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compound interest
1. Simple interestThe borrowed 1000$ is called the principal. Let's write P = 1000. If the (i.e. each) year interest rate is "r", then once the related year is finished, we will have to pay back T = P + (the interest of P) = P + I = P + P x r = P(1 + r). If we expect to pay back P in the coming "n" years, then the interest rate will be n x r and we will give back P + nrP = P(1 + nr). 1.1. Each monthFirst mounth: 100$ + 100$ x (12% /12) = 100 ( 1 + 1%) $ = 101 $Second mounth: 100 (1 + 2%) $ = 102 $ ... nth nouth : 100 ( 1 + n%) $ n = 1000/100 = 10 10th mounth: 100 (1 + 10%) $ = 110$ The total to pay back is : 101 + 102 + ... + 110 = 10 x 100 + ( 1 + 2 + ... + 10) = 1000 + (10 x 11 /2) = 1055 $ . Then 55$ extra. 1.2. At the end of the periodHere the period is equal to 10 month.10th mounth: 1000 (1 + 10%) $ = 1100$ Then 100$ extra. 2. Compound interestFor each mounth and RIGHT NOW: 100$ + 100$ x (12% /12) = 100 ( 1 + 1%) $ = 101 $ First mounth: 100 (1 + 1%) $ = 101 $ If the seconf amount (101$) of the second mounth is not payed NOW, it will be charged its iterest of 1% for the next month. Then, for the next mounth: 100 (1 + 1%)2 $ = 102.01 $ ... For the 10th mounth: 100 (1 + 1%)10 Let's write: r = 1 + 1% The total to pay back is : 100 (r + r2 + r3 + ... + r10) T = 100 r (1 + r + r2 + r3 + ... + r9 ) = 100 r [(1 - r10)/(1 - r)] = 100 r [(r10 - 1)/(r - 1)] We have: r - 1 = 1% 1. First approximation: (r10 - 1 = (1 + 1%)10 - 1 = (1 + 10%) - 1 = 10% Therefore: T = 100 (1 + 1%) (10%) /1% = 100 x 1.01 x 10 = 1010 $ 2. Second approximation: (1 + 1%)10 = 1 + 10% + 10 x 9 x 1% x 1%/2 + ... = 1 + 10% + 45%% + ... = 1.10 + 0.0045 = 1.1045 + ... (r10 - 1 = 0.1045 = 10.45% T = 100 (1 + 1%) (10.45%) /1% = 100 x 1.01 x 10.45 = 1055.45 $ 3. Third approximation: (1 + 1%)10 = 1 + 10% + 10 x 9 x 1% x 1%/2 + ... = 1 + 10% + 45%% + ... = 1.10 + 0.0045 = 1.1045 + ... (r10 - 1 = 0.1045 = 10.45% T = 100 (1 + 1%) (10.45%) /1% = 100 x 1.01 x 10.45 = 1055.45 $ 4. With Calculator: (r10 - 1 = 0.104622125 = 10.4622125% T = 100 (1 + 1%) (10.4622125%) /1% = 100 x 1.01 x 10.4622125 = 1056.68 $ 3. nominal and effective ratesThe nominal or stated rate is the unit rate: The interest rate "r" for a related period "n" is called the effective rate. Then the nominal rate is "r/n". The interest becomes I = P x n x (r/n), and the repayment is R = P + I = P(1 + r) for the simple interest and R = P [1 + (r/n)]n for the compound interest. The relationship between the two rates is: P(1 + effective) = P [1 + (nominal)]n Therefore: effective = [1 + (nominal)]n - 1 (1 + nominal rate)period - 1 Annex: 1. arithmetic series: 1 = 2 + 3 + ... + n = n(n + 1)/2, indeed: Sn = 1 + 2 + 3 + ... + n Sn = n + n-1 + n-2 + ... +1 2Sn = (1+n) + (1+n) + ... + (1+n) , n times. Therefore: 2Sn = n (1+n). Then: Sn = n(n+1)/2 More generally: First term: t1 = a Second term: t2 = a + r Third term: t3 = a + r + r = a + 2r ... nth term: tn = a + r + r + r + ... + r = a + (n - 1)r The sum is: Sn = n x a + r [1 + 2 + 3 + ... + (n - 1)] = n x a + r x (n - 1) x n/2 Sn = n [t1 + r(n - 1)/2] Example: For n = 10, tn = 1, and r = 1, we have: Sn = 10 [1 + 9/2] = 5 x 11 = 55. 2. Maclaurin polynomial: (1 + x)n = 1 + n x + n(n - 1)x2/2! + n(n - 1)(n - 2)x3/3! + ... + 3. geometric series: (1 + r + r2 + r3 + ... + rn) = [1 - rn+1]/(1 - r) = [rn+1 - 1]/(r - 1) 1 + r + r2 + r3 + ... + rn = (rn+1 - 1)/(r - 1) More generally: First factor: f1 = a Second factor: f2 = a x r Third factor: f3 = a x r x r = a x r2 ... nth factor: fn = a x r x r x r x ... x r = a x rn - 1 The sum is: Sn = a x r [1 + r + ... + rn - 2] = a x r x (1 - rn - 2)/(1 - r) Sn = f1 r (1 - rn-1)/(1-r) Example: For n = 10, fn = 1, and r = 2, we have: Sn = 1 x 2 (1 - 29)/(1 - 2) = 2 (29 - 1)/(2 - 1) = 2 (29 - 1) (29 - 1) = 1022 ≈ 210 = 1024. 2. Maclaurin polynomial: (1 + x)n = 1 + n x + n(n - 1)x2/2! + n(n - 1)(n - 2)x3/3! + ... + Application compound interest: p x r p x r2 p x r3 ... p x rn Therefore, the total is: T = p x r (1 + r + r2 + r3 + ... + rn) = p x r x (rn+1 - 1)/(r - 1) Interesr rate = P r n P: Principal, r: interest rate, n: priod Simple interest: P → P ( 1 + n r) Compound interest: P → P ( 1 + r)n effective rate = (1 + nominal rate)period - 1 arithmetic series: tn = a + (n - 1)r Sn = n [t1 + r(n - 1)/2] geometric series: fn = a x rn - 1 Sn = f1 r (1 - rn - 1)/(1-r) |
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