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LECTURE NOTES ON Mathematics of Finance Financial Institution Department Edited & Prepared by Dr. Mary Rafik 5 th Edition (2021)

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LECTURE NOTESON

Mathematics of Finance

Financial Institution Department

Edited & Prepared byDr. Mary Rafik

5th Edition (2021)

LECTURE NOTESON

Mathematics of Finance

Financial Institution Department

Edited & Prepared byDr. Mary Rafik

3rd Edition (2019)

Across the Disciplines WHY MATHEMATICS OF FINANCE MATTERS TO YOU…

Finance: need to understand Mathematics of Finance as Financial managers and investors are always confronted with opportunities to earn positive rates of return on their funds, whether through investment in attractive projects or in interest- bearing securities or deposits.

Accounting: need to understand Mathematics of Finance in order to account for certain transactions such as loan amortization, lease payments, and bond interest rates.

Information systems: need to understand Mathematics of Finance in order to design systems that optimize the firm’s cash flow.

Management: need to understand Mathematics of Finance so that you can plan cash collections and disbursements, in a way that will enable the firm to get the greatest value of its money.

Marketing: need to understand Mathematics of Finance because funding of new programs and products must be justified financially using all the financial math concepts and techniques.

Chapter One: The Concept of Time Value of Money ......................1

Chapter Two: The Concept of Interest rate ………………………… 12

Chapter Three: The Simple Interest......................................................17

Chapter Four: Bank Discount and the Promissory Notes …… 31

Chapter Five: The Compound Interest …………………………………… 42

Chapter six: Annuities and Loans ……………………………………………… 53

F Learning Outcomes:

÷ Knowledge and understanding� Appreciate the role Financial math and their contribution

to the attainment of corporate and bankers’ goals;� Explain What is Financial math;� Introduce the concept of Simple interest;� Introduce the concept of Interest-Bearing Note;� Introduce the concept of Discount;� Introduce the concept of Compound interest.

÷ Practical/ Professional skills� Application of the concepts of financial mathematics;� Make use of the techniques of financial mathematics;� The uses of financial mathematics concepts and techniques on

managerial decision making.

÷ Cognitive skills

� Apply appropriate knowledge, analytical techniques and concepts to problems and issues arising from both familiar and unfamiliar situations;

� Locate, extract and analyze data and information from a variety of different sources;

� Synthesize and evaluate data and information from multiple sources;

� Think critically, examine problems and issues from a number of perspectives, Challenge viewpoints, ideas and concepts, and make well-reasoned judgements.

÷ Key transferable skills� Ability to evaluate research and a variety of types of

information and evidence critically;� Ability to utilize problem solving skills;� Ability to analyze, evaluate and interpret evidence critically;� Ability to apply skills in communication and presentation;� Ability to apply numeracy and quantitative skills;� Ability to conduct research into business and

management issues.

Chapter 1

The Concept of Time Value of Money

The Basic Concept of Time Value of Money:Money has time value. A pound today is more valuable than a year hence. It is on this concept “the time value of money” is based. The recognition of the time value of money and risk is extremely vital in financial decision making.Most financial decisions such as the purchase of assets or procurement of funds, affect the firm’s cash flows in different time periods. For example, if a fixed asset is purchased, it will require an immediate cash outlay and will generate cash flows during many future periods. Similarly, if the firm borrows funds from a bank or from any other source, it receives cash and commits an obligation to pay interest and repay principal in future periods. The firm may also raise funds by issuing equity shares. The firm’s cash balance will increase at the time shares are issued, but as the firm pays dividends in future, the outflow of cash will occur. Sound decision-making requires that the cash flows which a firm is expected to give up over period should be logically comparable. In fact, the absolute cash flows which differ in timing and risk are not directly comparable. Cash flows become logically comparable when they are appropriately adjusted for their differences in timing and risk. The recognition of the time value of money and risk is extremely vital in financial decision-making. If the timing and risk of cash flows is not considered, the firm may make decisions which may allow it to miss its objective of maximizing the owner’s welfare. The welfare of owners would be maximized when Net Present Value is created from making a financial decision. It is thus, time value concept which is important for financial decisions.Thus, we conclude that time value of money is central to the concept of finance. It recognizes that the value of money is

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different at different points of time. Since money can be put to productive use, its value is different depending upon when it is received or paid. In simpler terms, the value of certain amount of money today is more valuable than its value tomorrow. It is not because of the uncertainty involved with time but purely on account of timing. The difference in the value of money today and tomorrow is referred as time value of money.

Reasons for Time Value of Money:Money has time value because of the following reasons:1. Risk and Uncertainty: Future is always uncertain and risky.

Outflow of cash is in our control as payments to parties are made by us. There is no certainty for future cash inflows. Cash inflows are dependent out on our Creditor, Bank etc. As an individual or firm is not certain about future cash receipts, it prefers receiving cash now.

2. Inflation: In an inflationary economy, the money received today, has more purchasing power than the money to be received in future. In other words, a pound today represents a greater real purchasing power than a pound a year hence.

3. Consumption: Individuals generally prefer current consumption to future consumption.

4. Investment opportunities: An investor can profitably employ a pound received today, to give him a higher value to be received tomorrow or after a certain period of time.

Thus, the fundamental principle behind the concept of time value of money is that, a sum of money received today, is worth more than if the same is received after a certain period of time. For example, if an individual is given an alternative either to receive10,000 L.E. now or after one year, he will prefer 10,000 L.E. now. This is because, today, he may be in a position to purchase more

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goods with this money than what he is going to get for the same amount after one year.Thus, time value of money is a vital consideration in making financial decision. Let us take some examples:

Example: A project needs an initial investment of \$1,00,000. It is expected to give a return of \$20,000 per annum at the end of each year, for six years. The project thus involves a cash outflow of \$1,00,000 in the ‘zero year’ and cash inflows of \$20,000 per year, for six years. In order to decide, whether to accept or reject the project, it is necessary that the Present Value of cash inflows received annually for six years is ascertained and compared with the initial investment of \$1,00,000.The firm will accept the project only when the Present Value of cash inflows at the desired rate of interest exceeds the initial

investment or at least equals the initial investment of ≤ 1,00,000.

Example: A firm has to choose between two projects. One involves an outlay of \$ 1,000,000 with a return of 12% from the first year onwards, for ten years. The other requires an investment of \$1,000,000with a return of 14% per annum for 15 years commencing with the beginning of the sixth year of the project. In order to make a choice between these two projects, it is necessary to compare the cash outflows and the cash inflows resulting from the project. In order to make a meaningful comparison, it is necessary that the two variables are strictly comparable. It is possible only when the time element is incorporated in the relevant calculations. This reflects the need for comparing the cash flows arising at different points of time in decision-making.

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Timelines and Notation:When cash flows occur at different points in time, it is easier to deal with them using a timeline. A timeline shows the timing and the amount of each cash flow in cash flow stream. Thus, a cash flow stream of \$10,000 at the end of each of the next five years can be depicted on a timeline like the one shown below.

