Dean's Office Official Model Question Paper

Expected Return and Variance

• Expected Return ($E(R)$): The probability-weighted average of all possible future returns across different economic states: $E(R) = \sum (p_i \times R_i)$.
• Variance ($\sigma^2$): A statistical measure of the dispersion or volatility of actual returns around the expected return: $\sigma^2 = \sum [p_i \times (R_i - E(R))^2]$.

Systematic vs. Unsystematic Risk

• Systematic Risk (Market Risk): Macroeconomic factors (interest rate swings, inflation, recessions) affecting all securities simultaneously. Measured by Beta ($\beta$); cannot be diversified away.
• Unsystematic Risk (Firm-Specific Risk): Microeconomic events (strikes, product failure, management scandals) unique to a company. Can be eliminated through broad portfolio diversification.

Security Market Line (SML)

The SML is the graphical representation of the Capital Asset Pricing Model (CAPM) showing the linear relationship between expected return and systematic risk (Beta). It plots: $E(R_i) = R_f + \beta_i [E(R_m) - R_f]$.

Sharpe Ratio vs. Treynor Ratio

• Sharpe Ratio: Measures excess return per unit of total risk (Standard Deviation): $S_p = \frac{R_p - R_f}{\sigma_p}$.
• Treynor Ratio: Measures excess return per unit of systematic risk (Beta): $T_p = \frac{R_p - R_f}{\beta_p}$.

Investment Policy Statement (IPS)

An IPS is a formal governing document between an investor and portfolio manager establishing guidelines for portfolio management.

Two Core Components:

1. Objectives: Return requirements and risk tolerance.
2. Constraints: Liquidity requirements, time horizon, tax considerations, and legal/regulatory limits.

Markowitz Two-Asset Portfolio Optimization

1. Given Data

• $w_X = 0.60$, $E(R_X) = 14\%$, $\sigma_X = 20\%$
• $w_Y = 0.40$, $E(R_Y) = 18\%$, $\sigma_Y = 30\%$

2. Expected Portfolio Return ($E(R_p)$)

$$E(R_p) = (w_X \times E(R_X)) + (w_Y \times E(R_Y)) = (0.60 \times 14) + (0.40 \times 18) = 8.4\% + 7.2\% = \mathbf{15.60\%}$$

(Note: Expected return is invariant to correlation).

3. Portfolio Variance and Standard Deviation Formula

$$\sigma_p = \sqrt{w_X^2 \sigma_X^2 + w_Y^2 \sigma_Y^2 + 2 w_X w_Y \sigma_X \sigma_Y \rho_{XY}}$$

Component Calculations:

• $w_X^2 \sigma_X^2 = (0.60)^2 \times (20)^2 = 0.36 \times 400 = 144$
• $w_Y^2 \sigma_Y^2 = (0.40)^2 \times (30)^2 = 0.16 \times 900 = 144$
• $2 w_X w_Y \sigma_X \sigma_Y = 2 \times 0.60 \times 0.40 \times 20 \times 30 = 288$

(a) When Correlation $\rho_{XY} = +1.0$ (Perfect Positive):

$$\sigma_p = \sqrt{144 + 144 + 288(1.0)} = \sqrt{576} = \mathbf{24.00\%}$$

• Note: Weighted average: $(0.60 \times 20) + (0.40 \times 30) = 24.0\%$. Zero diversification benefit.

(b) When Correlation $\rho_{XY} = 0.0$ (Uncorrelated):

$$\sigma_p = \sqrt{144 + 144 + 288(0)} = \sqrt{288} = \mathbf{16.97\%}$$

• Note: Risk drops significantly from 24% to 16.97% while maintaining 15.6% return.

(c) When Correlation $\rho_{XY} = -1.0$ (Perfect Negative):

$$\sigma_p = \sqrt{144 + 144 - 288} = \sqrt{0} = \mathbf{0.00\%}$$

• Note: Total portfolio risk is completely eliminated.

4. Diversification Takeaway

As long as the correlation coefficient is less than $+1.0$ ($\rho < 1.0$), combining assets creates diversification benefits by reducing portfolio standard deviation below the weighted average of individual component risks.

