Figure 1
Adopted from “Money saving growth image” by nattanan is licensed under CC BY 3.0

Take a look at the image above. This image illustrates the benefits of saving and the importance to stablise it in life. How to maintain the stabilisation? In this course, we will learn a few ways to make a financial decision by considering the economic aspects such as inflation and any uncertainty. However, note that every decision has its opportunity cost or missed opportunity. But it can be minimised through better financial decision-making.

Figure 2

Adopted from “Picture att note in standard chartered bank” by Mobergefinance is licensed under CC BY 3.0

Having a huge amount of savings doesn’t mean having financial freedom. A good financial management is needed. Figure 2 shows a choice for an individual to apply for a loan from a company or other financial institution. What elements should be considered before making the choice? Is it more profitable to choose Figure 2’s option or institutions with better bank discounts? Is the term of payment offered fruitful?

In this book, and each chapter, you will begin to explore the definition of the terminology of the title, the components required for the terminology, the calculation, and finally the application in finance and economics aspects.

Financial mathematics helps in evaluating any financial decision that may be related to economic conditions. For example, borrowers need to consider the interest rate given when making a loan. Meanwhile, the interest rate is closely related to the inflation rates. The inflation rate increases if the currency rate for the home country depreciates caused by economic uncertainties such as political instability in the country. Finally, the depreciation causes economic gains for individuals or firms to decrease.

Financial mathematics also is needed for many kinds of decision-making. For instance, buying a car. Given two alternatives, $18,000 now, or $600 per month for 3 years. Which is better? It depends! The issue is how much is money now worth compared to money in the future. It leads to the key concept of the time value of money.

Finally, financial mathematics applies in your personal life. In general, approaches to problem solving are: i) understand the problem; ii) collect all relevant data or information; iii) define the feasible alternatives; iv) evaluate each alternative; v) select the “best” alternative; vi) implement and monitor the decision. The major role of financial mathematics is applied to approach vi which is to evaluate different alternatives. It involves techniques of formulating, estimating, and evaluating economic outcomes where choices or alternatives are available. Using specific mathematical relationships, individuals will compare the cash flows of the different alternatives. As a resolver, a sensitivity analysis can be performed to explore how the decision changes as the estimates change.

Depreciation

Economic gains

Inflation

Interest rate

Opportunity cost

Saving

Time value of money

Interest is the compensation one gets for lending a certain asset. For example, suppose that you put some money in a bank account for a year. Then, the bank can do whatever it wants with the money for a year. To reward you for that, it pays you some interest. The asset being lent out is called the capital.

In general, both the capital and interest are expressed in money. However, that is not necessary. For instance, a farmer may lend his tractor to a neighbour, and get a part of the grain harvested in return. In this course, the capital is always expressed in money, or called capital.

Interest is the reward for lending capital to somebody for a period of time. There are various methods for computing the interest. For simple interest, is it the amount of interest in the product of three quantities which are: i) the rate of interest; ii) the principal; iii) the time period.

From a lender’s perspective, when money is borrowed for a loan, interest accumulates as a reward for the lenders. While, from the borrower’s perspective, accumulated interest is a charge to the borrower for the financial transaction to take place. The amount of interest incurred by the borrower depends on the amount of money borrowed or invested, the principal, the interest rate, and time. The simple interest, , accumulated on the principal, , over an interval of t years at an annual interest rate of , can be written as follows:

 (1)    

Where the rate of interest must be expressed as a decimal for calculations, the interest rate is expressed in year or annually, and the interest rate is a flat rate where there are no changes happens for the interest rate during the tenure period. To note that, p.a. represents per annum or annual interest rate.

2.2.1 Future Value and Present Value

The total amount of money that must be repaid on a loan or the total value of an investment can be called the future value, . The future value can be calculated using , where is the principal or money borrowed for a loan or invested, and is the interest accumulated. The future value also can be calculated using the following formula, given information on the accumulated amount of the principal and interest after years:

(2)  

The principal is also called the present value of the discounted value of . In Equation (2), is called the simple interest factor and  is called the present value discount factor at simple interest. The time, , must be in years. When the time is given in months, then

 (3)    

When the time is given in weeks, then divide the weeks with 52, when the time is given in days, then divide the days with 365 days.

