Have you ever wondered? Because those people who acquire a bank loan and have been paying in a timely manner for 3 or 4 years, the total debt seems not to decrease significantly? This is due to the fact that an important part of the total payment that is made is used to pay interest that the bank charges for having granted the loan and what is left is assigned to settle the initial debt or unpaid balance.

But what is the repayment of a loan?

When a loan is requested and it is intended to pay in installments, each of these credits is composed of interest and part of capital. The action to cancel part of the capital in each one of the payments is called amortization of the credit.

The breakdown of payments can be clearly seen in an amortization table. This table is calculated by the banks according to the characteristics of the credit (amount, term and interest rate).

It is important to mention that there are different ways to amortize a loan and these depend mainly on the way in which the amount of capital is reduced; However, before explaining some of the amortization methods, it is important to review the following topics: geometric progressions, annuities and the present value of an annuity; amortization.

Geometric progressions

If ayr are fixed numbers not equal to zero, then the infinite list of numbers a, ar, ar 2 , ar 3 , ar 4 ,. . . . it is called a geometric sequence or geometric progression. For example, if a = 5 and r = 2, we have the sequence: 5, (5) (2), (5) (2) 2 , (5) (2) 3 , (5) (2) 4 ,. . . . or 5, 10, 20, 40, 80

Example $ 100.00 is deposited into an account that pays 10% interest compounded annually. How much is in the account at the beginning of each of the first five years?

According to the formula for compound interest (with P = 100 and i = 0.01), the value of the account at the beginning of each of the first 5 years is given by the first five terms of the geometric progression with a = 100 yr = 1.1. The first 5 terms are:

100, 100 (1.1), 100 (1.1) 2 , 100 (1.1) 3 , 100 (1.1) 4

So the value of the account at the beginning of each of the five years is:

$ 100.00, $ 110.00, $ 121.00, $ 133.10, $ 146.41

Annuities

An annuity is a succession of equal payments that are made in equal periods and this may be ordinary, anticipated or deferred.

Ordinary annuities: Annuities begin to be paid from period 1.

Example. Suppose that $ 1,500.00 is deposited at the end of the year during the following years in an amount that pays 8% per year compounded annually.

1500 (1.08) 5

The 5 is used as an exponent, instead of the 6, since the money is deposited at the end of the first year and earns interest for only 5 years.

1500 (1.08) 5 + 1500 (1.08) 4 + 1500 (1.08) 3 + 1500 (1.08) 2 + 1500 (1.08) + 1500

$ 2,203.99; $ 2,040.73; $ 1,889.57; $ 1,749.60; $ 1,620.00; $ 1,500.00

$ 11,003.89

To perform the calculation easily, the following formula can be applied:

Advance Annuities : Annuities begin to be paid from period 0.

To find the future value of an anticipated annuity, each payment is treated as if it were done at the end of the preceding period. No payment is made at the end of each period, so to compensate for this, the amount of a payment is subtracted.

Considering the previous problem and applying the following formula:

Is obtained:

Present value of an annuity

The present value of an annuity is the sum of the present values of the different payments, each one discounted at the beginning of the term.

It is important to remember the way in which the present value of compound interest is calculated.

Capitalize: add interest

Update: Discount, remove interest 1

Example. Find the current value of an ordinary annuity of $ 1,000.00 per month, for 4 months to 5% per month.

To perform the calculation easily, the following formula can be applied:

As previously mentioned, and taking up the main theme, there are various ways to amortize a loan and these depend mainly on the way in which the amount of capital is reduced. Here are two ways of repaying a loan: fixed and declining rate.

one). Fixed fee : In this method, as the name indicates, all fees or payments must be equal. To calculate the quota, the formula of the present value of an annuity (VPA) reviewed in the previous section is used.

For a better understanding of the fixed fee method, a real amortization table of a bank loan will be constructed; Therefore, it is important to mention that when paying off a loan, and in accordance with Mexican tax legislation, you must pay 16% of the Value Added Tax (VAT) calculated on the interest that will be paid for the credit contracted.

Example. Build a table of amortization of a bank loan for an amount of $ 15,912.12 to a term of 60 months and a rate of 23.72% per year, considering a VAT of 16%.