Part A

0 12%1

12% 2 12% 3 12% 4 12% 510.000 10.000 10.000 10.000 10.000

Part B

0 12%1

12% 2 12% 3 12% 4 12% 5

10.000 10.000 10.000 10.000 10.000

As shown above, 0 refers to the present time. A cash flow that occurs at time 0 is already in present value terms and hence does not require any adjustment for time value of money. You must distinguish between a period of time and a point of time. Period 1 which is the first year is the portion of timeline between point 0 and point 1. The cash flow occurring at point 1 is the cash flow that occurs at the end of period 1. Finally, the discount rate, which is 12 percent in our example, is specified for each period on the timeline and it may differ from period to period. If the cash flow occurs at the beginning, rather than the end of each year, the timeline would be as shown in Part B. Note that a cash flow occurring at the end of the year 1 is equivalent to a cash flow occurring at the beginning of year 2. Cash flows can be positive or negative. A positive cash flow is called a cash inflow; and a negative cash flow, a cash outflow.

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We begin our discussion of valuing cash flows lasting several periods with some basic vocabulary and tools. We refer to a series of cash flows lasting several periods as a stream of cash flows. We can represent a stream of cash flows on a timeline, a linear representation of the timing of the expected cash flows. Timelines are an important first step in organizing and then solving a financial problem.

Constructing a Timeline:To illustrate how to construct a timeline, assume that a friend owes you money. He has agreed to repay the loan by making two payments of \$10,000 at the end of each of the next two years. We represent this information on a timeline as follows:

Date 0 represents the present. Date 1 is one year later and represents the end of the first year. The \$10,000 cash flow below date 1 is the payment you will receive at the end of the first year. Date 2 is two years from now; it represents the end of the second year. The \$10,000 cash flow below date 2 is the payment you will receive at the end of the second year.

Identifying Dates on a Timeline:To track cash flows, we interpret each point on the timeline as a specific date. The space between date 0 and date 1 then represents the time period between these dates—in this case, the first year of the loan. Date 0 is the beginning of the first

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year, and date 1 is the end of the first year. Similarly, date 1 is the beginning of the second year, and date 2 is the end of the second year. By denoting time in this way, date 1 signifies both the end of year 1 and the beginning of year 2, which makes sense since those dates are effectively the same point in time.

Distinguishing Cash Inflows from Outflows:In this example, both cash flows are inflows. In many cases, however, a financial decision will involve both inflows and outflows. To differentiate between the two types of cash flows, we assign a different sign to each: Inflows (cash flows received) are positive cash flows, whereas outflows (cash flows paid out) are negative cash flows.To illustrate, suppose you have agreed to lend your brother \$10,000 today. Your brother has agreed to repay this loan in two installments of \$6000 at the end of each of the next two years. The timeline is

Notice that the first cash flow at date 0 (today) is represented as –\$10,000 because it is an outflow. The subsequent cash flows of \$6000 are positive because they are inflows.

Representing Various Time Periods:So far, we have used timelines to show the cash flows that occur at the end of each year. Actually, timelines can represent cash flows that take place at any point in time. For example, if you pay rent each month, you could use a timeline such as the one in our

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first example to represent two rental payments, but you would replace the “year” label with “month.”Many of the timelines included in this chapter are very simple. Consequently, you may feel that it is not worth the time or trouble to construct them. As you progress to more difficult problems, however, you will find that timelines identify events in a transaction or investment that are easy to overlook. If you fail to recognize these cash flows, you will make flawed financial decisions. Therefore, approach every problem by drawing the timeline as we do in this chapter.

Cash Flow and Equivalence:

Cash flow:The sums of money recorded as receipts or disbursements in a project's financial records are called cash flows. Examples of cash flows are deposits to a bank, dividend interest payments, loan payments, operating and maintenance costs, and trade -in salvage on equipment. Whether the cash flow is considered to be a receipt or disbursement depends on the project under consideration. For example, interest paid on a sum in a bank account will be considered a disbursement to the bank and a receipt to the holder of the account.Because of the time value of money, the timing of cash flows over the life of a project is an important factor. Although they are not always necessary in simple problems (and they are often unwieldy in very complex problems), cash flow diagrams can be drawn to help visualize and simplify problems that have diverse receipts and disbursements.So, we can say that Cash Flow is a representation of all Money Out and all Money In throughout a specified period.

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Cash Flow Table:Time Period Cash Flow (\$)

0 -1001 +402 +403 +40

0: Represent Now1: End of First Period Which is at the same Time the beg. Of 2nd

period.2: End of Second Period Which is at the same Time the beg. Of

3nd period.3: End of Third Period

Minus Sign: Indicates disbursement of money (Giving a WayMoney)Positive Sign: Indicates receipt of money (Gaining Money)

The above Cash Flow Table represents the Payment of 100 \$ now.Receipt of 40 \$ al the end of 1st, 2nd and 3ed periods.The above Cash Flow Table Could be presented in different Way, Cash Flow diagram as Follows.

40 40 40

0 1 2 3

100

Downward Arrows Indicate Disbursement of Money Upward Arrows indicate receipt of money.

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For example, a simple problem might be to portray the year-by-year consequences of purchasing a used car as shown in the following:

Year Cash Flow

Beginning of first - 4500 Car purchased "now" forYear (Year 0) \$4500 cash. The minus sign

indicates a disbursement ofmoney

End of year 1

End of year 2 Maintenance Costs are

End of year 3 \$350 per Year

End of year 4

The Car is sold at the end

of the 4th year for \$2000.The Plus sign represents areceipt of money.

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The Same Cash Flow May Be Represented Graphically:

Receipt of money\$2000

0 1 2 3 4

\$350 \$350 \$350 \$350

\$4500Disbursement of money

The upward arrow represents a receipt of money, and the downward arrows represent disbursements. The X-axis represents the passage of time.

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Chapter 2

The Concept of Interest rate

The Concept of Interest Rate:Interest is the compensation one gets for lending a certain asset. For instance, suppose that you put some money on a bank account for a year. Then, the bank can do whatever it wants with that money for a year. To reward you for that, it pays you some interest.The asset being lent out is called the capital. Usually, both the capital and the interest is expressed in money. However, that is not necessary. For instance, a farmer may lend his tractor to a neighbor, and get 10% of the grain harvested in return. In this course, the capital is always expressed in money, and in that case, it is also called the principal.Perhaps the easiest way to build wealth is to use money to earn interest. If you can earn a good rate of interest, compounded continuously, and keep the investment for a long time, it is amazing how large an investment can grow. In this chapter, we will discuss this important money-making tool: interest.When money is borrowed, the lender expects to be paid back the amount of the loan plus an additional charge for the use of the money. This additional charge is called interest. When money is deposited in a bank, the bank pays the depositor for the use of the money. The money the deposit earns is also called interest.Interest can be computed in two ways: either as simple interestor as compound interest.

Simple interest:Interest is the reward for lending the capital to somebody for a period of time. There are various methods for computing the interest. As the name implies, simple interest is easy to

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understand, and that is the main reason why we talk about it here. The idea behind simple interest is that the amount of interest is the product of three quantities: the rate of interest, the principal, and the period of time.

Example: Suppose you put £1000 in a savings account paying simple interest at 9% per annum for one year. Then, you withdraw the money with interest and put it for one year in another account paying simple interest at 9%. How much do you have in the end?

Answer: In the first year, you would earn 1 · 0.09 · 1000 = 90 pounds in interest, so you have £1090 after one year. In the second year, you earn 2 · 0.09 · 1000 = 180 pounds in interest, so you have £1180 at the end of the two years.

