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I am very confused about compound interest rates , i know for the most basic example if we are given a yearly rate of say 10% , to calculate the nominal value of whatever your deposit would be ,it would be PV*(1+r/100)^t , t being the number of years r being the interest rate as a percentage and PV being the value you deposited . This makes perfect sense

The confusion arises with the following formulas

enter image description here

this forumla is used when the annual rate is split up over the year so in the case of the annual 10% rate it can be given as 5% in the first half of the year and 5% at the second half of the year so r would be 0.1 , n would be 2 (because the rate is split and given at 2 points in the year)

and t is how many years this happens for this cannot make sense if i had an investment or a deposit at the start of the year say £240 and this compounds 5% for the first half of the year then 5% compound again for the second half of the year (compounding on from the value halfway through the year ) the annual compound wont be 10% from the start of the year il give an example as to why this doesnt make sense P=£240

r=10% n=2 t=9, plug these numbers in il get 577 as the result using the second formula

if i used the first formula instead (PV*(1+r/100)^t) and plugged the numbers in, the result will obviously be different as I get 565

so why is it people use this formula(image above), when getting 5% compound return twice a year isnt the same as getting 10% annually ,which is basically what the formula says

enter image description here

People also use this formula as well which I'm assuming is just do show the actually return or profit form the initial deposit by misusing the original investment .

j jose
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  • For $r=10%$ and $t=9$ (years), you got two different $A$ depending on the compounding frequency $n$: $$n=2\implies A=£577\ n=1\implies A=£565$$ Isn't it reasonable that the amount can increase if the compounding frequency increases? – peterwhy Feb 24 '24 at 15:33
  • What you have focused on amounts to noticing that $~(1.05)^2 > 1.10.~$ So, when determining the amount of interest accrued each year, you need to know two things: [1] the annual interest rate. [2] the number of times per year that the interest is compounded. ...see next comment – user2661923 Feb 24 '24 at 16:55
  • Unfortunately, I must add to your confusion. To the best of my knowledge (and I could be mistaken), the phrase "annual interest rate" is ambiguous. The phrase could be used to represent the meaning that you have given it in your posted question. The alternative meaning would be (when the interest is compounded twice a year) that if the annual interest rate is $~10%~$, this implies that the semi-annual rate is that value $~i~$ such that $~(1 + i)^2 = 1.10.~$ You have to be very careful, anytime that you speak to a banker, to get the exact definition (from them) re the syntax they use. – user2661923 Feb 24 '24 at 16:59
  • @peterwhy yes what you said is true if the compounding frequency increase its crucial you add this into the formula , my worry is that in the formula they say r as the annual rate and divide it by the value n for how many time it compound in a year .So in my example r=10 ,n=2 which means it compounds 5% twice in a year which however isn't the same as compounding 10% annually even the formula claims r to be the annual compound value . – j jose Feb 24 '24 at 20:24
  • If your question is about such format, then I can't explain why conventionally the annual percentage rate is presented as a product $r = 5%\cdot 2=10%$ in both cases. (Why not the effective rate $(1+5%)^2-1$?) But at least the difference is clear: in one case $n=2$, and in another $n=1$. – peterwhy Feb 24 '24 at 22:33
  • @peterwhy yes that essentially was my confusion since the effective rate (ie the real annual rate ) isn't the same what the equation claims to be the annual rate ,this "negligence" of terminology really bothered me because I was starting to think I was wrong . – j jose Feb 29 '24 at 02:17
  • @user2661923 thank you very much for clearing that up , I find it strange that the financial industry would intentionally use terminology like that even though its wrong ,when even the smallest of percentages can represent millions of pounds . – j jose Feb 29 '24 at 02:19

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