Chapter 2: Financial Fundamentals¶

Core concepts every quant trader must understand.

Returns¶

Simple Returns¶

The percentage change in price:

\[R_t = \frac{P_t - P_{t-1}}{P_{t-1}} = \frac{P_t}{P_{t-1}} - 1\]

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signal simple_return:
  ret_1d = (prices - lag(prices, 1)) / lag(prices, 1)
  emit ret_1d

Log Returns¶

Natural logarithm of price ratio:

\[r_t = \ln\left(\frac{P_t}{P_{t-1}}\right)\]

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signal log_return:
  ret_1d = ln(prices / lag(prices, 1))
  emit ret_1d

Why Log Returns?¶

Property	Simple	Log
Additive over time	No	Yes
Symmetric	No	Yes
Normal distribution	Less	More
Multi-period	Multiply	Add

sigc default:

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// ret() uses simple returns by default
ret_60 = ret(prices, 60)  // 60-day simple return

Annualization¶

Convert daily to annual:

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// Volatility: multiply by sqrt(252)
annual_vol = daily_vol * sqrt(252)

// Returns: multiply by 252
annual_ret = daily_ret * 252

Volatility¶

Standard Deviation¶

Most common volatility measure:

\[\sigma = \sqrt{\frac{1}{n-1}\sum_{i=1}^n(r_i - \bar{r})^2}\]

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signal volatility:
  daily_ret = ret(prices, 1)
  vol = rolling_std(daily_ret, 60) * sqrt(252)  // Annualized
  emit vol

Realized Volatility¶

Historical volatility from actual returns:

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signal realized_vol:
  // Sum of squared returns
  daily_ret = ret(prices, 1)
  squared_ret = daily_ret * daily_ret
  rv = sqrt(rolling_sum(squared_ret, 21) * 252 / 21)
  emit rv

Implied Volatility¶

Forward-looking volatility from options prices. Often proxied by VIX.

Risk-Adjusted Returns¶

Sharpe Ratio¶

Return per unit of risk:

\[\text{Sharpe} = \frac{R_p - R_f}{\sigma_p}\]

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// In backtest results
sharpe = annual_return / annual_volatility

Information Ratio¶

Active return per unit of active risk:

\[\text{IR} = \frac{R_p - R_b}{\sigma_{p-b}}\]

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// Return vs benchmark, divided by tracking error
information_ratio = active_return / tracking_error

Sortino Ratio¶

Uses downside deviation instead of total volatility:

\[\text{Sortino} = \frac{R_p - R_f}{\sigma_d}\]

More appropriate when return distribution is asymmetric.

Correlation¶

Pearson Correlation¶

Linear relationship between two variables:

\[\rho_{xy} = \frac{\text{Cov}(x,y)}{\sigma_x \sigma_y}\]

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signal correlation:
  ret_aapl = ret(prices[AAPL], 1)
  ret_msft = ret(prices[MSFT], 1)
  corr = rolling_corr(ret_aapl, ret_msft, 60)
  emit corr

Why Correlation Matters¶

Diversification: Low correlation = better diversification
Risk: High correlation in crisis = increased risk
Factor exposure: Correlation reveals common factors

Drawdowns¶

Maximum Drawdown¶

Largest peak-to-trough decline:

\[\text{MaxDD} = \max_t\left(\frac{\text{Peak}_t - P_t}{\text{Peak}_t}\right)\]

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signal drawdown:
  peak = rolling_max(prices, 252)
  dd = (peak - prices) / peak
  emit dd

Drawdown Duration¶

Time underwater: - Drawdown start: When price falls from peak - Drawdown end: When new peak is reached - Recovery time: Duration of drawdown

Beta and Alpha¶

Beta¶

Sensitivity to market movements:

\[\beta = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}\]

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signal beta:
  stock_ret = ret(prices, 1)
  market_ret = ret(market_index, 1)
  beta = rolling_cov(stock_ret, market_ret, 60) / rolling_var(market_ret, 60)
  emit beta

Alpha¶

Return unexplained by market:

\[\alpha = R_i - \beta \cdot R_m\]

CAPM¶

\[E[R_i] = R_f + \beta_i(E[R_m] - R_f)\]

Expected return = Risk-free rate + Beta × Market risk premium

Statistical Concepts¶

Z-Score¶

Standardized value (number of standard deviations from mean):

\[z = \frac{x - \mu}{\sigma}\]

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signal momentum_zscore:
  raw_momentum = ret(prices, 60)
  // zscore() cross-sectionally normalizes
  emit zscore(raw_momentum)

Percentile Rank¶

Position in distribution:

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signal percentile:
  // Rank relative to history
  percentile = ts_rank(prices, 252) / 252
  emit percentile

Cross-Sectional vs Time-Series¶

Operation	Cross-Sectional	Time-Series
zscore	Across assets today	Single asset over time
rank	Across assets today	Single asset over time
mean	Average of all assets	Average over time

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// Cross-sectional (default)
cs_zscore = zscore(momentum)

// Time-series
ts_zscore = (momentum - rolling_mean(momentum, 252)) / rolling_std(momentum, 252)

Practical Example¶

Computing Key Metrics¶

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data:
  source = "prices.parquet"
  format = parquet

signal returns:
  emit ret(prices, 1)

signal volatility:
  daily_ret = ret(prices, 1)
  emit rolling_std(daily_ret, 60) * sqrt(252)

signal sharpe_estimate:
  daily_ret = ret(prices, 1)
  annual_ret = rolling_mean(daily_ret, 252) * 252
  annual_vol = rolling_std(daily_ret, 252) * sqrt(252)
  emit annual_ret / annual_vol

signal drawdown:
  peak = rolling_max(prices, 252)
  emit (peak - prices) / peak

signal beta:
  stock_ret = ret(prices, 1)
  market_ret = ret(market, 1)
  emit rolling_cov(stock_ret, market_ret, 60) / rolling_var(market_ret, 60)

Key Takeaways¶

Returns: Use simple returns for most cases, log returns for compounding
Volatility: Annualize daily vol by multiplying by √252
Sharpe Ratio: Primary measure of risk-adjusted performance
Correlation: Critical for portfolio construction
Drawdowns: Maximum drawdown shows worst-case scenario
Z-scores: Standardize signals for comparison

Exercises¶

Calculate the 60-day rolling Sharpe ratio for a stock
Compute the correlation between two stocks
Find the maximum drawdown in a price series
Calculate beta relative to SPY

Next Chapter¶

Continue to Chapter 3: Building Signals to learn how to construct trading signals.