Compound interest:Most bank accounts use compound interest. The idea behind compound interest is that in the second year, you should get interest on the interest you earned in the first year. In other words, the interest you earn in the first year is combined with the principal, and in the second year you earn interest on the combined sum.What happens with the example from the previous section, where the investor put £1000 for two years in an account paying 9%, if we consider compound interest? In the first year, the investor would receive £90 interest (9% of £1000). This would be credited to his account, so he now has £1090. In the second year, he would get £98.10 interest (9% of £1090) so that he ends up with £ 1188. 10;. The capital is multiplied by 1.09 every year: 1.09 · 1000 = 1090 and 1.09 · 1090 = 1188.1.

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Comparing Simple and Compound Interest:

2.5

2

1.5

1

0.5

2 4 6 8 100

time (in years)

2.5

2

1.5

1

0.5

0

0

….. simple

___ compound

51015 interest rate (%)

Comparison of simple interest and compound interest. The left figure plots the growth of capital in time at a rate of 9%. The right figure plots the amount of capital after 5 years for various interest rates.The left plot shows how a principal of 1 pound grows under interest at 9%. The dashed line is for simple interest and the solid curve for compound interest. We see that compound interest pays out more in the long term. A careful comparison shows that for periods less than a year simple interest pays out more, while compound interest pays out more if the period is longer than a year.A capital of £1000, invested for half a year at 9%, grows to

£1045 under simple interest and to £1044.03 under compound interest, while the same capital invested for two years grows to £1180 under simple interest and £1188.10 under compoundinterest. The difference between compound and simple interest get bigger as the period gets longer.

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As Albert Einstein once remarked, “I don’t know what the seven wonders of the world are, but I know the eighth – the compound interest. You may be wondering why your ancestors did not display foresight. Hopefully, you will show concern for your posterity.”

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Chapter 3

Simple Interest

Simple Interest:Simple interest is computed by finding the product of theprincipal (the amount of money on deposit), the rate of interest (usually written as a decimal), and the time (usually expressed in years).The Sum of Money borrowed in an interest transaction is called the principal. The amount of dollars received by the borrower is the present value.The Time is the period during which the borrower has the use of all or part of the borrowed money.The percentage of a Simple interest is expressed as an interest rate and is a fixed portion of the principal.So, Simple Interest is defined as the product of principal, rate, and time, This definition leads to the simple Interest formula

I = P r t

I = simple Interest in dollars (or another monetary unit)P = Principal in dollars (or other monetary unit consistent with

Interest)r = Interest rate or percent of the principal that is to be paid per

unit of time.i = time in units that correspond to the rate

N.B. The interest rate and time must be consistently stated. That is, if the rate is an annual rate, the time must be stated in years, or if the rate is a monthly rate, the time must be stated in months, before signing a contract, a person should be sure that the time interval associated with an interest rate is stated in writing. In practice, simple interest transactions are often made on a yearly or

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per annum basis. To avoid repetition in this text, rates are annual rates unless the problem states otherwise.

Simple Interest Rate:

Time period Cash Flow

+ P + P * I = P (1+i)

- P

P (1+i) + Pi = P + P i + P i= P (1+2i)

P (1+2i) + Pi = P +2 P i + P i= P (1+3i)

\ For n PeriodsS or (F) = P (1 + I n)

Where:S: Is The single amount of money That one gets at the end of (n)

Periods.P: Is The single amount of money That is disbursed at Present.N: Is The number of periods.

Simple Interest Applications:The first experience most people have with simple Interest is atime deposit in a bank. A time deposit is money held in the bank account of a person or firm for which the bank can require advance notice of withdrawal. There are three types of time deposits: savings accounts, open accounts, and certificates of deposit.

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Computing Simple Interest for Days:January 31February 28/29 (Leap Year)March 31April 30May 31June 30July 31August 31September 30October 31November 30December 31

N.B. The leap year is that which number divided 4 equal a number with no decimals.

For Example;� 2012˜ 4 = 503\2012 is a leap year

� 2016˜ 4 = 504\2016 is a leap year

� 2000˜ 4 = 500\2000 is a leap year

�2013˜ 4 = 503.25 \2013 is a simple year

�2017˜ 4 = 504.25 \2017 is a simple year

�2015˜ 4 = 503.75 \2015 is a simple year

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Example:Find the simple interest on \$8600 for 3 months at 8%.

Solution:Substituting P = 8600\$, r = .08, and t =3/12

I = P *r * tI = 8600¥ 0.08¥3/12

= \$ 172

Example:A bank pays 8% per annum on savings accounts. A man opens an account with a deposit of \$ 300 on March 1. How much interest will that man receive on June 1?

Solution:Since the rate is an annual rate, the time must be expressed in

years. The time is 3 months or year. We also have P =300 and r = 80. Substituting these values, we find that:

I = P * r * t

I=300¥0.08¥

= \$ 6.00

Example:A married man and a woman bought a studio and got a loan for \$50,000. The bank annual interest rate is 12%. The term of the loan is 30 years, and the monthly payment is \$514.31. At the end of the month, one spouse says, Dear, I am going down to make the first payment on our new studio. Find the interest for the first month and the amount of house purchased with the first payment.

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Solution:

Substituting P = 50,000, r = .12, and t = , we have:I= P * r * t

I = 50,000¥ 0.12¥= \$ 500.00

Although the studio owning eliminates rent on property, the rent on borrowed money can be Substantial. In this case the \$ 514.31 Payment buys only 514.31 – 500.00 = \$ 14.31 worth of studio

Amount:The sum of the principal and the interest is called the amount, designated by the symbol S. This definition leads to the formula

S=P+I= P + P r t

Factoring, we have

S = P (1 + r t)

Example:Find the simple interest on 750\$ for 2 months at 7% and the amount.

Solution:Substituting P = 750\$, r = .07, and t =2/12

I= P *r * tI= 750¥ 0.07¥2/12 = \$8.75S=P+IS= 750 + 8.75S= 758.75\$

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Example:A lady borrows \$30,000 to buy a mobile home, the interest rate is 12% and the monthly payment is 308.59\$. How much of the first payment goes to the interest and how much goes to the principal?

Solution:I= P r tI= 30,000 * 0.12 * 1/12I= 300\$S=P+I308.59= P + 300P= 8.59\$

Example:A young woman borrows \$350 for 6 months at 15%. What amount must she repay?

Solution:

Substituting P = 350, r = .15, and t = in formula, we find that:S = P (1 + r t)

S=350(1+0.15¥ )= 350 (1.075)= \$ 376.25

N.B. You can solve many problems in the mathematics of finance by more than one method. Look for the easiest way, there by reducing both labor and the risk of numerical errors. Working a problem more than one way is often desirable as a check.

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Exact and Ordinary Interest:When the Time is in days and the rate is an annual rate, it is necessary to convert the days to a fractional part of a year when Substituting in the simple Interest formulas. Interest Computed using a divisor of 360 is called ordinary Interest. When the divisor is 365 or 366, the result is known as exact interest.

Example:Find the ordinary and exact interest on \$6080.50 for 60 days at 12.5%.

Solution:P= 6080.5\$ t=60 days r= 12.5%Ordinary Interest: 6080.5 * 60/360 * 12.5/100= 126.67\$Exact Interest: 6080.5 * 60/365 * 12.5/100= 124.94\$

Example:Figure the ordinary and exact interest on a 60-day loan of \$300 if the rate is 15%.

