Why Simple and Relative Interest Compounding Are Not Equivalent

Summary

The document compares a simple-interest adjustment based on the difference between two rates with a compounded ratio of their gross rates over a day-count fraction. It shows that the expressions are not generally equal. Taking the natural logarithm of the compounded ratio turns it into a difference of logarithms; the answer then approximates that difference by the rate difference, yielding the simple adjustment.

This is an approximation, not an identity: the logarithm step is exact, while replacing the log difference with the rate difference relies on a small-rate approximation. The discussion is brief and gives no error analysis or numerical example, so it does not establish when the approximation is sufficiently accurate for a particular instrument or calculation.

Key ideas

• The simple adjustment using a rate difference is not generally equal to the ratio of compounded gross rates.
• Taking the natural logarithm converts the compounded ratio into a difference of logarithms.
• Approximating the log difference by the rate difference produces the simple-interest expression.
• The approximation’s accuracy depends on the rates, and the document does not quantify its error.

Tags

# Compounding Equivalence


# Compounding Equivalence












I am trying to understand under what circumstances or transformations would $[1+(E_2-E_1)*\frac{d}{360}]$ equal to $(\frac{1+E_2}{1+E_1})^{\frac{d}{360}}$.

For context, $E_2, E_1$ are interest rates.

Any help is very much appreciated.

## Answer by Chris (score 1)

https://quant.stackexchange.com/a/58147

They're not equivalent, but you can use log identities to derive something similar after applying a log.

Eg,

$ln\left(\left(\frac{1+E_2}{1+E_1}\right)^{\frac{d}{360}}\right)$

$\frac{d}{360}*\left(1+E_2-\left(1+E_1\right)\right) $

$\left(E_2-E_1\right)*\frac{d}{360}$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.