Portfolio Theory

Q1 (10 marks)

Derivation (5): $\sigma_p^2 = \mathrm{Var}(w_A R_A + w_B R_B)$ $= w_A^2\mathrm{Var}(R_A) + w_B^2\mathrm{Var}(R_B) + 2w_A w_B\mathrm{Cov}(R_A,R_B)$ (bilinearity of variance) (3) $= w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_A w_B\rho_{AB}\sigma_A\sigma_B$ (since $\mathrm{Cov}=\rho\sigma_A\sigma_B$) (2)

Computation (4): $= 0.25(0.04) + 0.25(0.09) + 2(0.5)(0.5)(0.2)(0.2)(0.3)$ $= 0.01 + 0.0225 + 0.006 = 0.0385$ $\sigma_p = \sqrt{0.0385} = 0.19621 = 19.62\%$ (4)

Explanation (1): Because $\rho_{AB} < 1$, the covariance term is smaller than the cross term of a perfect correlation, so imperfect correlation cancels part of the risk — diversification benefit. Weighted-average SD $= 0.5(20)+0.5(30)=25\% > 19.62\%$.

Q2 (10 marks)

Derivation (6): Set $w_B = 1-w_A$, differentiate $\sigma_p^2$ w.r.t. $w_A$, set to 0: $w_A^* = \frac{\sigma_B^2 - \rho_{AB}\sigma_A\sigma_B}{\sigma_A^2 + \sigma_B^2 - 2\rho_{AB}\sigma_A\sigma_B}$ Computation (3): Numerator $= 0.09 - 0.2(0.2)(0.3) = 0.09 - 0.012 = 0.078$ Denominator $= 0.04 + 0.09 - 2(0.012) = 0.13 - 0.024 = 0.106$ $w_A^* = 0.078/0.106 = 0.7358 = 73.58\%$ (so $w_B^*=26.42\%$). Comment (1): Both weights positive → no short-selling required.

Q3 (12 marks)

(a) $E(R_i) = R_f + \beta_i(E(R_m) - R_f)$; $R_f$=risk-free rate, $\beta_i$=sensitivity to market, $E(R_m)$=expected market return, $(E(R_m)-R_f)$=market risk premium. (4) (b) $4\% + 1.4(10\%-4\%) = 4 + 1.4(6) = 4 + 8.4 = 12.4\%$ (3) (c) $\alpha = 14\% - 12.4\% = 1.6\%$; positive alpha → lies above the SML (underpriced/outperforming). (3) (d) Unsystematic risk can be diversified away for free, so the market does not compensate investors for bearing it; only non-diversifiable (systematic/market) risk earns a premium. (2)

Q4 (12 marks)

(a) Excess return $= 12\%-3\% = 9\%$.

• Sharpe $= 9/18 = 0.5$ (2)
• Sortino $= 9/10 = 0.9$ (2)
• Treynor $= 9\%/0.9 = 10\%$ (or 0.10) (2)

(b) Sharpe uses total volatility (all risk); Sortino replaces the denominator with downside deviation, penalizing only harmful downside moves (better for asymmetric returns); Treynor uses beta, measuring reward per unit of systematic risk only. (4)

(c) Treynor — within a well-diversified portfolio unsystematic risk is already eliminated, so systematic risk (beta) is the relevant risk measure. (2)

Q5 (10 marks)

import numpy as np
def portfolio_stats(weights, mean_returns, cov_matrix, rf):
    w = np.array(weights)                       # weight vector
    ret = np.dot(w, mean_returns)               # expected portfolio return w·μ
    var = np.dot(w, np.dot(cov_matrix, w))      # variance wᵀΣw
    std = np.sqrt(var)                          # portfolio std dev
    sharpe = (ret - rf) / std                   # excess return per unit risk
    return ret, std, sharpe

Marks: correct return dot product (2), quadratic form for variance (3), sqrt (1), Sharpe (2), returns/structure (2).

Q6 (6 marks)

Total risk $=$ systematic (market/undiversifiable) $+$ unsystematic (specific/diversifiable) (2). As $N$ increases with roughly equal weights, unsystematic variance falls ~$1/N$ while covariance (systematic) terms dominate; risk declines steeply then flattens (2). Limiting value = average covariance = systematic risk floor; it cannot be diversified below that (2).

[
{"claim":"Q1 portfolio std = 0.19621","code":"var=0.25*0.04+0.25*0.09+2*0.5*0.5*0.2*0.2*0.3; result=abs(float(var)**0.5-0.19621)<1e-3"},
{"claim":"Q2 minvar weight = 0.7358","code":"num=0.09-0.2*0.2*0.3; den=0.04+0.09-2*0.2*0.2*0.3; result=abs(num/den-0.7358)<1e-3"},
{"claim":"Q3b required return = 12.4%","code":"r=4+1.4*(10-4); result=r==12.4"},
{"claim":"Q4 Sharpe=0.5, Sortino=0.9, Treynor=10","code":"result=(9/18==0.5) and (9/10==0.9) and (9/0.9==10.0)"}
]