# Which is better 10% interest compounded quarterly or 9.8% compounded continuously?

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We note the following formulas:

`A = P(1+r/n)^(nt)`

`A = Pe^(rt)`

where r is the interest rate, P is the principal amount, n is the interest rate periods, t is time, and A is the amount after a certain time. The first formula corresponds to a compound interest, while the second to continuous compounding.

A 10% interest compounded quarterly gives us:

`A_(q) = P(1+(0.10)/4)^4 = P(1.025)^4 = 1.104P`

On the other hand, continuous compounding gives us:

`A_c = Pe^0.098 = 1.103P`

Notice, that after a year, the 10% compounded quarterly gives 1.104 times the initial amount (or 10.4% effective interest rate) while the continuous case gives only 1.103 times the initial (only 10.3% effective interest rate). Hence, 10% interest compounded quarterly is better (i.e. has higher effective interest rate).

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