Portfolio Theory

Question 1. [4 marks]

• Definition: Diversification is the practice of spreading investment across multiple assets to reduce risk without necessarily sacrificing expected return. (2)
• Explanation: When assets have low/negative correlation, their price movements partially offset one another; losses in one are cushioned by gains (or smaller losses) in another, so portfolio variance is less than the weighted average of individual variances. (2)

Question 2. [4 marks] Formula: $\text{Cov}(A,B) = \rho_{AB}\,\sigma_A\,\sigma_B$ (1) $\text{Cov}(A,B) = 0.4 \times 0.20 \times 0.30 = 0.024$ (3) (Equivalently $2.4\%^2$ or $0.024$.)

Question 3. [6 marks] Formula: (1) $\sigma_p = \sqrt{w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_A w_B \rho_{AB}\sigma_A\sigma_B}$ Substitute: (2)

• $w_A^2\sigma_A^2 = 0.36 \times 0.04 = 0.0144$
• $w_B^2\sigma_B^2 = 0.16 \times 0.09 = 0.0144$
• $2w_A w_B \rho\sigma_A\sigma_B = 2(0.6)(0.4)(0.024) = 0.011520$

Sum $= 0.0144 + 0.0144 + 0.01152 = 0.04032$ (2) $\sigma_p = \sqrt{0.04032} \approx 0.2008 = 20.08\%$ (1)

Question 4. [5 marks] CAPM: $E(R_i) = R_f + \beta_i\,(E(R_m) - R_f)$ (2) where $R_f$ = risk-free rate, $\beta_i$ = beta (sensitivity to market), $E(R_m)$ = expected market return, $(E(R_m)-R_f)$ = market risk premium. (1) Calculation: (2) $E(R_i) = 5\% + 1.2\times(11\% - 5\%) = 5\% + 1.2\times6\% = 5\% + 7.2\% = 12.2\%$

Question 5. [4 marks]

• Systematic risk: market-wide risk affecting all assets, cannot be diversified away (e.g. interest-rate changes, recession, inflation). (2)
• Unsystematic risk: firm/industry-specific risk, can be eliminated through diversification (e.g. a company lawsuit, product failure). (2) Award marks for correct examples and stating unsystematic risk is diversifiable.

Question 6. [6 marks] Sharpe ratio: $S = \dfrac{R_p - R_f}{\sigma_p}$ (1) $S = \frac{0.14 - 0.04}{0.20} = \frac{0.10}{0.20} = 0.5$ (2) Treynor ratio: $T = \dfrac{R_p - R_f}{\beta_p}$ (1) $T = \frac{0.14 - 0.04}{1.25} = \frac{0.10}{1.25} = 0.08$ (2)

Question 7. [5 marks]

• Definition: The efficient frontier is the set of optimal portfolios offering the highest expected return for each level of risk (or lowest risk for each return level). (2)
• A portfolio on the frontier is efficient — no other portfolio gives higher return for the same risk. (2)
• A portfolio below the frontier is sub-optimal (dominated); another portfolio exists with higher return at the same risk (or same return at less risk). (1)

Question 8. [6 marks] Expected (CAPM) return: (2) $E(R) = R_f + \beta(R_m - R_f) = 3\% + 1.4\times(12\% - 3\%) = 3\% + 1.4\times9\% = 3\% + 12.6\% = 15.6\%$ Alpha: (2) $\alpha = \text{Actual} - E(R) = 16\% - 15.6\% = 0.4\%$ Interpretation: $\alpha > 0$, so the fund outperformed its benchmark on a risk-adjusted basis. (2)

Question 9. [4 marks]

• Sharpe ratio uses total standard deviation (both upside and downside volatility) as the risk measure. (1)
• Sortino ratio replaces total $\sigma$ with downside deviation (only returns below a target/MAR). (2)
• Advantage: it does not penalize upside volatility, giving a truer picture of harmful (downside) risk that concerns loss-averse investors. (1)

Question 10. [6 marks]

• SML is a graphical representation of CAPM. (1)
• X-axis: beta ($\beta$, systematic risk); Y-axis: expected return $E(R)$. (2)
• Intercept: risk-free rate $R_f$ (at $\beta = 0$); Slope: market risk premium $(E(R_m) - R_f)$. (2)
• An asset plotting above the SML is undervalued — it offers more return than CAPM requires for its beta, hence a positive alpha (a buy). (1)

[
  {"claim": "Q2 covariance = 0.024", "code": "result = (0.4*0.20*0.30 == 0.024)"},
  {"claim": "Q3 portfolio std dev approx 0.2008", "code": "sd = sqrt(Rational(36,100)*Rational(4,100)+Rational(16,100)*Rational(9,100)+2*Rational(6,10)*Rational(4,10)*0.4*0.20*0.30); result = abs(float(sd)-0.2008) < 0.001"},
  {"claim": "Q4 CAPM expected return = 12.2%", "code": "result = (0.05 + 1.2*(0.11-0.05) == 0.122)"},
  {"claim": "Q6 Sharpe 0.5 and Treynor 0.08", "code": "result = ((0.14-0.04)/0.20 == 0.5) and (abs((0.14-0.04)/1.25 - 0.08) < 1e-9)"},
  {"claim": "Q8 alpha = 0.4%", "code": "capm = 0.03 + 1.4*(0.12-0.03); result = abs((0.16 - capm) - 0.004) < 1e-9"}
]