Why Continuously Compounded Rates Are Convenient in Finance

Summary

The document asks why finance commonly expresses interest and growth with continuous compounding, despite the idealization of infinitely frequent compounding. Its answer emphasizes that continuous and discrete return conventions can be converted into one another through logarithms and exponentials. Thus, continuous compounding can serve as a mathematical representation without implying that real interest is literally compounded infinitely often.

The response also points to calculus as a reason this convention is convenient: exponential growth is easy to manipulate with differentiable functions. It offers no broader history of the convention, empirical comparison, or worked numerical example, and its brief treatment does not discuss day-count conventions or how quoted market rates are defined. The main practical takeaway is that continuous compounding is a useful equivalent convention for analysis, rather than a claim about the actual frequency of cash interest payments.

Key ideas

• A discrete return convention can be converted to a continuously compounded rate using logarithms.
• The reverse conversion uses the exponential function.
• Continuous compounding is mathematically convenient for calculus and exponential growth models.
• The convention does not require real-world interest payments to compound infinitely often.

Tags

# Why continuously compounded interest a standard in finance?


# Why continuously compounded interest a standard in finance?












Why is the "continuously compounded interest" the standard in finance? Many finance textbooks use the formula e^rt without justification.

The assumption that the interest frequency is approaching infinity, it is very unrealistic but how did it become a standard in finance?

## Answer by AKdemy (score 2)

https://quant.stackexchange.com/a/64004

It's a Duplicate which was in turn closed because it is a basic financial question. Reading it will "back" up how they are related: You asked rubikscube09 if he has a toy example to back up his answer. He answered that any non-continuously compounded return can be turned into a continuously compounded one: $1+r_d = e^{r_c}$ shows this in the post I refer to (in this case use the natural logarithm to solve for $r_c$; or if the other way around, exponential as noob2 pointed out).

If you refer to a toy example for convenient manipulation; it's because you can use calculus.

Edit: A differentiable function must be continuous.

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