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I'm trying to calculate the interest and total of money when :

Someone is loaning $3600 every year over 11 years with an interest of 10% ?

Like : 3600 + 10% = 3960 first year, 3960+3600+10% = 8316 second year etc.

Just that I need a formula to calculate it faster because it's a hurdle with the calculator.

wamp
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  • you mean each time you add 3960 ? – Cardinal Jul 19 '15 at 00:23
  • this doesn't make sense. you're paying interest on what you pay? – pancini Jul 19 '15 at 00:26
  • @Cardinal I mean 3600 are added to the total each year. So I'm starting at 3600 which then gets +10% which is 3960 in total. To these 3960 are 3600 added and 10% of the total so 8316 in total. To these 8316 are 3600 added again and 10% of the total over multiple years (11 in this example). – wamp Jul 19 '15 at 00:26
  • so, in the first you have 3600 and 3600 + 360 and 3600+(3600+0.13600)0.1 , do you try to finde the general rule as a series ? – Cardinal Jul 19 '15 at 00:31
  • @ElliotG I changed the post, you're right the example didn't made a lot of sense. I'm trying to calculate that someone needs to pay me 3600 yearly but the value he needs to pay me gets increased by 10% each year over 10 years. – wamp Jul 19 '15 at 00:33
  • @Cardinal first year I have 3600+(36000.1)=3960. Second year I have 3960+3600+((3960+3600)0.1)=8316. Third year I have : 8316+3600+((8316+3600)*0.1)=13107.6. This is not hard to calculate with a calculator but I'm hoping there might be a formula that can ease and speed up this process – wamp Jul 19 '15 at 00:39

1 Answers1

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The accumulated value of a cash flow under compound interest can be calculated by regarding each individual payment separately, then taking the sum of their accumulated values at the same time point.

In your case, the first payment of $3600$ has had $11$ years to compound interest at an $i = 0.10$ annual effective rate, thus its accumulated value at the end of the term is $3600(1.1)^{11}$. The second payment has had $10$ years to compound interest under the same rate, so its accumulated value is $3600(1.1)^{10}$; and so forth. The total accumulated value of $11$ such annual payments just before the eleventh would be paid, would then be $$3600\left((1.1) + (1.1)^2 + \cdots + (1.1)^{11}\right) = 3600 (1.1) \frac{ {1.1}^{11} - 1}{{1.1}-1}.$$ In actuarial terminology, this is the accumulated value of a 11-year level payment annuity-due of $3600$ per year with annual effective interest rate of $i = 0.10$, and has actuarial notation $$3600 \ddot s_{\overline{11}\rceil 0.10}.$$

Of course, I am making some assumptions because certain facts were not stated in the question. In particular,

  1. Interest is compounded annually at an effective annual rate of $i = 0.10$.
  2. We are interested in the accumulated value of the loan.
  3. Payments are made at the beginning of each payment period, and the valuation point is just prior to when a twelfth payment would be made.
heropup
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  • This does it perfectly. Thank you very much. I apologize for not describing the problem perfectly though. – wamp Jul 19 '15 at 01:17