Solution:Substituting P = 300 and r = .15, in formula (1), We have:Ordinary Interest = 300¥ 0.15¥

= \$ 7.50

Exact Interest = 300¥ 0.15¥= \$ 7.40

Exact and Approximate Time:There are two ways to compute the number of days between calendar dates. The more common method is the exact method, which includes all days except the first.

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The approximate method is based on the assumption that all the full months contain 30 days.

Example:Find the exact and approximate time between March 5 and September 28.

Solution:MarchApril 30May 31June 30July 31August 31September 28

To get the approximate time, we count the number of months from March 5 to September 5. Then we can figure that 6¥30=180 days. To this result we add the 23 days from September 5 to September 28, for a total of 203 days.

Commercial Practice:Since we have exact and ordinary interest and exact and approximate time, there are four ways to compute Simple Interest.

1- Ordinary interest and exact time (Bankers Rule).2- Exact interest and exact time.3- Ordinary interest and approximate time.4- Exact interest and approximate time.

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Example:The interest paid on a loan of \$500 for 2 months was \$12.50. What was the interest rate?

Solution:When a rate is required, ordinarily we get it correct to the nearest hundredth of 1%. We substitute I = 12.50, P = 500, and t

= , or in formula (P* r* t), It makes no difference whichmember of an equation is to the left of the equation is to the left of the equal sign.

50,000¥ r¥ = 12.50

r = = 0.15 = 15%

Present Value at Simple Interest:To find the amount of a principal invested at Simple Interest, We use the formula S = P (1 + r t). If we know the amount and want to obtain the principal, we solve the formula and want to obtain the principal, we solve the formula for P.

P =

Example:An investor gets \$36.75 every 6 months from an investment that pays 6% interest. How much money is invested?

Solution:

Substitute I = 36.75, r = .06, and t = in formula (1), We obtain:

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P¥ 0.06¥ = 36.75

P = = \$ 2125.00

Example:If money is worth 5%, What is the present value of \$105 due in 1 year?

Solution:Substituting S = 105, r = .05, and t = 1, We find that:

P =

= \$ 100.00

Example:An investor can discharge an obligation by paying wither \$200

now. Or \$208 in 6 months. If money is worth 5 % to her, what is the cash equivalent of choosing the better plan?

Solution:200 CashOR208 in 6 months, r = 5.5 % \P1=200

P = =

= = 202.433

\ The First offer is better and he can save 2.433 (202.433 –200)

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Partial Payments:First Step: Find the amount of the loan for the term of the

loan S = P + 1 = P + P r t = P (1 + r t)

Second Step: Find the amount of each payment from the date it is made to the date of maturity (due date). S1, S2, S3

Third Step: The Balance (The remaining of the loan for the due date).The Balance = Step (1) - Step (2)

Example:A man signs a note for L.E 1000 due in 1 year with interest at 12%. Three months after the debt is contracted, the holder of the note sells it to a third party who determines its value at 16% simple interest. How much does the seller of the note receive?

Solution:1 year

i = 12%

1000 3 months

P 9 months S

16%

S=1000(1+

= 1000¥ 1.125= 1120

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P =

== 1000 L.E.

Example:An Obligation of 650 L.E. is due on April 17, after which the borrower must pay interest at 18%. If the borrower pays 200 L.E. On June 17 and 100 L.E. On August, how much will need to be paid on October 17 to discharge the obligation.

6 months 18%

17/5 17/6 17/7 17/8 17/9 17/10

650 -200 -100

The amount of the original debt for 6 months

= 650(1+ ¥ )

= 650 (1 + 0.09)

= 650¥ 1.09

= 708.5

The amount of the two payments S1, S2

= 300(1+ ¥ )+1001+( ¥ )

= 200 (1 + 0.06)˜ 100 (1 + 0.03)

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= 200 (1 + 1.06)˜ (100¥ 1.03) (1 + 0.03)

= 212˜ 103

= 315

The Balance X = 708.5 - 315

= 393.5

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Chapter 4

Bank Discount and Promissory Notes

Bank Discount:The change for many loans is based on the final amount rather than the present value. This charge is called bank discount.As an illustration of a bank discount transaction, Consider the case of a person who wants to borrow \$100 for a year from a lender who uses a discount rate of 6%. The lender will take 6% of \$100 from the \$100 and give the borrower \$94. Thus, the computation of bank discount is exactly the same as the computation of simple interest except that it is based on the amount rather than the present value. This realization leads us to the bank discount formula:

D = S d tD = The bank discount in dollars S = The amount or maturity value

D = The discount rate per unit of time expressed as a decimal i = The time in units that correspond to the rate

Since the proceeds or present value of the loan is the difference between the amount and the discount, we can say that:

P=S–D= S – S d t

Factoring, we have:P = S (1 – d t)

Example:Determine the bank discount and the proceeds if \$ 4400 is discounted at a discount rate of 12% for 2 months.

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Solution:Substituting S = 4400, d = .12, and t = 2/12 in formula, we have:

D = S d tD = 4400¥ .12¥ 2/12

= \$ 88.00

Since the proceeds represent the difference between the maturity value and the discount, the borrower will get 4400 –88.00 = \$4312.00

Example:Find the discount on 3200\$ for 60 days at 13.5% discount. What are the proceeds?

Solution:Substituting S = 3200, d =13.5%, and t=60 days in formula, we have:

D = S d tD = 3200¥ 13.5/100¥ 60/360

= \$ 72.00Since the proceeds represent the difference between the maturity value and the discount, the borrower will get 3200 –72.00 = \$3128

Example:A debtor borrows \$600 for 6 months from a lender who uses a discount rate of 10% What is the discount and how much money does the borrower get?

Solution:

Substituting S = 600, d = .10, and t = in formula, We have

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D = S d t

D=600¥.10¥= \$ 30.00

Since the proceeds represent the difference between the maturity value and the discount, the borrower will get 600.00 –30.00 = \$570.00

Example:Find the present value of \$100 due in 1 year: at a simple rate of

12 %; at a bank discount rate 12 %.

Solution:

For a simple interest rate of 12 %, We Substitute S = 100, r =.125, and t = 1 in formula :

P =

P =

== \$88.89

For a bank discount rate of 12 %, We Substitute S = 100, d =.125, and t = 1 in formula:

D = S d tD=100¥.125¥1

= \$ 12.50Present value or proceeds = 100 – 12.50

= \$ 87.50

34

Note that the present value at 12 %, bank discount is \$1.39

less for the same maturity value than if it were based on a 12 % interest rate.

ORP = S (1 – d t)

= 100 (1 – 0.125¥ 1)= \$ 87.50

Promissory Notes:FA promissory note is a written promise to repay a borrowed

amount of money to the lender on an agreed date. The borrower of the amount is called the 'maker' of the note (as the borrower is making a promise to pay the amount) and the lender is called the 'payee' (as the borrower promises to pay the lender the amount). The amount shown on the promissory note is called the 'face value' of the note. The time period negotiated is known as the 'term or the maturity date', which may be in days, months, or years. The interest period is the time period from the date of issue to the legal due date.FThere are two types of promissory notes: interest-bearing

notes and non-interest-bearing notes.

