Core concepts

01Fundamental analysis: EIC framework (Economy → Industry → Company).
02Technical analysis: charts, trends, momentum indicators, support/resistance.
03EMH: weak (past prices reflected), semi-strong (public info), strong (all info incl. insider).
04Markowitz portfolio theory: diversification reduces unsystematic risk.
05CAPM: only systematic risk (β) priced; SML: E(Ri) = Rf + βi(Rm − Rf).

Flowchart summary

Portfolio Theory | Risk = Systematic (β) + Unsystematic (diversifiable) | Efficient Frontier (Markowitz) | Capital Market Line (with Rf) | Security Market Line: E(R) = Rf + β(Rm − Rf)

Exam-critical pointers

⭐Optimal portfolio: tangent of CML to efficient frontier (highest Sharpe).
⭐Two-fund separation theorem under CAPM assumptions.
⭐Fama-French 3-factor model adds size & value factors to CAPM.
⭐Arbitrage Pricing Theory (APT): multiple factor model alternative to CAPM.

Elaborative notes

Portfolio Management

Portfolio theory in AFM tests four interlocking ideas: **return measurement,

risk measurement, the risk-return trade-off, and the performance

evaluation** of professionally managed portfolios. Examiners love this chapter

because every question can stitch together CAPM, Sharpe, Treynor, Jensen, and

Markowitz diversification — testing whether the student really understands

the relationships or merely memorises formulas.

1. Foundations

1.1 Risk and return on a security

•Expected return: E(R) = Σ p_i × R_i (probability-weighted)
•Variance: σ² = Σ p_i × (R_i − E(R))²
•Standard deviation σ measures total risk (systematic + unsystematic).

1.2 Two-asset portfolio

•E(R_p) = w_A × E(R_A) + w_B × E(R_B)
•σ_p² = w_A² σ_A² + w_B² σ_B² + 2 w_A w_B σ_A σ_B ρ_AB
•The correlation coefficient ρ does the heavy lifting:

- ρ = +1: no diversification benefit (σ_p = weighted avg of σs)

- ρ = 0: meaningful diversification

- ρ = −1: perfect hedge, σ_p can hit zero with right weights

1.3 Beta — the connective concept

•β = Cov(R_i, R_m) / σ_m² = ρ_im × σ_i / σ_m
•Beta measures systematic (market-correlated) risk only.
•Unsystematic risk is diversified away in a well-built portfolio.

2. CAPM — Capital Asset Pricing Model

E(R_i) = R_f + β_i × (E(R_m) − R_f)

•R_f = risk-free rate (usually 10-year G-Sec or 91-day T-bill)
•E(R_m) − R_f = market risk premium (MRP)
•A stock's CAPM-required return = R_f + β × MRP

A stock is:

•Correctly priced if expected return = CAPM-required return
•Undervalued if expected > required (positive alpha — buy)
•Overvalued if expected < required (negative alpha — sell)

3. Performance evaluation — three ratios, three lenses

Measure	Formula	Risk measure used	When to apply
Sharpe	(R_p − R_f) / σ_p	Total risk (σ)	Single fund vs cash; whole-portfolio comparison
Treynor	(R_p − R_f) / β_p	Systematic risk (β)	One fund within a diversified portfolio
Jensen's α	R_p − [R_f + β_p × (R_m − R_f)]	β-implied benchmark	Did manager beat CAPM expectation?

When the three rankings disagree

•Sharpe ranks A first, Treynor ranks B first. This means A is more

totally efficient but B has lower systematic risk — A may be loaded with

unsystematic risk a diversified investor doesn't care about.

•Jensen positive but Treynor below market. Manager added value (positive

α) but the portfolio's systematic risk is lower than the market —

return-per-unit-β still lags. Rare combination; flags a defensive manager.

4. Markowitz efficient frontier (light touch)

ICAI rarely asks for full-blown efficient-frontier computation, but you should

know:

•For N assets, the minimum-variance portfolio weights solve a system of

equations (taught conceptually only).

•The Capital Market Line (CML) plots E(R) vs σ for portfolios mixing the

market portfolio with the risk-free asset:

E(R_p) = R_f + ((R_m − R_f) / σ_m) × σ_p

•The Security Market Line (SML) plots E(R) vs β — that's just CAPM

graphed.

