Chapter 3 Financial Mathematics BQT 133
2.1 Introduction
In this chapter we will study the financial mathematics problem involving the interest problem, annuity and depreciation. The objective of this chapter is to examine those procedures which used to answer questions associated with major financial transactions. These procedures are part of what is usually financial mathematics.
Most important financial transactions, such as repaying housing loan, involve a series of repayments. Before we learn how interest is charged on a series of payments, we must look at how interest is charged on a single payment. We will look both simple and compound interest in section 2.2 and 2.3. We will also look at what is called the effective interest rate.
To allows customers greater flexibility in the way in which they repay loans, financial institutions use a procedure called continuous compounding to calculate interest payments.
In section 2.4 we have a series of regular payments made at the end of each period and who’s compounding and payment periods coincide, this called an ordinary annuity. In this section, the formulae for the future and present value of an ordinary annuity have been discussed.
Section 2.5 examines the three procedures commonly used to calculate the depreciation on particular capital assets.
2.2 Simple Interest
The study of interest is very important and fundamental to the understanding of the economy of a country. For instance, when interest fall stock prices will increase and when interest rates increase, stock price will fall. Interest is money earned when money is invested or interest is charge incurred when a loan or credit is obtained. Interest usually expressed as per cent per annum. Simple interest is the interest calculated on the original principal for the entire period it is borrowed or invested.
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, then P is call principal. Usually we agree to repay this amount plus an extra amount. These extra amounts, which pay to the lender for the convenience of using lender money is called interest. Simple interest formula is given by the following formula:
In general, if a principal P is borrowed at a rate r, then 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. The formula relating A and P is as follows:
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.
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.
Example 2.2.3
Find the present value at 8% simple interest of a debt RM3000 due in ten months.
2.3 Compound Interest
Compound interest is based on the principal which 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. The formula to compute the amount of compound interest is given below
Some important terms
Some of the common terms used in relation to compound interest are:
· Principal, P
· Annual nominal rate, r
· Interest period or conversion period
· Frequency of conversions, m
· Periodic interest rate, i
· Number of interest periods in the investment period, n
These terms are best explained with the following example. Suppose RM9000 is invested for seven years at 12% compounded quarterly
Principal, P
The original principal, denoted by P is the original amount invested. Here the principal is P = RM9000
Annual nominal rate, r
Annual nominal rate denoted by r is 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
Interest period
Interest period is the length of time in which interest is calculated. Thus, the interest period is three month
Frequency of conversions, m
Frequency of conversion denoted by m is the number of times interest is calculated in a year. In other words, it is the number of interest periods in a year. In this case, m=4
Periodic interest rate, i
Periodic interest rate denoted by i is the interest rate for each interest period. Thus, the periodic interest rate in this case is given by
Number of interest periods in the investment period, n
The number of interest periods during the investment period denoted by n is the number of times interest is calculated. The number of interest periods is given by. Thus, .
Example 2.3.1
Find the accumulated amount after 3 years if RM1000 is invested at 8% per year compounded
a) Annually
b) Semi-annually
c) Quarterly
d) Monthly
e) Daily
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.
Example 2.3.3
What is the nominal rate compounded monthly that will make RM1000 become RM2000 in five years?
2.3.1 Effective Rate of Interest
Interest actually earned on an investment depends on the frequency with which the interest is compounded. Thus the nominal rate does not reflect the actual rate at which interest is earned. Now we need to find common basis for comparing interest rates. One such way of comparing interest rates is provided by the use of the effective rate. 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. The formula for computing the effective rate of interest is given below
Example 2.3.4
Find the effective rate of interest corresponding to a nominal rate of 8% per year compounded
f) Annually
g) Semi-annually
h) Quarterly
i) Monthly
j) Daily
Example 2.3.5
Kamal wishes to borrow some money to finance some business expansion. He has received two different quotes:
Bank A: charges 15.2% compounded annually
Bank B: charges 14.5% compounded monthly
Which bank provides a better deal?
