Dean's Office Official Model Question Paper

Systematic vs. Unsystematic Risk

• Systematic (Market) Risk: Macroeconomic risk factors affecting the entire financial market (interest rate changes, inflation, GDP shocks, political events). It cannot be eliminated through diversification (measured by Beta $\beta$).
• Unsystematic (Specific) Risk: Microeconomic risk factors unique to a specific firm or industry (strikes, management errors, patent expiration). It can be diversified away in a well-diversified portfolio.

CAPM Formula

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

Where:

• $E(R_i)$ = Expected return on security $i$
• $R_f$ = Risk-free rate of return
• $\beta_i$ = Beta coefficient (systematic risk measure of asset $i$ relative to market)
• $E(R_m)$ = Expected return on the market portfolio
• $[E(R_m) - R_f]$ = Market risk premium

Efficient Market Hypothesis (EMH)

EMH posits that asset prices fully and instantaneously reflect all available information.

1. Weak Form: Prices reflect all historical trading data and price trends (technical analysis cannot yield abnormal returns).
2. Semi-Strong Form: Prices reflect all publicly available information (fundamental analysis cannot yield abnormal returns).
3. Strong Form: Prices reflect all public and private insider information.

Sharpe Ratio

$$\text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}$$

It measures the excess return earned by a portfolio per unit of total risk (standard deviation $\sigma_p$). A higher Sharpe ratio indicates superior risk-adjusted performance.

CML vs. SML

• Capital Market Line (CML): Evaluates efficient portfolios only, graphing expected return against total risk (Standard Deviation $\sigma$).
• Security Market Line (SML): Evaluates any individual security or portfolio (efficient or inefficient), graphing expected return against systematic risk (Beta $\beta$).

Portfolio Risk and Return Computation

a. Expected Return and Standard Deviation of Stocks X and Y

Stock X:

$$E(R_X) = \sum P_i R_{X,i} = 0.30(25) + 0.50(15) + 0.20(-5) = 7.5 + 7.5 - 1.0 = \mathbf{14\%}$$

$$\begin{aligned} \sigma_X^2 &= 0.30(25 - 14)^2 + 0.50(15 - 14)^2 + 0.20(-5 - 14)^2 \\ &= 0.30(121) + 0.50(1) + 0.20(361) = 36.3 + 0.5 + 72.2 = 109 \end{aligned}$$

$$\sigma_X = \sqrt{109} \approx \mathbf{10.44\%}$$

Stock Y:

$$E(R_Y) = \sum P_i R_{Y,i} = 0.30(15) + 0.50(10) + 0.20(5) = 4.5 + 5.0 + 1.0 = \mathbf{10.5\%}$$

$$\begin{aligned} \sigma_Y^2 &= 0.30(15 - 10.5)^2 + 0.50(10 - 10.5)^2 + 0.20(5 - 10.5)^2 \\ &= 0.30(20.25) + 0.50(0.25) + 0.20(30.25) = 6.075 + 0.125 + 6.05 = 12.25 \end{aligned}$$

$$\sigma_Y = \sqrt{12.25} = \mathbf{3.50\%}$$

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b. Covariance, Correlation, and Portfolio Metrics ($w_X = 0.60, w_Y = 0.40$)

Covariance ($Cov_{XY}$):

$$\begin{aligned} Cov_{XY} &= \sum P_i [R_{X,i} - E(R_X)][R_{Y,i} - E(R_Y)] \\ &= 0.30(25 - 14)(15 - 10.5) + 0.50(15 - 14)(10 - 10.5) + 0.20(-5 - 14)(5 - 10.5) \\ &= 0.30(11)(4.5) + 0.50(1)(-0.5) + 0.20(-19)(-5.5) \\ &= 14.85 - 0.25 + 20.90 = \mathbf{35.50} \end{aligned}$$

Correlation Coefficient ($\rho_{XY}$):

$$\rho_{XY} = \frac{Cov_{XY}}{\sigma_X \sigma_Y} = \frac{35.50}{10.44 \times 3.50} = \frac{35.50}{36.54} \approx \mathbf{+0.9715}$$

Portfolio Expected Return ($E(R_p)$):

$$E(R_p) = w_X E(R_X) + w_Y E(R_Y) = 0.60(14) + 0.40(10.5) = 8.4 + 4.2 = \mathbf{12.60\%}$$

Portfolio Standard Deviation ($\sigma_p$):

$$\begin{aligned} \sigma_p^2 &= w_X^2 \sigma_X^2 + w_Y^2 \sigma_Y^2 + 2 w_X w_Y Cov_{XY} \\ &= (0.60)^2(109) + (0.40)^2(12.25) + 2(0.60)(0.40)(35.50) \\ &= 0.36(109) + 0.16(12.25) + 0.48(35.50) = 39.24 + 1.96 + 17.04 = 58.24 \end{aligned}$$

$$\sigma_p = \sqrt{58.24} \approx \mathbf{7.63\%}$$

(Note: Through portfolio diversification, portfolio risk of 7.63% is significantly lower than Stock X’s individual risk of 10.44%).

Markowitz Modern Portfolio Theory (MPT) and Efficient Frontier

Markowitz established that an asset’s risk should not be assessed in isolation, but by how it contributes to an overall portfolio’s risk-return profile.

