Factor & Behavioral Finance

Question 1

(a) OLS derivation (8 marks)

Minimise $S(\boldsymbol\beta)=\lVert y-X\boldsymbol\beta\rVert^2=(y-X\boldsymbol\beta)^\top(y-X\boldsymbol\beta)$. $S = y^\top y - 2\boldsymbol\beta^\top X^\top y + \boldsymbol\beta^\top X^\top X\boldsymbol\beta.$ Gradient: $\nabla_{\boldsymbol\beta}S = -2X^\top y + 2X^\top X\boldsymbol\beta = 0$. (2) Normal equations $X^\top X\hat{\boldsymbol\beta}=X^\top y \Rightarrow \hat{\boldsymbol\beta}=(X^\top X)^{-1}X^\top y$ (requires $X$ full column rank). (2) Hessian $2X^\top X\succeq 0$ ⇒ minimum. (1)

Gauss–Markov assumptions (any two, 2 marks): (i) $\mathbb{E}[\varepsilon\mid X]=0$ (exogeneity, zero conditional mean); (ii) homoskedasticity & no autocorrelation $\text{Var}(\varepsilon\mid X)=\sigma^2 I$. (Also linearity/full rank.) These give BLUE.

Economic interpretation of $\alpha$ (1 mark): $\alpha$ is the average return not explained by exposure to priced risk factors — risk-adjusted excess return (skill/mispricing). A positive significant $\alpha$ means outperformance beyond what factor betas justify.

(b) Alpha significance & tilts (8 marks)

Simple annualised alpha: $0.0025\times12 = 0.03 = 3.0\%$. (2) Compounded: $(1.0025)^{12}-1 = 0.030416 \approx 3.04\%$. (2) $t$-stat: $t=\alpha/SE = 0.0025/0.0011 = 2.2727$. (2) Since $2.27 > 2.00$, reject $H_0$: alpha is statistically significant at 5%. (1) Tilts (1 mark): $\beta_{SMB}=0.62>0$ ⇒ small-cap tilt (loads on small-minus-big); $\beta_{HML}=0.41>0$ ⇒ value tilt (loads on high book-to-market). Portfolio is a small-cap value fund.

(c) Five-factor extension (8 marks)

(i) Economic anomalies (3 marks):

• RMW (Robust-Minus-Weak): profitability premium — firms with robust operating profitability outperform weak ones.
• CMA (Conservative-Minus-Aggressive): investment premium — firms investing conservatively outperform aggressive investors.

(ii) Redundancy (2 marks): HML captures value; but value stocks tend to be low-investment/high-profitability. Once RMW and CMA are included they absorb the return variation previously attributed to HML, so $\beta_{HML}$ and its significance can shrink (HML becomes redundant in the FF5 for US data).

SMB pseudocode (3 marks):

# 2x3 sort: 2 size groups (S,B) x 3 value groups (L,N,H)
median_size = median(marketcap)
b30, b70 = quantiles(book_to_market, [0.3, 0.7])
for each group in {S,B}x{L,N,H}:
    port_return = value_weighted_mean(returns of stocks in that bucket)
SMB = (SL + SN + SH)/3  -  (BL + BN + BH)/3   # small minus big

Question 2

(a) Prospect theory (10 marks)

$v(+100)=100^{0.88}$. $\ln = 0.88\ln100 = 0.88(4.60517)=4.05255 \Rightarrow v(+100)=e^{4.05255}=57.54$. (3) $v(-100)=-2.25\cdot100^{0.88}=-2.25(57.54)=-129.46$. (3) Ratio $|v(-100)|/v(+100)=2.25$ (equals $\lambda$ since $\alpha=\beta$). (2)

Disposition explanation (2 marks): The value function is concave in gains (risk-averse over gains) so an investor is tempted to lock in a sure gain — sell winners early. It is convex in losses (risk-seeking over losses) so the investor gambles to recover, holding losers. Combined with reference-point dependence around purchase price, this produces the disposition effect: realise gains, defer losses.

(b) Herding fixed point (10 marks)

Cascade (2 marks): as $k\to\infty$, $\sigma(k(f-0.5))\to$ step function: any $f>0.5$ forces $p_i\to1$ (everyone buys), $f<0.5$ forces $p_i\to0$. Private signals are ignored — an information cascade where each trader copies the crowd.

Fixed point (3 marks): with common aggregate $f$, expected buy fraction is $g(f)=\sigma(k(f-0.5))$; equilibrium $f^\*=\sigma(k(f^\*-0.5))$.

