Backtesting Frameworks

Question 1

(a) CAGR derivation & compute (4 marks)

Compound growth: $V_{final} = V_{initial}(1+g)^n$. Solve for $g$: $\text{CAGR} = \left(\frac{V_{final}}{V_{initial}}\right)^{1/n} - 1$ (2 marks derivation)

Compute: $\left(\frac{18000}{10000}\right)^{1/3} - 1 = 1.8^{1/3} - 1$. $1.8^{1/3} \approx 1.21644$, so CAGR $\approx 0.2164 = \mathbf{21.64\%}$. (2 marks)

(b) Max Drawdown (4 marks)

Drawdown at time $t$: $DD_t = \frac{V_t - \max_{s\le t} V_s}{\max_{s\le t} V_s}$. Max DD is the most negative. (2 marks derivation)

Running peak before trough is $16{,}000$; trough $12{,}800$: $DD = \frac{12800 - 16000}{16000} = -0.20 = \mathbf{-20\%}$ (2 marks) The later recovery to 18,000 sets a new peak but does not change the max DD.

(c) Sharpe (4 marks)

Sharpe (daily) $= \mu_d / \sigma_d$. Annualize: multiply return by 252, std by $\sqrt{252}$, so $\text{Sharpe}_{ann} = \frac{\mu_d}{\sigma_d}\sqrt{252}$ (2 marks derivation)

$= \frac{0.0006}{0.012}\sqrt{252} = 0.05 \times 15.8745 \approx \mathbf{0.794}$. (2 marks)

Question 2

(a) Look-ahead bias (6 marks)

• Definition: using information not available at decision time to generate signals — inflates results. (2 marks)
• Mistake 1: computing signals on today's close then trading at today's close/open. Fix: signal = signal.shift(1) so you act on the next bar. (2 marks)
• Mistake 2: using full-sample statistics (e.g. mean/std, or .max() over entire series, or normalizing with future data) in indicators. Fix: use rolling/expanding windows only. (2 marks)

(b) Survivorship bias (4 marks)

• Testing only on companies that still exist today drops the delisted/bankrupt losers, so historical performance looks artificially strong (winners' selection). (2 marks)
• The dataset must be point-in-time / include delisted securities (a "survivorship-bias-free" universe reflecting the actual constituents at each historical date). (2 marks)

Question 3

(a) Cost drag (6 marks) Each round-trip = 2 sides. Cost rate per side = commission + slippage = 0.05% + 0.03% = 0.08%. (1) Per round-trip = $2 \times 0.08\% = 0.16\%$ of $5,000 = 0.0016 \times 5000 = \8$. *(2)* Annual:$120 \times $8 = \mathbf{$960}$. *(2)* As$50,000 account: $960/50000 = \mathbf{1.92\%}$ annual drag. (1)

(b) Slippage explain-out-loud (4 marks) Backtests fill at ideal prices (close/mid), but live orders move the market, cross spreads, and get partial fills — especially in illiquid names or fast markets. Optimistic slippage understates true entry/exit prices; the edge of many strategies is thinner than the real slippage, so a strategy profitable on paper becomes a loser once realistic slippage/spread is applied. (full 4 for capturing spread crossing + market impact + edge-vs-cost point)

Question 4

(a) max_drawdown (7 marks)

def max_drawdown(equity):
    running_max = equity.cummax()          # peak-to-date
    drawdown = equity / running_max - 1    # fractional DD, <=0
    return drawdown.min()                  # most negative value

(2 marks cummax, 2 marks drawdown formula, 2 marks .min(), 1 mark correctness/comments)

(b) MA crossover (7 marks)

df['fast'] = df['close'].rolling(20).mean()
df['slow'] = df['close'].rolling(50).mean()
df['raw'] = (df['fast'] > df['slow']).astype(int)  # 1 long, 0 flat
df['signal'] = df['raw'].shift(1)                  # act next bar: no look-ahead

(2 rolling means, 2 comparison signal, 2 shift(1), 1 correctness)

Question 5

(a) IS/OOS & walk-forward (6 marks)

• Split data: optimize/parameter-tune on in-sample; validate untouched on out-of-sample. (2)
• Over-optimizing fits parameters to noise in the in-sample data; the "edge" doesn't generalize, so OOS performance collapses (overfitting). (2)
• Walk-forward: repeatedly optimize on a rolling training window, test on the following window, then roll forward — approximating live re-optimization and giving a chain of OOS results. (2)

(b) Monte Carlo bootstrap (5 marks)

1. Take the 200 historical per-trade returns. (1)
2. Resample with replacement (bootstrap) 200 trades to form one simulated sequence. (1)
3. Compound them into an equity curve; compute its max drawdown. (1)
4. Repeat for e.g. 10,000 simulations, storing each max DD. (1)
5. Take the 5th percentile of the max-DD distribution as the estimated adverse-case drawdown. (1)

(c) Paper trading (3 marks) Backtests can't capture real-world execution: broker latency, order rejections, data feed differences, actual fills/slippage, and operational bugs. Paper trading in live market conditions surfaces these before real capital is risked. (3 for any valid execution/operational point)

[
 {"claim":"CAGR of 10000->18000 over 3yr is ~21.64%","code":"cagr=(18000/10000)**(sympy.Rational(1,3))-1; result=abs(float(cagr)-0.2164)<0.001"},
 {"claim":"Max DD from peak 16000 to trough 12800 is -20%","code":"dd=(12800-16000)/16000; result=dd==-0.20"},
 {"claim":"Annualized Sharpe = 0.0006/0.012*sqrt(252) ~ 0.794","code":"s=(0.0006/0.012)*sympy.sqrt(252); result=abs(float(s)-0.7937)<0.01"},
 {"claim":"Annual cost drag is $960 = 1.92% of 50000","code":"cost=120*2*0.0008*5000; result=cost==960 and abs(cost/50000-0.0192)<1e-9"}
]