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Chapter 6: Portfolio Management
CA Final AFM • Paper 2

Chapter 6: Portfolio Management

Exhaustive Master Notes covering Markowitz Modern Portfolio Theory, Minimum Variance Portfolio, Capital Market Theory & CML, Sharpe Single Index Model, Capital Asset Pricing Model (CAPM & SML), Arbitrage Pricing Theory (APT), Portfolio Performance Evaluation (Sharpe, Treynor, Jensen's Alpha, Modigliani-Modigliani M² & Information Ratio), Sharpe Optimal Portfolio Selection, and Formula Revision Plans (Constant Mix & CPPI).

📚 ICAI New Scheme Study Material ⏱️ Read Time: ~55 mins 🎯 Exam Weightage: 20 – 30 Marks

1. Portfolio Management: Objectives & Phases

Portfolio Management is the dynamic art and science of selecting and overseeing a collection of investments that meet long-term financial objectives and risk tolerance for an individual or institutional investor.

1.1 Core Objectives of Portfolio Management

1. Capital Appreciation

Maximizing capital growth of invested principal over time to beat inflation.

2. Stability of Income

Ensuring steady, predictable cash inflows via regular dividend and bond coupon disbursements.

3. Safety of Principal

Minimizing downside drawdown risk to safeguard initial invested capital.

4. Liquidity & Tax Efficiency

Maintaining sufficient liquidity for cash requirements and optimizing after-tax returns.

1.2 The Five Phases of Portfolio Management Process

  1. Security Analysis: Screening individual financial assets to assess risk-return profiles using fundamental and technical analysis.
  2. Portfolio Analysis: Determining expected returns, standard deviations, and covariances across various potential asset combinations.
  3. Portfolio Selection: Choosing the optimal combination of assets that maximizes expected return for a targeted risk level (Markowitz optimization).
  4. Portfolio Revision: Periodically adjusting asset allocations in response to changing market prices and economic conditions.
  5. Portfolio Evaluation: Measuring risk-adjusted performance against benchmarks (Sharpe, Treynor, Jensen's Alpha).

2. Risk Taxonomy: Systematic vs Unsystematic Risk

In modern finance, the total risk (variance \(\sigma^2\)) of any security or portfolio is cleanly decomposed into two distinct components:

Decomposition of Total Risk Risk Theory
$$\text{Total Risk} = \text{Systematic (Market) Risk} + \text{Unsystematic (Specific) Risk}$$ $$\sigma_i^2 = \beta_i^2 \sigma_m^2 + \sigma_{ei}^2$$
Feature Systematic Risk (Market / Undiversifiable) Unsystematic Risk (Company-Specific / Diversifiable)
Definition Risk inherent to the entire financial system and macroeconomy. Affects all securities simultaneously. Risk unique and specific to a particular firm or industry. Arises from internal business factors.
Causal Factors Interest rate changes, inflation, GDP growth shocks, wars, exchange rate fluctuations, central bank policy. Strikes, management fraud, litigation, loss of key patent, faulty products, marketing failure.
Diversifiability Cannot be eliminated through portfolio diversification. Remains present regardless of portfolio size. Can be completely eliminated by constructing a well-diversified portfolio of 25–30 uncorrelated stocks.
Measurement Measured by Beta (\(\beta\)). Measured by Residual Variance (\(\sigma_e^2\)).
Market Compensation Investors are compensated with higher expected returns for bearing systematic risk. Investors receive zero extra return for bearing unsystematic risk because it can be diversified away freely.

3. Statistical Foundations of Portfolio Theory

3.1 Expected Return & Standard Deviation of a Single Security

Expected Return & Variance Probability Distributions
$$E(R) = \sum_{i=1}^n p_i R_i \quad \text{or for historical series: } \bar{R} = \frac{\sum R_t}{n}$$ $$\sigma^2 = \sum_{i=1}^n p_i [R_i - E(R)]^2 \quad \implies \quad \sigma = \sqrt{\sum_{i=1}^n p_i [R_i - E(R)]^2}$$

3.2 Covariance & Correlation Coefficient

Covariance and Correlation Between Two Assets Co-movement Formulas
$$\text{Cov}_{12} = \sum_{i=1}^n p_i [R_{1i} - E(R_1)][R_{2i} - E(R_2)] = r_{12} \sigma_1 \sigma_2$$ $$r_{12} (\text{Correlation Coefficient}) = \frac{\text{Cov}_{12}}{\sigma_1 \sigma_2} \quad \text{where} \quad -1.0 \le r_{12} \le +1.0$$
• \(r = +1.0\) (Perfect Positive): Returns move in identical lockstep. No risk diversification benefit.
• \(r = -1.0\) (Perfect Negative): Returns move in completely opposite directions. Risk can be completely eliminated to zero.
• \(r < 1.0\): Any correlation below +1.0 generates diversification benefits, reducing portfolio risk below the weighted average risk!

