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May 16th, 2018, 01:34 AM   #1
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Ask About Annuity

Mr. Budi borrows 2,000,000IDR which will be amortized with 10 annuities. The first annuity will be paid in 1 year with 10% per year interest. Make the installment plan!
For the annuity I got A = $\displaystyle M\times\frac1{\sum_{n=1}^p(1+i)^{-n}}$ = $\displaystyle 2,000,000\times\frac1{\sum_{n=1}^{10}(1+0,1)^{-n}}$ = $\displaystyle 2,000,000\times\frac1{\sum_{n=1}^{10}(1,1)^{-n}}$ = $\displaystyle 2,000,000\times\frac1{\sum_{n=1}^{10}(1,1)^{-n}}$ = $\displaystyle 2,000,000\times\frac1{(1,1)^{-1}+(1,1)^{-2}+(1,1)^{-3}+(1,1)^{-4}+(1,1)^{-5}+(1,1)^{-6}+(1,1)^{-7}+(1,1)^{-8}+(1,1)^{-9}+(1,1)^{-10}}$
($\displaystyle (1,1)^{-1} + (1,1)^{-2} + … + (1,1)^{-10}$ is a geometric sequence with the first term $\displaystyle a = (1,1)^{-1}$ and ratio $\displaystyle r = (1,1)^{-1}$)
= $\displaystyle 2,000,000\times\frac1{\frac{1-1.1^{-10}}{0.1}}$ = $\displaystyle 2,000,000\times\frac{0.1}{1-1.1^{-10}}$ = 2,000,000 × 0.61445671 = 1,228,913.42
Thus, the annuity is 1,228,913.42.
On the end of first year:
Annuity = 1,228,913.42IDR
Interest : 10% × 2,000,000IDR = 200,000IDR
Installment : 1,228,913.42IDR – 200.000IDR = 1,028,913.42IDR
Remaining loan : 2,000,000IDR – 1,028,913.42IDR = 971,086.58IDR
On the end of second year:
Annuity = 1,228,913.42IDR
Interest : 10% × 971,086.58IDR = 97,108.66IDR
Installment : 1,228,913.42IDR – 97,108.66IDR = 1,131,804.76IDR
Remaining loan : 971,086.58IDR – 1,131,804.76IDR = -159.718,58IDR
Why am I seeing negatives already?

Last edited by Monox D. I-Fly; May 16th, 2018 at 01:40 AM.
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May 16th, 2018, 04:50 AM   #2
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Geezzzz Mr.Fly...your post gave me a headache!

Your problem is:
2000000 is borrowed at rate of 10% APR, over 10 years.
Calculate the annual payment.

Formula:
P = A*i / (1 - v) where v = 1 / (1 + i)^n

A = amount borrowed = 2000000
i = interest rate = .10
n = number of payments = 10
P = payment amount = ?
That'll give you P = 325490.79

The formula is of course the result of a geometric sequence:
present value of payments = 2000000
or
future value of payments = 2000000*1.10^10

Your loan statement will look like:
Code:
year  payment     interest       balance
  0                                      2000000.00
  1   325490.79  200000.00  1874509.21
  2   325490.79  187450.92  1736469.34
...
  9   325490.79   56490.14    295900.72
 10  325490.79   29590.07               .00
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