Part I · Chapter 2
Returns and Financial Data
Prices typically drift like a random walk and are non-stationary; returns (especially log returns) are usually much closer to stationary and are the standard object of financial modeling.
\( r_t^{\text{log}} = 100 \cdot \ln(P_t / P_{t-1}) \approx 100 \cdot (P_t - P_{t-1})/P_{t-1} \)
Learning objectives
Compute simple and log returns from price levels.
Understand why returns are preferred to prices for modeling.
Recognize stylized facts: volatility clustering, fat tails, weak return autocorrelation.
Convert between log returns (additive over time) and simple returns (multiplicative).
Price → Return Converter
Seed:
Live stats
Price end value: —
Mean return: — %
Std-dev (vol): — %
Excess kurtosis: —
Returns oscillate around the drift; prices wander. Excess kurtosis > 0 hints at fat tails.
📐 Return formulas
\[ r_t^{\text{simple}} = \frac{P_t - P_{t-1}}{P_{t-1}}, \qquad r_t^{\text{log}} = \ln \frac{P_t}{P_{t-1}} \]
\[ \text{Multi-period: } r_{0\to T}^{\text{log}} = \sum_{t=1}^{T} r_t^{\text{log}} \quad (\text{additive}) \]
P_t price at time t
simple ≈ log for small returns
log additive cumulates by summing
simple multiplicative (1+r₁)(1+r₂)…−1
🔍 What to look for
- Price chart drifts and wanders — characteristic of a random walk with drift.
- Return chart oscillates around zero (or a small mean) — much more stationary-looking.
- If σ is large, returns show occasional extreme spikes — fat tails / excess kurtosis.
⚠️ Pro Tip: What to Avoid
Student says
"A −40% return followed by a +40% return cancels back to the start."
Why this is wrong
Simple returns compound multiplicatively: 0.6 × 1.4 = 0.84 → still −16% from start. Log returns ADD over time but a −0.51 then +0.34 don't sum to 0.
Correct interpretation
Use log returns for time aggregation; recovering from a loss requires a larger percentage gain.
📝 Mini-quiz
📋 Key Takeaways
| Property | Price level | Return |
|---|---|---|
| Stationary? | Usually NO (I(1)) | Usually YES (I(0)) |
| Mean | Drifts | ≈ 0 (or small positive) |
| Variance | Grows over time | Roughly constant (but clusters!) |
| Model | Random walk | White noise / ARCH-GARCH for variance |