Trading vs Investing & Styles

Question 1

(a) Net edge = gross edge − round-trip cost (0.05%) (6 marks)

• Scalping: $0.08\% - 0.05\% = 0.03\%$ ✓ (1)
• Intraday: $0.35\% - 0.05\% = 0.30\%$ ✓ (1)
• Swing: $3.0\% - 0.05\% = 2.95\%$ ✓ (1)
• Positional: $9.0\% - 0.05\% = 8.95\%$ ✓ (1)

Rendered unprofitable: None are outright negative, BUT scalping's net edge (0.03%) is razor-thin — the slightest slippage/latency makes it negative. Award (2) for identifying scalping as marginal/fragile (cost consumes 62.5% of gross edge). (2)

(b) Expected daily net return (6 marks)

Compounding $n$ trades each with net edge $e$: $(1+e)^n - 1 \approx n\cdot e$ for small $e$. Full derivation credit (2).

• Scalping: $r_d = (1.0003)^{40} - 1 \approx 0.01207 = 1.207\%$/day ✓ (2)
• Intraday: $r_d = (1.003)^{4} - 1 \approx 0.01205 = 1.205\%$/day ✓ (2)

(Note: nearly identical daily return — but achieved with radically different effort.)

(c) Feasible & profitable style (5 marks)

• Scalping requires continuous full-session screen presence, instant execution, low-latency infrastructure — impossible in 45 min pre-market. (2)
• Intraday needs monitoring through the day → also violates constraint. (1)
• Swing trading fits: entries/reviews done pre-market in ~45 min; positions held days needing no intraday monitoring; net edge 2.95% is robust to costs. Swing is the answer. (2)

(d) Personality/discipline mismatch (3 marks) Forcing scalping under a time constraint means entering/exiting hurriedly without focus → execution errors, missed stops, overtrading, and emotional tilt. Scalping demands sustained concentration and rapid decision-making that a 45-min distracted window cannot supply, converting a thin +0.03% edge into net losses. (3)

Question 2

(a) Proof of mean reversion (6 marks)

$\mathbb{E}[x_{t+1}\mid x_t] = x_t + \theta(\mu - x_t) = (1-\theta)x_t + \theta\mu$ Distance to mean: $\mathbb{E}[x_{t+1}] - \mu = (1-\theta)(x_t - \mu)$. (3) Since $0<\theta<1 \Rightarrow 0<(1-\theta)<1$, the gap shrinks by factor $(1-\theta)$ → moves toward $\mu$. (2) Stability condition: need $|1-\theta|<1 \Rightarrow 0<\theta<2$; for non-oscillating (monotone) decay require $0<\theta<1$. (1)

(b) Sign of edge (6 marks) Mean-reversion buy at $x_t = \mu - k\sigma$: expected next-step drift $= \theta(\mu - x_t) = \theta k\sigma > 0$ → price expected to rise toward $\mu$ → positive edge. (3) Momentum buy at $x_t = \mu + k\sigma$: expected drift $= \theta(\mu - x_t) = -\theta k\sigma < 0$ → price expected to fall → negative edge (trend continuation contradicts the pull-to-mean). (3)

(c) Backtest code (6 marks)

import numpy as np
np.random.seed(0)
theta, mu, sigma, N, k = 0.3, 100.0, 2.0, 10000, 1.0
x = mu
xs = [x]
for _ in range(N):
    x = x + theta*(mu - x) + np.random.normal(0, sigma)
    xs.append(x)
 
rets, in_pos, entry = [], False, 0.0
for xt in xs:
    if not in_pos and xt < mu - k*sigma:   # entry
        in_pos, entry = True, xt
    elif in_pos and xt >= mu:              # exit at mean
        rets.append((xt - entry)/entry)
        in_pos = False
 
print("avg per-trade return:", np.mean(rets))

Marks: simulation loop (2), entry/exit logic (2), return calc + positive result (2). Expected avg per-trade return ≈ +2% (positive), confirming edge.

(d) Trending regime flips edge (4 marks) Modified drift with mean itself rising: $\mu_t = \mu_0 + g\,t$, so $x_{t+1} = x_t + \theta(\mu_0 + g\,t - x_t) + \varepsilon_t.$ Now $\mathbb{E}[\Delta x] = \theta(\mu_t - x_t) +$ the moving-target term; a price above the old mean may still lie below the rising mean, so buying strength (momentum) captures the drift $g$. When $g$ is large relative to $\theta\sigma$, momentum edge > mean-reversion edge. (4)

Question 3

(a) Expectancy (4 marks) $E = p(+1R) + (1-p)(-1R) = 0.55 - 0.45 = 0.10R \text{ per trade}$ (3) Positive edge (+0.10R). ✓ (1)

(b) Kelly (5 marks) $f^* = 2p - 1 = 2(0.55) - 1 = 0.10 = 10\%$ (2) Trader's $f=0.02$ is $1/5$ of Kelly ("fifth-Kelly"). Rational because: full Kelly gives brutal drawdowns (~50%+), estimation error in $p$ overshoots, and psychological tolerability + capital preservation matter more than maximal growth for realistic sustainable targets. (3)

(c) Doubling claim (6 marks) Per-trade expected geometric growth with $f=0.02$, symmetric $R$: Growth factor per trade $\approx p\ln(1+f) + (1-p)\ln(1-f)$ (log-utility) or arithmetically $1 + f\cdot E = 1 + 0.02(0.10) = 1.002$. (2) Over 20 trades: $(1.002)^{20} - 1 \approx 0.0408 = 4.08\%$ expected monthly. (2) Doubling (+100%) requires either $f$ near Kelly with huge variance or is simply unrealistic — the honest expectation is ~4% monthly (~60% annualized before variance drag), NOT 100%/month. The claim is fantasy. (2)

(d) Realistic contrast (3 marks) The day-trader might realistically target ~2–5%/month (30–60% annual) if skilled, but with high effort, steep drawdown risk, and inconsistency — most fail. The index investor accepts ~10–12% annual passively, with far lower effort and drawdown, and high consistency. The trade-off is effort + risk for higher potential return that is neither guaranteed nor easily sustained. (3)

[
  {"claim":"Scalping net edge = 0.03%","code":"result = round(0.08-0.05,2)==0.03"},
  {"claim":"Scalping daily return approx 1.207%","code":"result = round(((1.0003)**40-1)*100,3)==1.207"},
  {"claim":"Intraday daily return approx 1.205%","code":"result = round(((1.003)**4-1)*100,3)==1.205"},
  {"claim":"OU gap shrinks by (1-theta)=0.7","code":"result = (1-0.3)==0.7"},
  {"claim":"Expectancy = 0.10R","code":"result = round(0.55-0.45,2)==0.10"},
  {"claim":"Full Kelly = 10%","code":"result = 2*0.55-1==Rational(1,10)"},
  {"claim":"Monthly expected growth ~4.08%","code":"result = round(((1+0.02*0.10)**20-1)*100,2)==4.08"}
]