50  Portfolio Management, CAPM and APT

50.1 What is Portfolio Management?

Portfolio Management (PM) is the art and science of selecting and overseeing a group of investments that meet the long-term financial objectives and risk tolerance of an investor. Modern PM is founded on Harry Markowitz’s Modern Portfolio Theory (1952) — for which he won the Nobel Prize in 1990.

TipWorking definitions
Author Definition
Markowitz (1952) “The selection of portfolios that maximises expected return for a given level of risk, or minimises risk for a given expected return.”
Sharpe “Process of allocating funds among different assets to achieve diversification benefits and meet investor’s objectives.”
Fischer & Jordan “Combination of securities such that they provide the most favourable risk-return trade-off.”
CFA Institute “Continuous process of constructing, monitoring and rebalancing portfolios in line with the investment policy statement.”

50.2 Portfolio Management Process

flowchart LR
  IPS[1. IPS<br/>Objectives & Constraints] --> AA[2. Asset<br/>Allocation]
  AA --> SS[3. Security<br/>Selection]
  SS --> EX[4. Execution]
  EX --> MON[5. Monitor &<br/>Rebalance]
  MON --> EV[6. Evaluate<br/>Performance]
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

TipSix-step PM process
  1. Investment Policy Statement (IPS) — objectives, constraints, risk tolerance.
  2. Asset Allocation — across asset classes.
  3. Security Selection — within asset class.
  4. Execution — trading.
  5. Monitoring and Rebalancing.
  6. Performance Evaluation.

50.3 Risk and Return — Single Asset

TipSingle-asset measures
  • Expected Return: E(R) = Σ pᵢ × Rᵢ.
  • Variance: σ² = Σ pᵢ × (Rᵢ − E(R))².
  • Standard Deviation (σ): √Variance.
  • Coefficient of Variation (CV): σ / E(R) — risk per unit of return.

50.4 Portfolio Risk and Return

50.4.1 Portfolio Return

\[E(R_p) = w_1 E(R_1) + w_2 E(R_2) + \ldots + w_n E(R_n)\]

Weighted average of component returns — simple.

50.4.2 Portfolio Risk (Two-Asset Case)

\[\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2\]

Where ρ₁₂ = correlation coefficient between assets 1 and 2.

NotePower of diversification

Portfolio risk is not a simple weighted average; it depends on correlations. When ρ < 1, diversification reduces risk; when ρ = −1, perfect hedge possible (risk can theoretically be zero). When ρ = +1, no diversification benefit.

50.4.3 Multi-Asset Portfolio Variance

\[\sigma_p^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \sigma_{ij}\]

Where σᵢⱼ is covariance between assets i and j.

50.5 Markowitz’s Modern Portfolio Theory (1952)

Harry Markowitz’s “Portfolio Selection” (Journal of Finance, 1952) — investors should choose portfolios based on mean (expected return) and variance (risk) considerations. The set of optimal portfolios forms the Efficient Frontier.

50.5.1 MPT Assumptions

TipMarkowitz assumptions
  • Investors are rational and risk-averse.
  • Decisions based on mean and variance of returns.
  • Investors maximise utility for a given level of risk.
  • Markets are efficient; information freely available.
  • One-period investment horizon.
  • No transaction costs or taxes.
  • Returns are normally distributed.

50.5.2 The Efficient Frontier

TipEfficient Frontier

The set of portfolios that offer the maximum expected return for a given level of risk (or minimum risk for a given expected return). - Portfolios on the frontier are efficient. - Portfolios below the frontier are inefficient. - Investor chooses based on utility function (risk preference). - The frontier is upward-sloping and concave.

50.5.3 Capital Market Line (CML)

When a risk-free asset is introduced, all rational investors hold combinations of the Risk-Free Asset and the Market Portfolio (M). The locus of optimal portfolios becomes a straight line — the Capital Market Line:

\[E(R_p) = R_f + \frac{(E(R_m) - R_f)}{\sigma_m} \times \sigma_p\]

The slope is the Market Price of Risk (Sharpe ratio of the market portfolio).

