Portfolio Theory

Question 1

(a) Expected return: $E(R_p) = 0.6(12\%) + 0.4(8\%) = 7.2\% + 3.2\% = 10.4\%$ (2 marks)

Variance: $\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho\sigma_A\sigma_B$ $= 0.36(0.04) + 0.16(0.01) + 2(0.6)(0.4)(-0.30)(0.20)(0.10)$ $= 0.0144 + 0.0016 - 0.00288 = 0.01312$ $\sigma_p = \sqrt{0.01312} = 0.1145 = 11.45\%$ (3 marks: variance formula 1, covariance term 1, sqrt 1)

(b) Covariance: $\text{Cov}_{AB} = \rho\sigma_A\sigma_B = -0.30(0.20)(0.10) = -0.006$ (2 marks)

(c) Minimum-variance weight: $w_A = \frac{\sigma_B^2 - \text{Cov}_{AB}}{\sigma_A^2 + \sigma_B^2 - 2\text{Cov}_{AB}} = \frac{0.01 - (-0.006)}{0.04 + 0.01 - 2(-0.006)}$ $= \frac{0.016}{0.062} = 0.2581 \approx 25.8\%$ (3 marks)

With $w_A = 0.2581$, $w_B = 0.7419$: $\sigma_p^2 = 0.2581^2(0.04) + 0.7419^2(0.01) + 2(0.2581)(0.7419)(-0.006)$ $= 0.002664 + 0.005504 - 0.002298 = 0.005870$ $\sigma_p = 0.0766 = 7.66\%$ (2 marks)

Why: Negative correlation gives strong diversification — min-variance std dev (7.66%) is below both individual assets.

Question 2

(a) CAPM: $R = 4\% + 1.4(10\% - 4\%) = 4\% + 8.4\% = 12.4\%$ (3 marks)

(b) Alpha $= 13.6\% - 12.4\% = +1.2\%$. Positive alpha ⇒ plots above the SML ⇒ undervalued ⇒ buy. (4 marks: alpha 2, position 1, decision 1)

(c) On SML: $8.2\% = 4\% + \beta(6\%) \Rightarrow \beta = 4.2/6 = 0.7$ (3 marks)

(d) Why: The SML prices assets on beta (systematic risk), not total standard deviation. Two stocks with equal total σ may have very different systematic-risk shares; the one with higher beta contributes more non-diversifiable risk to a portfolio and thus requires a higher return. Unsystematic risk (which contributes to σ but not β) is uncompensated because it can be diversified away. (4 marks)

Question 3

(a) Systematic variance $= \beta^2\sigma_m^2 = 1.2^2(0.020) = 1.44(0.020) = 0.0288$ (3 marks)

(b) Unsystematic variance $= \sigma_{total}^2 - \text{systematic} = 0.040 - 0.0288 = 0.0112$ (3 marks)

(c) Fraction diversifiable $= 0.0112 / 0.040 = 0.28 = 28\%$. (2 marks) Interpretation: For a well-diversified investor this 28% unsystematic portion is already eliminated across holdings, so only the systematic 72% ($\beta^2\sigma_m^2$) matters for pricing/expected return. (2 marks)

Question 4

(a) Sharpe $= (R - R_f)/\sigma$:

• X: $(15-3)/18 = 12/18 = 0.667$
• Y: $(11-3)/9 = 8/9 = 0.889$

Y superior on Sharpe. (4 marks: 1.5 each + comparison 1)

(b) Sortino $= (R - R_f)/\text{downside dev}$:

• X: $12/10 = 1.20$
• Y: $8/6 = 1.333$ (4 marks)

(c) Treynor $= (R - R_f)/\beta$:

• X: $12/1.1 = 10.91\%$
• Y: $8/0.7 = 11.43\%$ (4 marks)

(d) Why: Y beats X on Sharpe and Sortino (better return per unit of total/downside risk). On Treynor Y is also higher here. If X ever wins on Treynor but loses on Sharpe, it signals X carries much unsystematic (diversifiable) risk that penalises its total-risk measures but not its beta measure — relevant only if held in isolation vs. within a diversified portfolio. (2 marks)

Question 5

(a) CAL slope = Sharpe of P $= (14-4)/22 = 10/22 = 0.4545$: $E(R_c) = 4\% + 0.4545\,\sigma_c$ (4 marks)

(b) $\sigma_c = w_P \times 22\% \Rightarrow w_P = 11/22 = 0.5$ (50% in P, 50% risk-free). $E(R_c) = 0.5(14\%) + 0.5(4\%) = 9\%$ (4 marks)

(c) Why: The tangency portfolio maximises the reward-to-risk (Sharpe) slope of the CAL, so combining it with the risk-free asset dominates every other risky portfolio — every investor's optimal risky mix is the tangency portfolio (two-fund separation). (2 marks)

[
  {"claim":"Q1a portfolio std dev = 11.45%","code":"sigma2=0.36*0.04+0.16*0.01+2*0.6*0.4*(-0.30)*0.20*0.10; s=sqrt(sigma2); result=abs(float(s)-0.11454)<0.001"},
  {"claim":"Q1c min-variance weight in A = 0.2581","code":"wA=(0.01-(-0.006))/(0.04+0.01-2*(-0.006)); result=abs(float(wA)-0.25806)<0.001"},
  {"claim":"Q1c min-variance std dev = 7.66%","code":"wA=0.016/0.062; wB=1-wA; v=wA**2*0.04+wB**2*0.01+2*wA*wB*(-0.006); result=abs(float(sqrt(v))-0.0766)<0.001"},
  {"claim":"Q2a CAPM required return = 12.4%","code":"r=0.04+1.4*(0.10-0.04); result=abs(float(r)-0.124)<1e-9"},
  {"claim":"Q2c beta from required return 8.2% = 0.7","code":"b=(0.082-0.04)/0.06; result=abs(float(b)-0.7)<1e-9"},
  {"claim":"Q3 unsystematic variance = 0.0112 and 28% diversifiable","code":"sysv=1.2**2*0.020; unsys=0.040-sysv; frac=unsys/0.040; result=abs(float(unsys)-0.0112)<1e-9 and abs(float(frac)-0.28)<1e-9"},
  {"claim":"Q4 Sharpe Y=0.889 > X=0.667","code":"sx=(15-3)/18; sy=(11-3)/9; result=(abs(float(sy)-0.8889)<0.001) and (sy>sx)"},
  {"claim":"Q4c Treynor X=10.91, Y=11.43","code":"tx=12/1.1; ty=8/0.7; result=abs(float(tx)-10.909)<0.01 and abs(float(ty)-11.4286)<0.01"},
  {"claim":"Q5b weight in P = 0.5 and return = 9%","code":"wp=11/22; rc=wp*0.14+(1-wp)*0.04; result=abs(float(wp)-0.5)<1e-9 and abs(float(rc)-0.09)<1e-9"}
]