4. Markowitz Modern Portfolio Theory (MPT) & MVP

Pioneered by Harry Markowitz in 1952 (for which he received the Nobel Prize), Modern Portfolio Theory demonstrated that an asset should not be selected solely based on its standalone risk and return, but on how it co-moves with other assets in a portfolio.

4.1 Two-Security Portfolio Return & Risk

Two-Security Portfolio Formulas Markowitz Two-Asset
$$E(R_p) = w_1 E(R_1) + w_2 E(R_2) \quad \text{where } w_1 + w_2 = 1$$ $$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \text{Cov}_{12} = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 r_{12} \sigma_1 \sigma_2$$ $$\sigma_p = \sqrt{w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 r_{12} \sigma_1 \sigma_2}$$

4.2 Minimum Variance Portfolio (MVP) for Two Securities

The Minimum Variance Portfolio represents the specific asset allocation weights that minimize portfolio standard deviation (\(\sigma_p\)):

MVP Weight Formula (Two Assets) Core ICAI Exam Formula
$$w_1 = \frac{\sigma_2^2 - \text{Cov}_{12}}{\sigma_1^2 + \sigma_2^2 - 2 \text{Cov}_{12}} = \frac{\sigma_2^2 - r_{12} \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 r_{12} \sigma_1 \sigma_2}$$ $$w_2 = 1 - w_1$$
Special Case: Perfect Negative Correlation (\(r = -1.0\))

When two securities are perfectly negatively correlated, portfolio risk can be driven strictly to zero (\(\sigma_p = 0\)) by setting: $$w_1 = \frac{\sigma_2}{\sigma_1 + \sigma_2} \quad \text{and} \quad w_2 = \frac{\sigma_1}{\sigma_1 + \sigma_2}$$

4.3 \(N\)-Security Portfolio Risk

General \(N\)-Asset Portfolio Variance Multi-Asset
$$\sigma_p^2 = \sum_{i=1}^n w_i^2 \sigma_i^2 + \sum_{i=1}^n \sum_{j \neq i}^n w_i w_j \text{Cov}_{ij}$$
• In an \(n\)-asset portfolio, there are \(n\) variance terms and \(n(n - 1)\) covariance terms.
• In an equally weighted portfolio (\(w_i = 1/n\)), as \(n \to \infty\), the contribution of individual variances \(\frac{1}{n} \bar{\sigma}^2 \to 0\).
• Profound Insight: Portfolio risk is dominated almost entirely by the average covariance between pairs of stocks, not by individual stock variances!

5. The Feasible Set & The Efficient Frontier

The Feasible Set (Opportunity Set) encompasses all attainable portfolio combinations of risk (\(\sigma_p\)) and expected return (\(E(R_p)\)).

The Efficient Frontier

The upper convex boundary of the feasible set extending upward from the Minimum Variance Portfolio (MVP). Portfolios lying on the Efficient Frontier dominate all other portfolios because they offer:
(1) The maximum expected return for a given level of risk, or
(2) The minimum risk for a specified target return.

Investor Utility & Optimal Portfolio

Every rational investor has an indifference curve (utility function) reflecting their individual degree of risk aversion. The Optimal Portfolio for a specific investor occurs at the point of tangency between their highest indifference curve and the Efficient Frontier.

6. Capital Market Theory & The Capital Market Line (CML)

When James Tobin and William Sharpe introduced a Risk-Free Asset (\(R_f\)) that investors can borrow or lend at without limit, the curved Markowitz efficient frontier transforms into a straight line called the Capital Market Line (CML).

Capital Market Line (CML) Equation Capital Market Theory
$$E(R_p) = R_f + \left[ \frac{E(R_m) - R_f}{\sigma_m} \right] \sigma_p$$
Where:
• \(E(R_p)\) = Expected return of the efficient portfolio
• \(R_f\) = Risk-free rate of return (e.g. 91-day T-Bill yield)
• \(E(R_m)\) = Expected return on the Market Portfolio (\(M\))
• \(\sigma_m\) = Standard deviation of the Market Portfolio
• \(\sigma_p\) = Total standard deviation of the efficient portfolio
• \(\text{Slope of CML} = \frac{E(R_m) - R_f}{\sigma_m} = \text{Sharpe Ratio of the Market Portfolio}\) (Market Price of Risk)
Critical CML Constraint for CA Final Exams

The Capital Market Line applies ONLY to completely efficient, well-diversified portfolios. It cannot be used to evaluate or price individual stocks or inefficient portfolios, because individual stocks contain diversifiable unsystematic risk which the CML's total risk metric (\(\sigma\)) inappropriately penalizes.

