When a stock increases in value, the holding period return is always greater than the continuously compounded return that would be required to generate that holding period return. For example, if a stock increases from $1 to $1.10 in a year, the holding period return is 10%. The continuously compounded rate needed to increase a stock's value by 10% is Ln(1.10) = 9.53%.

Key Concepts:

• Holding Period Return (HPR): Simple percentage return = (Ending Value - Beginning Value) / Beginning Value
• Continuously Compounded Return: r = ln(Ending Value / Beginning Value) = ln(1 + HPR)
• Mathematical Relationship: For any positive HPR, HPR > ln(1 + HPR) because the natural logarithm function grows slower than linear growth for positive values.
• Example: HPR = 10% = 0.10, Continuously compounded return = ln(1.10) ≈ 0.0953 = 9.53%

Why this occurs: The continuously compounded return represents the constant rate that, when compounded continuously, would produce the same ending value. Since continuous compounding is more efficient than simple interest, a lower continuously compounded rate is needed to achieve the same ending value as a higher simple rate.