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

1. Feb 18, 2014

### ainster31

Is there any way to derive an equation for compound interest based on effective interest rate instead of the nominal interest rate?

2. Feb 18, 2014

### phinds

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

3. Feb 18, 2014

### ainster31

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?

4. Feb 18, 2014

### phinds

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

5. Feb 18, 2014

### bahamagreen

6. Feb 18, 2014

### phinds

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.

7. Feb 18, 2014

### Staff: Mentor

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