Bank discount is used to determine the value of a promissory note at a stated point in time. The proceeds of a note are found as follows:

1- Find the maturity value of the note. This is the face value if it is non-interest-bearing. If the note is interest-bearing, the maturity value is the face value plus interest

35

at the stated rate for the term of the note, the time from the date of the note to the maturity date.

2- Discount the maturity value using the discount rate from the date the note is discounted to the maturity date.

3- Subtract the discount from the maturity value.

Example:On July 28, 2016, Diesel V. discounts the note in figure 1 at a bank that charges a discount rate of 12%. Find the proceeds.

Solution:Figure 1: A Non-interest-bearing Note:

3.000 June 15 2016

Six months after date I promise to pay to the order of Diesel

V. Three thousand and no/100

Payable at Trenton, New Jersey

No. 56 Due December 15, 2016

Maturity Date: December 15,2016 350(leap year)Discount Date: July 28,2016 210

Discount Period: 140 days

Substituting S = 3000, d = .12, and t = in formula, we have:D = S d t

D =3000¥ .12¥36

= \$ 140.00proceeds = 3000.00 – 140.00

= \$ 2860.00

Example:On July 28, 1996, John A. Blalock discounts the note in figure 2 at a bank that charges a discount rate of 12% . Find the proceeds.

Solution:First, to get the maturity value, we substitute P = 3000 , r = .11 ,

and t = in formula,I = P r t

I =3000¥ .11¥= \$ 165.00

S = 3000.00 + 165.00= \$3165.00

Figure 2: An Interest-bearing Note:

3.000 June 15 2016

Six months after date I promise to pay to the order of Diesel

V. Three thousand and no/100

Payable at Trenton, New Jersey

Value Received with interest at 11%

No. 56 Due December 15, 2016

37

The elapsed time from July 28 to December 15 is 140 days.

Substituting S = 3165.00 , d = .12 , and t = in formula, We obtain:

D = S d t

D =3165.00¥ 0.12¥= \$ 147.70

proceeds = 3165.00 – 147.70= \$ 3017.30

Example:A company holds the following non-interest-bearing notes. The notes are taken to the bank on November 5, 2014, and the proceeds deposited in the company’s account. Get the total proceeds if the discount rate is 12.5%

Date of the note Term Face value

90 Days 2000

60 days 2500

Solution:Date of Discount 5/11/90Discount rate = 12.5%For the first note:Due date is: t = 90 Days

Sep. Oct. Dec.\ t = 24 + 31 + 30 + 5 = 90 Days

The Discount time:Sep. Oct.

\ t = 24 + 31 + 5 = 60 Days38

Total Diff. t = 30 days.P = S (1 - d t)

= 2000 (1 – 12.5/100 ¥ 30/360) P1 = 1979.167\$

For the second note:Due date is: t = 60 Days

Oct. Nov. Dec.\ t = 15 + 30 + 15 = 60 Days

The Discount time:Oct. Nov.

\ t = 15 + 5 = 20 days

Total Diff. t = 40 days.

P = S (1 - d t)

= 2500 (1 – 12.5/100 ¥ 40/360)

P2 = 2465.27\$

\ The proceeds = P1 + P2

= 1979.167\$ + 2465.27\$

= 4,444.437\$

Example:Company ABC holds the following interest-bearing notes. They

are discounted at 14.2% on July 15. 2016. Find the total proceeds.

Date of the note Term Face value Rate

May 16, 2016 120 Days 4200 9%

June 10, 1991 6000 8%

39

Solution:

Date of Discount 15/7/2016 Discount rate = 14.2%

For the first note:

i = 9%

16/5 Due date 13/9

4200 tP = 4326

d = 14.2

Due date is: t = 120 Days

May June July Aug. Sep.

\ t = 15 + 30 + 31 + 31 + 13 = 120 Days

\ The due date is 13 September.

The Amount:S = P (1 + i t)

=4200 (1 + ¥ ) = 4200 (1.03) = 4326

The Discount time:July Aug. Sep.

\ t = 16 + 31 + 13 = 60 DaysP = S (1 + d t)

=4326(1-¥ )P1 = 4326¥ 0.97633

= 4223.618

40

For the second note:

i = 8%

10/6

S

PDue date is: t = 45 Days

June July\ t = 20 + 25 = 45 Days\The due date is 25 July.

The Amount:S = P (1 + i t)

= 6000(1+ ¥ )= 6000 (1.01)= 6060

The Discount time:t = 10 days (from 15/7 till 25/7)P2 = S (1 - d t)

=6060(1- ¥ )P2 = 6060 ¥ 0.99605

= 6036.097\ The proceeds = P1 + P2

= 4223.618 + 6036.097 = 10259.715

41

Chapter 5

Compound Interest

Compound Interest:As we have explained in chapter three (Simple Interest). The Simple Interest is based on the original principal only, and thus; It remains unchanged throughout the term of the loan (or investment).Here in Compound interest, the interest which is due is added to the principal at the end of each year (or at the end of each conversion period), then the sum (P+I) is considered to be a new principal for the next time period. This means that this interest as well as the principal will earn interest for the next period of time. Interest paid on an increasing principal in this way is known as compound interest.Simple interest is normally used for loans or investments of a year or less. For longer periods compound interest is used. With compound interest, interest is charged (or paid) on interest as well as on principal.

For example, if \$1000 is deposited at 5% interest for 1 year, at the end of the year the interest is the balance in the account is If this amount is left at 5% interest for another year, the interest is calculated on \$1050 instead of the original \$1000, so the amount in the account at the end of the second year is \$1050+ \$1050(.05) (1) = \$1102.50. Note that simple interest would produce a total amount of only \$1000[1 + (.05) (2)] = \$1100.

Compound Amount Formulas:In many business transactions, the interest is computed annually, semiannually, quarterly, monthly, daily, or at some other time interval. the time between successive interest computations is called the conversion, or interest, period. This basic unit of time is used in all compound interest problems. The is designated by

43

the symbol i, the symbol for the total number of conversion periods is n.In most business transactions, the practice is to quote an annualinterest rate and the frequency of conversion. From this information, the rate per period is determined. thus 6% compounded (or converted) semiannually means that 3% interest will be earned every 6 months. The quoted annual rate is called the nominal rate and is indicated by the symbol j. the number of interest conversion periods per year is indicated by the symbol m. he equation relating j, i, and m is I = j/m. Or j = i m ('Jim') The symbol j(m) means a nominal rate j converted m times a year. When no conversion period is stated in a problem, assume that the interest is compounded annually.

The basic formula for compound interest,S = P (1 + i) n

S = The amount at compound interest P = The principali = The rate per conversion periodn = The number of conversion periods

Compound Interest Rate:Time Period Cash Flow

- P

P + (P¥ i) = P (1 + i)

P (1 + i) + P (1 + i) i

P (1 + i) (1 + i) = P (1 + i )2

P (1 + i )2 + P (1 + i )2 iP (1 + i )2 (1 + i) = P (1 + i )3

44

\ For n periods:

F = P (1 + I) n

In functional from;F = P (F/P, I %, n)

(F/P, I %, n) is read as:

To Find F / given P as a function of i% / period and with n periods.

If you want to get F given P,

Therefore P = = F (1 + i)-n

P = F (P/F, i, n)

The Amount at Compound Interest:For the first year;The Principal at the start of the first year = P. Interest for the first year.

