1. Quantitative Foundations

The math you actually need. Not for academic purity — for making better trading decisions under uncertainty.

Returns: arithmetic vs log

Arithmetic return of a single period: $r_t = \frac{P_t - P_{t-1}}{P_{t-1}}$

Log return: $\ell_t = \ln\!\left(\frac{P_t}{P_{t-1}}\right)$

Use log returns for analysis because:

1. They’re time-additive: $\ell_{1\to N} = \sum \ell_t$.
2. They’re approximately symmetric (better for statistical models).
3. They handle compounding cleanly.

Convert: $r = e^\ell - 1$.

Distribution of returns

The textbook says returns are normally distributed. They aren’t.

Real-world equity returns exhibit:

• Fat tails (kurtosis > 3) — extreme moves happen far more often than normal predicts.
• Negative skew — crashes are sharper than rallies.
• Volatility clustering — quiet days follow quiet days, big days follow big days (GARCH effect).
• Autocorrelation in absolute returns but not in returns themselves.

Assuming normality is the #1 source of risk model failure. LTCM, 2008, COVID — all “5-sigma” events that statistically shouldn’t happen in the lifetime of the universe.

Volatility — the core concept

Standard deviation of returns over a window: $\sigma = \sqrt{\frac{1}{N-1}\sum (r_i - \bar{r})^2}$

Annualization (assuming 252 trading days): $\sigma_{annual} = \sigma_{daily} \cdot \sqrt{252}$

A daily $\sigma$ of 1% → annualized ~16%. Nifty’s long-term annualized vol ~17–20%.

Sharpe & friends

Metric	Formula	Use
Sharpe	$(R - R_f) / \sigma$	Risk-adjusted return
Sortino	$(R - R_f) / \sigma_{downside}$	Penalizes only downside vol
Calmar	$CAGR /	MaxDD
MAR	$CAGR /	MaxDD
Information Ratio	$(R_p - R_b) / \sigma(R_p - R_b)$	Active return per unit of tracking error
Omega	Probability-weighted ratio of gains/losses above threshold	More complete than Sharpe

Sharpe lies when returns aren’t normal. A strategy with high Sharpe and high negative skew (selling tail risk) looks great until the tail event. Always check distribution shape, not just Sharpe.

Hypothesis testing — does my edge exist?

You backtest and get an avg trade return of +0.4% over 200 trades. Is this real or noise?

t-statistic

$t = \frac{\bar{r}}{s / \sqrt{N}}$

Where $\bar{r}$ is mean return, $s$ is std dev, $N$ is # trades.

• $|t| > 2$ → ~95% confident edge is real.
• $|t| > 3$ → ~99.7% confident.

For 200 trades with mean 0.4% and std 2%: $t = \frac{0.4}{2 / \sqrt{200}} = \frac{0.4}{0.141} \approx 2.83$

Real edges are often weak ($t \approx 2$). Spectacular t-stats (> 6) usually mean overfitting or look-ahead bias.

Multiple-comparisons / data snooping correction

If you test 100 strategies, ~5 will appear significant at $p < 0.05$ by random chance. Use Bonferroni or Bayesian methods to adjust.

This is why “I found this edge by trying 50 indicator combinations” is almost always overfit.

Drawdowns — the real risk metric

A strategy with 30% CAGR and 50% drawdown is untradeable in practice. Most allocators (and humans) blow out at 20–30% drawdown.

• Max DD = worst peak-to-trough.
• Average DD duration = how long you spend underwater.
• Recovery time = peak → new peak. Often 2–5× the drawdown depth.

Plan capital and psychology around 2× your historical max DD. What hasn’t happened yet, will.

Expectancy & R-multiples

$E = (W \cdot A_w) - (L \cdot A_l)$

Where $W$ is win rate, $A_w$ avg win in R, $L$ loss rate, $A_l$ avg loss in R.

A system with 40% win rate, avg win 3R, avg loss 1R: $E = 0.4(3) - 0.6(1) = 1.2 - 0.6 = 0.6R \text{ per trade}$

Think in R, not rupees. R normalizes across position sizes and capital growth.

Optimal position sizing — Kelly criterion

The mathematically optimal fraction of capital to risk per trade:

$f^* = \frac{p \cdot b - q}{b}$

Where $p$ = win prob, $q = 1 - p$, $b$ = odds (avg win / avg loss).

Example: $p = 0.55$, $b = 1.5$: $f^* = \frac{0.55 \times 1.5 - 0.45}{1.5} = \frac{0.825 - 0.45}{1.5} = 0.25$

Kelly says risk 25% of capital per trade.

Why nobody uses full Kelly

• Real-world $p$ and $b$ are estimated with error → Kelly is too aggressive.
• Drawdowns are huge (you can lose 50% on a normal sequence of bad luck).

Practical: use ¼ Kelly to ½ Kelly. Or just stick with the 1% rule, which is roughly 1/10–1/20 Kelly for typical edges.

Correlation & covariance

If you have 5 trades open and they’re all in bank stocks, you don’t have 5 trades — you have one trade with 5 names.

Correlation between two return series: $\rho_{X,Y} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} \in [-1, +1]$

Portfolio variance with multiple positions: $\sigma_p^2 = \sum_i w_i^2 \sigma_i^2 + 2\sum_{i<j} w_i w_j \sigma_i \sigma_j \rho_{ij}$

If $\rho = 1$, “diversification” gives nothing. If $\rho = 0$, two equal positions reduce vol by $\sqrt{2}$.

In a crisis, correlations spike to 1. A “diversified” basket of cyclicals all crashes together. True diversification requires across-asset-class exposures (equity + bonds + gold + cash).

Bayes’ theorem — updating beliefs

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Trading example: a setup historically wins 55%. Today RSI also gives a buy signal. Should you increase your conviction?

Only if RSI confirmation is more frequent on winning trades than losing ones. If RSI fires equally on winners and losers, it adds zero information regardless of how cool it looks.

Ask: does this signal change my probability of winning, or does it just confirm what I already knew? If the latter, it’s noise.

Stationarity

A time series is stationary if its statistical properties (mean, variance, autocorrelation) are constant over time.

Most price series are non-stationary (they trend, vol changes). But returns are roughly stationary (if you squint).

For mean-reversion strategies (pairs, stat-arb), you need a stationary spread — verified with the Augmented Dickey-Fuller (ADF) test. We’ll use this in Stat Arb.

Random walks vs trends

If returns are i.i.d. (independent and identically distributed), price is a random walk and prediction is impossible.

Empirically, equity prices show:

• Short-term momentum (1m–12m): trends persist (basis of momentum strategies).
• Long-term reversal (3y–5y): mean reversion (basis of contrarian/value).
• Very short-term (intraday): mostly noise + microstructure effects.

These are the stylized facts of equity returns. Any strategy implicitly bets on one of these. Know which one yours uses.