The present value (or discounted value) of was calculated by using the present value factor at simple interest.  The present value can be written as Equation (4) :

 (4)    

2.2.2 The Time between Dates

Figure 2
“Calendar” by Nick Youngson is licensed under CC BY-SA 3.0

In any financial transaction, loan terms are an important aspect to be considered before signing off. These include the loan’s repayment period. The loan repayment period or time can be calculated using two ways which are i) the exact time, and ii) the approximate time.

Exact time is found as the exact number of days including all days except the first. The exact time can be refereed using the table of the number of each year of the day (see Table 1). It is obtained as the difference between serial numbers of the given dates. For example, to find the exact time from April 18 to November 3 of the same year, see Figure 1. May 18 is the 108th day of the year and November 3 is the 307th day of the year. The exact time is 307 – 108 = 199 days. Alternatively, use the Microsoft Excel.

Whereas, the approximate time is calculated by assuming that each month has 30 days.  Using the same example as above, see Table 1, for the solution.

Where we have borrowed 30 days from the 11th month. The approximate time is 6 months and 15 days, or (6 x 30 days) + 15 days = 195 days.

This is a video about simple interest using sequences. Please answer all questions in the video.

2.4.1 Simple Discount at an Interest Rate

In discounting at simple interest, the difference is called the simple discount (on ) at an interest rate (). We may interpret either as the interest on which when added to gives , or as the true discount on which when substracted from gives .

2.4.2 Simple Discount at a Discount Rate

The discount rate for a year is the ratio of the discount for the year to the amount on which the discount is given. The simple discount on an amount , also called bank discount, for years at the discount rate , which can be calculated as follows:

(1)  

And the discounted value, or proceeds, of is given by

(2)  

The charge for some short-term loans may be based on the final amount rather than on the present value. The lender calculates the bank discount on the final amount that must be paid on the due date and deducts it from ; the borrower receives the proceeds . For this reason, bank discount is sometimes called interest in advance. The following equation calculate the maturity value of a loan for specified proceeds/ principals/ present values of .

(3)  

1. At what rate of simple interest will $1200 accumulate interest of $72 in 6 months?
2. Eighty days after borrowing money, a person pays back exactly $850. How much was borrowed if the $850 payment includes principal and simple interest at 9 ¾ %?
3. Find the present value at 12% simple discount of $1000 due in 5 months. What is the simple discount?

4. A bank charges 12% simple interest in advance. A borrower needs $2000 cash, to be repaid with interest in 9 months. What size loan should he ask for, and how much interest will he pay?

Discount rate

Interest

Interest rate

Present value

Principal

Simple interest

Time period

An interest paid only on the principal is called simple interest. When the interest of each period is added to the principal in computing the interest for the next period, it is called compound interest.

Interest amount computed at the end of a period is added to a single principal sum. That is, at the end of each interest period the interest earned for that period is added to the principal so that the interest also earns interest over the next interest period. In the same way, interest due on a debt at the end of a period is subject to interest in the next period.

Suppose that is deposited at a rate of interest per year. The amount on deposit at the end of the first year is found by the simple interest formula with t = 1;

(1)  

If the deposit earns compound interest, the interest earned during the second year is paid on the total amount on the deposit at the end of the first year. Using the formula again, with replaced by and t = 1, gives the total amount on deposit at the end of the second year;

(2)  

In the same way, the total amount on deposit at the end of the third year is;

(3)  

Generalizing, in t years the total amount on deposit is;

(4)  

called the compound amount.

3.1.1 The Formula  

The compound amount can be calculated using the following formula:

Compound amount:

(5)  

where i = r/m and n = mt ;

S = future (maturity) value, or the compound amount of P, or the accumulated value of P

P = original principal, or the present value of P, or the discounted value of S

i = interest rate per period

r = annual interest rate

m = number of compounding periods per year

n = number of compounding periods

t = number of years

Let P is the principal sum and i the interest rate per annum, mathematically, we can calculate the total balance at the ends of successive years for n years as:

At the end of the 1^st year:

The interest due              = Pi, then the future value = P + Pi = P(1+i)

At the end of the 2^nd year:

The interest due              = [P(1+i)]i, then the future value = P (1+i) + [P(1+i)]i

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

= P(1+i)²

At the end of the 3^rd year:

The interest due              = [P(1+i)²]i, then the future value = P (1+i)² + [P(1+i)²]i

= P(1+i)²(1+i)

= P(1+i)³

and so on.

