INTEREST RATES An interest rate is the ‘rental’ price of money. When a resource or asset is borrowed, the borrower pays the lender for the use of it. The interest rate is the price paid for the use of money for a period of time. One type of interest rate is the yield on a bond.When you deposit your money in a bank, you are giving the bank permission to use the money for a specific period of time. You do this in exchange for an expected increase in future income. The expected increase in income (relative to the amount loaned) is the interest rate.In finance, interest has three general definitions.
- Interest is a surcharge on the repayment of debt (borrowed money).
- Interest is the return derived from an investment.
- Interest is the right to claim in a corporation such as that of an owner or creditor.
2.1.1 Simple InterestIn exchange for the use of a depositor’s money, banks pay a fraction of the account balance back to the depositor. This fractional payment is known as interest. The money a bank uses to pay interest is generated by investments and loans that the bank makes with the depositor’s money. Interest is paid in many cases at specified times of the year, but nearly always the fraction of the deposited amount used to calculate the interest is called the interest rate and is expressed as a percentage paid per year.For example a credit union may pay 6% annually on savings accounts. This means that if a savings account contains $100 now, then exactly one year from now the bank will pay the depositor $6 (which is 6% of $100) provided the depositor maintains an account balance of $100 for the entire year. In this chapter and those that follow, interest rates will be denoted symbolically by r. To simplify the formulas and mathematical calculations, when r is used it will be converted to decimal form even though it may still be referred to as a percentage. The 6% annual interest rate mentioned above would be treated mathematically as r = 0.06 per year. The initially deposited amount which earns the interest will be called the principal amount and will be denoted P. The sum of the principal amount and any earned interest will be called the compound amount and A will represent it symbolically. Therefore the relationship between P, r, and A for a single year period isA = P + Pr = P (l + r).The interest, once paid to the depositor, is the depositor’s to keep. Banks and other financial institutions "pay" the depositor by adding the interest to the depositor’s account. Unless the depositor withdraws the interest or some part of the principal, the process begins again for another interest period. Thus two interest periods (think of them as years) after the initial deposit the compound amount would beA = P (l + r) + P (l + r) r = P (1 + r) 2.Continuing in this way we can see that t years after the initial deposit of an amount P, the compound amount A will grow toA = P (l + r)t (1.1)This is known as the simple interest formula. A mathematical "purist" may wish to establish Eq. (1.1) using the principle of induction. Banks and other interest-paying financial institutions often pay interest more than a single time per year. The simple interest formula must be modified to track the compound amount for interest periods of other than one year.
| Example1Find the simple interest earned on $2800 deposited in a bank at 2% annual interest after 5 years.Simple interest is the interest paid only on the original deposit or the principal.The formula used to calculate simple interest is I = ptr, where I is the simple interest, p is the principal, t is the time in years and r is the interest rate per year.The rate of interest is 2% = 2 / 100.[Divide the interest rate by 100.]I = (2800 × 5 × 2100)[Substitute 2800 for p, 5 for t and 2 / 100 for r.]280[Simplify.]The interest after 5 years is $280. |
| Example 2 Marissa deposited $900 in her savings account. The rate of simple interest is 5% per year. Find the balance at the end of 4 years.Interest = p × t × r[Use the simple interest formula.]= 900 × 4 × 0.05[Substitute 0.05 for 5%.]= $180Balance = Interest + Principal = $180 + $900 = $1080. |
| Example 3Find the balance in Mrs. Charlie’s account after 6 years, if she had deposited $11000 in her account 4 years ago. The bank pays a simple interest of 8.4% annually.Total number of years t = 6 + 4 = 10 years.Interest = p x r x t[Use the simple interest formula.]= 11000 x 0.084 x 10 = $9240[Substitute 0.084 for 8.4% and simplify.]Balance = Principal + Interest= 11000 + 9240 = 20240[Substitute the values and add.]So, the balance in Mrs. Charlie’s account after 6 years is $20240. |
