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?

1. Stationarity: Returns usually fluctuate around a stable mean (often near zero), making them easier to model.
2. 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).

Comparison: Simple vs Log Returns

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Which one to use?

• Portfolio Management: Use Simple Returns.
• Risk Modeling / Econometrics / Options: Use Log Returns.

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