flowchart LR E[Efficient Frontier:<br/>Max return per unit of risk] --> O[Optimal Portfolio:<br/>Tangency with investor's<br/>indifference curve] ID[Investor's<br/>Indifference Curves] -. tangent .- O RF[Risk-free Asset] -. CML emerges .- O style E fill:#E8F5E9,stroke:#2E7D32 style O fill:#FCE4EC,stroke:#AD1457
49 Portfolio Management: CAPM and APT
49.1 What is a Portfolio?
A portfolio is a collection of financial assets — equities, bonds, cash, real estate, derivatives — held by an individual or institutional investor. Portfolio management is the art and science of selecting and managing the mix of assets to meet specified investment objectives within a chosen risk budget.
The discipline rests on a single, classical insight from Harry Markowitz’s 1952 paper “Portfolio Selection” — for which Markowitz received the Nobel Prize in 1990 (markowitz1952?). The insight: what matters is the risk-return profile of the portfolio, not of individual assets. Diversification reduces risk without proportionally reducing expected return.
TipThree Working Definitions
| Source | Definition | What it foregrounds |
|---|---|---|
| Markowitz (1952) | “Portfolio selection is the choice of an efficient combination of risky securities.” | Efficiency |
| Brealey-Myers-Allen | “Portfolio management is the systematic combination of securities to obtain the desired risk-return trade-off.” | Trade-off |
| Sharpe (1970) | “Portfolio theory provides a framework for the systematic combination of risky securities.” | Framework |
49.2 Markowitz’s Portfolio Theory
Markowitz’s framework rests on four assumptions:
TipMarkowitz’s Four Assumptions
| # | Assumption |
|---|---|
| 1 | Investors evaluate portfolios on the basis of expected return and risk (variance) over a single period |
| 2 | Investors are risk-averse |
| 3 | Investors prefer higher expected return for given risk and lower risk for given return |
| 4 | Markets are perfect — no taxes, no transaction costs, no information asymmetry |
49.2.1 Expected return and risk of a portfolio
For a portfolio of two assets:
\[E(R_p) = w_1 E(R_1) + w_2 E(R_2)\]
\[\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2\]
The correlation coefficient \(\rho_{12}\) between the two assets drives the diversification benefit:
TipDiversification by Correlation
| Correlation | Diversification effect |
|---|---|
| ρ = +1 | No diversification — risk is the weighted sum |
| 0 < ρ < 1 | Some diversification |
| ρ = 0 | Substantial diversification |
| ρ = −1 | Risk can be eliminated entirely |
49.2.2 Efficient frontier and the optimal portfolio
The efficient frontier is the set of portfolios that offer the maximum expected return for each level of risk (or, equivalently, minimum risk for each return). All other combinations are dominated — beneath the frontier.
49.2.3 Capital Market Line (CML)
When a risk-free asset is added, the efficient frontier becomes a straight line — the Capital Market Line — running from the risk-free rate, tangent to the efficient frontier of risky assets, at the market portfolio:
\[E(R_p) = R_f + \frac{E(R_m) - R_f}{\sigma_m} \cdot \sigma_p\]
The slope \((E(R_m) - R_f) / \sigma_m\) is the Sharpe ratio of the market — the price of risk.
49.3 Capital Asset Pricing Model (CAPM)
William Sharpe (Nobel 1990), John Lintner and Jan Mossin extended Markowitz to derive the Capital Asset Pricing Model in the mid-1960s (sharpe1964?):
\[E(R_i) = R_f + \beta_i (E(R_m) - R_f)\]
where \(\beta_i = \dfrac{\text{Cov}(R_i, R_m)}{\sigma_m^2}\) is the asset’s systematic risk.
CAPM is the single most-tested model in modern finance. Three implications:
TipThree Implications of CAPM
| Implication | What it says |
|---|---|
| Only systematic risk is priced | Investors are not rewarded for diversifiable risk |
| Linear relationship | Expected return rises linearly with β |
| Single-factor | One macro factor (the market) explains all systematic returns |
49.3.1 Security Market Line (SML)
The SML is the graphical representation of CAPM — expected return on the Y-axis, β on the X-axis. Securities priced fairly lie on the line; under-priced securities lie above; over-priced below.
TipCML vs SML
| Feature | CML | SML |
|---|---|---|
| X-axis | Total risk (σ) | Systematic risk (β) |
| Plots | Efficient portfolios | All securities and portfolios |
| Slope | (E(R_m) − R_f) / σ_m | (E(R_m) − R_f) — market risk premium |
49.3.2 Limitations of CAPM
TipLimitations of CAPM
| Limitation |
|---|
| Single-period model |
| β is unstable over time |
| Empirical tests show that beta alone is a weak predictor |
| Fama and French (1992) found that size (SMB) and value (HML) factors matter beyond beta |
| Roll critique (1977) — the true market portfolio is unobservable |
49.4 Fama-French and Multi-Factor Models
Eugene Fama and Kenneth French’s 1992-93 papers extended CAPM to a three-factor model (famafrench1993?):
\[E(R_i) - R_f = \beta_i (E(R_m) - R_f) + s_i \cdot \text{SMB} + h_i \cdot \text{HML}\]
where SMB = Small Minus Big (size factor) and HML = High Minus Low (value factor — book-to-market). A 2015 five-factor extension added profitability and investment factors. Carhart’s four-factor model adds a momentum factor.