7. Sharpe Single Index Model (SIM)

To overcome the massive computational complexity of the Markowitz model (which requires estimating \(\frac{n(n-1)}{2}\) covariances), William Sharpe developed the Single Index Model, assuming that stock co-movements are driven exclusively by a single common macroeconomic factor: a broad market index.

Sharpe Single Index Model Equations Linear Regression Model
$$R_i = \alpha_i + \beta_i R_m + e_i$$ $$\text{Total Variance: } \sigma_i^2 = \beta_i^2 \sigma_m^2 + \sigma_{ei}^2$$ $$\text{Systematic Risk} = \beta_i^2 \sigma_m^2 \quad \text{and} \quad \text{Unsystematic / Residual Risk} = \sigma_{ei}^2$$ $$\text{Covariance between Stock } i \text{ and Stock } j: \text{Cov}_{ij} = \beta_i \beta_j \sigma_m^2$$
Where:
• \(\alpha_i\) = Alpha / unique expected return when market return is zero
• \(\beta_i\) = Beta / sensitivity coefficient of stock \(i\) to market movements
• \(e_i\) = Random error term with mean 0 and variance \(\sigma_{ei}^2\)
• Drastically reduces required inputs from \(\frac{n(n+3)}{2}\) in Markowitz to just \((3n + 2)\) inputs!

8. Capital Asset Pricing Model (CAPM) & Security Market Line (SML)

The Capital Asset Pricing Model (CAPM) establishes the linear relationship between systematic risk and expected return for every asset (individual stocks, bonds, or portfolios) in equilibrium.

Security Market Line (SML) / CAPM Equation Foundational Pricing Model
$$E(R_i) = R_f + \beta_i [E(R_m) - R_f]$$ $$\beta_i (\text{Systematic Risk Beta}) = \frac{\text{Cov}(i, m)}{\sigma_m^2} = \frac{r_{im} \sigma_i \sigma_m}{\sigma_m^2} = r_{im} \frac{\sigma_i}{\sigma_m}$$
Where:
• \(E(R_i)\) = Required rate of return on security \(i\)
• \([E(R_m) - R_f]\) = Market Risk Premium
• \(\beta_i\) = Beta of security \(i\)
• If \(\beta = 1.0\): Security has identical systematic risk to the market.
• If \(\beta > 1.0\): Aggressive stock (amplifies market moves).
• If \(\beta < 1.0\): Defensive stock (less volatile than market).

8.1 CML vs SML: Comprehensive Comparison

Basis Capital Market Line (CML) Security Market Line (SML)
Risk Measure Total Risk: Standard Deviation (\(\sigma_p\)) Systematic Risk: Beta (\(\beta_i\))
Applicability Applies ONLY to efficient portfolios. Applies to ALL individual securities and all portfolios (efficient or inefficient).
Slope Sharpe Ratio of Market Portfolio: \(\frac{E(R_m) - R_f}{\sigma_m}\) Market Risk Premium: \([E(R_m) - R_f]\)
Pricing Role Sets required return for complete macro asset allocation. Sets required hurdle rate / cost of equity (\(K_e\)) for valuation.

8.2 Overvalued vs Undervalued Stocks (Jensen's Alpha \(\alpha\))

Undervalued Stock (Lies ABOVE SML)

• Condition: \(\text{Expected Return } E(R_i) > \text{Required Return (CAPM)}\)
• Alpha: \(\alpha_i > 0\) (Positive Alpha).
• Action: BUY / ACCUMULATE. As investors buy, price rises and expected return drops until it settles back onto the SML.

Overvalued Stock (Lies BELOW SML)

• Condition: \(\text{Expected Return } E(R_i) < \text{Required Return (CAPM)}\)
• Alpha: \(\alpha_i < 0\) (Negative Alpha).
• Action: SELL / SHORT. As selling pressure mounts, price drops and expected return rises back to the SML.

9. Levered Beta vs Unlevered Beta (Hamada's Equation)

A firm's equity beta reflects both business risk (operating volatility) and financial risk (debt leverage). When comparing firms with differing capital structures, analysts unlever and relever beta using Robert Hamada's formula:

Hamada's Levered and Unlevered Beta Formulas Capital Structure Adjustment
$$\beta_L = \beta_U \left[ 1 + (1 - t) \frac{D}{E} \right]$$ $$\beta_U (\text{Asset Beta}) = \frac{\beta_L}{1 + (1 - t) \frac{D}{E}}$$
Where:
• \(\beta_L\) = Levered equity beta (incorporates debt and financial risk)
• \(\beta_U\) = Unlevered asset beta (pure operational business risk of assets)
• \(t\) = Corporate income tax rate
• \(D / E\) = Debt-to-Equity ratio by market values

10. Arbitrage Pricing Theory (APT)

Formulated by Stephen Ross in 1976 as an alternative to CAPM, Arbitrage Pricing Theory (APT) is a multi-factor model requiring fewer restrictive assumptions (it does not require a market portfolio or normal return distributions).