50.5.4 Tobin’s Separation Theorem

James Tobin (1958) — under MPT with a risk-free asset, investors:

  1. First choose the same risky portfolio (the market portfolio M) — investment decision.
  2. Then choose how much to allocate to M vs Rf based on personal risk tolerance — financing decision.

Tobin won the Nobel Prize 1981.

50.6 Single Index / Market Model — Sharpe (1963)

William Sharpe (1963) — “A Simplified Model for Portfolio Analysis” — simplified Markowitz by relating each security’s return to a single common index (market):

\[R_i = \alpha_i + \beta_i R_m + e_i\]

Where: - αᵢ = constant. - βᵢ = sensitivity to market. - eᵢ = idiosyncratic / unsystematic random error.

This reduced computational complexity from N(N+3)/2 inputs (Markowitz) to 3N + 2 inputs.

50.7 Systematic vs Unsystematic Risk

TipTwo components of total risk
Type Source Diversifiable?
Systematic / Market Risk Macro factors — interest rates, inflation, war, recession No — cannot be diversified away
Unsystematic / Specific Risk Firm/industry-specific — strikes, lawsuits, mgmt change Yes — diversifiable

Total Risk = Systematic + Unsystematic Risk.

NoteThe 20-30 stock rule

Empirical research (Evans-Archer 1968; Statman 1987) shows that 20-30 well-diversified stocks eliminate most unsystematic risk. Beyond ~30, the marginal benefit of additional diversification is small.

50.8 Beta — Measure of Systematic Risk

\[\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \frac{\rho_{im} \sigma_i}{\sigma_m}\]

Beta measures the sensitivity of a security’s return to the market’s return. β of the market itself = 1.

TipInterpreting Beta
β value Interpretation
β = 1 Moves with market
β > 1 Aggressive; magnifies market moves (autos, tech)
β < 1 Defensive; cushioned (FMCG, utilities)
β = 0 Risk-free asset; no market sensitivity
β < 0 Counter-cyclical (e.g., gold sometimes)

50.9 Capital Asset Pricing Model (CAPM)

William Sharpe (1964), John Lintner (1965), Jan Mossin (1966) — independently developed the CAPM:

\[E(R_i) = R_f + \beta_i \times (E(R_m) - R_f)\]

The expected return on a security = risk-free rate + beta × market risk premium. Sharpe won the Nobel Prize 1990 along with Markowitz.

50.9.1 CAPM Assumptions

TipCAPM’s strong assumptions
  • All investors have homogeneous expectations.
  • All can borrow and lend at the risk-free rate.
  • No taxes or transaction costs.
  • All assets are infinitely divisible.
  • No information asymmetry.
  • Single-period horizon.
  • Mean-variance optimisers.
  • Markets are in equilibrium.

50.9.2 Security Market Line (SML)

The graphical representation of CAPM — plots Expected Return on Y-axis against Beta on X-axis. Slope = market risk premium (Rm − Rf).

TipSML interpretation
  • All correctly-priced securities lie on the SML.
  • Securities above SML are undervalued (offer return > required).
  • Securities below SML are overvalued (offer return < required).

50.9.3 CML vs SML

TipCML vs SML
Dimension CML SML
X-axis Total risk (σ) Systematic risk (β)
Applies to Efficient portfolios only All securities and portfolios
Slope (Rm − Rf) / σm (Rm − Rf)
Equilibrium point All on the line All correctly-priced on the line

50.10 Criticisms and Empirical Tests of CAPM

TipCriticisms of CAPM
  • Unrealistic assumptions — perfect markets, homogeneous expectations.
  • Single-period model — investments are multi-period.
  • Market portfolio unobservable (Roll’s Critique 1977).
  • Empirical failures — β alone doesn’t explain returns; small firms and value stocks outperform.
  • Fama-French (1992) — book-to-market and size matter; β has weak explanatory power.
  • Risk-free rate unobservable in some markets.
  • Static — ignores time variation.
NoteRoll’s Critique (1977)

Richard Roll (1977) — CAPM is fundamentally untestable because the true market portfolio is unobservable (must include all assets globally — stocks, bonds, real estate, human capital). Any test of CAPM is actually a joint test of CAPM and the proxy used for the market portfolio.

