Financial Mathematics

CHAPTER 2 Financial Mathematics

2.1 Introduction • What is financial math? - field of applied mathematics, concerned with financial markets. • Procedures which used to answer questions associated with major financial transactions • Example: Interest problem, annuity and depreciation

2.2 Simple Interest • The study of interest is very important and fundamental to the understanding of the economy of a country • Interest is money earned when money is invested or interest is charge incurred when a loan or credit is obtained.

If you deposit a sum of money P in a savings account or if you borrow a sum of money P from a lending agent such as financial agency or bank, then P is call principal. • Usually we have to repay this amount P plus an extra amount. These extra amounts, which pay to the lender for the convenience of using lender money is called interest.

In general, if the principal P is borrowed at a rate r after t years, the borrower will owe the lender an amount A that will include the principal P plus interest I.

Since P is the amount that is borrowed now and A is the amount that must be paid back in the future, P is often referred to as the present value and A as the future value.

Example 2.2.1 • RM1000 is invested for two years in a bank, earning a simple interest rate of 8% per annum. Find the simple interest earned

Exercise 1 • RM5000 is invested for 6 years in a bank, earning a simple interest rate of 5.7 % per annum. Find the simple interest earned • Solution

Example 2.2.2 • RM10000 is invested for 4 years 9 month in a bank earning a simple interest rate of 10% per annum. Find the simple amount at the end of the investment period. • Solution

Exercise 2 • If Bank A offers a simple interest rate of 8 % per annum, Ahmad invested RM 9000 for 4 years 6 months in a bank earning. Find the future value obtain by Ahmad at the end of the investment period. • Anwer

Example 2.2.3 Find the present value at 8% simple interest of a debt amount RM3000 due in ten months. • Solution

Example 3 Find the present value at 6% simple interest with total amount of debt RM 40000 due in 15 years. • Solution

2.3 Compound Interest • Based on the principal which interest changes from time to time. • Interest that is earned is compounded or converted into principal and earns thereafter. • Hence the principal increases from time to time.

Some important terms are best explained with the following example. Suppose RM9000 is invested for 7 years at 12% compounded quarterly Principal, P • The original principal, denoted by P is the original amount invested. Here the principal is P = RM 9000 Annual nominal rate, r • The interest rate for a year together with the frequency in which interest is calculated in a year. Thus the annual nominal rate is given by r = 12% compounded quarterly, that is four times a year.

Frequency of conversions/ number of compounding periods per year, m • The number of times interest is calculated in a year. The annual nominal rate is given by r = 12% compounded quarterly, that is four times a year. In this case, m=4 Interest period • Interest period is the length of time in which interest is calculated. Thus, the interest period is three month

Example 2.3.1 • Find the accumulated amount after 3 years if RM1000 is invested at 8% per year compounded • Annually • Semi-annually • Quarterly • Monthly • Daily

Solution

Example 2.3.2 • RM9000 is invested for 7 years 3 months. This investment is offered 12% compounded monthly for the first 4 years and 12% compounded quarterly for the rest of the period. Calculate the future value of this investment.

Solution

Example 2.3.3 What is the annual nominal rate compounded monthly that will make RM1000 become RM2000 in five years? Solution

2.3.1 Effective Rate of Interest • The effective rate is the simple interest rate that would produce same accumulated amount in 1 year as the nominal rate compounded m times a year. • The effective rate also called the effective annual yield.

Example 2.3.4

Solution

Example 2.3.5 Solution

2.3.2 Present Value The process for finding the present value is called discounting.

Solution RM 16713 should be invested so that we get accumulated RM 20000 after 3 years

Solution

2.3.3 Continuous Compounding of Interest

Example Find the accumulated value of RM1000 for six months at 10% compounded continuously. Solution

3.4 Annuities • Sequence of equal payments made at equal intervals of time. • Examples of annuity are shop rentals, insurance policy premiums, instalment payment, etc. • An annuity in which the payments are made at the end of each payment period is call ordinary annuity certain.

2.4.1 Future Value of Ordinary Annuity Certain • Future value of an ordinary annuity certain is the sum of all the future values of the periodic payments. • If you know how much you can invest per period for a certain time period, the future value of an ordinary annuity formula is useful for finding out how much you would have in the future by investing at your given interest rate. • If you are making payments on a loan, the future value is useful for determining the total cost of the loan.

Example 2.4.1 Find the amount of an ordinary annuity consisting of 12 monthly payments of RM100 that earn interest at 12% per year compounded monthly. Solution

Example 2.4.2 RM 100 is deposited every month for 2 years 7 month at 12% compounded monthly. What is the future value of this annuity at the end of the investment period? Solution

Example Lily invested RM100 every month for five years in an investment scheme. She was offered 5% compounded monthly for the first three years and 9% compounded monthly for the rest of the period. Determine the future value of this annuity at the end of five years and total amount money after 5 years.

Solution

2.4.2 Present Value of Ordinary Annuity Certain • In certain instances, we may try to determine the current value P of a sequence of equal periodic payment that will be made over a certain period of time. • After each payment is made, the new balance continues to earn interest at some nominal rate. • The amount P is referred to as the present value of ordinary annuity certain.

Example 2.4.5 After making a down payment of RM4000 for an automobile, Maidin paid RM400 per month for 36 month with interest charged at 12% per year compounded monthly on the unpaid balance. What was the original cost of the car?

Solution

2.4.3 Amortization • An interest bearing debt is said to be amortized when all the principal and interest are discharged by a sequence of equal payments at equal intervals of time.

Example 2.4.8 A sum of RM50000 is to be repaid over a 5 year period through equal instalments made at the end of each year. If an interest rate of 8% per year is charged on the unpaid balance and interest calculations are made at the end of each year, determine the size of each instalment so that the loan is amortized at the end of 5 years.

Solution

Example Andy borrowed RM120, 000 from a bank to help finance the purchase of a house. The bank charges interest at a rate 9% per year in the unpaid balance, with interest computations made at the end of each month. Andy has agreed to repay the loan in equal monthly instalments over 30 years. How much should each payment be if the loan is to be amortized at the end of the term?

Solution Andy has to pay RM 965.55 per month within 30 years

2.4.4 Sinking Fund • A sinking fund is an account that is set up for a specific purpose at some future date. • For example, an individual might establish a sinking fund for the purpose ofdischarging a debt at a future date. • A corporation might establish a sinking fund in order to accumulate sufficient capital to replace equipment that is expected to be obsolete at some future date.

Financial Mathematics