Mathematics of Finance

1. Mathematics of Finance 5.3 Compound Interest

2. 5.2 Compound Interest Objectives: • Find the compound interest and compound amount • Apply the concept of compounding period to find • compound amount • time and interest rate • present value • effective rate

3. Definition - Compound Interest If the interest due at the end of each payment period is added to the principal, so that the interest computed for the next payment period is based on this new amount of old principal plus the interest, then the interest is said to have been compounded. Compounding the interest means earning interest on interest and we call the result as compound interest.

4. Terms for payment period

5. Simple interest VS Compound interest

6. Assume that we have deposited RM8,000 in a credit union, which pays interest of 8% per year-compounded quarterly. Assume that we want to determine the amount of money we will have on deposit at the end of one year if all interest is left in the savings account. At the end of the first quarter interest is computed as I1= (RM8,000)(0.08)(0.25) = RM160.00

7. Note that t was defined as 0.25 year. With the interest left in the account, the principal on which interest is earned in the second quarter is the original principal plus the RM160 in interest earned during the first quarter, or P2 = P1 + I1 = RM 8160 Interest earned during the second quarter is I2= (RM8,160)(0.08)(0. 25) = RM163.20

8. Table below summarizes the computations for the four quarters. Note that for each quarter the accumulated principal plus interest to as the compound amount. Notice that the total interest earned during the four quarters equals RM659.46

9. In this example, simple interest for the year would have been equal to I = (8,000)(0.08)(1) = RM640.00 The difference between the simple interest and the compound interest is 659.46 – 640.00 = RM19.46

10. Accumulated Value Compound interest Simple interest Time The accumulated value for compound interest is greater than that of simple interest.

11. Compound amount The compound amount or future value or accumulated value is the final sum at the end of the period. For example, RM5,000 invested in a bank at the rate of 10%, compounded yearly. The first year accumulated value is: = 5,000(1 + 0.1) = RM5,500.00

12. For the second year, the calculation of the future value (S2) is = 5,500(1+0.1) = RM6,050 or

13. The S2amount is used as a principal to calculate the accumulated value for the third year. = 6,050(1+0.1) = RM6,655 or

14. Hence, the compound amount (future value) for the n-th period may be expressed as: Where S = compound amount P = original principal i = the yearly interest rate j = nominal rate m = frequency n = number of compounding period t = time or terms in years

15. Example 1 - compounding amount A long - term investment of RM 250,000 has been made by a small company. The interest rate is 12 percent per year, and the interest is compounded quarterly. What will the value of the investment be after 8 years? How much interest will be earned?

17. Example 2 Suppose that RM 1,000 is invested in a bank, which earns interest at a rate of 8% per year compounded annually. What will the account balance be after 10 years?

19. Example 3 Sara decided to invest RM1000 in a saving account at an annual interest rate of 10%. What is the amount after 5 years if the compounding takes place: a) annually b) monthly c) daily

21. Example 4 Khalidah deposited RM500 in a saving account for eleven years. The interest rate is 6% compounded semiannually for the five years and 8% compounded quarterly for the next six years. What is the amount in the saving account at the end of the 11th year?

23. Present Value Present value starts with the future (amount) and tries to calculate its worth in the present (now), at interest period. From the relationship, S = P( 1 + i )n The present value (P ) is obtained as follows or P = S(1 + i )-n

24. Example 5 Computing present value How much money should be invested now at 8% per annum so that after 2 years the amount will be RM10,000 when interest is compounded a) annually b) monthly c) daily

26. Example 6 A young man has recently received an inheritance of RM 200,000. He wants to take a portion of his inheritance and invest it for his later years. His goal is accumulated RM300,000 in 15 years. How much of the inheritance should be invested if the money will earn 12% per year compounded semiannually? How much interest will be earned over the 15 years?

28. Example 7 How much should Aimen invest now to get RM3,216.90 in 6 years if the interest rate is 8% compounded quarterly.

30. Min has been saving up to buy a Tanjung Print Co. The total cost will be RM10 million. Min currently has RM2.3 million. If Min can earn 5% p.a. compounded annually on his money, how long will Min has to wait? Example 8 Finding the period

32. Example 9 - Finding the period How long will it take to increase RM 1,800 to RM 3,002 at the rate of 9% per year compounded every three months?

34. Example 10 - interest rate A person wishes to invest RM 10,000 and wants the investment to grow to RM 20,000 over the next 10 years. At what annual interest rate would the RM 10,000 have to be invested for this growth to occur, assuming annual compounding?

36. Example 11 Rahman invests RM30,000 in a fund that pays a compound interest at a rate of 6.5% per year compounded monthly. (a) How long does it take for Rahman’s investment to triple? (b) Find the interest earned after 5 years.

38. Effective Rate For compound interest, if the interest is compounded m times per year at an annual rate i, the effective rate is the annual interest rate that yields the same interest after one year.

39. Example 12 - Effective rate Calculate the effective rate associated with an 8% annual rate when interest is compounded a) annually b) quarterly

41. Example 13 Find the effective rate for: (a) 16% compounded quarterly, (b) 15% compounded monthly.