Factor & Behavioral Finance

Question 1 (12)

(a) [6 marks]

Three-factor: $R_p - R_f = \alpha + \beta_{mkt}(R_m - R_f) + \beta_{s}\,\text{SMB} + \beta_{h}\,\text{HML} + \varepsilon$

• $(R_m-R_f)$ = market excess return (1)
• SMB (Small Minus Big) = return of small-cap portfolio minus large-cap (size) (1)
• HML (High Minus Low) = return of high book-to-market (value) minus low B/M (growth) (1)

Five-factor adds: $+\ \beta_r\,\text{RMW} + \beta_c\,\text{CMA}$

• RMW (Robust Minus Weak) = high profitability minus low profitability (1)
• CMA (Conservative Minus Aggressive) = low investment firms minus high investment firms (1)
• Correct full equation & alpha defined as intercept/abnormal return (1)

(b) [4 marks]

• $\beta_{mkt}=0.95$: slightly less than market risk / defensive-ish (1)
• SMB $=+0.40$: tilts toward small-cap (1)
• HML $=-0.30$: negative → tilts toward growth, not value (1)
• Overall: small-cap growth fund (1)

(c) [2 marks] Annualised: $(1.0025)^{12}-1 = 0.03042 \approx 3.04\%$ (1) Significant positive alpha implies returns not fully explained by factor risks → evidence against strong/semi-strong EMH (skill or unpriced risk) (1)

Question 2 (10)

(a) [5 marks]

• Value: buy cheap stocks (high B/M, low P/E, high earnings/dividend yield), short expensive (2)
• Momentum: buy past winners, short past losers (1)
• Look-back: past 12 months, skip the most recent 1 month (12-1) (1)
• Skip reason: short-term (1-month) reversal effect from bid-ask bounce/liquidity contaminates the signal (1)

(b) [3 marks] Value and momentum have negative correlation (~ -0.5 to -0.7 of factor returns) (1); combining diversifies — when value struggles momentum often works (1); improves portfolio Sharpe beyond either alone (1).

(c) [2 marks] Behavioral: investors under-react to news, so prices drift and trends persist (1). Risk-based: winners are riskier / time-varying risk exposure justifying premium (1).

Question 3 (12)

(a) [8 marks]

def long_short(df):
    out = []
    for date, g in df.groupby('date'):
        g = g.dropna(subset=['signal','ret_fwd'])
        g['decile'] = pd.qcut(g['signal'], 10, labels=False)
        long_ret  = g.loc[g.decile == 9, 'ret_fwd'].mean()
        short_ret = g.loc[g.decile == 0, 'ret_fwd'].mean()
        out.append({'date': date, 'ls_ret': long_ret - short_ret})
    return pd.DataFrame(out).set_index('date')['ls_ret']

Marks: groupby date (2), qcut into deciles (2), long top / short bottom equal-weight mean (2), return long-short series (2).

(b) [4 marks]

import numpy as np
def ann_sharpe(r):
    return (r.mean() / r.std(ddof=1)) * np.sqrt(12)

Marks: monthly mean/std ratio (2), annualise by $\sqrt{12}$ (2).

Question 4 (8)

(a) [4 marks] Low-vol anomaly: low-volatility/low-beta stocks earn higher risk-adjusted returns than high-vol stocks (2). Contradicts CAPM, which predicts return should rise with beta (positive risk-return line); here the empirical line is flat or inverted (2). (Causes: leverage constraints, lottery preference for high-vol names.)

(b) [4 marks] Any two:

• Premium concentrated in tiny/illiquid microcaps with high transaction costs (2)
• Post-publication decay / arbitrage after discovery (2)
• Once controlling for quality/junk, "size" largely disappears (2)
• Data/survivorship issues in original studies (2)

Question 5 (12)

(a) [4 marks] Disposition effect: tendency to sell winners too early and hold losers too long (2). Mechanism: prospect theory — investors are risk-averse in gains (lock in) but risk-seeking in losses (gamble to avoid realizing a loss); loss aversion makes realizing a loss psychologically painful (2).

(b) [4 marks]

• Anchoring: fixating on an irrelevant reference (e.g., purchase price or a 52-week high) when valuing (2)
• Confirmation bias: seeking/weighting only info that supports an existing view, ignoring contradicting data (2)

(c) [4 marks] Herding: following the crowd's trades rather than own analysis (1). Recency bias: over-weighting recent price rises as the norm (1). Together: recent gains → extrapolation → more buying → others herd in → self-reinforcing price spike detached from fundamentals = bubble (2).

Question 6 (6)

(a) [3 marks] Smart beta: rules-based, transparent strategies that weight by factors (value, low-vol, etc.) rather than market cap (2); systematic like passive (low cost, rules) but tilts away from cap-weight like active — a middle ground (1).

(b) [3 marks]

• Weak: prices reflect all past price/volume info (1)
• Semi-strong: prices reflect all public info (1)
• Strong: prices reflect all info incl. private (0.5); persistent factor premia most challenge semi-strong form (0.5)

[
  {"claim":"Annualised alpha of 0.25% monthly compounded ≈ 3.04%","code":"a=(1+Rational(25,10000))**12-1; result=abs(float(a)-0.03042)<0.0005"},
  {"claim":"Sharpe annualisation factor for monthly data is sqrt(12)","code":"result=abs(float(sqrt(12))-3.4641016)<1e-4"},
  {"claim":"HML loading -0.30 indicates growth tilt (negative sign)","code":"hml=-0.30; result=(hml<0)"},
  {"claim":"Value-momentum diversification: combined variance < average if correlation negative","code":"s=1; rho=-0.6; comb=0.5*0.5*(s*s)+0.5*0.5*(s*s)+2*0.5*0.5*rho*s*s; result=comb<0.5*(s*s)"}
]