Here, we can see that the total balance or the future balance value at the end of n years forms a geometric progression with first term A and common ratio (1+r). Thus, to obtain the total balance at the end of n years, we can use the above formula for the nth term of a geometric progression with n-1 replaced by n as follows:

S = P(1+i)ⁿ

Here, the total balance at the beginning of the nth year is given by:

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

However, to note that the above calculation is express as rates per annum or called as nominal rates. In some cases, the interest period need not be a year and for example, interest is “payable half-yearly’ or ‘compounded half-yearly’, interest period is payable half a year. Then, the interest rate need to be divided by two to give the effective rate per half year. The number of times you pay interest or compounding on your principal is important to calculate the total balance or future value.

3.3.1 Compound Discount at a Discount Rate

Let d^m be the nominal rate of discount compounded m times per year.

Then the discount rate per conversion period is d^m/m and the discounted value 𝑃 of a future amount 𝑆 due in 𝑛 periods is

(1)  

Thus, the accumulated value is:

(2)  

Effective interest rates can be used to compare investment opportunities. The effective rate of interest is the equivalent rate of simple interest  earned over one year for an interest rate that is compounded twice or more over the year. The annual simple interest rate will be greater than the annual compounding interest rate to earn the same amount of interest.

The annual interest rate, r, for compounding calculations, is often called the nominal rate.

Illustration: If RM1 is deposited at 4% compounded quarterly, the  compound amount is RM1.0406, an increase of 4.06% over the  original RM1. Here, the actual increase of 4.06% in the money is somewhat higher than  the stated increase of 4%. To differentiate between these two numbers, 4% is called the nominal  or stated rate of interest, r while 4.06% is called the effective rate, .

The effective rate corresponding to a stated rate of interest r compounded m times per year can be calculated as follows:

(1)  

Select TRUE or FALSE for each of the following statements.

Compound interest

Discount rate

Interest

Interest rate

Principal

Time period

A bank discount is a bank charge that is made for payment of a note at some point prior to maturation. It is applied at the time that the  note/loan is extended, and is automatically deducted from the loan amount that is used to calculate the schedule of payments on the loan. This means that the receiver of the loan simply repays the face value of the loan with little/no interest.

To receive a bank discount, an individual or a firm should possess a solid record of previous financing with the institution. If the past loan history shows no late payments and no complications with the loans, then the chances of receiving a bank discount are greatly improved.

The level of bank credit is also a factor. The eligibility for receiving a bank discount is impacted by the current assets and liabilities of the borrower. If there is a high credit rating, then the chances of getting a bank discount are enhanced. These types of customers are considered as good credit  risks with the bank can afford to extend a bank discount with the expectation of doing more business with the customers in the future.

4.1.1 The Formula

In section 2.4.2, we deal with a simple discount at a discount rate. Similarly, we can calculate the bank discount (D) as follows;

D = Sdt

where S is the maturity value, d is the discount rate, and t is time.

The money received for the discounted note is called the proceeds. The proceeds (P) are obtained by deducting the bank discount (D) from the maturity value (S) of the note:

P = S-D

From both formulas, we can calculate the proceeds as follows:

P = S-D = S-Sdt = S(1-dt)

A trade discount is a discount that is cut from the retail or published price of an item. It helps to increase the demand for the goods. Customers who purchase goods in larger quantities including wholesalers, retailers, and industrial users tend to receive trade discounts. In general, the larger the purchase, the bigger the trade discount.