| Example 4A credit card company charges a monthly simple interest of 3% on any unpaid debts. Mr. Isla used his wife’s new credit card at the beginning of January this year to get himself a television set worth $300 and at the beginning of February he bought a music system worth $150. Find the balance that Mr. Isla has to pay at the end of February to pay off the debt.= 300 x 0.03 x 1 = $9Interest at the end of January = p x r x t[Use the simple interest formula.][Substitute 0.03 for 3%.]= 300 + 9 = $309Amount outstanding at the end of January = Principal + Interest= 309 + 150 = $459Amount outstanding at the beginning of February = Amount outstanding at the end of January + new credit= 459 x 0.03 x 1 = $13.77Interest at the end of February = p x r x t[Use the simple interest formula.][Substitute 0.03 for 3%.]= 459 + 13.77= $472.77Amount outstanding at the end of February = Principal + Interest[Balance = Principal + Interest] |
2.1.2 Compound InterestThe typical interest bearing savings or checking account will be described an investor as earning a specific annual interest rate compounded monthly. In this section will compare and contrast compound interest to the simple interest case of the previous section. Whenever interest is allowed to earn interest itself, an investment is said to earn compound interest. In this situation, part of the interest is paid to the depositor more than once per year. Once paid, the interest begins earning interest. We will let the number of compounding periods per year be n. For example for interest "compounded monthly" n = 12. Only two small modifications to the simple interest formula (1.1) are needed to calculate the compound interest. First, it is now necessary to think of the interest rate per compounding period. If the annual interest rate is r, then the interest rate per compounding period is r/n. Second, the elapsed time should be thought of as some number of compounding periods rather than years. Thus with n compounding periods per year, the number of compounding periods in t years is nt. Therefore the formula for compound interest isA = P 
nt (1.2)
Example 1Suppose an account earns 5.75% annually compounded monthly. If the principal amount is $3104 then after three and one half years the compound amount will beA = 3104  (12) (3,5) = 3794.15 |
The reader should verify if the principal in the previous example earned a simple interest rate of 5.75% then the compound amount after 3.5 years would be only $3774.88. Thus, happily for the depositor, compound interest builds wealth faster than simple interest. Frequently it is useful to compare an annual interest rate with compounding to an equivalent simple interest, i.e. to the simple annual interest rate which would generate the sample amount of interest as the annual compound rate. This equivalent interest rate is called the effective interest rate. For the amounts and rates mentioned in the previous example we can find the effective interest rate by solving the equation3104 
(12) = 3104(1+ re)1.05904 = 1 + re0.05904 = reThus the annual interest rate of 5.75% compounded monthly is equivalent to an effective annual simple rate of 5.904%. Intuitively it seems that more compounding periods per year implies a higher effective annual interest rate. In the next section we will explore the limiting case of frequent compounding going beyond semi-annually, quarterly, monthly, weekly, daily, hourly, etc. to continuously.
| Example 2A credit card company charges compound interest annually on any unpaid debts at 5%. Mr. B now has outstanding debt of $4200. What will be his outstanding debt 3 years from now?Balance = p x (1 + r)t[Use the compound interest formula.]= 4200 x (1 + 0.05)3[Substitute 0.05 for 5%.]= 4200 x 1.16[Simplify.]= $4872Example 3An amount of $600 is deposited at 3% interest compounded annually. What would be the balance after 3 years?Balance, B = p x (1 + r)t[Use the compound interest formula.]= 600 x (1 + 0.03)3[Substitute p= 600 r = 0.03 and t = 3.]= 600 x 1.09[Cube of 1.03 = 1.09.]= $654 |
| ExercisesSuppose that $3659 is deposited in a savings account which earns 6.5% simple interest. What is the compound amount after five years?Suppose that $3993 is deposited in an account which earns 4.3% interest. What is the compound amount after two years if the interest is compounded? |
2.1.3 Hire purchaseWhen we buy assets on credit, we may enter into hire-purchase agreements with our creditors. A hire-purchase contract is a financing arrangement that enables a person to take possession of an expensive asset while making regular payments on the asset. The asset becomes the legal property of the person only once the asset has been paid for in full. This arrangement has certain cost implications for the person buying the asset on hire purchase. Hire purchase is an expensive exercise in the long term. Normally a deposit is required when buying an asset on hire purchase. Although the instalment of hire-purchase agreements is usually low, the total amount that you eventually pay back is often much higher than the cash price of the asset. We will illustrate this fact in the following example.