$f^\*=0.5$ check (2 marks): $\sigma(k(0.5-0.5))=\sigma(0)=0.5$. ✓ Always a fixed point.

Stability (3 marks): $g'(f)=k\,\sigma(k(f-0.5))\,[1-\sigma(k(f-0.5))]$. At $f=0.5$: $\sigma=0.5$, so $g'(0.5)=k(0.5)(0.5)=k/4$. Fixed point unstable when $|g'|>1 \Rightarrow k/4>1 \Rightarrow k>4$. Critical $k^\*=4$: above it the symmetric equilibrium is unstable and two asymmetric cascade equilibria (herd-buy / herd-sell) emerge.

VERIFY note: $k_{crit}=4$, ratio $=2.25$, $v(+100)\approx57.54$.

Question 3

(a) EMH & martingale proof (10 marks)

Three forms (3 marks): Weak — prices reflect all past price/volume info (technical analysis useless). Semi-strong — reflect all public info (fundamental analysis of public data useless). Strong — reflect all info incl. private/insider.

Martingale proof (5 marks): Let $M_t=P_t-\mu t$. Then $\mathbb{E}[M_{t+1}\mid\mathcal{F}_t]=\mathbb{E}[P_{t+1}-\mu(t+1)\mid\mathcal{F}_t]=P_t+\mu+\mathbb{E}[\varepsilon_{t+1}\mid\mathcal{F}_t]-\mu(t+1).$ Since $\mathbb{E}[\varepsilon_{t+1}\mid\mathcal{F}_t]=0$: $=P_t+\mu-\mu(t+1)=P_t-\mu t=M_t.$ Thus $\{M_t\}$ is a martingale. Excess return $P_{t+1}-P_t-\mu=\varepsilon_{t+1}$ has conditional mean zero ⇒ unforecastable from $\mathcal{F}_t$ (past prices). ∎

Reconciliation (2 marks): Persistent premia are consistent with EMH under the risk-based view — value/momentum returns are compensation for systematic risk (a joint-hypothesis with the asset-pricing model). The behavioral view holds they are mispricings from bias (overreaction/underreaction) that limits-to-arbitrage prevent from being eliminated. The debate is unresolved due to the joint-hypothesis problem (Fama): any efficiency test jointly tests the model of expected returns.

(b) Alpha vs exposure vs data-mining test (6 marks)

• Null model (2): Regress ETF excess returns on the appropriate multifactor benchmark including a momentum factor (UMD/WML): $R-R_f=\alpha+\beta_{MKT}MKT+\beta_{SMB}SMB+\beta_{HML}HML+\beta_{MOM}MOM+\varepsilon$. Genuine skill ⇔ $\alpha$ significant after controlling for MOM. If MOM absorbs the return, it is factor exposure not alpha.
• Multiple-testing / Deflated Sharpe (2): Apply the Deflated Sharpe Ratio (Bailey–López de Prado) or Bonferroni/Harvey-Liu haircut: adjust the significance threshold for the number $N$ of strategies tested; expected max Sharpe under the null grows with $\sqrt{2\ln N}$, so raw high SR may be selection bias.
• Out-of-sample safeguard (2): Reserve a holdout / walk-forward period never used in strategy design; require the alpha to persist out-of-sample (and/or across markets, cross-validation) to rule out overfitting.

[
  {"claim":"t-stat for alpha = 0.0025/0.0011 ~ 2.2727 > 2.00 (reject H0)", "code":"t = 0.0025/0.0011; result = bool(t > 2.00) and bool(abs(t - Rational(25,11)) < Rational(1,1000))"},
  {"claim":"Compounded annual alpha (1.0025)^12 - 1 ~ 0.03042", "code":"val = (1 + Rational(25,10000))**12 - 1; result = bool(abs(float(val) - 0.030416) < 1e-4)"},
  {"claim":"Loss aversion ratio |v(-100)|/v(+100) = 2.25 and v(+100) ~ 57.54", "code":"vp = 100**0.88; vn = 2.25*100**0.88; ratio = vn/vp; result = bool(abs(ratio - 2.25) < 1e-9) and bool(abs(vp - 57.544) < 0.02)"},
  {"claim":"Critical herding intensity k_crit = 4 from g'(0.5)=k/4=1", "code":"k = symbols('k', positive=True); g = 1/(1+exp(-k*(Symbol('f')-Rational(1,2)))); gp = diff(g, Symbol('f')).subs(Symbol('f'), Rational(1,2)); sol = solve(Eq(gp, 1), k); result = bool(4 in [simplify(s) for s in sol])"}
]