# Is compound interest better than simple interest at all times????

I was recently calculating compound interest for one of my questions and I came to one quite intruiging problem

so from what I understand simple interest is! when you have the same amount of interest every year so e.g Principal amount is $$100$$ and rate is $$10\%$$ then every year you get $$10$$.

Compound interest on other hand is when you get interest on the new value e.g. Principal amount is $$100$$ and rate is $$10$$ then first year you get $$10$$ interest second year you get $$\11$$ interest etc and so on!

Now for the question I was working on! Principal amount is $$\11,000$$ and the rate is $$12\%$$ compounded quarterly for 5 years.

So Simple interest would be $$\text{Interest} = \text{PRT}= \11000\cdot0.12\cdot20= \26400$$

And compound interest would be! $$\text{Interest} = \11000(1+ \frac{0.12}4)^{20}$$ that is $$= \19,867.22$$ HOW?? HOW is compound interest less than Simple interest while the interest each quarter for compound interest is larger than the simple interest????

• As there are only $5$ years involved...please check your simple interest computation.
– lulu
Sep 27 '18 at 14:40
• Why use four question marks? Sep 27 '18 at 14:44
• no real reason for 4 question marks it's just my style Sep 27 '18 at 14:51
• @lulu it's 5 years but as it is interested quarterly it's 4 times per year right? so it should be 5*4=20 right?? Sep 27 '18 at 14:52
• Well, in that case you have $12\%$ interest quarterly...which would make even the Mafia blush. But, sure. $12\%$ quarterly simple interest is a better deal then $12\%$ annual interest compounded quarterly.
– lulu
Sep 27 '18 at 14:57

1. The $$11000\cdot 0.12\cdot 20$$ calculation only finds the amount of interest accrued, while $$11000\cdot (1+\frac{0.12}4)^{20}$$ is the final balance of the account. The comparison between these two is meaningless. You should instead be comparing $$11000(1+0.12\cdot 20)$$ with $$11000\cdot (1+\frac{0.12}4)^{20}$$.
2. 12% is the annual interest rate (it is always annual unless otherwise stated), so the simple interest calculation should be $$11000(1+0.12\cdot \color{red}{5})$$.
Let's generalize. Suppose the principle is $$P$$, the annual interest rate is $$i$$, the time interval is $$t$$ (in years), and the compound interest divides the time interval into $$n$$ equal subintervals. The final account balance under simple interest is $$P(1+it)$$ and under compound interest is $$P\left(1+\frac{it}n\right)^n$$ The question is why the latter is always larger than the former. This follows from Bernoulli's inequality, which says $$(1+x)^n\ge 1+nx$$ whenever $$n\ge 0$$ is an integer and $$x\ge 0$$. Therefore, $$\text{compound} = P\left(1+\frac{it}n\right)^n\ge P\left(1+n\cdot \frac{it}n\right)=P(1+it)=\text{simple}$$