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

by ainster31
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ainster31
#1
Feb18-14, 10:39 AM
P: 154
Is there any way to derive an equation for compound interest based on effective interest rate instead of the nominal interest rate?
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phinds
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Feb18-14, 10:42 AM
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Why would the equation for the effective rate be any different than the equation for the nominal rate ?
ainster31
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Feb18-14, 10:45 AM
P: 154
Quote Quote by phinds View Post
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?

phinds
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Feb18-14, 12:35 PM
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Is there any way to derive an equation for compound interest based...

Quote Quote by ainster31 View Post
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
#5
Feb18-14, 05:07 PM
P: 543
See if this helps...

Difference Between Nominal & Effective Interest Rates

http://www.ehow.com/info_8149388_dif...est-rates.html
phinds
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Feb18-14, 06:28 PM
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Quote Quote by bahamagreen View Post
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.
Chestermiller
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Feb18-14, 07:35 PM
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Quote Quote by ainster31 View Post
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:

[tex]Pe^r=P(1+I)[/tex]
So, [tex]I=e^r-1[/tex]

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

[tex]F=P(1+\frac{r}{n})^{nt}[/tex]
So, [tex](1+\frac{r}{n})^{n}=(1+I)[/tex]
So,[tex]I=(1+\frac{r}{n})^{n}-1[/tex]


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