50.11 Multi-Factor Models

50.11.1 Fama-French Three-Factor Model (1992)

Eugene Fama and Kenneth French (1992, 1993) — single β is insufficient. Add two more factors:

\[R_i - R_f = \alpha + \beta_M (R_m - R_f) + \beta_{SMB} \times SMB + \beta_{HML} \times HML\]

  • SMB — Small Minus Big (size effect)
  • HML — High Minus Low (value effect)

50.11.2 Fama-French Five-Factor Model (2015)

Adds profitability (RMW) and investment (CMA) factors:

\[R_i - R_f = \alpha + \beta_M (R_m-R_f) + \beta_{SMB} SMB + \beta_{HML} HML + \beta_{RMW} RMW + \beta_{CMA} CMA\]

50.11.3 Carhart Four-Factor Model (1997)

Mark Carhart added momentum (UMD = Up Minus Down) to Fama-French three-factor.

50.12 Arbitrage Pricing Theory (APT)

Stephen Ross (1976) — “The Arbitrage Theory of Capital Asset Pricing” — relaxed CAPM’s assumptions. APT says expected return is a linear function of multiple macroeconomic factors:

\[E(R_i) = R_f + \beta_{i,1} \lambda_1 + \beta_{i,2} \lambda_2 + \ldots + \beta_{i,k} \lambda_k\]

Where λⱼ = risk premium for factor j; βᵢ,ⱼ = sensitivity of asset i to factor j.

50.12.1 APT Common Factors (Chen-Roll-Ross 1986)

TipCommon APT factors
  • Unexpected changes in inflation.
  • Unexpected changes in industrial production.
  • Unexpected shifts in risk premia (credit spreads).
  • Unexpected shifts in the yield curve (term structure).
  • Industry factors.
  • Market factors.

50.12.2 APT vs CAPM

TipAPT vs CAPM
Dimension CAPM APT
Factors Single (market) Multiple
Foundation Equilibrium No-arbitrage
Assumptions Strong Weaker
Identification of factors Specified (market) Not specified — empirical
Risk premium Market risk premium Multiple factor premia

50.13 Portfolio Performance Evaluation

Same risk-adjusted measures introduced in Topic 45:

TipPortfolio performance measures
Measure Formula Risk metric
Sharpe Ratio (Rp − Rf) / σp Total risk (SD)
Treynor Ratio (Rp − Rf) / βp Systematic risk (β)
Jensen’s Alpha Rp − [Rf + βp(Rm − Rf)] Excess over CAPM
Information Ratio Active Return / Tracking Error vs benchmark
Sortino Ratio (Rp − Rf) / Downside Deviation Downside risk
M² (Modigliani²) Sharpe × σm + Rf Risk-adjusted to market σ

50.13.1 Fama Decomposition

Eugene Fama (1972) decomposed portfolio return into:

TipFama’s components of return
  • Selectivity — return from picking under-priced securities (within risk class).
  • Diversification — return from broader diversification.
  • Net Selectivity = Selectivity − Diversification.
  • Risk = Return from bearing additional risk.