49.5 Arbitrage Pricing Theory (APT)
Stephen Ross’s APT (1976) replaces CAPM’s single market factor with multiple macroeconomic factors (ross1976?):
\[E(R_i) = R_f + \beta_{i,1} F_1 + \beta_{i,2} F_2 + \dots + \beta_{i,n} F_n\]
where each \(F_j\) is the risk premium associated with factor \(j\) (e.g., inflation, interest-rate term structure, industrial production, default spread).
TipCAPM vs APT
| Feature | CAPM | APT |
|---|---|---|
| Number of factors | One (market) | Many (macroeconomic) |
| Foundation | Mean-variance optimisation | Arbitrage |
| Identification of factors | Theory specifies the market | Theory is silent — empirics decide |
| Empirical fit | Weaker single-factor fit | Stronger multi-factor fit |
APT does not identify the factors — that is left to empirics. Common factors used in studies include unanticipated changes in inflation, GDP growth, term-spread, default-spread, and oil price.
49.6 Performance Measurement
Three classical risk-adjusted return measures — covered in every textbook on portfolio management.
TipThree Classical Risk-Adjusted Performance Measures
| Measure | Formula | What it captures |
|---|---|---|
| Sharpe ratio | (R_p − R_f) ÷ σ_p | Excess return per unit of total risk |
| Treynor ratio | (R_p − R_f) ÷ β_p | Excess return per unit of systematic risk |
| Jensen’s alpha | R_p − [R_f + β_p(R_m − R_f)] | Return in excess of CAPM prediction |
TipWhen to Use Which
| If the portfolio is | Use |
|---|---|
| Diversified — only systematic risk left | Treynor or Jensen |
| Less than fully diversified | Sharpe |
| Comparing manager skill against benchmark | Jensen’s α |
A more recent measure — the Information Ratio — divides active return by tracking error; widely used in active management.
49.7 Portfolio Management Process
TipStandard Portfolio Management Process
| # | Step | Activity |
|---|---|---|
| 1 | Investment policy statement | Risk tolerance, return goal, horizon, constraints |
| 2 | Strategic asset allocation | Long-run mix across asset classes |
| 3 | Tactical asset allocation | Short-run tilts |
| 4 | Security selection | Stock and bond picking |
| 5 | Portfolio implementation | Trade, settle, custody |
| 6 | Performance evaluation and rebalancing | Sharpe / Treynor / Jensen; periodic rebalance |
The single most important decision is strategic asset allocation — empirical research (Brinson, Hood and Beebower, 1986) finds that ≈ 90 per cent of the variance in portfolio returns is explained by asset allocation rather than security selection.
49.8 Practice Questions
Q 01
Markowitz
Easy
Modern portfolio theory was founded by:
View solution
Correct Option: B
Harry Markowitz's "Portfolio Selection" (1952) — Nobel Prize 1990. Sharpe extended Markowitz to derive CAPM.
Q 02
Diversification
Medium
For two risky assets, perfect diversification (zero portfolio variance) is theoretically possible when the correlation coefficient is:
View solution
Correct Option: C
When ρ = −1, the assets move in perfectly opposite directions; risk can be hedged away exactly. ρ = +1 → no diversification benefit.
Q 03
CAPM
Medium
Under CAPM, the expected return on a security depends on:
View solution
Correct Option: B
CAPM: E(Ri) = Rf + βi(Rm − Rf). Only systematic risk is priced.
Q 04
APT
Medium
The Arbitrage Pricing Theory (APT) was developed by:
View solution
Correct Option: B
Stephen Ross's 1976 paper. APT generalises CAPM to multiple factors, founded on arbitrage.
Q 05
Sharpe Ratio
Medium
The Sharpe ratio measures:
View solution
Correct Option: A
Sharpe = (Rp − Rf) ÷ σp — total risk. Treynor uses β; Jensen's α uses CAPM-predicted return; Information Ratio uses tracking error.
Q 06
Jensen Alpha
Medium
Jensen's alpha measures:
View solution
Correct Option: B
Jensen's α = Rp − [Rf + βp(Rm − Rf)] — excess return over the CAPM benchmark, attributed to manager skill.
Q 07
Fama French
Medium
Fama and French's three-factor model adds which two factors to the market factor?
View solution
Correct Option: A
Fama-French (1993): market + SMB (Small Minus Big — size) + HML (High Minus Low — value, book-to-market). The 2015 five-factor model adds profitability and investment.
Q 08
Brinson
Medium
The Brinson, Hood and Beebower (1986) study found that the largest share of variance in portfolio returns comes from:
View solution
Correct Option: A
Brinson et al. (1986) attributed roughly 90 per cent of variance in portfolio returns to strategic asset allocation — far more than security selection or market timing.
ImportantQuick recall
- Portfolio = collection of assets. Foundation: Markowitz (1952) — risk-return at the portfolio level, not the asset level.
- Diversification works through correlation: ρ = +1 no benefit; ρ = −1 can eliminate risk entirely.
- Efficient frontier = max return for each level of risk. Adding a risk-free asset gives the Capital Market Line (CML).
- CAPM (Sharpe-Lintner-Mossin): E(R) = Rf + β(Rm − Rf). Only systematic risk is priced.
- CML uses total risk (σ) for efficient portfolios; SML uses β for all securities.
- Fama-French 3-factor: market + SMB (size) + HML (value). 5-factor adds profitability and investment.
- APT (Ross 1976): multi-factor; theory silent on identity of factors.
- Performance measures: Sharpe (total risk), Treynor (β), Jensen’s α (CAPM benchmark), Information Ratio (tracking error).
- Brinson et al. (1986): ~90% of variance in returns explained by strategic asset allocation.