Multi-Factor APT Return Equation Multi-Factor Pricing
$$E(R_i) = R_f + \beta_{i1} \lambda_1 + \beta_{i2} \lambda_2 + \beta_{i3} \lambda_3 + \dots + \beta_{ik} \lambda_k$$
Where:
• \(R_f\) = Expected return on a zero-systematic-risk asset
• \(\beta_{ik}\) = Sensitivity of security \(i\) to macroeconomic factor \(k\)
• \(\lambda_k\) = Risk premium associated with factor \(k\)
• Typical Macro Factors: GDP surprises, unexpected inflation shifts, term structure / yield curve shifts, crude oil price shocks, default credit spreads.

11. Portfolio Performance Evaluation Ratios

Evaluating whether a fund manager generated excess return through superior skill or simply by taking on excess risk requires standardized risk-adjusted performance measures:

1. Sharpe Ratio (Reward-to-Variability) Total Risk Adjusted
$$S_p = \frac{R_p - R_f}{\sigma_p}$$
• Measures excess return per unit of Total Risk (\(\sigma_p\)).
• Best Used For: Investors whose total wealth is concentrated in a single non-diversified fund or portfolio.
2. Treynor Ratio (Reward-to-Volatility) Systematic Risk Adjusted
$$T_p = \frac{R_p - R_f}{\beta_p}$$
• Measures excess return per unit of Systematic Risk (\(\beta_p\)).
• Best Used For: Evaluating a fund that forms only a sub-part of a well-diversified broader portfolio (where unsystematic risk is already eliminated).
3. Jensen's Alpha (\(\alpha_p\)) Excess Absolute Return
$$\alpha_p = R_p - [R_f + \beta_p (R_m - R_f)]$$
• Measures the absolute abnormal return generated over and above the CAPM benchmark return.
• If \(\alpha_p > 0\): Manager demonstrated superior market timing or stock selection skill.
• If \(\alpha_p < 0\): Manager underperformed the risk-adjusted benchmark.
4. Modigliani-Modigliani Measure (\(M^2\)) Risk-Standardized Return
$$M^2 = R_f + S_p \times \sigma_m = R_f + \left( \frac{R_p - R_f}{\sigma_p} \right) \sigma_m$$
• Leverages or deleverages the portfolio to match the market's standard deviation (\(\sigma_m\)).
• Yields a return in percentage terms that is directly comparable to the market benchmark return \(R_m\)!
• If \(M^2 > R_m \implies\) Portfolio beat the market.
5. Information Ratio (Appraisal Ratio) Active Management Skill
$$IR = \frac{R_p - R_b}{\sigma_{(p - b)}} = \frac{\text{Active Return}}{\text{Tracking Error}}$$
• Measures an active manager's ability to generate excess returns relative to a benchmark per unit of active tracking risk.

12. Sharpe's Optimal Portfolio Selection (Cut-Off Rate \(C^*\))

William Sharpe formulated an elegant algorithm to select the single optimal risky portfolio without constructing quadratic programming models:

The 4-Step Sharpe Optimal Portfolio Algorithm

Step 1: Calculate the excess return to beta ratio for each security and rank securities in descending order: $$\text{Ratio} = \frac{R_i - R_f}{\beta_i}$$ Step 2: Compute the cumulative cut-off value \(C_i\) for each ranked security: $$C_i = \frac{\sigma_m^2 \sum_{j=1}^i \frac{(R_j - R_f)\beta_j}{\sigma_{ej}^2}}{1 + \sigma_m^2 \sum_{j=1}^i \frac{\beta_j^2}{\sigma_{ej}^2}}$$ Step 3: Identify the highest value of \(C_i\), designated as the Cut-Off Rate \(C^*\). All securities with \(\frac{R_i - R_f}{\beta_i} > C^*\) are selected in the optimal portfolio!
Step 4: Determine the investment weight for each selected security: $$Z_i = \frac{\beta_i}{\sigma_{ei}^2} \left[ \frac{R_i - R_f}{\beta_i} - C^* \right] \quad \implies \quad w_i = \frac{Z_i}{\sum Z_i}$$

13. Portfolio Revision & Formula Plans

Formula plans are predefined mechanical trading rules that enforce disciplined portfolio rebalancing without relying on subjective market timing forecasts.