50.14 Portfolio Strategies

TipPortfolio management strategies
Strategy Description
Active Management Beat the benchmark; security selection + market timing
Passive Management Track the benchmark; index funds, ETFs
Buy & Hold Long-term; minimal rebalancing
Constant Weighting Periodic rebalancing to target weights
Tactical Asset Allocation Adjust based on market conditions
Strategic Asset Allocation Long-run target mix
Core-Satellite Passive core + active satellite
Smart Beta Rules-based factor-tilted strategies
Risk Parity Equal risk contribution across asset classes (Bridgewater All-Weather)
120/Age Rule Equity % = 120 − age
Black-Litterman (1990) Combines market equilibrium with investor views

50.15 Behavioural Portfolio Theory

Hersh Shefrin and Meir Statman (2000) — investors hold mental accounts with different risk tolerances; portfolios shaped like pyramids (low-risk base + speculative top), not mean-variance efficient.

50.16 Indian Portfolio Management Industry

TipIndian PM industry — key elements
  • Portfolio Management Services (PMS) — minimum investment ₹50 lakh (SEBI 2019).
  • Mutual Funds (MFs) — regulated by SEBI (MF) Regulations 1996.
  • AMFI — Association of Mutual Funds in India.
  • Alternative Investment Funds (AIFs) — Category I, II, III; minimum ₹1 cr.
  • NPS — National Pension System; managed by PFs under PFRDA.
  • Insurance ULIPs — regulated by IRDAI.
  • PMS Categories: Discretionary, Non-Discretionary, Advisory.
  • Smallcase — modern Indian thematic portfolios.
  • Investment Advisers (IAs) — registered under SEBI IA Regulations 2013.

50.18 Practice Questions

Q 01 Markowitz Easy

Modern Portfolio Theory was introduced in 1952 by:

  • AHarry Markowitz
  • BWilliam Sharpe
  • CStephen Ross
  • DEugene Fama
View solution
Correct Option: A
Harry Markowitz — *Portfolio Selection*, JF 1952; Nobel 1990.
Q 02 CAPM Easy

The CAPM equation is:

  • ARi = Rf + β(Rm − Rf)
  • BRi = α + β Rm
  • CRi = Σ βᵢ λᵢ
  • DRi = D/P + g
View solution
Correct Option: A
**CAPM**: Rf + β × Market Risk Premium.
Q 03 Beta Easy

A stock with β = 1.5:

  • AMoves opposite to market
  • BLess volatile than market
  • CMore volatile than market
  • DRisk-free
View solution
Correct Option: C
β > 1 → aggressive; 1.5x market sensitivity.
Q 04 Systematic Easy

Systematic risk is:

  • AFirm-specific and diversifiable
  • BMarket-wide and non-diversifiable
  • CAlways zero
  • DIndustry-specific
View solution
Correct Option: B
Macro factors → cannot be diversified away.
Q 05 Efficient Frontier Medium

The Efficient Frontier represents:

  • AAll possible portfolios
  • BPortfolios with max return for given risk (or min risk for given return)
  • CRisk-free portfolios only
  • D100 % equity portfolios
View solution
Correct Option: B
Optimal trade-offs — concave, upward-sloping.
Q 06 CML Medium

The Capital Market Line (CML) plots:

  • AReturn vs Beta
  • BReturn vs Total Risk (σ)
  • CBeta vs Variance
  • DReturn vs Dividend Yield
View solution
Correct Option: B
CML — return vs **total risk (σ)**; SML — return vs **β**.
Q 07 Tobin Hard

The Separation Theorem in portfolio theory was given by:

  • AJames Tobin
  • BModigliani & Miller
  • CSharpe
  • DMarkowitz
View solution
Correct Option: A
James Tobin (1958); Nobel 1981.
Q 08 APT Medium

Arbitrage Pricing Theory was developed in 1976 by:

  • AStephen Ross
  • BEugene Fama
  • CRobert Merton
  • DJack Treynor
View solution
Correct Option: A
Stephen Ross (1976); multi-factor; no-arbitrage foundation.
Q 09 Fama-French Medium

The Fama-French 3-factor model adds which factors to the market premium?

