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Financial Mathematics on Interest and Annuity

Interest and annuity

Interest will occur when we do financial business, it may be a deposit. There are two main types of interest-bearing loans:

1) Simple interest based on the current value of money P (Present value), in this case, is the initial deposit or initial loan, and

2) Compound interest, which Interest is charged on a proportion of the current period’s money value. which compound interest looks like a recurring relationship.

In this section we will study simple interest. installment payment In addition, each example illustrates an analysis of loan investments. and deposit as well as planning to get the desired return.

Simple interest, let P be the present money value of the investment. There is an interest rate r per year, if invested t years, simple interest I can be calculated from

I = Prt

The future value of money Means the present money plus future earnings is the interest, so the future value F

F = P + I = P + Prt = P(1 + rt)

Installment Payments Generally, when we take out a loan based on present value plus interest, we know the total amount to be paid, which is the future value of the money. If n installments are made, the amount to be paid in each installment is calculated from

Pay = F/n

Example 1 Take a loan to invest from a financial institution in the amount of 200,000 baht. The financial institution charges interest at 2% per annum by entering into a loan contract for monthly installments for 3 years. Calculate the interest rate. Loan amount to be paid and monthly installments

How to do it?

Take a loan in the value of P = 200,000 baht at an interest rate of 2% per annum for 3 years.

Interest

I = Prt = 200,000 x 0.02 x 3 = 12,000

Future value of money

F = P + I = 212,000

Installment Payment made through 12 monthly installments per year for 3 years, n = 12 x 3 = 36 installments.

Payment = 212,000 / 36

= 5888.89

In conclusion, if borrowing 200,000 baht with an interest rate of 2% per annum for 3 years, there will be interest of 12,000 baht and the total amount to be paid is 212,000 baht.

Example 2:

Deposit money to a bank in the amount of 200,000 baht without additional deposits. If the bank gives an interest rate of 1.5% per annum, if the future deposit value is not less than 210,000 baht at the end of the year, is it possible?

Procedure Current account balance:

P = 200,000 Interest rate r = 1.5% = 0.015 Future account balance F = 210,000 Time t can be obtained by considering F = P(1 + rt)

t = (F / P – 1) / r = (210000 / 200000 – 1) / 0.015 = 3.33 years

It can be seen that the deposit must be at least 4 years, which is not possible.

Example 3:

If you want the interest of 10,000 baht from a bank deposit for a period of 1 year and the bank gives an interest rate of 1.5% per year, how much is the initial deposit without depositing more money in the account throughout the year?

How to do it, interest value = 10,000, interest rate r = 0.015, and deposit period of 1 year, from the formula I = Prt, current investment P can be found from

P = I / (rt) = 10000 / (0.015 x 1) = 666,667 baht

In conclusion, if wanting to return interest on investment by depositing money with a bank at an interest rate of 1.5% per annum for 1 year, an initial deposit of approximately 666,667 baht is required.

Example 4:

If you want to plan an investment by borrowing money from the fund, you can pay into the fund 5,000 baht per month and pay off within 2 years. If the interest rate is 20 percent per year, how much money can you borrow?

How to make payment per installment pay = 5000 per month, the number of installments is 12 installments per year, period t = 2 years, will get n = 24 with an interest rate of 20 % r = 0.2 from pay = F / n will get nx pay = P (1 + rt) therefore

P = (n x pay) / (1 + rt) = (5000 x 24) / (1 + 0.2 x 2) = 85,715

In conclusion, it can be borrowed in the amount of 85,715 at an interest rate of 20 percent per year, the installment period is 2 years, and monthly installments of 5,000 baht.

Example 5:

If an investment amounts to 200,000 baht but wants the value of money to increase to 300,000 baht within 10 years, what percentage of the investment unit must be invested in the average interest per year?

Method Present value of money

P = 200,000 Future value of money F = 300,000 Investment period of 10 years From the formula F = P(1 + rt) will get

r = (F / P – 1) / t = (300000 / 200000 – 1) / 10 = 0.05

In conclusion, if wanting to return from an investment of 200,000 baht to 300,000 baht in 10 years, one must choose investment units that give interest or an average profit of 5% per year.

However, in depositing an account with the bank, in the case of depositing more than 1 year, interest returns will look like compound interest because when the 1-year period is reached, the bank will combine the interest with the principal. and used as money for the next installment It can be said that compounding interest charges causes the value of money to change over time.

Source:

• Hillman AP and Alexanderson GL (1971). Algebra to Problem Solving. Allyn and Bacon, Inc. USA.
• Smith KJ, (1992). Collage Mathematics and Calculus with Applications to Management, Life and Social Sciences 2nd edition. Brooks/Cole Publishing company. The USA.
• Yanapol Sangsan and others Lecture notes investment management Ramkhamhaeng University