I = P¥ i¥ t= P¥ i¥ 1 year= P¥ i

S1: Amount at the end of the first year = P + 1S1=P+1

= P + PiS1 = P (1+ i)

For the Second year;The Principal at the start of the second year = P (1+i)Interest for the first year:

I = P (1+1)¥ i¥ 1 year45

S2: Amount at the end of the second yearS2 = P (1+i) + 1

= P (1+i) + P (1+i)¥ i= P (1+i)¥ (1+i)

= P (1+i)2 and so on ……

S3: at the end of the 3rd year

S3 = P (1+i)3

For n periods or yearsS = P (1 + i) n

And hence;I = AMOUNT – PRINCIPAL

= S - P= P (1 + i) n - P

I = P [(1 + i) n - 1]Where;

P: The PrincipalI: Rate of compound interest (per conversion period)S: Amount of compound interestn: Total number of conversion periods(1+i): The amount of one pound.

Compounding Interest for More Than Once Use:M: number of conversion period per yearJ: Annual nominal ratei : Interest rate per conversion period

Interest may be compounded annually, semiannually, quarterly, monthly, weekly, daily.

46

1 - The rate of interest per conversion period;

i =

2 - The total number of conversion period per year:n = m¥ t

How to calculate m:FIf the interest rate is compounded annually⟶ m = 2.FIf the interest rate is compounded quarterly⟶ m = 4.FIf the interest rate is monthly⟶ m = 12.FIf the interest rate is weekly⟶ m = 52.FIf the interest rate is daily⟶ m = 360.

The present value:

P =

\ P = S (1 + i)-n

Example:Suppose \$1000 is deposited for 6 years in an account paying 4.25% per year compounded annually.(a) Find the compound amount.

Solution:In the formula above P = 1000, i = .0425/1, and n = 6(1) = 6.The compound amount is:

S= P (1 + i)n

S = 1000(1.0425)6, Using a calculator, we get:

S= \$1283.68,The compound amount.

47

(b) Find the amount of interest earned.

Solution:Subtract the initial deposit from the compound amount. Amount of interest = \$1283.68 - \$1000 = \$283.68

Example B:Find the amount of interest earned by a deposit of \$2450 for 6.5 years at 5.25% compounded quarterly.

Solution:

Interest compounded quarterly is compounded 4 times a year. In 6.5 years, there are 6.5(4) = 26 periods. Thus, n = 26. Interest of5.25% per year is 25%/4 per quarter, so I = .0525/4. Now use the formula for compound amount.

S = P (1 + i) n

S= 2450 ( 1 + .0525/4)26 = 3438.78Rounded to the nearest cent, the compound amount is \$3438.78, so the interest is \$3438.78 - \$2450 = \$988.78.

Caution:As shown in Example B, compound interest problems involve two rates—the annual rate r and the rate per compounding period i. Be sure you understand the distinction between them. When interest is compounded annually, these rates are the same. In all other cases, i ≠ r.

Example:Find the amount of \$ 6000 for 8 years at 8 % nominal,Compounded:

(a) annually (b) semiannually (c) quarterly (d) monthly. 48

Solution:P = 6000, t = 8 years, j = 8%

a- Annually:

S = P (1 + i) n

= 6000 (1 + 0.08 )8 = 6000¥ 1.85093021= 11105.581

b- Semiannually: m = 2i = j + m= 8%˜2=4% n =

t ¥ m= 8¥2=16

\ S = 6000 (1 + 0.04) 16 = 6000¥ 1.8729812 = 11237.887

c- Quarterly: m = 4\ i = 8%˜ 4 = 2 % n = 8¥ 4 = 32

\ S = 6000 (1 + 0.02)32 = 6000¥ 1.88454059= 11307.244

d- Monthly: m = 12\ i = 8%˜ 12 = 0.667 % n = 8 ¥ 12 = 96

\ S = 6000 (1.00667)96 = 6000 ¥ 1.8924572= 11354.743

Example:Find the compound amount and the compound interest, if \$1000 is invested 10 years at 7%.

49

Solution:P = 1000, t = 10 years,M= 1, n = 10,S= P (1 + i) n

= 1000 (1 + 0.07 )10

= 6000¥ 1.96715= 1967.15

i =S-P= 1967.15 – 100= 967.15

Example:Find the compound amount and the compound interest, if \$24,500 is borrowed for 3 years at 9% converted monthly.

Solution:P = 24500, t = 3 years, j = 9%M=12I=9%˜ 12

= 0.75 %= 0.0075

n = 3 ¥ 12= 36

= 24500¥ 1.30864537= 32061.812

Example:On June 30, 2000, a man put \$15,000 in a deferred savings account paying 8% converted quarterly. Find the amount in the account when it matures on June 30, 2006.

50

Solution:30 June 2000‡ 30 June 2006P = 15000t = 6 yearsj = 8%M = 4\ n = 6¥ 4

= 24

i = = 2%

S = P ( 1 + i )n

\ S = 15000 (1+0.02)24

= 15000¥ 1.60843724= 24126.558

Example:A note of \$ 80,000 is due in 5 years with interest at 8%. At the end of 3 years the note is discounted at 9%. What are the proceeds at the time of discounting?

Solution:5 Years

8 %3 years

80000 2years

P = S

9 %

51

S = P (1 + i) n

S =8000 (1 + 0.08 )5

= 80000¥ 1.469328= 117546.246

S =117546.246 (1 + 0.09)-2

= 117546.246¥ 0.84167999= 98936.324

52

Chapter 6

◌ِAnnuities and Loans

The Concept of Annuity:An annuity is a sequence of payments with fixed frequency. The term “annuity” originally referred to annual payments (hence the name), but it is now also used for payments with any frequency. Annuities appear in many situations; for instance, interest payments on an investment can be considered as an annuity. An important application is the schedule of payments to pay off a loan.The word “annuity” refers in everyday language usually to a life annuity. A life annuity pays out an income at regular intervals until you die. Thus, the number of payments that a life annuity makes is not known. An annuity with a fixed number of payments is called an annuity certain, while an annuity whose number of payments depends on some other event (such as a life annuity) is a contingent annuity.

Types of Annuities:Ordinary annuities:As mentioned before, a sequence of equal payments made at equal periods of time is called an annuity. If the payments are made at the end of the time period, and if the frequency of payments is the same as the frequency of compounding, the annuity is called an ordinary annuity. The time between payments is the payment period, and the time from the beginning of the first payment period to the end of the last period is called theterm of the annuity. The future value of the annuity, the final sum on deposit, is defined as the sum of the compound amounts of all the payments, compounded to the end of the term.

54

Future Value of An Ordinary Annuity:Where

S: is the future value;R: is the payment;i: is the interest rate per period;n: is the number of periods.

S = R(1 + i)n – 1 or

i

Annuities Due:The formula developed above is for ordinary annuities — those with payments made at the end of each time period. These results can be modified slightly to apply to annuities due-annuities in which payments are made at the beginning of each time period. To find the future value of an annuity due, treat each payment as if it were made at the end of the preceding period. That is, find snּךi

for one additional period; to compensate for this, subtract the amount of one payment.Thus, the future value of an annuity due of n payments of Rdollars each at the beginning of consecutive interest periods, with interest compounded at the rate of i per period, is

S = R– 1

or S = R . sn+1 i – Ri – R

Two common uses of annuities:FFirst, is to accumulate funds for some goal or to withdraw

funds from an account.