Rebalancing Strategy Operating Rule Market Regime Outperformed Risk-Return Profile
1. Buy and Hold No rebalancing; initial portfolio of stocks and bonds is held passively. Equity proportion drifts with market. Strong continuous bull trend. Linear payoff; minimum transaction and tax costs.
2. Constant Mix Maintains a fixed target ratio (e.g. 50:50 Equity to Debt). Rebalances periodically by selling appreciating assets and buying depreciating assets. Oscillating, range-bound, volatile markets. (Contrarian strategy: buys low, sells high). Underperforms in strong unidirectional trending bull markets.
3. Constant Proportion Portfolio Insurance (CPPI) Allocates equity dynamically based on cushion above a guaranteed minimum floor: $$\text{Stock Investment} = m \times (\text{Portfolio Value} - \text{Floor})$$ where \(m > 1\) is the multiplier. Strong bull markets with severe crash protection. (Momentum strategy: buys rising stocks, sells falling stocks). Guarantees portfolio value will never breach the specified Floor. Suffers in choppy whip-saw markets.

14. Asset Allocation Strategies

Strategic Asset Allocation (SAA)

Establishes long-term target asset mix (e.g. 60% Equity / 40% Debt) based on investor risk tolerance, horizon, and expected asset returns.

Tactical Asset Allocation (TAA)

Allows moderate, short-term opportunistic deviations from SAA targets to exploit temporary macroeconomic mispricings.

Dynamic Asset Allocation

Continuously adjusts portfolio weights based on systematic rules as market conditions and asset risk-return profiles fluctuate.

Insured Asset Allocation

Sets a hard wealth floor and shifts into risk-free assets whenever portfolio capital approaches the designated floor (e.g. CPPI).

15. Fixed Income Portfolio Management Strategies

ICAI categorizes bond portfolio strategies into passive and active disciplines:

Passive Strategies

• Buy and Hold: Selects bonds matching investment horizon and holds to maturity.
• Bond Indexing: Replicates a target bond index (e.g. NIFTY Sovereign Bond Index) through full replication or stratified sampling.

Active Structural Strategies

• Bullet Strategy: Concentrates all bond maturities around a single intermediate point (e.g. all 7-year bonds).
• Barbell Strategy: Concentrates maturities at extreme ends (short-term 1-year and long-term 20-year bonds), balancing liquidity and yield.
• Ladder Strategy: Distributes maturities equally across years (e.g. 10% maturing every year for 10 years). Automatically replenishes reinvestment.

16. Alternative Investment Avenues

Modern multi-asset portfolios integrate alternative assets to achieve superior diversification beyond traditional equities and bonds:

Asset Class Characteristics & Role in Portfolio Primary Valuation Approach
Real Estate Tangible physical asset, inflation hedge, low correlation to public equities, steady rental yields. • Sales Comparison Approach
• Cost Approach
• Income Capitalization: \(\text{Value} = \frac{\text{Net Operating Income (NOI)}}{\text{Cap Rate}}\)
Gold Global store of value, universal hedge against sovereign currency debasement and geopolitical crises. Zero credit risk. Negative correlation to equities during panics. Supply-demand dynamics, real interest rates (opportunity cost of holding non-yielding gold), and currency movements.
Private Equity & Venture Capital Direct equity investment in unlisted private firms; long lock-in periods (5–10 years), illiquid, high return potential. Discounted Cash Flow (DCF), Venture Capital Method, Precedent M&A transactions.
REITs & InvITs Publicly traded trusts owning income-generating real estate/infrastructure. Mandated to distribute 90% of cash flows. Net Asset Value (NAV) per unit, Discounted Cash Flow of lease agreements, Distribution Yield.
Distressed Securities Bonds or debt of companies in bankruptcy or near-default, purchased at deep discounts with turnaround/liquidation upside. Liquidation waterfall analysis, enterprise valuation under Chapter 11 / IBC insolvency restructuring.

17. Comprehensive Solved ICAI Practical Problems

Solved Problem 1: Two-Asset Portfolio Return, Risk & Minimum Variance Portfolio (MVP)

Problem Statement: An investor is considering two stocks, A and B, with the following parameters:

  • Stock A: Expected Return \(E(R_A) = 16\%\), Standard Deviation \(\sigma_A = 20\%\)
  • Stock B: Expected Return \(E(R_B) = 12\%\), Standard Deviation \(\sigma_B = 15\%\)
  • Coefficient of Correlation between A and B is \(r_{AB} = 0.30\)

Required:
(i) Calculate the expected return and standard deviation of a portfolio with 60% in Stock A and 40% in Stock B.
(ii) Determine the Minimum Variance Portfolio (MVP) weights and compute the minimum portfolio risk.

