Unit 4: Index models and performance evaluation
Security Analysis and Portfolio Management notes · PTU syllabus (MCOP402-18)
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Unit summary
Index and factor models simplify portfolio analysis and help judge managers fairly. This unit covers the single index model — total, market and unique risk — arbitrage pricing theory and arbitrage portfolios, two-factor and multi-factor models, portfolio performance measures, risk-adjusted evaluation, market timing and evaluation criteria.
After this unit you can
- Explain the single index model and decompose risk
- Explain APT and arbitrage portfolios and multi-factor models
- Evaluate portfolio performance with risk-adjusted measures
- Explain market timing and performance evaluation criteria
PTU syllabus topics
- Single index model — total
- market and unique risk
- arbitrage pricing theory and arbitrage portfolios
- two-factor and multi-factor models
- portfolio performance measures
- risk-adjusted evaluation
- market timing and evaluation criteria
Sharpe ratio
(Rp − Rf) / σp
Return per unit of total risk
Treynor ratio
(Rp − Rf) / βp
Return per unit of market risk
Jensen's alpha
Rp − [Rf + βp (Rm − Rf)]
Positive means outperformance
Topic 1
Single index model (Sharpe)
- Assumption: securities move together only because of a common factor — the market index.
Return equation
Ri = αi + βi Rm + ei
Total risk of a security
σi² = βi² σm² + σei² (market risk + unique risk)
Portfolio beta
βp = Σ wi βi
Portfolio variance
σp² = βp² σm² + Σ wi² σei²
Example
β = 1.2, σm = 15%, σe = 10%: total variance = 1.44 × 225 + 100 = 424; systematic share = 324/424 ≈ 76%.
- Advantage over Markowitz: needs only 3n + 2 estimates instead of n(n + 3)/2.
- Sharpe's optimal portfolio: rank stocks by excess return to beta (Ri − Rf)/βi and include those above a cut-off rate C*.
Topic 2
Arbitrage pricing theory and arbitrage portfolios
- Stephen Ross (1976): expected return depends on several systematic factors, not just the market.
Expected return
E(Ri) = Rf + βi1 λ1 + βi2 λ2 + … + βik λk — λ factor risk premiums, β factor sensitivities
- Factors: inflation, industrial production, interest rate term structure, default risk premium, oil prices, exchange rates.
- Based on no-arbitrage — two portfolios with the same factor exposures must offer the same return.
- Vs CAPM: fewer assumptions, multiple factors, no need for the market portfolio; but factors are not specified by theory.
- Arbitrage portfolio: requires no investment (weights sum to zero), has zero sensitivity to factors, yet earns a positive return — its existence means securities are mispriced; arbitrage trading removes it.
Topic 3
Two-factor and multi-factor models
- Two-factor model: Ri = ai + bi1 F1 + bi2 F2 + ei (e.g., GDP growth and inflation).
- Fama–French three-factor model: market, size (SMB — small minus big), value (HML — high minus low book-to-market); extended to five factors (profitability, investment); Carhart adds momentum.
- Macroeconomic factor models (Chen, Roll, Ross): industrial production, inflation, term structure, default premium.
Topic 4
Portfolio performance measures and risk-adjusted evaluation
Sharpe ratio
(Rp − Rf) ÷ σp — reward per unit of total risk
Treynor ratio
(Rp − Rf) ÷ βp — reward per unit of systematic risk
Jensen's alpha
αp = Rp − [Rf + βp (Rm − Rf)]
| Fund | Return | σ | β | Sharpe | Treynor |
|---|---|---|---|---|---|
| A | 15% | 18% | 1.1 | 0.50 | 8.18 |
| B | 13% | 12% | 0.8 | 0.58 | 8.75 |
| Market | 12% | 14% | 1.0 | 0.43 | 6.00 |
Example
With Rf = 6%: Fund B ranks higher on both Sharpe and Treynor. Jensen's alpha for A = 15 − [6 + 1.1 × 6] = 2.4%; for B = 13 − [6 + 0.8 × 6] = 2.2% — both beat the market on a risk-adjusted basis.
- Sharpe for undiversified portfolios (total risk matters); Treynor and Jensen for well-diversified portfolios.
- Fama's decomposition: return due to selectivity and due to risk.
- Information ratio: active return ÷ tracking error.
Topic 5
Market timing and evaluation criteria
- Market timing: shifting between equities and cash/bonds based on market forecasts — a good timer has higher beta in rising markets and lower beta in falling markets.
- Tests: Treynor–Mazuy (quadratic term in the characteristic line), Henriksson–Merton (dummy variable for up and down markets).
- Evaluation criteria: returns relative to benchmark and peers, risk-adjusted measures, consistency over time, style analysis, costs (expense ratio), survivorship bias awareness; SEBI's riskometer and benchmark disclosures for mutual funds.
Key terms
- Single index model
- Model linking security returns to a market index
- Unique risk
- Firm-specific risk reducible by diversification
- Arbitrage portfolio
- Zero-investment, zero-risk portfolio with positive return
- Fama–French model
- Three-factor model of market, size and value
- Market timing
- Shifting exposure based on market forecasts
Quick revision
- SIM: R = α + βRm + e; total risk = market + unique risk.
- Sharpe's cut-off method for optimal portfolios.
- APT and arbitrage portfolios; two-factor and multi-factor models.
- Sharpe, Treynor, Jensen, information ratio.
- Market timing tests: Treynor–Mazuy, Henriksson–Merton.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.State the single index model.
- Q2.Decompose total risk under the single index model.
- Q3.What is an arbitrage portfolio?
- Q4.Name the Fama–French factors.
- Q5.What is the information ratio?
- Q6.What is market timing?
Long-answer questions
- Q1.Explain the single index model and Sharpe's optimal portfolio construction.
- Q2.Explain APT, arbitrage portfolios and multi-factor models.
- Q3.Explain risk-adjusted measures of portfolio performance.
- Q4.Explain market timing and the criteria for evaluating portfolio performance.
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