For example, an annuity may be used to save money for a large purchase, such as an automobile, an expensive trip, or a down payment on a home.

55

FSecond, used to provide monthly payments for retirement.

For example, suppose \$1500 is deposited at the end of each year for the next 6 years in an account paying 8% per year compounded annually.To find the future value of the annuity, look separately at each of the \$1500 payments. The first of these payments will produce a compound amount of:

1500(1 + .08)5 = 1500(1.08)5,

Term of annuity

End of year1 2 3 4 5 6

Period 1 Period 2 Period 3 Period 4 Period 5 Period 6

\$1500 \$1500 \$1500 \$1500 \$1500

Use 5 as the exponent instead of 6 since the money is deposited at the end of the first year and earns interest for only 5 years. The second payment of \$1500 will produce a compound amount of 1500(1.08)4. As shown in the next figure, the future value of the annuity is

1500(1.08)5 +1500(1.08)4 +1500(1.08)3 +1500(1.08)2 +1500(1.08)1 +1500.

(The last payment earns no interest at all.)

56

Year 1 2 3 4 5 6

Deposit \$1500 \$1500 \$1500 \$1500 \$1500 \$1500

\$1500

\$1500(1.08)

\$1500(1.08)2

\$1500(1.08)3

\$1500(1.08)4

\$1500(1.08)5

Reading this sum in reverse order, we see that it is the sum of the first six terms of a geometric sequence, with α = 1500, r = 1.08 and n = 6. Thus, the sum equals

= = \$11,003.89.

To generalize this result, suppose that payments of R dollars each are deposited into an account at the end of each period for nperiods, at a rate of interest i per period. The first payment of Rdollars will produce a compound amount of R (1 + i)n-1 dollars, the second payment will produce R(1 + i)n-2 dollars, and so on; the final payment earns no interest and contributes jus R dollars to the total. If S represents the future value (or sum) of the annuity, then (as shown in the figure below),

S = R (1 + i) n-1 + R (1 + i)n-2 + R(1 + i)n-3 + …. + R (1 + i) +R

or, written in reverse order,

S = R + R (1 + i)1 + R(1 + i)2 + …. + R (1 + i) n-1.57

Period 1 2 3 n – 1 n

Deposit SR SR SR SR SR

A deposit of SRbecomes

R

R( + i)

R(1 + i)n-3

R(1 + i)n-2

R(1 + i)n-1

The sum of these is the amount of the annuity.

This result is the sum of the first n terms of the geometric sequence having first term R and common ratio 1 + i. Using the formula for the sum of the first n terms of a geometric sequence,

S = R[(1 + i)n – 1] = = R[(1 + i) – 1] i i

A formula for the future value of an annuity S of n payments of Rdollars each at the end of each consecutive interest period, with interest compounded at a rate i per period, follows. Recall that this type of annuity, with payments at the end of each time period, is called an ordinary annuity.

58

◌Aِnnuities

Ordinary Due

The periodic payment is made The periodic payment is madeat the end of each period at the beginning of each period

Amount Sn1 = R ( ) Sn(dut) = R(1+i) ( )

Present value An= R( ) A n(dut)= R(1+i) ( )

Sn is : The amount of ordinary annuity of (n) payments.Sn(dut) : Is the amount of a due annuity of (n) paymentsR : Is the periodic paymentI: Is the rate period.

In Case of Equal Payment / Receipt of Money:

F

0 1 2 3

A A A

59

For 3 periods:

F = A + A(1+i) + A(1+i)2

Note that the last term ends with n-1 and not n where n = 3\

\ F = A + A(1+i) + A(1+i)2 + A(1+i)3 + A(1+i)4 + ……. A(1+i)n-1 ...1Multiply both sides by (1+i) so that the last term of the equation ends with A(1+i)n instead of A(1+i)-n

F(1+i) = A(1+i) + A(1+i)2 + A(1+i)3 + A(1+i)4 + ……. A(1+i)n-1 + A(1+i)n …2

Subtract equation (1) from (2):

\ Fi = - A + A (1+i)n

Fi = - A (1+i)n - A

Fi = A [ (1+i)n – 1 ]

\F=A[ ]

F = A (F/A , I % , n)

To Find F given A as a function of i% , n .To Find A given F:

A=F[ ]

A = F (A/F , i% , n)

Example:Starting 1 year from now, an investor will deposit \$ 500 a year in an account paying 6% interest compounded annually. What amount

is in the account just after the 4th deposit is made?

Solution:

S=500[ ]60

= 500[ ]

= \$ 2187.31

Example:A young man gets a loan for \$15.000 to be repaid in monthly

payments over a period of 25 years. If the interest rate is 10 %converted monthly, what is the size of the monthly payment and the total interest?

Solution:The loan will be repeals in monthly payment15000 = An m = 12 J = 10.5% t=15years i = 10.5%˜ 12

= 0.875 %= 0.00875

n = 25¥ 12

= 300

An = R [ ]

1500=R[ ]

= R [ ]

= R [ ]

1500 = R¥ 105.911817R = 15000˜ 105.911817

= 141.627The monthly payment = 141.627

61

The total interest= 42488.1767 – 15000= 27488.177

Example:An investment of \$200 is made at the beginning of each for 10 years. If interest is 6% effective, how much will the investment be worth at the end of 10 years?

Solution:R = 200, n = 10, and i = 6%

Sn(due) = 200 (

= \$ 2794.33

Example:The premium on life insurance policy is \$60 a quarter in advance. Find the cash equivalent of year's premiums if the insurance company charges 6% converted quarterly for the privilege of paying every three months instead of all at once for the year.

Solution:

Substituting R = 60, n=4, and I = 1 %

) (1.015)= \$ 234.73

62

Is there a series of payments beingmade at equal time intervals?

YesNopayment period

Ordinary annuity

Simple or compoundinterest problem

How is interestcalculated?

Interest only applies Interest is compounded at a certainto original amount frequency (annually, semi-annually,

AnnuityWhen are the

At the beginning ofpayment period

Annuity Due

(the principal) quarterly, monthly). Interest appliesto the original principal AND to the

Simple interest

Compound interest

Applications:- Promissory notes

Applications:- Demand loans/notes- Discounting promissory- Treasury bills (T-bills)

- Replacement notespayments (equivalent - Replacement payments

Is the conversion period (C/Y) equal to the

payment interval (P/Y)?

Are the beginning and

ending dates known?

Yes.C/Y = p/Y Simple annuity

No.C/Y = p/Y General annuity

BeginningAND ending Annuity certaindate known

Beginning date

known; payments Perpetuity

continue forever

values) (equivalent values) Contingent annuitydate OR both are

References:M. J. Alhabeeb (2012). Mathematical Finance, 1st ed., John Wiley & Sons, Inc.

Zima, P., and R. Brown (2001). Mathematics of Finance. McGraw-Hill, New York.

Cissell, R., H. Cissel, and D. Flaspohler (1990). Mathematics of Finance, 8th ed. Houghton Mifflin, Boston.

Reading list:Stephen G. Kellison, (2009). The Theory of Interest,3rd ed., McGraw-Hill

Margaret L. Lial, Raymond N. Greenwell, and Nathan P. Ritchey (2005). Finite Mathematics, Eighth Edition, Pearson Education, Inc.