| ExampleYou want to buy furniture by means of a hire-purchase agreement. The cash price of the furniture amounts to R10 000. The furniture store requires a 10% deposit on the cash price, and equal payments of R800 per month over24 months. How much do you actually pay for the furniture? How much interest do you pay in total over the 24 months?We have that:total paid = deposit + (instalment × no. of periods)= R1 000 + R800 × 24= R20 200 total interest paid = total paid – cash price= R20 200 – R10 000= R10 200 |
As you can see, the amount of interest you will pay over 24 months is more than the price of the furniture itself!We will examine savings more closely later on in this unit. First, let’s see how to calculate an instalment on a hire-purchase agreement. To avoid paying such excessive amounts of money unnecessarily, we have to avoid buying on credit. It is usually much cheaper for us to buy assets for cash, but not all of us have the money readily at hand. We need to save money in order to buy the things that we need. By saving, we earn interest on our money and watch it increase over time!Calculating instalment payments when buying on creditThe formula for calculating an instalment (S ) when buying on credit is:S =where: A is the initial amount to be borrowedi is the interest rate charged per periodn is the number of time periods over which instalments are to be paidAlthough this formula may look complicated, it becomes more manageable when used in practice a few times. Notice that the interest rate (i) can be given as a yearly, bi-annually, quarterly or monthly rate. This has implications on your number of time periods (n). Make sure that i and n are always consistent. For instance, when your rate is given per month, your time periods must be in months. When your rate is given per quarter, your time period must be four times per year.Look at the following example as an illustration of how to apply the formula for calculating an instalment.
| ExampleSuppose you want to buy a car for R65 000. The authorised motor dealer offers you financing and tells you he will give you the special rate of 1% per month over 54 months. Although you know this is no special rate, you need to know the costs involved. You ask the following questions:1. What is the monthly instalment?2. What is the total I would have paid over 54 months?3. How much interest have I paid in total by the end of the 54 months?We can now work out the answers to these questions ourselves!1. You have the following figures:A = R65 000i = 1% per monthn = 54 monthsYou can use the formula to calculate the monthly instalment:R1 563,68Now you know that your monthly instalment for your new car will beR1 563, 68.2. total paid = instalment (S) × no. of periods (n)= R1 563,68 × 54 months= R84 438,72Therefore, the total amount you will pay for your car worth R65 000 willbe R84 438,72.3. You can now easily calculate the total interest you will have to pay by subtracting the cash price of the car from the total amount that you will be charged.total interest paid = total amount paid – cash price= R84 438,72 – R65 000= R19 438,72 |
As you can see, when you are able to do these calculations, you are able to decide for yourself where you would like to buy your car, and what the real costs involved are.Let’s now return to the subject of saving our money and watching it grow, rather than paying large amounts of interest in credit arrangements!The future value of regular savings.All of us have different reasons for saving. Some of us may be saving to put down a deposit on a home. Others may be saving for their education, or their children’s education. We should all start saving for our retirement early, to ensure a reasonable income in old age. On a lighter note, we may want to save to splash out on a luxury item, such as a new bicycle or a dream holiday to an idyllic island!The best way to save is to put aside some money on a regular basis. Even if it isn’t a lot, regular monthly payments into a savings account or unit trust investment will grow faster and faster as time goes by. It is wise to put aside some money each month as an investment for the future.Whatever our reasons for saving, we will want to know how fast our money is growing and what the increased amount will be after a certain time period. Let’s take a look at the way we can calculate this. Remember, most financial institutions calculate interest on savings (and loans) using compound interest. Therefore, we will use compound interest in our calculations. Remember that this is a positive factor when it comes to savings, as compound interest grows faster than simple interest!The formula we use to calculate the future value (F) of our savings is:F = s ×where: s is the regular savings payment we make each period i is the interest rate we earn on our savingsn is the number of periods over which we saveOnce again, it is important to remember to keep the interest rate (i) and the number of time periods (n) consistent. The time period you use for n depends on how often the interest is compounded. If the interest rate is quoted per year but the interest is compounded monthly, we need to divide the rate by 12 in order to get the monthly rate charged.The following example shows you how to use this formula to calculate the future value of savings.
