Comparing Discrete and Continuous Compounding Rates

Summary

The document examines how annual discrete and continuous interest rates relate when they produce the same accumulation, then extends the comparison to a period of length t. By expanding the exponential function as a Taylor series and equating the resulting accumulation expressions, the response derives an approximation: the discrete rate is the continuous rate plus a positive term proportional to the square of the continuous rate and the period, with higher-order terms omitted.

This expansion supports two observations for positive rates and positive time: the discrete rate is greater, and the gap is small for short periods, growing as the period length increases. The result is presented as a local approximation, so the omitted higher-order terms matter when rates or periods are larger. The discussion does not explore negative rates or spell out broader assumptions about the accumulation conventions, which limits how widely its stated inequality should be applied.

Key ideas

• Equating discrete and continuous accumulation over a period allows their rates to be compared.
• A Taylor expansion gives an approximation for the discrete rate in terms of the continuous rate and period length.
• For positive rates, the leading correction makes the discrete rate higher than the continuous rate.
• The rates are closer over short periods, while the gap grows with period length.
• The approximation omits higher-order terms and does not discuss negative-rate cases.

Tags

# Properties of difference between continuous and discrete compounding of interest rate


# Properties of difference between continuous and discrete compounding of interest rate












The relationship between annual discrete and continuous compounding interest rates is given as:

$$1+r_d = e^{r_c}$$

My question is what are the properties of the difference between $r_d$ and $r_c$?

For example, it should hold $r_d>r_c$ because more compounding should have lower interest to arrive at the same value. Can you show this mathematically?

I am not sure what other properties could exist?

## Answer by StackG (score 3, accepted)

https://quant.stackexchange.com/a/55945

Let's add a time variable to extend to non-annual periods $$1 + r_d t = e^{r_c t}$$

The taylor expansion of exponential is \begin{align} e^{r_c t} &= \sum_{n=0}^\infty {\frac {(r_c t)^n} {n!}}\\ &= 1 + r_c t + {\frac 1 2}(r_c t)^2 + \cdots \end{align}

so by equating the two equations, we see that $$r_d = r_c + {\frac 1 2}r_c^2 t + O(t^2)$$

Two things we can see from this:

- $r_d > r_c$

- for $t$ small, the two rates are almost the same. As $t$ gets bigger, and the rates are no longer the same

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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