  • ASMB and HML
  • BRMW and CMA
  • CUMD and Quality
  • DBeta and Alpha
View solution
Correct Option: A
SMB (size) and HML (value). 5-factor adds RMW + CMA.
Q 10 Carhart Hard

The Carhart 4-factor model adds which factor to Fama-French?

  • AQuality
  • BMomentum (UMD)
  • CVolatility
  • DDividend
View solution
Correct Option: B
Momentum (Up-Minus-Down) — Carhart 1997.
Q 11 Roll's critique Hard

Roll's Critique of CAPM (1977) is that:

  • AThe risk-free rate is wrong
  • BThe true market portfolio is unobservable
  • Cβ is always zero
  • DCAPM has too few factors
View solution
Correct Option: B
**Richard Roll (1977)** — true market portfolio unobservable; CAPM untestable.
Q 12 Portfolio risk Medium

Diversification reduces portfolio risk because:

  • AReturns are negatively correlated
  • BAsset correlations are less than +1
  • CAll returns are random
  • DVariance is zero
View solution
Correct Option: B
When ρ < 1, diversification reduces risk; benefits maximised when ρ = −1.
Q 13 Jensen's Alpha Medium

Jensen's Alpha measures:

  • AExcess return over CAPM-predicted return
  • BTotal return
  • CTracking error
  • DSharpe ratio
View solution
Correct Option: A
α = Rp − [Rf + β(Rm − Rf)]. Positive α → out-performance.
Q 14 PMS minimum Medium

SEBI minimum investment for PMS in India is:

  • A₹5 lakh
  • B₹25 lakh
  • C₹50 lakh
  • D₹1 crore
View solution
Correct Option: C
₹50 lakh — raised from ₹25 lakh in 2019.
Q 15 Sharpe model Medium

Sharpe's Single Index Model relates a security's return to:

  • AAll other securities
  • BA single market index
  • CMultiple macro factors
  • DDividend yields only
View solution
Correct Option: B
Ri = αᵢ + βᵢ Rm + eᵢ — simplifies Markowitz drastically.
Q 16 Above SML Medium

A security plotted *above* the SML is:

  • ACorrectly priced
  • BUndervalued
  • COvervalued
  • DRisk-free
View solution
Correct Option: B
Offers higher return than CAPM-required → undervalued; buy.
Q 17 Black-Litterman Hard

The Black-Litterman model (1990) combines:

  • AMarket equilibrium with investor views
  • BBonds with equities
  • CCAPM with APT
  • DFundamental with technical analysis
View solution
Correct Option: A
Fischer Black and Robert Litterman (Goldman 1990) — Bayesian blend.
Q 18 Diversification Medium

Empirical research (Evans-Archer, Statman) suggests most unsystematic risk can be eliminated with:

  • A5 stocks
  • B20-30 stocks
  • C100 stocks
  • D500 stocks
View solution
Correct Option: B
20-30 well-diversified stocks.
Q 19 Risk parity Hard

"Risk Parity" portfolio strategy is associated with:

  • AVanguard
  • BBridgewater (All-Weather)
  • CBlackRock
  • DBerkshire Hathaway
View solution
Correct Option: B
Bridgewater's All-Weather fund — Ray Dalio's risk-parity approach.
Q 20 Match models Hard

Match the model with its founder:

(i) MPT (a) Sharpe-Lintner-Mossin
(ii) CAPM (b) Stephen Ross
(iii) APT (c) Harry Markowitz
(iv) Separation Theorem (d) James Tobin
  • A(i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
  • B(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
  • C(i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
  • D(i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
View solution
Correct Option: A
MPT — Markowitz; CAPM — Sharpe-Lintner-Mossin; APT — Ross; Separation — Tobin.

50.18.1 Advanced Format Questions

AR 1Assertion-ReasonHard

A: CAPM uses beta to price securities.
R: Beta measures unsystematic risk.

