Asset Returns – Types & Measurement
In Financial Analytics, we rarely analyze prices directly because they act like a "Random Walk" (Non-Stationary). Instead, we analyze Returns.
Why Returns?
- Stationarity: Returns usually fluctuate around a stable mean (often near zero), making them easier to model.
- Scale Free: Returns allow us to compare a ₹10 stock with a ₹10,000 stock.
1. Simple Returns (Rt) (Arithmetic Return)
This is the standard percentage change.
R_t = (P_t - P_{t-1}) / P_{t-1} = (P_t / P_{t-1}) - 1
- Usage: Best for communicating with clients ("Your portfolio is up 5%").
- Property: Simple returns across assets in a portfolio can be weighted linearly (Portfolio return = Weighted average of asset returns).
2. Log Returns (rt) (Continuously Compounded)
This is the natural logarithm of the price ratio.
r_t = ln(P_t / P_{t-1}) = ln(P_t) - ln(P_{t-1})
- Usage: Best for statistical modeling and time-series analysis.
- Property: Log returns are Time-Additive. The log return over a year is simply the sum of the daily log returns. (Simple returns do not add up this way).
Note
Approximation: For very small changes, Rt approx rt. But for large moves, they diverge.
Comparison: Simple vs Log Returns
Simple Returns vs Log Returns
Simple Returns
- Formula: (Pt - Pt-1) / Pt-1
- Portfolio Calc: Easy (Weighted Average works).
- Time Aggregation: Hard (Must multiply).
- Range: Limited (Can't be < -100%).
- Assumption: Often skewed.
VS
Log Returns
- Formula: ln(Pt / Pt-1)
- Portfolio Calc: Hard (Weighted Avg not exact).
- Time Aggregation: Easy (Just Add).
- Range: -Infinity to +Infinity.
- Assumption: More likely to be Normal.
Which one to use?
- Portfolio Management: Use Simple Returns.
- Risk Modeling / Econometrics / Options: Use Log Returns.
Test Your Knowledge
Question 1 of 5
1. Which return type is 'Time Additive' (you can sum them up over time)?