Solution & Working Notes

Part (i): Portfolio with \(w_A = 0.60\) and \(w_B = 0.40\):
• Expected Portfolio Return: $$E(R_p) = w_A E(R_A) + w_B E(R_B) = (0.60 \times 16\%) + (0.40 \times 12\%) = 9.60\% + 4.80\% = \mathbf{14.40\%}$$ • Covariance Between A and B: $$\text{Cov}_{AB} = r_{AB} \times \sigma_A \times \sigma_B = 0.30 \times 20 \times 15 = 90$$ • Portfolio Variance (\(\sigma_p^2\)): $$\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}_{AB}$$ $$\sigma_p^2 = (0.60)^2(20)^2 + (0.40)^2(15)^2 + 2(0.60)(0.40)(90)$$ $$\sigma_p^2 = (0.36 \times 400) + (0.16 \times 225) + (0.48 \times 90) = 144 + 36 + 43.20 = 223.20$$ • Portfolio Standard Deviation: $$\sigma_p = \sqrt{223.20} = \mathbf{14.94\%}$$ (Notice that \(\sigma_p = 14.94\%\) is lower than the weighted average of standard deviations: \(0.60 \times 20 + 0.40 \times 15 = 18\%\)!)

Part (ii): Minimum Variance Portfolio (MVP) Weights:

$$w_A = \frac{\sigma_B^2 - \text{Cov}_{AB}}{\sigma_A^2 + \sigma_B^2 - 2 \text{Cov}_{AB}} = \frac{15^2 - 90}{20^2 + 15^2 - 2(90)} = \frac{225 - 90}{400 + 225 - 180} = \frac{135}{445} = \mathbf{0.3034} \quad (\mathbf{30.34\%})$$ $$w_B = 1 - w_A = 1 - 0.3034 = \mathbf{0.6966} \quad (\mathbf{69.66\%})$$

• Minimum Portfolio Risk (\(\sigma_{\text{min}}\)):
$$\sigma_{\text{min}}^2 = (0.3034)^2(400) + (0.6966)^2(225) + 2(0.3034)(0.6966)(90)$$ $$\sigma_{\text{min}}^2 = (0.09205 \times 400) + (0.48525 \times 225) + (0.4227 \times 90) = 36.82 + 109.18 + 38.04 = 184.04$$ $$\sigma_{\text{min}} = \sqrt{184.04} = \mathbf{13.57\%}$$ (Remarkable: The minimum portfolio risk of 13.57% is lower than the risk of either individual stock alone, since \(\sigma_B = 15\%\)!)

Solved Problem 2: Sharpe, Treynor & Jensen's Alpha Fund Evaluation

Problem Statement: Five mutual fund schemes experienced the following performance over a 5-year evaluation period. The Risk-Free Rate (\(R_f\)) is 6% and the Market Portfolio Return (\(R_m\)) is 12% with a market standard deviation (\(\sigma_m\)) of 16%.

Fund Average Return (\(R_p\)) Standard Deviation (\(\sigma_p\)) Beta (\(\beta_p\))
Fund P14%18%1.10
Fund Q11%12%0.80
Fund R16%22%1.25
Fund S13%15%0.90
Fund T10%10%0.60

Required: Compute the Sharpe Ratio, Treynor Ratio, Jensen's Alpha, and Modigliani-Modigliani (\(M^2\)) measure for each fund, and rank the funds under each metric.

Solution & Working Notes

1. Market Benchmark Reference Values:
• Excess Market Return \(= R_m - R_f = 12\% - 6\% = 6\%\)
• Market Sharpe Ratio \(S_m = \frac{12\% - 6\%}{16\%} = \frac{6}{16} = 0.375\)
• Market Treynor Ratio \(T_m = \frac{12\% - 6\%}{1.0} = 6.0\)