Expected Return E(R)
   ^
   |               EFFICIENT FRONTIER
   |                 *  *  *
   |              *           Optimal Portfolio
   |            *           /
   |           * <---------+
   |          *
   |  MVP -> *
   |          *  (Feasible Set)
   |           *
   +-------------------------------------> Total Risk (Sigma)

1. Core Principles of MPT

• Diversification and Correlation: As long as asset returns are not perfectly positively correlated ($\rho < +1$), combining assets reduces overall portfolio variance without sacrificing expected return.
• Minimum Variance Portfolio (MVP): The portfolio on the efficient frontier with the lowest possible risk.
• The Efficient Frontier: The set of optimal portfolios that offer the highest expected return for a given level of risk, or the lowest risk for a given level of expected return.

2. Investor Utility Optimization

• An investor’s optimal portfolio occurs at the tangency point where their indifference curve touches the Efficient Frontier.

Dividend Discount Models (DDM) in Equity Valuation

DDM values a share of common stock as the present value of all expected future cash dividends.

1. Constant Growth DDM (Gordon Growth Model)

Assumes dividends grow indefinitely at a constant rate $g < r$:

$$P_0 = \frac{D_1}{r - g} = \frac{D_0 (1 + g)}{r - g}$$

Where:

• $P_0$ = Intrinsic value per share today
• $D_0$ = Current dividend paid
• $D_1$ = Expected dividend in year 1
• $r$ = Required rate of return on equity (via CAPM)
• $g$ = Constant sustainable growth rate ($g = b \times ROE$, where $b$ is retention ratio)

2. Two-Stage / Multistage Growth Model

Used for high-growth enterprises that experience a temporary phase of supernormal growth ($g_s$) followed by long-term stable growth ($g_n$):

$$P_0 = \sum_{t=1}^n \frac{D_t}{(1 + r)^t} + \frac{P_n}{(1 + r)^n}, \quad \text{where } P_n = \frac{D_{n+1}}{r - g_n}$$

Portfolio Performance Evaluation Measures

Comprehensive Security and Portfolio Analysis

a. CAPM Required Rate of Return Computation

$$\text{CAPM Formula: } k_i = R_f + \beta_i [E(R_m) - R_f]$$

Given: $R_f = 6.0\%$, $E(R_m) = 13.0\%$, Market Risk Premium = $13.0 - 6.0 = 7.0\%$.

• Bank A: $k_A = 6.0 + 1.30(7.0) = 6.0 + 9.10 = \mathbf{15.10\%}$
• Bank B: $k_B = 6.0 + 0.85(7.0) = 6.0 + 5.95 = \mathbf{11.95\%}$
• Bank C: $k_C = 6.0 + 1.10(7.0) = 6.0 + 7.70 = \mathbf{13.70\%}$
• Bank D: $k_D = 6.0 + 0.65(7.0) = 6.0 + 4.55 = \mathbf{10.55\%}$

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b. Valuation on the Security Market Line (SML)

All four stocks generate positive alpha and plot above the SML, offering excess risk-adjusted returns relative to the market.

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c. Intrinsic Value via Gordon Growth Model ($P_0 = \frac{D_0(1 + g)}{k_i - g}$)

• Bank A:

  $$D_1 = 18(1 + 0.08) = 19.44$$
  $$V_A = \frac{19.44}{0.1510 - 0.08} = \frac{19.44}{0.0710} \approx \mathbf{\text{Rs } 273.80}$$
  Comparison: Market price is Rs 420. Despite positive CAPM alpha, under fundamental DDM dividend capitalization, Bank A is Market Overvalued ($P_0 > V_A$).

• Bank B:

  $$D_1 = 14(1 + 0.06) = 14.84$$
  $$V_B = \frac{14.84}{0.1195 - 0.06} = \frac{14.84}{0.0595} \approx \mathbf{\text{Rs } 249.41}$$
  Comparison: Market price is Rs 280. Moderately overvalued on DDM.

• Bank C:

  $$D_1 = 15(1 + 0.07) = 16.05$$
  $$V_C = \frac{16.05}{0.1370 - 0.07} = \frac{16.05}{0.0670} \approx \mathbf{\text{Rs } 239.55}$$

• Bank D:

  $$D_1 = 10(1 + 0.05) = 10.50$$
  $$V_D = \frac{10.50}{0.1055 - 0.05} = \frac{10.50}{0.0555} \approx \mathbf{\text{Rs } 189.19}$$

Strategic Takeaway: The discrepancy highlights that NEPSE banking stocks trade at a premium over dividend yields due to stock-dividend (bonus share) expectations.

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d. Target Portfolio Allocation (Portfolio $\beta_p = 1.00$)

To achieve a balanced market-risk profile ($\beta_p = 1.00$) while tilting toward highest alpha:

• Overweight Bank A (High alpha $+1.40\%$, $\beta = 1.30$) and Bank B (High alpha $+1.05\%$, $\beta = 0.85$):
  $$\beta_p = w_A(1.30) + w_B(0.85) = 1.00$$
  $$w_A + w_B = 1.00 \implies w_B = 1 - w_A$$
  $$1.30 w_A + 0.85(1 - w_A) = 1.00$$
  $$0.45 w_A = 0.15 \implies w_A = \frac{0.15}{0.45} = \mathbf{33.33\%}$$
  $$w_B = 1 - 0.3333 = \mathbf{66.67\%}$$

Optimal Allocation: Invest Rs 166.67 million (33.33%) in Bank A and Rs 333.33 million (66.67%) in Bank B to lock in a portfolio beta of exactly 1.00 with superior blended expected return.