  • ABoth true; R explains A
  • BBoth true; R does not explain A
  • CA true, R false
  • DA false, R true
View solution
Correct Option: C
Beta measures systematic (market) risk, not unsystematic.
S 1Statement-basedMedium

Portfolio risk measures: (i) Variance. (ii) SD. (iii) Beta. (iv) VaR.

  • AAll four
  • B(i) and (ii) only
  • C(iii) and (iv) only
  • D(iii) only
View solution
Correct Option: A
N 1NumericalMedium

Rf = 6%; Rm = 14%; β = 1.5. CAPM expected return:

  • A18 %
  • B12 %
  • C14 %
  • D20 %
View solution
Correct Option: A
E(R) = 6 + 1.5(14−6) = 6 + 12 = 18%.
N 2NumericalHard

Portfolio: 60% in A (β=1.2), 40% in B (β=0.8). Portfolio beta:

  • A1.04
  • B1.00
  • C2.00
  • D0.96
View solution
Correct Option: A
0.6 × 1.2 + 0.4 × 0.8 = 0.72 + 0.32 = 1.04.

50.19 Quick Recall

ImportantQuick recall
  • MPT — Markowitz (1952, Nobel 1990): mean-variance optimisation; Efficient Frontier.
  • 6-step PM process: IPS → AA → SS → Execute → Monitor → Evaluate.
  • Single asset: E(R), σ², σ, CV.
  • Portfolio risk (2-asset): σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂; ρ < 1 → diversification benefits.
  • CML — Capital Market Line: introduce Rf → Rf + (Rm − Rf)/σm × σp; all efficient portfolios; total risk.
  • Tobin’s Separation Theorem (1958, Nobel 1981): separate investment (M) from financing (Rf vs M).
  • Single Index — Sharpe (1963): Ri = αᵢ + βᵢ Rm + eᵢ; simplifies Markowitz.
  • Risk types: Systematic (market, non-diversifiable) + Unsystematic (firm-specific, diversifiable).
  • Empirical: 20-30 stocks eliminate most unsystematic risk (Evans-Archer 1968, Statman 1987).
  • Beta: Cov(Ri, Rm)/σm²; β=1 market; >1 aggressive; <1 defensive; <0 counter-cyclical.
  • CAPM — Sharpe (1964), Lintner (1965), Mossin (1966), Nobel 1990: Ri = Rf + β(Rm − Rf).
  • SML — Security Market Line: Y = E(R), X = β; above SML → undervalued; below → overvalued.
  • CML vs SML: CML uses σ (efficient portfolios); SML uses β (all assets).
  • Roll’s Critique (1977): true market portfolio unobservable → CAPM untestable.
  • Fama-French 3-factor (1992): Market + SMB (size) + HML (value).
  • Fama-French 5-factor (2015): + RMW (profitability) + CMA (investment).
  • Carhart 4-factor (1997): + UMD (momentum).
  • APT — Ross (1976): multi-factor, no-arbitrage; Chen-Roll-Ross (1986) common factors: inflation, IP, risk premia, yield curve.
  • Performance measures: Sharpe · Treynor · Jensen’s α · Information ratio · Sortino · M² · Fama decomposition.
  • PM strategies: Active · Passive · Buy-Hold · Constant Weight · TAA · SAA · Core-Satellite · Smart Beta · Risk Parity (Bridgewater) · 120/Age · Black-Litterman (1990).
  • Behavioural PM (Shefrin-Statman 2000): mental accounts; pyramid portfolios.
  • India: PMS ≥ ₹50 lakh (SEBI 2019) · MF Regs 1996 · AMFI · AIFs Cat I/II/III ≥ ₹1 cr · NPS/PFRDA · ULIPs/IRDAI · IA Regs 2013 · Smallcase.
  • Modern trends: Robo-advisors · Smart beta / factor ETFs · ESG · Direct indexing · Thematic · Crypto · AI portfolios · Risk parity · Black-Litterman · Tax-loss harvesting · PE/VC · Liquid alts.