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1. BUSINESS MATHEMATICS INTERESTS K ristine D. Buña

2. COMPOUND INTEREST Let P be the principal amount, r the rate, and n the number of compoundment per annum, and t for term in years. The compound amount An, at the end of the nth period is:   Compound interest is also the amount earned for one year calculated by multiplying the principal by the interest rate. They are usually used in long-term transactions.

3. Examples: Compare the maturity value of a principal value of Php 100,000.00 subjected to 6% interest in 60 days using simple interest. Compute for the maturity value if the principal amount is rolled at 6% every 60 days for one year.

4. PERIOD PER YEAR IN COMPOUND INTEREST Annual – 1 payment in one year/1 period Semiannually – 2 payments in one year/2 periods Quarterly - 4 payments in one year/4 periods Monthly - 12 payments in one year/12 periods

5. ANNUITY Annuity is another common business practice of payment. House rental, life insurance premiums, bond dividends, installment payments, and labor wages are form of annuities. Payments made annually. Forms of Annuity (depends on the mode of payment): 1. Annuity certain – an annuity with fixed dates for both the first and the last payments; 2. Contingent annuity – an annuity with indefinite dates for either the first or last payments; 3. Simple annuity – an annuity with same interest conversion dates and payment dates; 4. General annuity – an annuity where the payment dates do not coincide with the interest conversion periods; 5. Ordinary annuity – an annuity for which the payments are made at the end of the interest conversion periods; and 6. Annuity due – an annuity for which payments are made at the beginning of each interest conversion rates.

6. ORDINARY ANNUITY VS ANNUITY DUE An ordinary annuity is when a payment is made at the end of a period. An annuity due is when a payment is due at the beginning of a period. While the difference may seem meager, it can make a significant impact on your overall savings or debt payments. Which Annuity Is Best? In general, an ordinary annuity is most advantageous for consumers when they are making payments. Conversely, an annuity due is most advantageous for people when they are collecting payments. The payments made on an annuity due have a higher present value than an ordinary annuity due to inflation and the time value of money.

7. FUTURE VALUE OF ORDINARY ANNUITY The future value of an ordinary annuity for n periods at interest i per compounding period is Where p is the fixed periodic payments.  

8. Examples: Dr. Boiser deposited part of her royalty in a bank at the end of each year for 8 years. The bank paid 2.5% interest compounded annually. If she deposited Php 200,000 every year, how much did she have deposited including the interest after her 8 th deposit?

9. PRESENT VALUE OF ORDINARY ANNUITY The PRESENT value of an ordinary annuity for n periods at interest i per compounding period is Where p is the fixed periodic payments.  

10. Examples: Determine the future value and present value of an ordinary annuity for Php 300,000 investment at 2% quarterly for 5 years.

11. FUTURE VALUE OF ANNUITY DUE The future value of an annuity due for n periods at interest i per compounding period is Where p is the fixed periodic payments.  

12. Examples: Mr. Torres invested Php 10,000 in an annuity due on January 1, 2010 until December 31, 2014. The bank credits 2.2% interest compounded annually to Mr. Torres’ account. Find the future value of Mr. Torres’ Annuity.

13. PRESENT VALUE OF ANNUITY DUE The PRESENT value of an annuity due for n periods at interest i per compounding period is Where p is the fixed periodic payments.  

14. Examples: Mr. Torres signed a least contract with the owner of a commercial space worth Php 100 000 per year for five years, and made a first payment on January 1. Evaluate the present value of Mr. Torres’ 5-year lease on the same day as the first payment was made assuming a 2.2% annual compound interest rate.

15. LOAN, AMORTIZATION, AND MORTGAGE Loans are debt contract entered by two parties, organization or individual, through a note which states the principal amount, interest rate, the mode of payment and due date. Loans usually need collaterals, guarantors, and documentations. Amortization is a process of paying loan and its interest through series of regular equal payments. Mortgage is a type of secured loan that makes use of real property such a house or lot as collateral. This type of loan is usually entered into by real property buyers who do not have enough cash on hand to purchase the property.

16. AMORTIZATION AND AMORTIZATION SCHEDULE Amortization is a process of paying loan and its interest through series of regular equal payments. Formula: P = amortized loan R = periodic payment i = Interest n = period  

17. Examples: An amortized car loan of Php 1,000,000 for purchasing a brand new car is granted to Mr. Enriquez by a bank. If the loan is to be paid in 1 years at an annual interest rate of 12%, find the monthly amortization and create the amortization schedule. Payment Number Periodic Payment Interest Payment Amount Repaid Balance