Roman, S. (2004). Introduction to the Mathematics of Finance.Springer-Verilog, New York.

Mathematics of Finance

WORK SHEETS

Chapter One

Work Sheet

THE CONCEPT OF TIME VALUE OF MONEY

Student Name:

Semester:1

1: What do you mean by time value of money?

2

2: What are the key elements of a timeline?

How can you distinguish cash inflows from outflows on a timeline?

3

3: In January 2010, a firm purchased a used typewriter for \$500. Repairs cost nothing in 2010 and 2001. In 2012 they are \$85, \$130 in 2013 and \$140 in 2014. It is sold in 2014 for \$300. Construct the cash table.a) Prepare a cash flow table.b) Prepare a cash flow Diagram.

4

4: A man purchased a used car for \$19,000. Maintenance cost nothing in the first year; the maintenance did cost \$600 in the second year, \$800 in the third year, and \$900 in the fourth year. The car was sold for \$16,000 at the end of the fourth year.1. Prepare a Cash Flow Table.2. Prepare a Cash Flow Diagram.

5

5: A woman purchased a used mobile home for \$16,000. Maintenance cost nothing in the first year; the maintenance did cost \$400 in the second year, \$500 in the third year, and \$600 in the fourth year. The car was sold for \$12,000 at the end of the fourth year.1. Prepare a Cash Flow Table.2. Prepare a Cash Flow Diagram.

6

Chapter Two

Work Sheet

THE CONCEPT OF INTEREST RATE

Student Name:

Semester:7

1: Compare between the simple Interest and the Compound Interest” illustrate giving a practical example”

8

Chapter Three

Work Sheet

THE SIMPLE INTEREST

Student Name:

Semester:9

1: Find the simple interest gained on \$ 9000 for 5 months at 7% simple interest rate.

10

2: A \$ 40,000 car loan is to be repaid with monthly payments of \$ 505,78. If the interest rate is 15% simple, how much of the first payment goes to interest and how much to principal.

11

3: A young woman gets 900\$ every 3 months from an investment that pays 16% interest. How much money is invested?

12

4: A man had a gambling debt of 150\$. He repaid it in 1 month with interest of 15\$. Find the rate.

13

5: A mechanic borrowed \$125 from a licensed loan company and at the end of 1 month paid off the loan with \$128.75. What annual rate of simple interest was paid?

14

6: A debtor borrows \$ 7000 on December 15, 2016, and repays the debt on March 3, 2017, with simple interest at 15%. Find the amount repaid.

15

7: On May 4, 2014, an investor borrows \$ 1850 and promises to repay the debt in 120 days with simple interest at 12%. If the loan is not paid on time the contract requires the borrower to pay 10% simple interest on the unpaid amount for the time after the due date. Determine how much this person must pay to settle the debt. Determine how much this investor must pay to settle the debt on the December 15, 2014.

16

8: A man borrows \$ 3000 at 16% simple interest on August15. She pays \$600 September 15, \$ 800 on October 15, \$600 on December 15. If makes a final settlement on February 15 of the following year how much will she have to pay?

17

9: Suppose that you borrow \$ 100 at 12% simple interest but the lender refuses to take payment until the time when you must repay \$200. How long will it take?

18

10: The Jones Company owes Dakota Supply Company \$3000 on January 15, 2014, and is required to pay 16% interest from that date. After making payments of \$1000 on April 15, 2014, and \$ 1500 on October 15, 2014, what payment on January 15, 2015 will retire the debt.

19

Chapter Four

Work Sheet

BANK DISCOUNT AND THE

PROMISSORY NOTES

Student Name:

Semester:

20

1: Determine the bank discount and the proceeds if 1500\$ is discounted for 3 months at a discount rate 0f 10%

21

2: Find the discount on \$ 568.30 for 120 days at 10 %discount. What are the proceeds?

22

3: Find the bank discount and the proceeds if \$240,000 is discounted for 60 days at 15%.

23

4: An obligation of \$650 is due on August 10. What is its value on April 6 if it is discounted at a discount rate of 12%?

24

5: A merchant receives a note for \$ 1245.40 that is due in 60 days with interest at 15% simple. The note is discounted immediately at a bank that charges 16% discount. What are the proceeds?

25

6: A company holds the following interest-bearing notes. They are discounted at 10% on June 1. 2016. Find the total proceeds.

Date of the note

April 2, 2016 90 Days 2000 8%

June 11, 1991 1000 9%

26

7: On December 20, 2006, Dave A. discounts the note in the following Figure at a bank that charges a discount rate of 12 %. How inch does he receive?

3.000 October 1 2006

Ninety days

Dave A.

Three thousand and no/100:

Trenton, New Jersey:

No.58 December 30. 2006 10%

27

8: On August 5, 2015, Alice discounts the note in the following Figure at a bank that charges a discount rate of 12.5 %. How inch does he receive?

800 July 1 2015

Sixty days

Alice A.

Eight thousand and no/100:

No.22 August 30. 2015 12%

28

Chapter Five

Work Sheet

THE COMPOUND INTEREST

Student Name:

Semester:29

1: What amount of money will be required to repay a loan of 6000\$ on December 31, 2014, if the loan is made on December 31,2008, at rate of 10% compounded semiannually?

30

2: The sales of a business have been increasing at the rate of 3% per year if the sales in 2000 were \$250.000. What are the estimated sales to the nearest thousand dollars for 2005.

31

3: On June 2009, a woman incurs a debt of \$3000 that is to be repaid on demand of the lender with interest at 9% converted.

32

4: As part of her retirement program, a woman puts \$12.000 in a deferred saving account paying 7% converted quarterly. The investment is made on the day she becomes 57. What will be the maturity value of this account if it matures when she becomes 62?

33

5: What amount of money will be required to repay a loan of \$1835.50 on July 1, 2007, If the loan is made on October 1.2003, at an interest rate of 12%. Compounded quarterly?

34

6: What is the amount of \$40.000 for 6 years and 3 months at 10% converted semiannually?

35

7: An investment of \$ 4000 is made for 12 years. During the first 5 years the interest rate is 9% converted semiannually. Then the rate drops to 8% converted semiannually for the remainder of the time. What is the final amount?

36

8: On a girl’s 10th birthday, her parents place \$150,000 in her name in an investment paying 8% compounded quarterly. How much will she have to her credit on her 21st birthday?

37

Chapter SIX

Work Sheet

ANNUITIES AND LOANS

Student Name:

Semester:

38

1: Find the amount of an annuity of \$5000 per. Year for 10 years at; (a) 6%; (b) 7%; Interest is compounded annually.

39

2: Find the amount of an annuity of \$1200 at the end of each 6 months for 5 years if money is worth; (a) 5%; (b) 6%; All rates are converted semiannually.

40

3: A mobile home was purchased for \$ 6000 down and \$1000 at the end of each 6 months for 8 years. If the payments are based on 14% converted semiannually, what was the cash price of the mobile home now?

41

4: A donor wants to provide a \$3000 scholarship every year for 4 years with the first to be awarded 1 year from now. If the school can get 9% return on its investment, how much money should the donor give now?

42

Mathematics of Finance

QUIZZES

43

QUIZ ONE

Student Name:

44

QUIZ TWO

Student Name:

45

QUIZ THREE

Student Name:

46

QUIZ FOUR

Student Name:

47

QUIZ FIVE

Student Name: