Two payment methods for mortgages

 

As we all know, the installment payment methods of bank housing loans are divided into two types: the same amount of principal payment and the same amount of payment. The monthly payment and calculation methods of the two payment methods are different. There are two types of online payment calculation formulas. However, the sources of these two formulas are rarely explained, or the explanations are rough or wrong. After some time of thinking, I finally understood the principles and used the High School Mathematics Theory to derive these two formulas. This article explains the principles of the two repayment methods and the derivation process of the formula.

Either method of repayment has one thing in common: the monthly repayment amount (also called monthly subscription) contains two parts: principal repayment and interest repayment:

Monthly repayment amount = current month's principal repayment + current month's interest 1

The principal is actually used to repay the loan. After monthly repayment, the remaining principal of the loan will be reduced accordingly:

Remaining current month principal = remaining last month Principal-current month Principal Repayment

Until the last month, all the principal has been paid off.

Interest repayment is used to repay the interest generated by the remaining principal in this month. The interest generated by the current month's principal must be paid in monthly repayment:

Monthly interest = remaining principal of the previous month × monthly interest rate 2

The monthly interest rate = the annual interest rate is 12. It is said that the monthly interest rate of some banks, such as ICBC, used a great dealAlgorithm.

As shown in the above interest repayment formula, monthly interest is directly proportional to the remaining principal of the previous month. Since there are many remaining principal at the beginning of the loan period, the monthly interest at the early stage of the loan period is large, the portion of the monthly repayment amount is heavy. As the number of payments increases, the remaining principal will gradually decrease, and the monthly interest will also decrease accordingly. Until the last month, the principal will be paid off and the interest will be paid for the last time, there will be no principal and no interest next month. Now, all loans have been paid off.

The repayment principles of the two types of loans are described above. The above two formulas are the basic formulas for monthly repayment. Other formulas can be derived from them. The following two formulas are used to derive the specific calculation formulas for the two repayment methods.

1. Same amount of principal repayment method

The same amount of principal is relatively simple. As the name suggests, in this way, the number of each repayment of the principal is the same. Therefore:

Current month's principal repayment = total number of loans; times of repayment

Monthly interest = remaining principal of the previous month × monthly interest rate

= Total number of loans × (1-(number of repayment months-1) times of repayment) × monthly interest rate

Monthly repayment amount for the current month = current month's principal repayment + current month's interest

= Total number of loans × (1 ÷ repayment times + (1-(number of repayment months-1) ÷ repayment times) × monthly interest rate)

Total interest = sum of all interest

= Total number of loans × monthly interest rate × (repayment times-(1 + 2 + 3 +... + Repayment times-1) renewal times)

Among them, 1 + 2 + 3 +... + Repayment frequency-1 is an arithmetic difference series, and the sum is (1 + repayment times-1) × (repayment times-1)/2 = repayment times × (repayment times-1) /2

Therefore, after sorting out, we can conclude that:

Total interest = total loan count × monthly interest rate × (repayment times + 1) limit 2

Since the amount of the same principal is fixed each month, and the monthly interest is decreasing, the amount of the same principal is different each month. It starts to be much more, and then decreases by month.

2. Same amount of principal and interest repayment methods

The formula for the repayment of the same amount of principal and interest is complicated, but you don't have to worry about it. As long as you have the knowledge of the high school series, you can deduce it.

The same amount of principal is fixed as the name suggests. Since the repayment interest is reduced by month, the amount of the principal paid in the monthly repayment is increased by month.

First, let's make some settings:

Set: total loan amount =

Repayment Times = B

Monthly Repayment Rate = C

Monthly repayment amount = x

Monthly principal repayment = yn (n = monthly repayment)

First, let's talk about the first month. The current month's principal is all loan amount = A. Therefore:

Interest for the first month = A × C

The principal amount of the first month

Y1 = x-interest for the first month

= X-A x C

Remaining principal in the first month = total loan amount-principal repayment amount in the first month

= A-(X-A x C)

= A × (1 + C)-x

In the second month, the amount of interest in the current month = the remaining principal of the previous month × monthly interest rate

Interest for the second month = (a × (1 + C)-x) × C

The second month's principal repayment amount

Y2 = x-interest for the second month

= X-(A x (1 + C)-x) x C

Remaining principal in the second month = remaining principal in the first month-remaining principal amount in the second month

= A × (1 + C)-X-(x-(A × (1 + C)-x) × C)