Fund Sharpe: \(\frac{R_p - 6}{\sigma_p}\) (Rank) Treynor: \(\frac{R_p - 6}{\beta_p}\) (Rank) CAPM Req: \(6 + \beta(6)\) Alpha: \(R_p - \text{Req}\) (Rank) \(M^2\): \(6 + S_p(16)\) (Rank)
Fund P \(\frac{14 - 6}{18} = \mathbf{0.444}\) (3) \(\frac{14 - 6}{1.10} = \mathbf{7.27}\) (3) \(6 + 1.1(6) = 12.60\%\) \(14 - 12.6 = \mathbf{+1.40\%}\) (3) \(6 + 0.444(16) = \mathbf{13.10\%}\) (3)
Fund Q \(\frac{11 - 6}{12} = \mathbf{0.417}\) (4) \(\frac{11 - 6}{0.80} = \mathbf{6.25}\) (5) \(6 + 0.8(6) = 10.80\%\) \(11 - 10.8 = \mathbf{+0.20\%}\) (5) \(6 + 0.417(16) = \mathbf{12.67\%}\) (4)
Fund R \(\frac{16 - 6}{22} = \mathbf{0.455}\) (2) \(\frac{16 - 6}{1.25} = \mathbf{8.00}\) (1) \(6 + 1.25(6) = 13.50\%\) \(16 - 13.5 = \mathbf{+2.50\%}\) (1) \(6 + 0.455(16) = \mathbf{13.28\%}\) (2)
Fund S \(\frac{13 - 6}{15} = \mathbf{0.467}\) (1) \(\frac{13 - 6}{0.90} = \mathbf{7.78}\) (2) \(6 + 0.9(6) = 11.40\%\) \(13 - 11.4 = \mathbf{+1.60\%}\) (2) \(6 + 0.467(16) = \mathbf{13.47\%}\) (1)
Fund T \(\frac{10 - 6}{10} = \mathbf{0.400}\) (5) \(\frac{10 - 6}{0.60} = \mathbf{6.67}\) (4) \(6 + 0.6(6) = 9.60\%\) \(10 - 9.6 = \mathbf{+0.40\%}\) (4) \(6 + 0.400(16) = \mathbf{12.40\%}\) (5)

Evaluation Insights:
• Fund S is ranked #1 under the Sharpe Ratio (0.467) and \(M^2\) (13.47%) because it has exceptional total risk diversification.
• Fund R is ranked #1 under Treynor Ratio (8.00) and Jensen's Alpha (+2.50%) because of its massive market outperformance per unit of systematic risk.
• All five funds beat the market Sharpe ratio (0.375) and achieved positive Jensen's Alpha!

Solved Problem 3: Constant Proportion Portfolio Insurance (CPPI) Simulation

Problem Statement: An investor has a portfolio corpus of ₹1,00,000. He establishes a minimum floor value of ₹75,000. The CPPI multiplier is \(m = 2.0\).
Required: Determine the initial asset allocation in Equity and Debt (Floor). If the equity market drops by 20%, calculate the necessary rebalancing action.

Solution & Working Notes

Step 1: Initial Allocation:
• Total Portfolio Value \(= ₹1,00,000\)
• Floor Value \(= ₹75,000\)
• Cushion \(= \text{Portfolio Value} - \text{Floor} = 1,00,000 - 75,000 = ₹25,000\)
• Target Equity Investment \(= m \times \text{Cushion} = 2.0 \times 25,000 = \mathbf{₹50,000}\)
• Debt / Risk-Free Investment \(= 1,00,000 - 50,000 = \mathbf{₹50,000}\)

Step 2: Market Event: Equity Drops by 20%:
• New Value of Equity \(= 50,000 \times (1 - 0.20) = ₹40,000\)
• Value of Debt \(= ₹50,000\) (assumed constant for short period)
• New Total Portfolio Value \(= 40,000 + 50,000 = ₹90,000\)

Step 3: Rebalancing Requirement:
• New Cushion \(= \text{New Portfolio Value} - \text{Floor} = 90,000 - 75,000 = ₹15,000\)
• New Target Equity Investment \(= m \times \text{Cushion} = 2.0 \times 15,000 = \mathbf{₹30,000}\)
• Current Equity Holding \(= ₹40,000\)
• Rebalancing Action: Sell ₹10,000 of Equity (\(40,000 - 30,000\)) and invest ₹10,000 in Debt, increasing Debt to ₹60,000. This mechanical rebalancing ensures that even if equities continue to collapse, the total portfolio value will never penetrate the ₹75,000 floor!