ExampleYou plan to save R300 per month. Your bank offers a nominal interest rate on savings of 8% p.a. compounded monthly (p.a. stands for per annum, which means per year). How much will your savings be worth after four years?Because you are making your payments on a monthly basis, we will convert the interest rate (i) and the periods (n) to monthly units.A nominal rate of 8% p.a. gives us  = 0,67% per month.A period of 4 years gives us 4 × 12 months = 48 months.We have: s = R300 saved per monthi = 0,67% per monthn = 48 monthsRemember that 0,67% =  = 0,0067By substituting these figures into the formula, we get:F = s ×= R300 × 56, 39616= R16 918,85So, after saving R300 per month for four years or 48 months, we end up with a healthy amount of R16 918,85 when the interest is compounded monthly. |
Let’s consider the negative side of this by calculating the interest we pay on a hire-purchase contract in the example below.
| ExampleGo to a furniture store. Find one item that you like and find out thehire- purchase option of payment. Compare this to the cash price and calculate how much interest you will be paying over the period.Of course each person will have a different answer to this question. However, we’re giving you a model answer to compare your answer against.cash price of TV: R2 999hire-purchase agreement: R300 per month for 12 monthstotal paid = R300 12 months= R3 600interest paid = total paid – cash price= R3 600 – R2 999= R601Therefore, the total amount paid for the TV that is worth R2 999 will beR3 600, because a total of R601 of interest has been charged over 12 months |
After you have worked through Activity, you should be able to: use mathematics to analyse, describe and represent financial situations, and to identify and solve a variety of numerical and financial problems using critical and creative thinking. Identify and solve problems using critical and creative thinking ; and gather, organise, evaluate and interpret financial information to plan and make provision for monitoring budgets and other financial situations.2.1.4 Mortgage loansA mortgage loan is also known as a home loan. Since most of us either dream about owning a home or own a home already, home loans are of interest to us A mortgage loan is a loan that is taken out for the specific purpose of buying fixed property (for example, a piece of land or a dwelling built on it).Most home loans are repaid over a term of 20 years. This is a long time to commit to regular monthly payments and a huge amount of interest is charged by the bank over this period. Therefore, we are very interested in the calculations of interest, monthly repayments and the term (payment period) of a home loan.It is important to understand that when we sign a home loan agreement with a bank, the bank retains the right to seize the house and sell it if we do not make the required monthly repayments. For this reason, too, it is important and useful to understand how the repayments on a mortgage loan are calculated.The formula for calculating the monthly repayment on a mortgage loan is the same as the one we used to calculate instalments on credit purchases. However, we adapt it slightly to suit the purposes of a home loan.Let’s remind ourselves of the formula for calculating instalments on a credit purchase:F = s ×In the case of mortgage loans we have:A = the amount that is borrowed, usually the price of the housei = the interest rate charged per period(Normally the rate must be converted to a monthly interest rate)n = the number of periods over which instalments are to be paid(Normally the periods are months)The formula for calculating the instalment (S) is:S =Let’s now look at the example below where we calculate monthly repayments on a home loan and also demonstrate how we are able to save on interest by paying off our home loan more quickly.
| ExampleYou apply for a mortgage loan of R400 000 to buy a house. The bank grants you the full loan amount of R400 000 at an interest rate of 1% per month, to be paid off over a term of 20 years. It will be very useful for your financial planning to be aware of the following:1. What is your monthly repayment?2. What is the total interest paid by you over the 20-year period?3. What would you have to pay each month to pay off the mortgage loan in 15 years?4. What is the total interest paid by you over the 15-year period?5. How much interest do you save by paying over 15 years rather than 20 years?Before we proceed to answer your questions, we need to make sure that all the variables consistent. The repayments are monthly, so we need to make sure that the interest rate (i) and the number of periods (n) are in monthly units.i = 1% per month (already in % per month)20 years = 20 × 12 months = 240 monthsTherefore, n = 240 in months.1. You have the following figures:A = R400 000i = 1% per monthn = 240 monthsNow the monthly repayment (S) is:S = R4 404, 342. Total paid = monthly instalment × no. of months= R4 404,34 × 240= R1 057 041,60Interest paid = total paid – house price= R1 057 041, 60 – R400 000= R657 041, 60Therefore, the interest you will pay at 1% per month after 20 years for buying a house worth R400 000 is R657 041, 60 – which is more than the value of the house!3. All the variables are the same in this scenario except the term over which the mortgage is repaid. You want to pay it off in 15 years rather than 20 years.Therefore:A = R400 000i = 1% per monthn = 15 years × 12 months = 180 monthsNow the monthly repayment (S) is:S = R4 800,674. Total paid = monthly instalment × no. of months= R4 800,67 × 180= R864 120,60interest paid = total paid – house price= R864 120,60 – R400 000= R464 120,605. interest saved = interest paid over 20 years – interest paid over 15 years= R657 041,60 − R464 120,60= R192 921,00 |
This example illustrates that, by paying R4 800 per month rather than R4 404 per month, you can save R192 921 in interest! You will also have finished paying off your house five years earlier. You can then put the R4 800 per month towards another investment or even a dream holiday!Now try to do some home loan calculations yourself.Imagine you have been granted a mortgage loan of R300 000 with interest charged at 1% per month. Your monthly repayment is R3 300. After the first month’s payment, what part of the R3 300 is used to pay the bank interest and by what amount is the R300 000 home loan reduced?interest = 1% × R300 000= 0,01 × R300 000= R3 000So R3 000 of the R3 300 repayment goes to the bank as an interest charge.Capital reduction = total repayment – interest amount= R3 300 – R3 000= R300Therefore, our home loan of R300 000 is reduced by only R300 to R299 700 after the first month’s repayment.After you have worked through the activity, you should be able to: use mathematics to analyse, describe and represent financial situations, and to identify and solve a variety of numerical and financial problems using critical and creative thinking. Identify and solve problems using critical and creative thinking and communicate effectively by using everyday and mathematical language to describe relationships, processes and problem-solving methods2.1.5 Retirement annuitiesWe should all be saving for our retirement. Although it may seem far away, it takes a long time to build up a savings or investment account that is big enough to support us in our retirement. You could live another 30 years after you’ve retired. You might want to do things you’ve never had the time to do when you were younger. You could also require special medical care. That is why you have to start making provision for your retirement when you are young.The most popular form of investment used to make provision for retirement is a retirement annuity.Retirement annuities are an effective means of providing an income during our later years in life. The size of the regular monthly income payments depends directly on the size of the contributions we can afford. It is very important to remember, then, that we need to save hard now to build up a lump sum large enough to buy a reasonable stream of monthly income payments.SummaryIn this lesson, we have shown you a few examples of the influence that interest can have on your finances. Whenever you borrow or invest money, you will either pay or receive interest. It is therefore important for you to be aware of interest and how and when it is calculated.Saving up for something and earning interest as opposed to buying it on credit and paying interest can make a huge different to the final amount we pay for an item we want or need. It is up to you to decide whether you can wait or not. As long as you are aware of the real price you pay.We have also looked at providing for your retirement. The more you invest now in your retirement, the more money you will have to make life easier for you when you are not earning any more.We trust that this lesson has given you new insight into the workings of borrowing and investing.
| ExampleYou apply for a mortgage loan of R400 000 to buy a house. The bank grants you the full loan amount of R400 000 at an interest rate of 1% per month, to be paid off over a term of 20 years. It will be very useful for your financial planning to be aware of the following:1. What is your monthly repayment?2. What is the total interest paid by you over the 20-year period?3. What would you have to pay each month to pay off the mortgage loan in 15 years?4. What is the total interest paid by you over the 15-year period?5. How much interest do you save by paying over 15 years rather than 20 years?Before we proceed to answer your questions, we need to make sure that all the variables consistent. The repayments are monthly, so we need to make sure that the interest rate (i) and the number of periods (n) are in monthly units.i = 1% per month (already in % per month)20 years = 20 × 12 months = 240 monthsTherefore, n = 240 in months.1. You have the following figures:A = R400 000i = 1% per monthn = 240 monthsNow the monthly repayment (S) is:S = R4 404, 342. Total paid = monthly instalment × no. of months= R4 404,34 × 240= R1 057 041,60Interest paid = total paid – house price= R1 057 041, 60 – R400 000= R657 041, 60Therefore, the interest you will pay at 1% per month after 20 years for buying a house worth R400 000 is R657 041, 60 – which is more than the value of the house!3. All the variables are the same in this scenario except the term over which the mortgage is repaid. You want to pay it off in 15 years rather than 20 years.Therefore:A = R400 000i = 1% per monthn = 15 years × 12 months = 180 monthsNow the monthly repayment (S) is:S = R4 800,674. Total paid = monthly instalment × no. of months= R4 800,67 × 180= R864 120,60interest paid = total paid – house price= R864 120,60 – R400 000= R464 120,605. interest saved = interest paid over 20 years – interest paid over 15 years= R657 041,60 − R464 120,60= R192 921,00 |
| ExerciseTest your knowledge of this lesson by completing the self-assessment questions below. When you answer the questions, don’t look at the suggested answers that we give. Look at them only after you’ve written your answers and then compare your answers with our answers.1. What is the future value of an investment of R5 000 if you earn simple interest at a rate of 12% per year for 14 years?2. What is the future value of an investment of R5 000 if you earn compound interest at a rate of 12% per year for 14 years?3. Suppose you want to buy furniture by means of a hire-purchase agreement. The cash price of the furniture amounts to R10 000. The furniture store calculates instalments over 24 months and charges 2% interest per month. Calculate your monthly instalment and the total you will pay over the 24 months.4. You want to start saving for a deposit on a house. You decide to invest R400 in a unit trust account every month. If the unit trust earns 1% per month compounding, how much money will be saved at the end of three years? (Ignore the costs normally charged by unit trust companies.)5. Why do we need to save for retirement now to be able to invest in a retirement annuity on retirement?6. A bank grants you a R500 000 loan to buy a house. It charges an effective interest rate of 0,8% per month on the loan. If you are planning to repay the loan over 20 years, what will your monthly repayment be?Answers to exercise1. A = R5 000i = 12%n = 14 yearsThe future value (F) after n time periods using simple interest is:F = A + (A × i × n)= R5 000 + (R5 000 × 0,12 × 14)= R5 000 + R8 400= R13 400Therefore, the future value using simple interest will be R13 400.2. A = R5 000i = 12%n = 14 years |
|
Tsakani Stella Rikhotso | Monitoring & Evaluation OfficerSayProWebsite: www.saypro.onlineCell: 27 (0) 713 221 522Email: tsakaniStudy and Qualifications www.saypro.onlineOur Company www.saypro.online |
SayPro is a group of brands leading in Africa’s development, building innovative online solutions and a strategic institution on child growth, youth empowerment and adult support programmes, applications in Africa.SayPro Core Skills and Expertise:
- SayPro Artificial Intelligence, Graphics, Online Design and Web Development
- SayPro Higher Education, Certification, eLearning Development, Qualification Design and Online Training.
- SayPro Community Development in Youth Unemployment, HIV/AIDS, Human Rights and Gender-Based Violence
- SayPro Company Registrations, Tax, VAT and Website Designs.
- SayPro Research, Opportunity Sharing in Tenders, Funding and Contact Directories.
- SayPro Monitoring, Evaluation, Knowledge Management, Learning and Sharing.
SayPro is providing international and global opportunities for African youth. Partner with SayPro now by sending an email to info or give us a call at + 27 11 071 1903Please visit our website at www.saypro.online Email: info@saypro.online Email: info@saypro.online Call: + 27 11 071 1903 WhatsApp: + 27 84 313 7407. Comment below for any questions and feedback. For SayPro Courses, SayPro Jobs, SayPro Community Development, SayPro Products, SayPro Services, SayPro Consulting, and SayPro Advisory visit our website to www.saypro.online[attach 14]
Leave a Reply
You must be logged in to post a comment.