operatorname{var}(R_A)}}=\text{IC}\sqrt{\text{BR}} $$

Optimal active risk $=\frac{\text{IR}}{\text{SR}_B}\sigma_B$; Treynor-Black weights $\propto\alpha_i/\sigma_{\varepsilon i}^2$; the fundamental law is skill × breadth (× transfer coefficient under constraints). See Lesson 5.

International investing

Foreign return and the parities (Lesson 6)

$$ r^B\approx r^A+r_{A/B}, \qquad F_{A/B}=X_{A/B}\frac{1+i_B\frac{\tau}{360}}{1+i_A\frac{\tau}{360}}\ \text{(CIP)}, \qquad E(X_1)=\frac{1+i_{base}}{1+i_{high}}X_0\ \text{(UIP)} $$

Low cross-region correlations improve the frontier (despite home bias); cross rates obey $\text{A/C}=\text{A/B}\times\text{B/C}$ (else triangular arbitrage); the carry trade earns the rate spread but bears crash risk. See Lesson 6.

The IPS and investor objectives

An IPS fixes objectives, constraints and governance. Return/risk objectives are absolute or relative and must be consistent — required return is often inferred from a future goal by inflating it and solving $C_0(1+\mu)^T=\text{goal}$. Risk tolerance combines ability (objective) and willingness (psychological), erring conservative on conflict. Constraints: liquidity, horizon, legal/tax, unique needs. When return and risk objectives can't both be met on the CAL, an explicit compromise is required. See Lesson 7.

Trade execution and technology

Market orders guarantee speed (paying market impact); limit orders guarantee price (not fill). Large orders are sliced by VWAP/TWAP/POV algos and benchmarked by $\text{VWAP}=\sum n_ip_i/\sum n_i$. HFT tightens spreads but fuels a latency arms race; big data / machine learning (regression → neural nets → reinforcement learning) extend the toolkit, with over-fitting and opacity as the key risks. See Lesson 8.

Exam approach

For calculation questions, write the formula first, define the strategy, and explain step by step (realise profit in the same currency as the initial investment, per unit invested). For written questions, answer clearly and concisely with bullet points. Lecture material and problem sets are sufficient; extra readings help with open-ended questions.

Key takeaway

The course is one arc: mean-variance efficiency → equilibrium (CAPM) and no-arbitrage (APT/factors) pricing → risk (VaR/ES) and performance measurement → active value (IR, fundamental law) → international and applied implementation → execution. Master the handful of formulas above and the intuition behind each, and you have the whole syllabus.

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