# Is there any way to derive an equation for compound interest based

• ainster31
In summary, the equation for compound interest based on effective interest rate is F=(1+I)^t, where I=(1+\frac{r}{n})^{n}-1. This differs from the equation for the nominal rate because the effective rate takes into account the compounding frequency, while the nominal rate assumes continuous compounding.
ainster31
Is there any way to derive an equation for compound interest based on effective interest rate instead of the nominal interest rate?

Why would the equation for the effective rate be any different than the equation for the nominal rate ?

phinds said:
Why would the equation for the effective rate be any different than the equation for the nominal rate ?

I am aware of this equation for compound interest based on nominal interest:

$$F=P{ e }^{ rt }\\ where\quad r=nominal\quad annual\quad interest\\ and\quad t=number\quad of\quad years$$

How would I modify it for effective interest?

ainster31 said:
I am aware of this equation for compound interest based on nominal interest:

$$F=P{ e }^{ rt }\\ where\quad r=nominal\quad annual\quad interest\\ and\quad t=number\quad of\quad years$$

How would I modify it for effective interest?

Why would the equation for the effective rate be any different than the equation for the nominal rate ?

bahamagreen said:
See if this helps...

Difference Between Nominal & Effective Interest Rates

http://www.ehow.com/info_8149388_difference-nominal-effective-interest-rates.html

That's interesting. I was interpreting "effective" in this context to mean "real", which is not at all what it means. Basically the "effective" rate is just the nominal rate plus a very small amount, it has nothing to do with the real rate.

ainster31 said:
I am aware of this equation for compound interest based on nominal interest:

$$F=P{ e }^{ rt }\\ where\quad r=nominal\quad annual\quad interest\\ and\quad t=number\quad of\quad years$$

How would I modify it for effective interest?

This equation assumes that there is continuous compounding at the nominal interest rate. The relationship between the nominal interest rate in this equation and the effective interest rate I is found by calculating the principal after 1 year:

$$Pe^r=P(1+I)$$
So, $$I=e^r-1$$

If we substitute this into your original equation, we obtain:
$$F=(1+I)^t$$
More generally, if there are n compounding periods a year, and r is the nominal interest rate,

$$F=P(1+\frac{r}{n})^{nt}$$
So, $$(1+\frac{r}{n})^{n}=(1+I)$$
So,$$I=(1+\frac{r}{n})^{n}-1$$

## 1. How is compound interest calculated?

Compound interest is calculated by multiplying the principal amount by the interest rate, raising it to the number of compounding periods, and adding any additional contributions or withdrawals.

## 2. Can you explain the compound interest formula?

The compound interest formula is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

## 3. What is the difference between simple and compound interest?

Simple interest is calculated by multiplying the principal amount by the interest rate and the number of years. Compound interest takes into account the interest earned in previous periods and adds it to the principal for future interest calculations.

## 4. How does the compounding frequency affect compound interest?

The more frequent the compounding, the higher the interest earned. For example, a 5% annual interest rate with monthly compounding will yield more interest than annual compounding.

## 5. Can compound interest be negative?

No, compound interest cannot be negative. It is always a positive value that represents the interest earned on the principal amount.

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