Capital Asset Pricing Model (CAPM) & Jensen’s Alpha

1. Formula

• CAPM Required Return: $k_i = R_f + \beta_i [E(R_m) - R_f]$
• Market Risk Premium: $E(R_m) - R_f = 13\% - 5\% = 8\%$
• Jensen’s Alpha: $\alpha_i = \text{Expected Return } E(R_i) - \text{Required Return } k_i$

2. Calculations for Each Stock

3. Managerial Interpretation

• Stock P: Generates an expected return (12.0%) higher than required by its risk profile (11.4%). Its positive alpha (+0.6%) signals an attractive buying opportunity.
• Stock R: Fails to deliver enough return (16.0%) to compensate for its high systematic risk ($\beta=1.5$, requiring 17.0%). Its negative alpha (-1.0%) makes it an overvalued asset that should be sold or shorted.

Capital Market Line (CML) vs. Security Market Line (SML)

Both lines originate from Modern Portfolio Theory and CAPM, but serve distinct analytical roles:

     Expected Return                   Expected Return
            ^\                                 ^\
            |         / CML                   |         / SML
            |       /                         |       /
            |     /                           |     /
        R_f |---/                         R_f |---/
            +--------------------->           +--------------------->
             Total Risk (Std Dev sigma)        Systematic Risk (Beta beta)

1. Comparative Matrix

2. Strategic Portfolio Insight

• An investor evaluating an entire multi-asset pension fund portfolio looks to the CML to verify overall risk-adjusted efficiency.
• An analyst evaluating whether an individual banking stock on the Nepal Stock Exchange (NEPSE) is a worthwhile buy relies on the SML.

Portfolio Performance Evaluation Measures

Evaluating investment performance requires adjusting raw returns for risk taken:

1. The Three Classic Measures

1. Sharpe Ratio (Total Risk):
   $$S_p = \frac{R_p - R_f}{\sigma_p}$$
   • Best suited when evaluating an investor’s entire, non-diversified wealth.
2. Treynor Ratio (Systematic Risk):
   $$T_p = \frac{R_p - R_f}{\beta_p}$$
   • Best suited when evaluating a fund that will be held as one component of a well-diversified broader portfolio.
3. Jensen’s Alpha (Abnormal Risk-Adjusted Excess Return):
   $$\alpha_p = R_p - [R_f + \beta_p (R_m - R_f)]$$
   • Measures the fund manager’s ability to beat the CAPM market benchmark. $\alpha > 0$ indicates superior managerial skill.

2. Decomposing Performance: Market Timing vs. Security Selection

• Security Selection (Micro-Forecasting): The manager’s skill in identifying undervalued individual stocks within an industry. Measured directly by positive Jensen’s Alpha ($\alpha_p$).
• Market Timing (Macro-Forecasting): The manager’s ability to adjust the portfolio’s overall Beta ($\beta$) ahead of market cycles—raising portfolio Beta to $> 1.0$ before a bull run, and lowering Beta to $< 0.5$ (or holding cash) before an anticipated market crash.
• Models like Treynor-Mazuy ($R_p - R_f = \alpha + \beta(R_m - R_f) + \gamma(R_m - R_f)^2$) evaluate market timing through the curvature coefficient $\gamma$.

Comprehensive Solution: Portfolio Performance & IPS

(a) Performance Metrics Calculation & Ranking (7 Marks)

Given: $R_f = 5.0\%$, Market Return $R_m = 14.5\%$, Market Std Dev $\sigma_m = 16.0\%$, Market Risk Premium $R_m - R_f = 14.5 - 5.0 = 9.5\%$.

1. Metric Formulas:

• Sharpe Ratio: $S_p = \frac{R_p - R_f}{\sigma_p}$
• Treynor Ratio: $T_p = \frac{R_p - R_f}{\beta_p}$
• Jensen’s Alpha: $\alpha_p = R_p - [R_f + \beta_p(R_m - R_f)]$

2. Fund-by-Fund Calculations:

• Fund Alpha:

  • $S_{Alpha} = \frac{16.5 - 5.0}{18.0} = \frac{11.5}{18.0} = \mathbf{0.639}$
  • $T_{Alpha} = \frac{16.5 - 5.0}{0.90} = \frac{11.5}{0.90} = \mathbf{12.78\%}$
  • $\alpha_{Alpha} = 16.5 - [5.0 + 0.90(9.5)] = 16.5 - [5.0 + 8.55] = 16.5 - 13.55 = \mathbf{+2.95\%}$

• Fund Beta:

  • $S_{Beta} = \frac{19.2 - 5.0}{24.0} = \frac{14.2}{24.0} = \mathbf{0.592}$
  • $T_{Beta} = \frac{19.2 - 5.0}{1.25} = \frac{14.2}{1.25} = \mathbf{11.36\%}$
  • $\alpha_{Beta} = 19.2 - [5.0 + 1.25(9.5)] = 19.2 - [5.0 + 11.875] = 19.2 - 16.875 = \mathbf{+2.325\%}$

• Fund Gamma:

  • $S_{Gamma} = \frac{14.0 - 5.0}{15.0} = \frac{9.0}{15.0} = \mathbf{0.600}$
  • $T_{Gamma} = \frac{14.0 - 5.0}{0.75} = \frac{9.0}{0.75} = \mathbf{12.00\%}$
  • $\alpha_{Gamma} = 14.0 - [5.0 + 0.75(9.5)] = 14.0 - [5.0 + 7.125] = 14.0 - 12.125 = \mathbf{+1.875\%}$

• NEPSE Market Benchmark:

  • $S_{Market} = \frac{14.5 - 5.0}{16.0} = \mathbf{0.594}$
  • $T_{Market} = \frac{14.5 - 5.0}{1.00} = \mathbf{9.50\%}$
  • $\alpha_{Market} = 14.5 - 14.5 = \mathbf{0.00\%}$

3. Tabulation and Relative Rankings:

• Evaluation: Fund Alpha is the undisputed top performer, ranking #1 across all three risk-adjusted performance measures.

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(b) Minimum Variance Portfolio Weights (7 Marks)

To find the portfolio weight $w_A$ that minimizes variance for two risky assets with correlation $\rho_{AB}$:

$$w_A^* = \frac{\sigma_B^2 - \text{Cov}(A,B)}{\sigma_A^2 + \sigma_B^2 - 2\text{Cov}(A,B)}$$

Where:

• $\sigma_A = 18.0 \implies \sigma_A^2 = (18)^2 = 324$
• $\sigma_B = 24.0 \implies \sigma_B^2 = (24)^2 = 576$
• $\rho_{AB} = 0.25$
• $\text{Cov}(A,B) = \rho_{AB} \times \sigma_A \times \sigma_B = 0.25 \times 18 \times 24 = 108$

Calculation:

$$w_A^* = \frac{576 - 108}{324 + 576 - 2(108)} = \frac{468}{900 - 216} = \frac{468}{684} = \mathbf{0.6842\ (68.42\%)}$$

$$w_B^* = 1 - w_A^* = 1 - 0.6842 = \mathbf{0.3158\ (31.58\%)}$$

• Result: To minimize total risk, the client should allocate 68.42% to Fund Alpha and 31.58% to Fund Beta.
• Resulting Minimum Portfolio Variance:
  $$\sigma_p^2 = (0.6842)^2(324) + (0.3158)^2(576) + 2(0.6842)(0.3158)(108) = 151.68 + 57.44 + 46.68 = 255.80$$
  $$\sigma_p = \sqrt{255.80} = \mathbf{15.99\%}$$
  (Portfolio risk of 15.99% is lower than both individual funds: 18% and 24%).

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(c) Investment Policy Statement (IPS) Framework for Endowment Fund (6 Marks)

An Investment Policy Statement (IPS) for an educational or healthcare endowment in Nepal must detail:

+----------------------------------------------------------------------+
|                     ENDOWMENT IPS SPECIFICATIONS                     |
+-------------------+--------------------------------------------------+
| 1. Return Obj.    | Preserve real purchasing power + 4.5% net annual |
|                   | spending distribution (Target nominal return 11%)|
+-------------------+--------------------------------------------------+
| 2. Risk Tolerance | Moderate; maximum allowable 1-year drawdown 10%; |
|                   | long investment horizon permits equity volatility|
+-------------------+--------------------------------------------------+
| 3. Liquidity Req. | Maintain 12 months of operating expenses in short-|
|                   | term bank fixed deposits and money market funds. |
+-------------------+--------------------------------------------------+
| 4. Time Horizon   | Perpetual horizon (> 20 years).                  |
+-------------------+--------------------------------------------------+
| 5. Regulatory /   | Comply with SEBON mutual fund investment caps,   |
|    Legal Limits   | NRB bank exposure limits; zero tobacco/arms stock|
+-------------------+--------------------------------------------------+

• Asset Allocation Benchmarks: 50% Equity / Mutual Funds, 35% Nepal Government Bonds & Debentures, 15% Cash & Bank Fixed Deposits.