In general, there are two types of merchants or demanders of goods in the market, the wholesaler and the retailer. Wholesalers are those who usually purchase goods from manufacturers and producers and sell them to retailers and large consumers. While, retailer merchants are those who purchase goods from several sources, including wholesalers, producers, manufacturers, and agent middlemen such as brokers and manufacturers’ agents. They mainly sell goods to the ultimate consumers.

Wholesalers normally tend to buy in larger amounts than the retailers, and the retailers tend to buy in larger amounts than the consumers. Manufacturers, wholesalers, or other types of sellers frequently grant substantial reductions from the list price quoted in their catalogs to allow for price differentials. Such reductions based on the list price are called trade discounts. Merchants do not need to reprint the catalog when offering trade discounts, and the trade discount is not fixed as it depends on the market situations that control the price fluctuation.

Besides being offered to customers who purchase in bulk, trade discounts also are offered based on customer location, order size, and the customer credit rating. Customers may have the opportunity to receive multiple trade discounts known as chain discounts. When a chain discount or series discount is applied, each subsequent discount is deducted from the balance of the previous discount.

Figure 1: Example of trade discount offered. “Borders Books Ad 2 | The second going out of business notice… |” by Mark Mathosian is licensed under CC BY 4.0

Among economic reasons, merchants offer trade discounts are:

1. 1. To create healthy competition between firms
   2. To finish old/ unused stocks.
   3. To have a quick sale of the stocks before the expiry date.
   4. To encourage sales in terms of bulk or large amounts.

4.2.1 The Formula

The trade discount for a single discount can be computed using the following formula:

Trade discount = List price x Discount rate

Where the selling price is the difference between the list price and the trade discount given, it can be written as:

Selling price = List price – Trade discount

Whereas, the trade discount for chain discount, assumed for three discounts, can be computed as:

Trade discount = List price x [1- ((1-First discount rate)(1-Second discount rate)(1-Third discount rate))]

With the following selling price:

Selling price = List price – Trade discount

There are considerable variations in the methods of payment among different types of businesses. Some firms require immediate payment, while others allow their customers to pay bills within a specified period known as the credit period. If used properly, cash discounts also known as purchase discounts can improve the days-sales-outstanding aspect of a business’s cash conversion cycle.

Many companies offer their customers a discount from the selling price called a cash discount. A cash discount is also called a sales discount by the seller, and a purchase discount by the buyer. Such a discount is to induce an early payment before the expiration of the credit period.

Among reasons to offer cash discount are:

1. To collect receivables quickly. Providing a small cash discount would be beneficial for the seller as it would allow him to have access to the cash sooner. The sooner a seller receives the cash, the earlier he can put the money back into the business to buy more supplies and/or grow the company further.

2. Faster collections reduce debts.
3. Sellers can pay their own bills and hence reduce interest costs.
4. Competitiveness.
5. To finish old stocks/unsalable goods/poor conditioned stocks.
6. To encourage goods sold in bulk/large quantities.

In general, a cash discount term is written as “2/10, n/30”. It means if a payment is made within 10 days from the date of the invoice, a 2% cash discount is allowed, although the debtor is permitted a period of 30 days to pay the bill. However, if the bill is paid after the end of 10 days but on or before the end of the 30-day period, the net amount of the invoice must be paid. After 30 days, the bill will be considered overdue and may be subjected to an interest charge.

However, note that freight costs are not subject to cash discounts and need to be deducted from the invoice.

4.3.1 The Formula

The amount of cash discount is given as:

Cash Discount = Invoice amount (or selling price) x Cash discount rate (%)

Subject to a few conditions, such as:

Normal date: 2/10, n/30.

End of the Month (E.O.M.): 2/10 E.O.M.

Received of Goods (R.O.G.): 2/10 R.O.G.

E.O.M means the discount period starts from the end of the month instead of the invoice date.

R.O.G. means the discount period starts from the date of receipt of the goods.

A negotiable instrument is a document guaranteeing the payment of a specific amount of money, either on demand or at a set time, with the payer named on the document. It is a document contemplated by or consisting of a contract, which promises the payment of money without condition, which may be paid  either on demand or at a future date. Examples: promissory notes, bills of exchange, banknotes, and cheques.

