Portfolio Theory

Question 1

(a) For two assets, portfolio variance is $\sigma_p^2 = w_A^2\sigma_A^2 + (1-w_A)^2\sigma_B^2 + 2w_A(1-w_A)\sigma_{AB},$ where $\sigma_{AB} = \rho_{AB}\sigma_A\sigma_B$. Setting $\frac{d\sigma_p^2}{dw_A}=0$: $w_A^{GMVP} = \frac{\sigma_B^2 - \sigma_{AB}}{\sigma_A^2 + \sigma_B^2 - 2\sigma_{AB}}. \quad (2\text{ marks for derivation})$

Compute $\sigma_{AB} = 0.20\cdot0.20\cdot0.30 = 0.012$. $w_A = \frac{0.09 - 0.012}{0.04 + 0.09 - 2(0.012)} = \frac{0.078}{0.106} = 0.7358.$ $w_B = 0.2642$. (2 formula + 2 correct numbers = 6)

(b) $\mu_p = 0.7358(0.10) + 0.2642(0.16) = 0.11585 \approx 0.1159$. (1) $\sigma_p^2 = 0.7358^2(0.04) + 0.2642^2(0.09) + 2(0.7358)(0.2642)(0.012)$ $= 0.021658 + 0.006283 + 0.004666 = 0.032607 \Rightarrow \sigma_p = 0.1806.$ (2) Since $0.1806 < 0.20 = \min(\sigma_A,\sigma_B)$, diversification lowers risk. (1) Why: because $\rho_{AB}=0.2 < 1$, the covariance term is smaller than it would be under perfect correlation; imperfectly correlated assets partially offset each other's fluctuations, so combined variance falls below the individual variances. (1)

(c) With $\rho=-1$, $\sigma_{AB} = -\sigma_A\sigma_B$ and $\sigma_p^2 = (w_A\sigma_A - (1-w_A)\sigma_B)^2.$ This is a perfect square, so $\sigma_p=0$ when $w_A\sigma_A = (1-w_A)\sigma_B$, giving $w_A = \frac{\sigma_B}{\sigma_A+\sigma_B} = \frac{0.30}{0.50} = 0.60.$ (2 for perfect-square form + 2 for weight = 4)

(d)

import numpy as np
def frontier(mu, cov, n):
    ws = np.linspace(0, 1, n)
    pts = []
    for wA in ws:
        w = np.array([wA, 1-wA])
        mp = w @ mu
        vp = w @ cov @ w
        pts.append((np.sqrt(vp), mp))
    return np.array(pts)

Efficient portion: find the GMVP (minimum $\sigma_p$); all points with $\mu_p \ge \mu_{GMVP}$ form the efficient frontier — for any given risk level an investor prefers the higher return, so the lower branch (below GMVP return) is dominated. (3 code + 2 explanation = 5)

Question 2

(a) $\beta_P = \dfrac{\text{Cov}(P,M)}{\sigma_M^2} = \dfrac{0.0243}{0.18^2} = \dfrac{0.0243}{0.0324} = 0.75$. (2) SML: $r_P^{CAPM} = r_f + \beta_P(\mu_M - r_f) = 0.04 + 0.75(0.11-0.04) = 0.04 + 0.0525 = 0.0925$. (3)

(b) Jensen's alpha $= \mu_P - r_P^{CAPM} = 0.14 - 0.0925 = 0.0475$ (4.75%). (3) Positive alpha ⇒ the manager earned 4.75% above the systematic-risk-adjusted required return ⇒ adding value. (1)

(c) Systematic variance $= \beta_P^2\sigma_M^2 = 0.75^2(0.0324) = 0.018225$. (2) Total variance $= \sigma_P^2 = 0.25^2 = 0.0625$. (1) Unsystematic variance $= 0.0625 - 0.018225 = 0.044275$. (1) Fractions: systematic $= 0.018225/0.0625 = 0.2916$ (29.16%); unsystematic $= 70.84\%$. (1)

(d) Sharpe$_P = \dfrac{\mu_P - r_f}{\sigma_P} = \dfrac{0.10}{0.25} = 0.40$. Sharpe$_M = \dfrac{0.11-0.04}{0.18} = \dfrac{0.07}{0.18} = 0.3889$. (2) Treynor$_P = \dfrac{\mu_P - r_f}{\beta_P} = \dfrac{0.10}{0.75} = 0.1333$. Treynor$_M = \dfrac{0.07}{1} = 0.07$. (1) So $P$ beats market on both here; but in general why divergence occurs: Treynor divides only by systematic risk ($\beta$), Sharpe by total risk ($\sigma$). A portfolio with large unsystematic risk has high $\sigma$ relative to $\beta$, so it can look strong on Treynor (which ignores diversifiable risk) yet weak on Sharpe. Divergence therefore flags poor diversification / high idiosyncratic risk. (2)