When you clear R, you get that the amount of the fixed payment is $ 490.79. The corresponding amortization table is shown below:

Amortization table by the fixed installment method | |||||

Payment | Insolute Capital | Capital payment | Subscription Interest | VAT | Fixed payment |

one | $ 15,912.12 | $ 125.94 | $ 314.53 | $ 50.32 | $ 490.79 |

two | $ 15,786.18 | $ 128.83 | $ 312.04 | $ 49.93 | $ 490.79 |

3 | $ 15,657.35 | $ 131.78 | $ 309.49 | $ 49.52 | $ 490.79 |

4 | $ 15,525.57 | $ 134.80 | $ 306.89 | $ 49.10 | $ 490.79 |

5 | $ 15,390.77 | $ 137.89 | $ 304.22 | $ 48.68 | $ 490.79 |

6 | $ 15,252.88 | $ 141.06 | $ 301.50 | $ 48.24 | $ 490.79 |

7 | $ 15,111.82 | $ 144.29 | $ 298.71 | $ 47.79 | $ 490.79 |

8 | $ 14,967.53 | $ 147.60 | $ 295.86 | $ 47.34 | $ 490.79 |

9 | $ 14,819.93 | $ 150.98 | $ 292.94 | $ 46.87 | $ 490.79 |

10 | $ 14,668.95 | $ 154.44 | $ 289.96 | $ 46.39 | $ 490.79 |

eleven | $ 14,514.51 | $ 157.99 | $ 286.90 | $ 45.90 | $ 490.79 |

12 | $ 14,356.52 | $ 161.61 | $ 283.78 | $ 45.40 | $ 490.79 |

13 | $ 14,194.91 | $ 165.31 | $ 280.59 | $ 44.89 | $ 490.79 |

14 | $ 14,029.60 | $ 169.10 | $ 277.32 | $ 44.37 | $ 490.79 |

fifteen | $ 13,860.50 | $ 172.98 | $ 273.98 | $ 43.84 | $ 490.79 |

16 | $ 13,687.52 | $ 176.95 | $ 270.56 | $ 43.29 | $ 490.79 |

17 | $ 13,510.57 | $ 181.01 | $ 267.06 | $ 42.73 | $ 490.79 |

18 | $ 13,329.56 | $ 185.16 | $ 263.48 | $ 42.16 | $ 490.79 |

19 | $ 13,144.41 | $ 189.40 | $ 259.82 | $ 41.57 | $ 490.79 |

twenty | $ 12,955.01 | $ 193.74 | $ 256.08 | $ 40.97 | $ 490.79 |

twenty-one | $ 12,761.26 | $ 198.19 | $ 252.25 | $ 40.36 | $ 490.79 |

22 | $ 12,563.08 | $ 202.73 | $ 248.33 | $ 39.73 | $ 490.79 |

2. 3 | $ 12,360.35 | $ 207.38 | $ 244.32 | $ 39.09 | $ 490.79 |

24 | $ 12,152.97 | $ 212.13 | $ 240.22 | $ 38.44 | $ 490.79 |

25 | $ 11,940.83 | $ 217.00 | $ 236.03 | $ 37.76 | $ 490.79 |

26 | $ 11,723.83 | $ 221.97 | $ 231.74 | $ 37.08 | $ 490.79 |

27 | $ 11,501.86 | $ 227.06 | $ 227.35 | $ 36.38 | $ 490.79 |

28 | $ 11,274.80 | $ 232.27 | $ 222.87 | $ 35.66 | $ 490.79 |

29 | $ 11,042.53 | $ 237.60 | $ 218.27 | $ 34.92 | $ 490.79 |

30 | $ 10,804.93 | $ 243.04 | $ 213.58 | $ 34.17 | $ 490.79 |

31 | $ 10,561.89 | $ 248.62 | $ 208.77 | $ 33.40 | $ 490.79 |

32 | $ 10,313.27 | $ 254.32 | $ 203.86 | $ 32.62 | $ 490.79 |

33 | $ 10,058.95 | $ 260.15 | $ 198.83 | $ 31.81 | $ 490.79 |

3. 4 | $ 9,798.81 | $ 266.11 | $ 193.69 | $ 30.99 | $ 490.79 |

35 | $ 9,532.69 | $ 272.22 | $ 188.43 | $ 30.15 | $ 490.79 |

36 | $ 9,260.48 | $ 278.46 | $ 183.05 | $ 29.29 | $ 490.79 |

37 | $ 8,982.02 | $ 284.84 | $ 177.54 | $ 28.41 | $ 490.79 |

38 | $ 8,697.18 | $ 291.37 | $ 171.91 | $ 27.51 | $ 490.79 |

39 | $ 8,405.81 | $ 298.05 | $ 166.15 | $ 26.58 | $ 490.79 |

40 | $ 8,107.75 | $ 304.89 | $ 160.26 | $ 25.64 | $ 490.79 |

41 | $ 7,802.86 | $ 311.88 | $ 154.24 | $ 24.68 | $ 490.79 |

42 | $ 7,490.99 | $ 319.03 | $ 148.07 | $ 23.69 | $ 490.79 |

43 | $ 7,171.96 | $ 326.35 | $ 141.77 | $ 22.68 | $ 490.79 |

44 | $ 6,845.61 | $ 333.83 | $ 135.31 | $ 21.65 | $ 490.79 |

Four. Five | $ 6,511.78 | $ 341.48 | $ 128.72 | $ 20.59 | $ 490.79 |

