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

by ainster31
Tags: based, compound, derive, equation
 P: 154 Is there any way to derive an equation for compound interest based on effective interest rate instead of the nominal interest rate?
 PF Gold P: 6,276 Why would the equation for the effective rate be any different than the equation for the nominal rate ?
P: 154
 Quote by phinds 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?

PF Gold
P: 6,276
Is there any way to derive an equation for compound interest based...

 Quote by 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?
Why would the equation for the effective rate be any different than the equation for the nominal rate ?
 P: 542 See if this helps... Difference Between Nominal & Effective Interest Rates http://www.ehow.com/info_8149388_dif...est-rates.html
PF Gold
P: 6,276
 Quote by bahamagreen See if this helps... Difference Between Nominal & Effective Interest Rates http://www.ehow.com/info_8149388_dif...est-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.
HW Helper
Thanks
PF Gold
P: 5,177
 Quote by 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?
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,$$I=(1+\frac{r}{n})^{n}-1$$

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