(e) For a fully diversified portfolio, $\sigma_P = \beta_P\sigma_M$ (zero residual). Then $\text{Sharpe}_P = \frac{\mu_P-r_f}{\beta_P\sigma_M} = \frac{1}{\sigma_M}\cdot\frac{\mu_P-r_f}{\beta_P} = \frac{\text{Treynor}_P}{\sigma_M}.$ Since $\sigma_M>0$ is a common positive constant, Sharpe and Treynor are proportional across all such portfolios, giving identical rankings. (3)

Question 3

(a) Downside deviation relative to $\tau$: $\sigma_d = \sqrt{\frac{1}{N}\sum_{i=1}^N \big(\min(r_i-\tau,\,0)\big)^2}.$ Sortino $= \dfrac{\bar r - \tau}{\sigma_d}$. It differs from Sharpe because Sharpe penalises all volatility (upside and downside) via $\sigma$, whereas Sortino only penalises returns below the target — investors dislike downside, not upside dispersion. (2 def + 2 why = 4)

(b) Mean $\bar r = (0.05-0.02+0.03-0.06+0.04+0.01-0.03+0.02)/8 = 0.04/8 = 0.005$. (1) Shortfalls (negatives only): $-0.02, -0.06, -0.03$. Squares: $0.0004, 0.0036, 0.0009$; sum $=0.0049$. (2) $\sigma_d = \sqrt{0.0049/8} = \sqrt{0.0006125} = 0.024749$. (2) Sortino $= 0.005/0.024749 = 0.2020$. (1)

(c)

import numpy as np
def sortino(returns, target, periods_per_year):
    r = np.asarray(returns, dtype=float)
    excess_mean = r.mean() - target
    downside = np.minimum(r - target, 0.0)
    dd = np.sqrt(np.mean(downside**2))
    if dd == 0:
        return np.inf
    ann_mean = excess_mean * periods_per_year
    ann_dd = dd * np.sqrt(periods_per_year)
    return ann_mean / ann_dd

(4 — must annualise mean by periods and dd by √periods, and guard dd=0)

(d) If no observation is below target, every $\min(r_i-\tau,0)=0$, so $\sum(\cdot)^2 = 0 \Rightarrow \sigma_d = 0$. The Sortino ratio $\frac{\bar r-\tau}{0}$ is undefined; with positive excess mean it tends to $+\infty$. (2) Implication: the metric cannot rank such a strategy against peers with finite downside risk — the ratio saturates at infinity for any strategy that never breaches the target, erasing differences among them. In practice one must either use a longer/finer sample, raise the target, or fall back to Sharpe for comparison. (2)

[
  {"claim":"GMVP weight wA = 0.7358","code":"sA,sB,rho=0.20,0.30,0.20; sAB=rho*sA*sB; wA=(sB**2-sAB)/(sA**2+sB**2-2*sAB); result=abs(float(wA)-0.7358)<0.001"},
  {"claim":"GMVP volatility 0.1806","code":"sA,sB,rho=0.20,0.30,0.20; sAB=rho*sA*sB; wA=(sB**2-sAB)/(sA**2+sB**2-2*sAB); wB=1-wA; var=wA**2*sA**2+wB**2*sB**2+2*wA*wB*sAB; result=abs(float(var**0.5)-0.1806)<0.001"},
  {"claim":"rho=-1 risk-free weight wA=0.60","code":"sA,sB=0.20,0.30; wA=sB/(sA+sB); result=abs(float(wA)-0.60)<1e-9"},
  {"claim":"beta=0.75 and Jensen alpha=0.0475","code":"cov,sM=0.0243,0.18; beta=cov/sM**2; rf,muM,muP=0.04,0.11,0.14; req=rf+beta*(muM-rf); alpha=muP-req; result=abs(float(beta)-0.75)<1e-9 and abs(float(alpha)-0.0475)<1e-9"},
  {"claim":"systematic variance fraction 0.2916","code":"beta,sM,sP=0.75,0.18,0.25; frac=(beta**2*sM**2)/sP**2; result=abs(float(frac)-0.2916)<0.001"},
  {"claim":"downside deviation 0.024749 and Sortino 0.2020","code":"import numpy as np; r=np.array([0.05,-0.02,0.03,-0.06,0.04,0.01,-0.03,0.02]); dd=float(np.sqrt(np.mean(np.minimum(r,0)**2))); s=float(r.mean()/dd); result=abs(dd-0.024749)<1e-4 and abs(s-0.2020)<1e-3"}
]