= A × (1 + C)-X-X + (A × (1 + C)-x) × C

= A × (1 + C) × (1 + C)-[x + (1 + C) × x]

= A × (1 + C) ^ 2-[x + (1 + C) × x]

(1 + C) ^ 2 indicates the power of (1 + C)

The third month,

Interest for the third month = remaining principal for the second month × monthly interest rate

Interest for the third month = (a × (1 + C) ^ 2-[x + (1 + C) × x]) × C

The principal amount of the third month

Y3 = x-interest for the third month

= X-(A x (1 + C) ^ 2-[x + (1 + C) x]) x C

Remaining principal of the third month = remaining principal of the second month-repayment of the principal of the third month

= A × (1 + C) ^ 2-[x + (1 + C) × x]

-(X-(A x (1 + C) ^ 2-[x + (1 + C) x]) x C)

= A × (1 + C) ^ 2-[x + (1 + C) × x]

-(X-(A × (1 + C) ^ 2 × C + [x + (1 + C) × x]) × C)

= A × (1 + C) ^ 2 × (1 + C)

-(X + [x + (1 + C) x] x (1 + C ))

= A × (1 + C) ^ 3-[x + (1 + C) × x + (1 + C) ^ 2 × x]

The preceding formula can be divided into two parts:

Part 1: A × (1 + C) ^ 3.

Part 2: [x + (1 + C) × x + (1 + C) ^ 2 × x]

= X × [1 + (1 + C) + (1 + C) ^ 2]

By summarizing the remaining principal formula of the previous three months, we can see the rule:

The first part of the remaining principal = the total loan amount × (1 + monthly interest rate) The Npower (where n = the number of months of repayment)

The second part of the remaining principal is an equal-ratio series, which uses (1 + monthly interest rate) as the proportional coefficient, the monthly repayment amount as the constant coefficient, and the number of items as the number of months for repayment n.

Promotion to any month:

The remaining principal of the nth month = A × (1 + C) ^ N-X × Sn (the first n of the proportional sequence with Sn (1 + C) and)

According to the first n items and formulas of the proportional sequence:

1 + Z + Z2 + Z3 +... + Zn-1 = (1-z ^ N)/(1-z)

We can conclude that

X Sn = x (1-(1 + C) ^ N)/(1-(1 + C ))

= X × (1 + C) ^ N-1)/C

Therefore, the remaining principal for the nth month is a × (1 + C) ^ N-X × (1 + C) ^ N-1)/C

Since the principal of the last month will be completely paid back, when n is equal to the number of payments, the remaining principal is zero.

Set n = B (repayment times)

Remaining principal = A × (1 + C) ^ B-X × (1 + C) ^ B-1)/C = 0

So that

Monthly repayment amount

X = A × c × (1 + C) ^ B ÷ (1 + C) ^ B-1)

= Total loan amount × monthly interest rate × (1 + monthly interest rate) ^ number of payments limit [(? Number of remaining payments-1]

Bringing the X value back to the remaining principal formula for the nth month

Remaining principal of the nth month = A × (1 + C) ^ N-[A × c × (1 + C) ^ B/(1 + C) ^ B-1)] × (1 + C) ^ N-1)/C

= A × [(1 + C) ^ N-(1 + C) ^ B × (1 + C) ^ N-1)/(1 + C) ^ B-1)]

= A × [(1 + C) ^ B-(1 + C) ^ N]/(1 + C) ^ B-1)

Interest for the nth month = remaining principal for the nth month-the monthly interest rate

= A × c × [(1 + C) ^ B-(1 + C) ^ (n-1)]/(1 + C) ^ B-1)

Principal repayment amount for the nth month = x-interest for the nth month

= A × c × (1 + C) ^ B/(1 + C) ^ B-1)-A × c × [(1 + C) ^ B-(1 + C) ^ (n-1)]/(1 + C) ^ B-1)

= A × c × (1 + C) ^ (n-1)/(1 + C) ^ B-1)

Total repayment amount = x B

= A × B × c × (1 + C) ^ B × (1 + C) ^ B-1)

Total interest = total repayment amount-total loan amount = X × B-

= A × [(B × C-1) × (1 + C) ^ B + 1]/(1 + C) ^ B-1)

The repayment amount of the same amount is fixed every month. Due to the large interest at the early stage of repayment, the amount of the original principal is very small. Compared with the same amount of principal, the total interest for repayment is much higher.