A stock moves from $80 to $92. What is the continuously compounded return to two decimal places, and how does it compare with the simple return?

A stock moves from $80 to $92. What is the continuously compounded return to two decimal places, and how does it compare with the simple return?

Approach: Compute the simple return first, then take its logarithm with a short series expansion and describe the direction and size of the gap between the two measures.

13.98%. The simple return is 12/80, which is 15%, and the continuously compounded figure is the log return ln(1.15) = 0.13976, so 13.98%. The Taylor series ln(1 + x) = x - x^2/2 + x^3/3 gives 0.15 - 0.01125 + 0.001125 = 0.13988, within 0.012 percentage points of the truth. The log return sits below the simple return on a gain and exceeds it in magnitude on a loss, because ln(1 + x) is concave. That concavity explains why a plus 15% followed by a minus 15% leaves you down 2.25%: the log returns are 13.98% and -16.25%, which sum to -2.27%. Exponentiating that gives 0.9775, and the direct product is 1.15 * 0.85 = 0.9775.

Follow-up: A price series has log returns with mean 0.5% and standard deviation 2% per day. What is the expected simple return per day?

Key concepts: log return, simple return, concavity, Taylor series.