2.3.2 Present Value
The present value at i% per interest period of an amount, A due in n interest periods is that value, P which will yield the sum A at the same interest rate after n interest periods. Hence from, we get
The process for finding the present value is called discounting.
Example 2.3.6
How much money should be deposited in a bank paying interest at the rate of 6% per year compounded monthly so that, at the end of 3 years, the accumulated amount will be RM20000?
Example 2.3.7
Find the present value of RM49, 158.60 due in 5 years at an interest of 10% per year compounded quarterly.
Example 2.3.8
A debt of RM3000 will mature in three years time. Find
a) The present value of this debt
b) The value of this debt at the end of the first year
c) The value of this debt at the end of four years
Assuming money is worth 14% compounded semi-annually
2.3.3 Continuous Compounding of Interest
Thus far, we have been discussing compounding of interest on discrete time intervals (daily, monthly, etc). If compounding of interest is done on a continuous basis the formula is
Example 2.3.9
Find the accumulated value of RM1000 for six months at 10% compounded continuously.
Example 2.3.10
Find the amount to be deposited now so as to accumulate RM1000 in eighteen months at 10% compounded continuously.
2.4 Annuities
Annuity is a sequence of equal payments made at equal intervals of time. Example of annuity are shop rentals, insurance policy premiums, annual dividends received, instalment payment, etc. In this section we shall mainly discuss ordinary annuity certain. 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. Formula for calculate the future value of ordinary annuity certain is
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.
Example 2.4.2
RM100 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?
Example 2.4.3
RM100 was invested every month in an account that pays 12% compounded monthly for two years. After the two years, no more deposit was made. Find the amount of the account at the end of five years.
Example 2.4.4
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. Find the future value at the end of five years.
2.4.2 Present Value of Ordinary Annuity Certain
In certain instances, you may want 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. Formula to calculate this present value is given below
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?
Example 2.4.6
As a savings program toward Alfian college education, his parents decide to deposit RM100 at the end of every month into a bank account paying interest at the rate of 6% per year compounded monthly. If the saving program began when Alfian was 6 years old, how much money would have accumulated by the time he turn 18?
Example 2.4.7
Ray has to pay RM300 every month for twenty-four months to settle loan at 12% compounded monthly.
a) What is the original value of the loan?
b) What is the total interest that he has to pay?
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.
Example 2.4.9
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?
2.4.4 Sinking Fund
Sinking funds are another important application of the annuity formulas. Simply stated, 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 of discharging 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. The formula for finding the sinking fund is given below
Example 2.4.10
A debt of RM1000 bearing interest at 10% compounded annually is to be discharged by the sinking fund method. If five annual deposits are made into a fund which pays 8% compounded annually,
a) Find the annual interest payment
b) Find the size of the annual deposit into the sinking fund
c) What is the annual cost of this debt?
Example 2.4.11
The proprietor of Carling Hardware has decided to set up a sinking fund for the purpose of purchasing a truck in 2 years time. It is expected that the truck will cost RM30000. If the fund earns 10% interest per year compounded quarterly, determine the size of each quarterly instalment the proprietor should pay into the fund.
Example 2.4.12
Harris, a self employed individual who is 46 years old, is setting up a defined benefit retirement plan. If he wishes to have RM250000 in this retirement account by age 65, what is the size of each yearly instalment he will be required to make into a saving account earning interest at % yr?
2.5 Depreciation
Depreciation is an accounting procedure for allocating the cost of capital assets, such as buildings, machinery tools and vehicles over their useful life. It is important to note that depreciation amount are estimates and no one can estimate such amounts with certainty. Depreciation expenses allow firms to recapture the amount of money to provide for replacement of the assets and to recover the original investments. Depreciation can also be viewed as decline in value of assets because of age, wear or decreasing efficiency. Many properties such as buildings, machinery, vehicles and equipment depreciate in value as they get older.