Why Compounded Interest Rates Use Growth Factors

Summary

The document explains why effective interest formulas add one to a periodic rate, multiply the resulting growth factors, and subtract one at the end. The added one represents retaining the original principal as well as earning interest during each period. Multiplication captures reinvestment, because each period’s interest is earned on the balance accumulated so far.

A deposit earning a nominal annual rate split into two equal compounding periods illustrates the process: the first period’s interest increases the balance, and the next period earns interest on that larger amount. The example decomposes the final growth factor into principal, interest from each period, and interest earned on earlier interest. It offers an intuitive explanation for discrete compounding; it does not address other conventions such as continuous compounding, fees, taxes, or changes in rates over time.

Key ideas

• Adding one to a rate forms a growth factor that includes principal and interest.
• Multiplying periodic growth factors models reinvestment between periods.
• Subtracting one converts the accumulated growth factor back into an effective rate.
• Later-period interest includes interest earned in earlier periods.
• The example explains discrete compounding under a fixed periodic rate.

Tags

# Why do most interest rate formulas, and indeed finance in general, add 1 to a rate and then subtract afterwards?


# Why do most interest rate formulas, and indeed finance in general, add 1 to a rate and then subtract afterwards?












For example, in the formula that shows the relationship between the nominal and effective interest rate shown below, 1 is first added to in/m and then 1 is subtracted from the result. What is the intuitive explanation for this?

## Answer by Lliane (score 1)

https://quant.stackexchange.com/a/42934

The formula is multiplicative because interests are compounded (re-invested).

Let's say you have 10% interest on a 1 USD deposit, compounded (calculated and paid) twice a year. That would be 5 cents of interest for the first semester. Then you have a deposit of 1.05 USD during the next semester. After one year, your interests are

```
1 * 0.05 + 1.05 * 0.05 = 0.05 + 0.0525 = 0.1025 = 10.25% per annum
```

Using the formula you provided

```
1.05 * 1.05 - 1 = (1 + 0.05) * (1 + 0.05) = 1 + 0.05 + 0.05 + 0.05*0.05 - 1
= principal + interest[principal, S1] + interest[principal, S2] + interest[interests of S1, S2]
```

Where interest[amount, period] denotes the interests on "amount" during the "period"

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