46 | $ 6,170.30 | $ 349.31 | $ 121.97 | $ 19.51 | $ 490.79 |

47 | $ 5,820.99 | $ 357.32 | $ 115.06 | $ 18.41 | $ 490.79 |

48 | $ 5,463.66 | $ 365.52 | $ 108.00 | $ 17.28 | $ 490.79 |

49 | $ 5,098.15 | $ 373.90 | $ 100.77 | $ 16.12 | $ 490.79 |

fifty | $ 4,724.25 | $ 382.47 | $ 93.38 | $ 14.94 | $ 490.79 |

51 | $ 4,341.78 | $ 391.24 | $ 85.82 | $ 13.73 | $ 490.79 |

52 | $ 3,950.54 | $ 400.21 | $ 78.09 | $ 12.49 | $ 490.79 |

53 | $ 3,550.33 | $ 409.39 | $ 70.18 | $ 11.23 | $ 490.79 |

54 | $ 3,140.95 | $ 418.77 | $ 62.09 | $ 9.93 | $ 490.79 |

55 | $ 2,722.17 | $ 428.38 | $ 53.81 | $ 8.61 | $ 490.79 |

56 | $ 2,293.80 | $ 438.20 | $ 45.34 | $ 7.25 | $ 490.79 |

57 | $ 1,855.60 | $ 448.25 | $ 36.68 | $ 5.87 | $ 490.79 |

58 | $ 1,407.35 | $ 458.52 | $ 27.82 | $ 4.45 | $ 490.79 |

59 | $ 948.83 | $ 469.04 | $ 18.76 | $ 3.00 | $ 490.79 |

60 | $ 479.79 | $ 479.79 | $ 9.48 | $ 1.52 | $ 490.79 |

To start the construction of the table, once you have the amount of the fixed payment, the interest payment is first calculated and for this the unpaid capital is multiplied by 0.01976 (15,912.12 X 0.01976 = 314.53). 0.01976 is the result of dividing 23.72% between twelve months. Subsequently, the interest payment is multiplied by 16% to obtain VAT (314.53 X 0.16 = 50.32), and finally, the fixed payment is subtracted from the sum of the interest payment plus VAT to obtain the capital payment (490.79 – 364.85 = 125.94). For the next period the same procedure is performed; but the initial insolute capital is no longer used, but the capital payment is subtracted to obtain the unpaid capital to be used in the following period (15,912.12 – 125.94 = 15,786.18).

But where do you get 0.0229? You get 16% of the rate of 23.72% and the result obtained is added to this rate (23.72 X 1.16 = 27.5252) and the final result is divided by twelve to convert the annual rate into monthly (27.5252 / 12 = 2.29).

It is important to mention that the fixed fee method is the most used by banks in Mexico, so it was decided to construct the amortization table in a real way. Through this exercise, and as mentioned at the beginning, it can be observed that part of the total payment that is made is used to pay interest and what is left is assigned to settle the initial debt or unpaid balance.

The process of building an amortization table corresponding to a mortgage loan or a credit for the purchase of a car is similar to the one developed in this case.

two). Decreasing share : In this method the depreciation is the same for all periods, the depreciation must be calculated first.

Formula to calculate the amortization

Example. Build a table of amortization of a debt of $ 1000.00 that must be repaid in 4 months at an effective rate of 5% per month.

Amortization table by the declining installment method | ||||

Payment | Insolute capital | Capital payment | Subscription Interest | Fixed payment |

one | $ 1,000.00 | $ 250.00 | $ 50.00 | $ 300.00 |

two | $ 750.00 | $ 250.00 | $ 37.50 | $ 287.50 |

3 | $ 500.00 | $ 250.00 | $ 25.00 | $ 275.00 |

4 | $ 250.00 | $ 250.00 | $ 12.50 | $ 262.50 |