Figure 2: Example of a negotiable instrument

From the figure above, the total of principal and interest is called the maturity value of the note. Given the term of the note is in months, ordinary simple interest is used to calculate the maturity value. An important feature of a promissory note is that it is negotiable. Negotiable, that is, it can be transferred to another payee by the endorsement of the present payee.

Bank discount

Cash discount

Negotiable instrument

Trade discount

An annuity is a sequence of equal payments made at equal periods of time. Insurance premiums, mortgage payments, interest payments on bonds, payments of rent, payments on hire purchases,  and dividends are a few examples of annuities.

When the payments are made at the end of the time period, the annuity is called an ordinary annuity. Examples include loan repayments and interest payments on bonds. When the payments are paid at the beginning of each payment period, the annuity is called an annuity due. Examples include insurance premiums.

The future value (or accumulated value) of an annuity is the total value of the set of payments at the end of the term. Similarly, the present value (or discounted value) of an annuity is the equivalent value of the set of payments due at the beginning of the term.

The future value of an ordinary annuity is defined as the amount due at the end  of the term that is equivalent to the sum of the future value of all the payments comprising the annuity, with the date of the last payment as the focal date. An ordinary annuity is shown on the time diagram below.

Figure 1: Illustration of an ordinary annuity timeline

To calculate the future value of this annuity, we need to accumulate each payment at the end of the term. This may be expressed as follows:

= (1 + i)ⁿ⁻¹ + (1 + i)ⁿ⁻² + … + (1 + i)² + (1 + i)¹ + 1

Written this in the other way around, we have:

= 1 + (1 + i)¹ + (1 + i)² + … + (1 + i)ⁿ⁻² + (1 + i)ⁿ⁻¹

This expression is a geometric progression with n terms, where the first term is 1 and with a common factor of (1 + i). Hence, using the formula of the sum of a geometric progression, we can generate the future value as follows:

=

=

=

The future value at i of RM1 paid at each period for n periods is often  written as:

, read “s angle n at i” where;

The future value of an ordinary annuity of n payments of RMR each, multiply  by R and is written as:

Future value =

The present value (or discounted value) of an annuity is the equivalent value of the set of payments due at the beginning of the term which is one period before the first payment and is equivalent to the sum of the present values of all the payments comprising the annuity.

The relationship between the present value and future value can be written as:

Present value, = Future value × (1 + i)⁻ⁿ

Substitute the formula of future value in section 5.2 we have:

Annuity due is an annuity whose periodic payments are paid at the beginning of each payment period. The term of an annuity due starts at the time of the first payment and ends one payment period after the date of the last payment. This can be illustrated as figure below.

Figure 2: Illustration of an annuity due timeline

Annuity due also known as annuities payable in advance and normally arise in respect of insurance premiums, rents, etc.

We know that the future value of the payments at the end of the (n − 1)th period is R. We then accumulate it for 1 interest period, to obtain the future value of an annuity due, at the end of the term as;

Future value,

While the present value of the payments one period before the first payment is R . To obtain the present value of an annuity due, R (i.e. on the date of the first payment), we accumulate R for one interest period so that:

Present value, = R

Select TRUE or FALSE for each of the following statements.

Annuity

Annuity due

Future value

Ordinary annuity

Present value

Economic life of most assets is limited. After being used for a number of years, it will come to the end of their lives. As the time of retirement, assets may have a small trade-in or scrap value, or even may be worthless.

Computation of depreciation may involve compound interest and ordinary annuity. The computation of the periodic depreciation include allocating the cost of the property as an expense/ a cost of goods manufactured to the proper business operating periods. The amount of accumulated depreciation is known as accumulated depreciation or allowance for depreciation.

The difference between the original cost of an asset and its total amount in the accumulated depreciation is known as book value. Among reasons for depreciation include time factor, decay, worn out, etc.

In this chapter, we will use the following notation:

C          =   original cost of an asset

T          =   estimated trade-in value or scrap value

C – T   =  total depreciation charges or expenses

n     =  useful life of the asset estimated in years, service-hours, or product-units.

r    =  rate of depreciation expense per year, per service-hour, or per product-units.