18. Self-Test Practice Questions (ICAI TYK)

Q1: Explain the Capital Asset Pricing Model (CAPM) and state its relevant underlying assumptions.
Model Answer: Developed by William Sharpe, John Lintner, and Jan Mossin, CAPM specifies that the expected return of any financial asset is determined solely by its systematic risk (Beta \(\beta\)) relative to the market: \(E(R_i) = R_f + \beta_i [E(R_m) - R_f]\).
Key Assumptions:
1. Investors are risk-averse, mean-variance optimizers seeking to maximize expected terminal utility;
2. Frictionless capital markets: no transaction costs, no taxes, assets are infinitely divisible;
3. Homogeneous expectations: all investors possess identical forecasts regarding future expected returns, variances, and covariances;
4. Unlimited risk-free borrowing and lending at a single constant rate \(R_f\);
5. All investors plan over a single identical time horizon.
Q2: Distinguish between the Sharpe Ratio and the Treynor Ratio. When should an analyst prefer one over the other?
Model Answer:
• Sharpe Ratio: \(S_p = \frac{R_p - R_f}{\sigma_p}\). It evaluates excess return per unit of Total Risk (Standard Deviation \(\sigma_p\)), thereby incorporating both systematic and unsystematic risk.
• Treynor Ratio: \(T_p = \frac{R_p - R_f}{\beta_p}\). It evaluates excess return per unit of Systematic Risk (Beta \(\beta_p\)), completely ignoring unsystematic risk.
• When to Use:
- Use the Sharpe Ratio when evaluating an investor's entire standalone wealth or a non-diversified fund, because unsystematic risk has not been eliminated and poses a real threat.
- Use the Treynor Ratio when evaluating a specialized fund that will be combined as a minor constituent inside an already well-diversified macro portfolio, because its unsystematic risk will be diversified away.
Q3: Differentiate between Constant Mix Strategy and Constant Proportion Portfolio Insurance (CPPI) as portfolio revision formula plans.
Model Answer:
• Constant Mix Strategy: Maintains a strictly constant ratio between aggressive equity and conservative debt (e.g. 50:50). It operates as a contrarian strategy: as equities rise, it sells equities to buy debt; as equities fall, it buys equities. It outperforms in oscillating, sideways, range-bound markets but underperforms in persistent bull or bear trends.
• CPPI Strategy: Allocates equity as a multiple of the cushion above a predetermined floor: \(\text{Equity} = m \times (\text{Portfolio} - \text{Floor})\). It operates as a momentum strategy: as equities rise, the cushion expands, so it aggressively buys more equities; as equities fall, it sells equities to protect the floor. It guarantees capital preservation at the designated floor while capturing runaway bull market upside.
Q4: Explain how an investor can eliminate portfolio risk completely using two securities. What condition must be met?
Model Answer: Total portfolio risk can be driven strictly to zero (\(\sigma_p = 0\)) if and only if the two securities are perfectly negatively correlated (\(r_{12} = -1.0\)).
Under perfect negative correlation, portfolio variance simplifies to: $$\sigma_p^2 = (w_1 \sigma_1 - w_2 \sigma_2)^2 \quad \implies \quad \sigma_p = |w_1 \sigma_1 - w_2 \sigma_2|$$ Setting \(\sigma_p = 0\) and using \(w_1 + w_2 = 1\), the exact weights required are: $$w_1 = \frac{\sigma_2}{\sigma_1 + \sigma_2} \quad \text{and} \quad w_2 = \frac{\sigma_1}{\sigma_1 + \sigma_2}$$ This creates a synthetic risk-free portfolio with guaranteed zero variance.

19. Master Quick Revision Summary

Model / Metric Core Formula Crucial Exam Takeaway
2-Asset Portfolio Risk $$\sigma_p = \sqrt{w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 r_{12} \sigma_1 \sigma_2}$$ Risk reduced whenever \(r_{12} < 1.0\).
Minimum Variance Portfolio (MVP) $$w_1 = \frac{\sigma_2^2 - \text{Cov}_{12}}{\sigma_1^2 + \sigma_2^2 - 2 \text{Cov}_{12}}$$ Weights that yield the lowest possible \(\sigma_p\).
Capital Market Line (CML) $$E(R_p) = R_f + \left[ \frac{E(R_m) - R_f}{\sigma_m} \right] \sigma_p$$ Applies only to efficient portfolios.
CAPM (SML) $$E(R_i) = R_f + \beta_i [E(R_m) - R_f]$$ $$\beta_i = \frac{\text{Cov}(i, m)}{\sigma_m^2} = r_{im} \frac{\sigma_i}{\sigma_m}$$. Applies to all assets.
Hamada Formula $$\beta_L = \beta_U [1 + (1 - t) D/E]$$ Converts asset beta to equity beta under financial leverage.
Arbitrage Pricing (APT) $$E(R_i) = R_f + \sum \beta_{ik} \lambda_k$$ Multi-factor macro sensitivity pricing without market portfolio.
Sharpe & Treynor Ratios $$S = \frac{R_p - R_f}{\sigma_p}, \quad T = \frac{R_p - R_f}{\beta_p}$$ Sharpe uses total risk \(\sigma\); Treynor uses systematic risk \(\beta\).
Jensen's Alpha $$\alpha_p = R_p - [R_f + \beta_p (R_m - R_f)]$$ \(\alpha > 0 \implies\) Undervalued (Buy); \(\alpha < 0 \implies\) Overvalued (Sell).
Modigliani-Modigliani (\(M^2\)) $$M^2 = R_f + S_p \times \sigma_m$$ Directly benchmark-comparable percentage return.
CPPI Revision $$\text{Equity} = m \times (\text{Portfolio} - \text{Floor})$$ Dynamic momentum strategy providing floor protection.
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