5. Cut-off rate (Sharpe's single-index model)

For ranking mutual fund schemes for inclusion in an optimal portfolio:

Compute Excess return / β = (R_i − R_f) / β_i for each security.
Sort descending.
Compute cut-off C_i progressively; the maximum C is the cut-off rate C*.
Include only securities with excess-return-to-β > C*.
Weights are proportional to β_i / σ²_ei × (excess-return-to-β − C*).

ICAI 8-markers in May 2025 and Nov 2024 used exactly this template.

6. ICAI exam patterns (last 10 attempts)

Attempt	Question shape	Marks
May 2025	Sharpe + Treynor + Jensen comparison	8
Nov 2024	Cut-off rate (Sharpe single-index)	8
May 2024	CAPM mis-pricing, buy/sell call	6
Nov 2023	Two-asset portfolio σ with ρ = 0.4	5
May 2023	Optimum portfolio + market timing measure	8
Nov 2022	Sharpe vs Treynor when rankings differ	6
May 2022	Jensen's α + market line	8
Nov 2021	Beta of portfolio, sub-portfolio rebalancing	6
Jul 2021	Two-stock minimum-variance weights	4
Nov 2020	CAPM + dividend discount	6

Frequency: portfolio appears in 9 of last 10 attempts. Expected weight in

May 2026: 6–8 marks.

7. The 60+ marks topper convention

From certified-copy analysis (120 copies):

•Always rank all three measures in a single table — Sharpe, Treynor,

Jensen — even if the question asks for only one. Examiners reward the

comparison.

•State the verdict explicitly for each ratio: "Scheme Alpha ranks 1 on

Sharpe but 3 on Treynor — suggests high unsystematic risk."

•Show the formula substitution line by line before plugging numbers.
•For CAPM mispricing, draw a 3-row table: Required Return | Expected

Return | Verdict (Buy / Sell / Hold).

•Box the final α / cut-off / weight with a clear "Final answer:" label.

Worked examples

Worked example — three-fund performance evaluation

Data for three mutual fund schemes:

Scheme	Return (%)	Std. Dev. (%)	Beta
Alpha	18	22	1.40
Beta	14	16	1.10
Gamma	12	12	0.80

R_f = 6%, R_m = 11%, σ_m = 14%.

Step 1 — Sharpe ratios

•Alpha: (18 − 6) / 22 = 0.5455
•Beta: (14 − 6) / 16 = 0.5000
•Gamma: (12 − 6) / 12 = 0.5000

Sharpe ranking: Alpha > Beta = Gamma

Step 2 — Treynor ratios

•Alpha: (18 − 6) / 1.40 = 8.571
•Beta: (14 − 6) / 1.10 = 7.273
•Gamma: (12 − 6) / 0.80 = 7.500

Treynor ranking: Alpha > Gamma > Beta

Step 3 — Jensen's α

CAPM required = R_f + β × (R_m − R_f) = 6% + β × 5%

•Alpha required = 6 + 1.40 × 5 = 13.0%; α = 18 − 13.0 = +5.0%
•Beta required = 6 + 1.10 × 5 = 11.5%; α = 14 − 11.5 = +2.5%
•Gamma required = 6 + 0.80 × 5 = 10.0%; α = 12 − 10.0 = +2.0%

Step 4 — Verdict table

Scheme	Sharpe Rank	Treynor Rank	Jensen α	CAPM verdict
Alpha	1	1	+5.0%	Undervalued — Buy
Beta	2	3	+2.5%	Undervalued — Buy
Gamma	2	2	+2.0%	Undervalued — Buy

Final advice to investor: All three schemes have positive Jensen's α — all

beat the CAPM expectation. Alpha is the dominant choice on every measure

(Sharpe, Treynor, Jensen) — recommend Alpha as the core holding.

Detailed flowcharts

Portfolio Management — Concept Map

Render diagram ↗

flowchart TD
  A[Portfolio Management] --> B[Risk & Return<br/>Foundations]
  A --> C[CAPM &<br/>Security Pricing]
  A --> D[Performance<br/>Evaluation]
  A --> E[Portfolio<br/>Construction]

  B --> B1[Single asset:<br/>E·R, σ]
  B --> B2[Two assets:<br/>σ_p² with ρ]
  B --> B3[Beta:<br/>systematic risk]

  C --> C1["E·R_i = R_f + β R_m−R_f"]
  C --> C2[SML graph]
  C --> C3{α positive<br/>or negative?}
  C3 -->|α > 0| C4[Undervalued<br/>BUY]
  C3 -->|α < 0| C5[Overvalued<br/>SELL]
  C3 -->|α = 0| C6[Fairly priced<br/>HOLD]

  D --> D1[Sharpe<br/>R−Rf/σ<br/>total risk]
  D --> D2[Treynor<br/>R−Rf/β<br/>systematic]
  D --> D3[Jensen's α<br/>R − CAPM]
  D1 --> D4[Whole portfolio<br/>vs cash]
  D2 --> D5[Fund within<br/>diversified portfolio]
  D3 --> D6[Manager<br/>skill]

  E --> E1[Markowitz<br/>efficient frontier]
  E --> E2[Sharpe single-index<br/>cut-off rate C*]
  E2 --> E3[Include if<br/>R-Rf/β > C*]
  E --> E4[Weights<br/>∝ β/σ²_e × R-Rf/β − C*]

  style C4 fill:#dcfce7,stroke:#15803d
  style C5 fill:#fee2e2,stroke:#dc2626
  style C6 fill:#f3f4f6,stroke:#6b7280

When Sharpe and Treynor Disagree

Render diagram ↗

flowchart TD
  A[Compute Sharpe<br/>and Treynor<br/>for each fund] --> B{Same ranking?}
  B -->|Yes| C[Use either —<br/>both confirm<br/>the verdict]
  B -->|No| D{Sharpe higher<br/>but Treynor lower?}
  D -->|Yes| E[High unsystematic<br/>risk]
  D -->|No| F[High systematic<br/>risk]

  E --> G{Is investor<br/>diversified?}
  G -->|Yes — only β matters| H[Use Treynor<br/>ranking]
  G -->|No — whole exposure| I[Use Sharpe<br/>ranking]

  F --> J{Is investor<br/>diversified?}
  J -->|Yes| K[Sharpe<br/>understates<br/>useful risk]
  J -->|No| L[Both signal<br/>caution]

  style H fill:#dbeafe,stroke:#1d4ed8
  style I fill:#dbeafe,stroke:#1d4ed8

Pitfalls examiners flag

Common pitfalls

Using the wrong risk denominator. Sharpe uses σ (total), Treynor uses β.

Students apply σ everywhere — that's wrong when comparing a single fund

inside a diversified portfolio.

Forgetting to subtract R_f before dividing. "Sharpe = R_p / σ" is the

single most common error — costs 2 marks every time.

Treating undervalued = high return. A stock undervalued by CAPM has

positive α — i.e., expected return > required. Students invert this.

Mis-applying correlation in the two-asset formula. ρ_AB is a number,

not σ_AB. The cross-term is 2 w_A w_B σ_A σ_B ρ_AB — not

2 w_A w_B σ_A σ_B and not 2 w_A w_B ρ_AB.

Mixing arithmetic and geometric mean returns. For ex-post performance,

geometric. For ex-ante CAPM, arithmetic. Confusing them flips answers.

Quoting "buy because α > 0" without showing the CAPM line computation.

Examiners want the required-return number, the expected-return number, and

the verdict — not just the conclusion.

30-second revision card

Portfolio Management — 30-second recap

•σ_p² = w_A² σ_A² + w_B² σ_B² + 2 w_A w_B σ_A σ_B ρ
•ρ closer to −1 ⇒ more diversification benefit
•CAPM: E(R) = R_f + β(R_m − R_f)
•Sharpe = (R_p − R_f) / σ (total risk)
•Treynor = (R_p − R_f) / β (systematic risk)
•Jensen's α = R_p − CAPM-required
•Cut-off C: include if (R_i − R_f)/β_i > C
•Always tabulate, always box final, always state verdict.

Make it click