6.1.1 Method of Averages

6.1.1.1 The Straight-Line Method

The simplest and most popular method of accounting for depreciation is the straight-line method. The depreciation charges are equal for each year. If R is the depreciation expense per year and n is the number of years, then R is the total depreciation charges divide by n, the number of years or can be written as:

R =

6.1.1.2 Service Hours Method

The service hours method relates depreciation to the estimated productive capacity of the asset in terms of its hours of useful service. The depreciation rate per service hour r is:

r =

Where n is the number of service hours.

The depreciation charges for a given year is equal to r multiplies the number of service hours of that year.

6.1.1.3 Product Units Method

The product units method defined depreciation to the estimated number of units produced by each asset during its useful life. The depreciation rate per unit product r is:

r =

Where n is the number of product units. The depreciation charges for a given period equal to r multiplies the number of units produced in that period.

6.1.2 Reducing Charge Method

6.1.2.1 Sum of the years-digit method

The total depreciation is fixed. It is the difference between the original cost and the trade-in or scrap value (C-T). The rate of depreciation is expressed in a changing fraction which becomes smaller each year. The numerator of the fraction is the number of remaining years of life of the asset. Whereas the denominator is the sum of digits that represent the years of life.

6.1.2.2 Constant/ Fixed rate method on diminishing book value

The depreciation expense for earlier year is higher than the later years. The depreciation expense is equal to fixed/ constant annual rate multiply the diminishing book value of an asset. There are two ways of computation: i) the declining balance method, ii) the fixed rate method.

For the declining balance method, first we need to find the annual depreciation rate by dividing 100% by the number of years of useful life of the asset. Then, find the maximum annual depreciation rate by multiplying the annual depreciation rate obtained by 2.

For the fixed rate method, the book value at the end of the life of the asset is equal to the scrap or trade-in value. The rate of annual depreciation charges (r) can be written as:

r =

6.1.3 Compound Interest Method

6.1.3.1 Annuity method

The annuity method resembles the method of amortizing a debt. The depreciation charges are equal and include not only a part of the cost of the asset but also the interest on the book value for each operating period. The step by step calculation are as follows:

Step 1: Find  the present value of the total depreciation charges. The present value is, P=T(1+i)^-n

Step 2: The annual depreciation charges are required to be equal. Thus, we can assume that the present value of an annuity (A_n) with payments consisting of equal depreciation charges as:

C-P = C- T(1+i)^-n = A_n = R

Step 3: R represents the annual depreciation charges, and can be obtained as follows:

R =

6.1.3.2 Sinking fund method

It is assumed that a sinking fund is established for the purpose of replacing an asset at the end of its useful life. The periodic depreciation charges are exactly the same as the periodic increases including the periodic deposit and interest) in the sinking fund. Hence, the depreciation charges are not equal. But the total of the depreciation charges is equal to the amount in the sinking fund (S_n) at the end of the useful life of the asset. The size of each deposit (R) made in the sinking fund is given by the annuity formula:

= R or R  =  

6.1.4 Composite Rate Method

The composite rate method is used for computing  the depreciation charges of a group of assets. The composite rate is obtained by dividing the total annual depreciation charges by the total cost  of  the  group  of assets. The annual depreciation charges of each asset are obtained by the straight line method. The composite rate may be used to compute depreciation charges  during  the  later  years  if  there  is  no  significant changes in the values and useful lives of assets. The work of computing the depreciation expense of each item can be avoided.  The composite rate of an asset can be calculated as:

The composite rate =

The composite life of a group of assets is the average life of the group. In computing the composite life, the annual depreciation charges of the group of assets may be obtained by using: i) the composite rate, ii) the depreciation charges are assumed to be equal for each year.

The composite life =

An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://openbook.ums.edu.my/financialmathematicsineconomics/?p=110#h5p-33

6.1.4.1 The sinking fund method

If the depreciation charges are not equal for each year, it is not categorised as a fixed composite rate. The annual deposits in the fund for each asset are equal during the life of the asset. The composite life is the time necessary for the total annual deposits at the given interest rate to accumulate to the total depreciation charges of the group of assets.

Accumulated depreciation

Book value

Depreciation