ICAI Notes- Mathematics of Finance | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


MATHEMATICS OF FINANCE
4
CHAPTER
After studying this chapter students will be able to understand:
? The concept of interest, related terms and computation thereof;
? Difference between simple and compound interest;
? The concept of annuity;
? The concept of present value and future value;
? Use of present value concept in Leasing, Capital expenditure and Valuation of Bond.
Interest
Annuity Effective rate of
Interest
Simple
Interest
Applications of Annuities Sinking Funds
Leasing Valuation
of Bond
Annuity Types
Capital Expenditure
(Investment
decision)
Future value
of the annuity
regular
Present value
 of annuity
regular
Future value of
annuity due
Present value of
annuity due
Compound
Interest
First Payment or/
Receipt at the end
of the period
First Payment/ Receipt in
the first (beginning) of
the period
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
Page 2


MATHEMATICS OF FINANCE
4
CHAPTER
After studying this chapter students will be able to understand:
? The concept of interest, related terms and computation thereof;
? Difference between simple and compound interest;
? The concept of annuity;
? The concept of present value and future value;
? Use of present value concept in Leasing, Capital expenditure and Valuation of Bond.
Interest
Annuity Effective rate of
Interest
Simple
Interest
Applications of Annuities Sinking Funds
Leasing Valuation
of Bond
Annuity Types
Capital Expenditure
(Investment
decision)
Future value
of the annuity
regular
Present value
 of annuity
regular
Future value of
annuity due
Present value of
annuity due
Compound
Interest
First Payment or/
Receipt at the end
of the period
First Payment/ Receipt in
the first (beginning) of
the period
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
4.2
People earn money for spending it on housing, food, clothing, education, entertainment etc.
Sometimes extra expenditures have also to be met with. For example there might be a marriage
in the family; one may want to buy house, one may want to set up his or her business, one may
want to buy a car and so on. Some people can manage to put aside some money for such expected
and unexpected expenditures. But most people have to borrow money for such contingencies.
From where they can borrow money?
Money can be borrowed from friends or money lenders or Banks. If you can arrange a loan from
your friend it might be interest free but if you borrow money from lenders or Banks you will
have to pay some charge periodically for using money of money lenders or Banks. This charge is
called interest.
Let us take another view. People earn money for satisfying their various needs as discussed
above. After satisfying those needs some people may have some savings. People may invest
their savings in debentures or lend to other person or simply deposit it into bank. In this way
they can earn interest on their investment.
Most of you are very much aware of the term interest. Interest can be defined as the price paid by
a borrower for the use of a lender’s money.
We will know more about interest and other related terms later.
Now question arises why lenders charge interest for the use of their money. There are a variety of
reasons. We will now discuss those reasons.
1. Time value of money: Time value of money means that the value of a unity of money is
different in different time periods. The sum of money received in future is less valuable than
it is today. In other words the present worth of money received after some time will be less
than money received today. Since money received today has more value rational investors
would prefer current receipts to future receipts. If they postpone their receipts they will
certainly charge some money i.e. interest.
2. Opportunity Cost: The lender has a choice between using his money in different investments.
If he chooses one he forgoes the return from all others. In other words lending incurs an
opportunity cost due to the possible alternative uses of the lent money.
3. Inflation: Most economies generally exhibit inflation. Inflation is a fall in the purchasing
power of money. Due to inflation a given amount of money buys fewer goods in the future
than it will now. The borrower needs to compensate the lender for this.
4. Liquidity Preference: People prefer to have their resources available in a form that can
immediately be converted into cash rather than a form that takes time or money to realize.
5. Risk Factor: There is always a risk that the borrower will go bankrupt or otherwise default
on the loan. Risk is a determinable factor in fixing rate of interest.
A lender generally charges more interest rate (risk premium) for taking more risk.
© The Institute of Chartered Accountants of India
Page 3


MATHEMATICS OF FINANCE
4
CHAPTER
After studying this chapter students will be able to understand:
? The concept of interest, related terms and computation thereof;
? Difference between simple and compound interest;
? The concept of annuity;
? The concept of present value and future value;
? Use of present value concept in Leasing, Capital expenditure and Valuation of Bond.
Interest
Annuity Effective rate of
Interest
Simple
Interest
Applications of Annuities Sinking Funds
Leasing Valuation
of Bond
Annuity Types
Capital Expenditure
(Investment
decision)
Future value
of the annuity
regular
Present value
 of annuity
regular
Future value of
annuity due
Present value of
annuity due
Compound
Interest
First Payment or/
Receipt at the end
of the period
First Payment/ Receipt in
the first (beginning) of
the period
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
4.2
People earn money for spending it on housing, food, clothing, education, entertainment etc.
Sometimes extra expenditures have also to be met with. For example there might be a marriage
in the family; one may want to buy house, one may want to set up his or her business, one may
want to buy a car and so on. Some people can manage to put aside some money for such expected
and unexpected expenditures. But most people have to borrow money for such contingencies.
From where they can borrow money?
Money can be borrowed from friends or money lenders or Banks. If you can arrange a loan from
your friend it might be interest free but if you borrow money from lenders or Banks you will
have to pay some charge periodically for using money of money lenders or Banks. This charge is
called interest.
Let us take another view. People earn money for satisfying their various needs as discussed
above. After satisfying those needs some people may have some savings. People may invest
their savings in debentures or lend to other person or simply deposit it into bank. In this way
they can earn interest on their investment.
Most of you are very much aware of the term interest. Interest can be defined as the price paid by
a borrower for the use of a lender’s money.
We will know more about interest and other related terms later.
Now question arises why lenders charge interest for the use of their money. There are a variety of
reasons. We will now discuss those reasons.
1. Time value of money: Time value of money means that the value of a unity of money is
different in different time periods. The sum of money received in future is less valuable than
it is today. In other words the present worth of money received after some time will be less
than money received today. Since money received today has more value rational investors
would prefer current receipts to future receipts. If they postpone their receipts they will
certainly charge some money i.e. interest.
2. Opportunity Cost: The lender has a choice between using his money in different investments.
If he chooses one he forgoes the return from all others. In other words lending incurs an
opportunity cost due to the possible alternative uses of the lent money.
3. Inflation: Most economies generally exhibit inflation. Inflation is a fall in the purchasing
power of money. Due to inflation a given amount of money buys fewer goods in the future
than it will now. The borrower needs to compensate the lender for this.
4. Liquidity Preference: People prefer to have their resources available in a form that can
immediately be converted into cash rather than a form that takes time or money to realize.
5. Risk Factor: There is always a risk that the borrower will go bankrupt or otherwise default
on the loan. Risk is a determinable factor in fixing rate of interest.
A lender generally charges more interest rate (risk premium) for taking more risk.
© The Institute of Chartered Accountants of India
4.3 MATHEMATICS OF FINANCE
Now we can define interest and some other related terms.
4.3.1 Interest
Interest is the price paid by a borrower for the use of a lender’s money. If you borrow (or lend)
some money from (or to) a person for a particular period you would pay (or receive) more money
than your initial borrowing (or lending). This excess money paid (or received) is called interest.
Suppose you borrow (or lend) ` 50,000 for a year and you pay (or receive) ` 55,000 after one year
the difference between initial borrowing (or lending) ` 50,000 and end payment (or receipts) `
55,000 i.e. ` 5,000 is the amount of interest you paid (or earned).
4.3.2 Principal
Principal is initial value of lending (or borrowing). If you invest your money the value of initial
investment is also called principal. Suppose you borrow (or lend) ` 50,000 from a person for one
year. ` 50,000 in this example is the ‘Principal.’ Take another example suppose you deposit `
20,000 in your bank account for one year. In this example ` 20,000 is the principal.
4.3.3 Rate of Interest
The rate at which the interest is charged for a defined length of time for use of principal generally
on a yearly basis is known to be the rate of interest. Rate of interest is usually as expressed as
percentages. Suppose you invest ` 20,000 in your bank account for one year with the interest rate
of 5% per annum. It means you would earn ` 5 as interest every ` 100 of principal amount in a
year.
Per annum means for a year.
4.3.4 Accumulated amount (or Balance)
Accumulated amount is the final value of an investment. It is the sum total of principal and
interest earned. Suppose you deposit ` 50,000 in your bank for one year with an interest rate of
5% p.a. you would earn interest of ` 2,500 after one year. (method of computing interest will be
illustrated later). After one year you will get ` 52,500 (principal+ interest), ` 52,500 is accumulated
amount here.
Amount is also known as the balance.
Now we can discuss the method of computing interest. Interest accrues as either simple interest
or compound interest. We will discuss simple interest and compound interest in the following
paragraphs:
4.4.1 Simple Interest
Now we would know what is simple interest and the methodology of computing simple interest
© The Institute of Chartered Accountants of India
Page 4


MATHEMATICS OF FINANCE
4
CHAPTER
After studying this chapter students will be able to understand:
? The concept of interest, related terms and computation thereof;
? Difference between simple and compound interest;
? The concept of annuity;
? The concept of present value and future value;
? Use of present value concept in Leasing, Capital expenditure and Valuation of Bond.
Interest
Annuity Effective rate of
Interest
Simple
Interest
Applications of Annuities Sinking Funds
Leasing Valuation
of Bond
Annuity Types
Capital Expenditure
(Investment
decision)
Future value
of the annuity
regular
Present value
 of annuity
regular
Future value of
annuity due
Present value of
annuity due
Compound
Interest
First Payment or/
Receipt at the end
of the period
First Payment/ Receipt in
the first (beginning) of
the period
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
4.2
People earn money for spending it on housing, food, clothing, education, entertainment etc.
Sometimes extra expenditures have also to be met with. For example there might be a marriage
in the family; one may want to buy house, one may want to set up his or her business, one may
want to buy a car and so on. Some people can manage to put aside some money for such expected
and unexpected expenditures. But most people have to borrow money for such contingencies.
From where they can borrow money?
Money can be borrowed from friends or money lenders or Banks. If you can arrange a loan from
your friend it might be interest free but if you borrow money from lenders or Banks you will
have to pay some charge periodically for using money of money lenders or Banks. This charge is
called interest.
Let us take another view. People earn money for satisfying their various needs as discussed
above. After satisfying those needs some people may have some savings. People may invest
their savings in debentures or lend to other person or simply deposit it into bank. In this way
they can earn interest on their investment.
Most of you are very much aware of the term interest. Interest can be defined as the price paid by
a borrower for the use of a lender’s money.
We will know more about interest and other related terms later.
Now question arises why lenders charge interest for the use of their money. There are a variety of
reasons. We will now discuss those reasons.
1. Time value of money: Time value of money means that the value of a unity of money is
different in different time periods. The sum of money received in future is less valuable than
it is today. In other words the present worth of money received after some time will be less
than money received today. Since money received today has more value rational investors
would prefer current receipts to future receipts. If they postpone their receipts they will
certainly charge some money i.e. interest.
2. Opportunity Cost: The lender has a choice between using his money in different investments.
If he chooses one he forgoes the return from all others. In other words lending incurs an
opportunity cost due to the possible alternative uses of the lent money.
3. Inflation: Most economies generally exhibit inflation. Inflation is a fall in the purchasing
power of money. Due to inflation a given amount of money buys fewer goods in the future
than it will now. The borrower needs to compensate the lender for this.
4. Liquidity Preference: People prefer to have their resources available in a form that can
immediately be converted into cash rather than a form that takes time or money to realize.
5. Risk Factor: There is always a risk that the borrower will go bankrupt or otherwise default
on the loan. Risk is a determinable factor in fixing rate of interest.
A lender generally charges more interest rate (risk premium) for taking more risk.
© The Institute of Chartered Accountants of India
4.3 MATHEMATICS OF FINANCE
Now we can define interest and some other related terms.
4.3.1 Interest
Interest is the price paid by a borrower for the use of a lender’s money. If you borrow (or lend)
some money from (or to) a person for a particular period you would pay (or receive) more money
than your initial borrowing (or lending). This excess money paid (or received) is called interest.
Suppose you borrow (or lend) ` 50,000 for a year and you pay (or receive) ` 55,000 after one year
the difference between initial borrowing (or lending) ` 50,000 and end payment (or receipts) `
55,000 i.e. ` 5,000 is the amount of interest you paid (or earned).
4.3.2 Principal
Principal is initial value of lending (or borrowing). If you invest your money the value of initial
investment is also called principal. Suppose you borrow (or lend) ` 50,000 from a person for one
year. ` 50,000 in this example is the ‘Principal.’ Take another example suppose you deposit `
20,000 in your bank account for one year. In this example ` 20,000 is the principal.
4.3.3 Rate of Interest
The rate at which the interest is charged for a defined length of time for use of principal generally
on a yearly basis is known to be the rate of interest. Rate of interest is usually as expressed as
percentages. Suppose you invest ` 20,000 in your bank account for one year with the interest rate
of 5% per annum. It means you would earn ` 5 as interest every ` 100 of principal amount in a
year.
Per annum means for a year.
4.3.4 Accumulated amount (or Balance)
Accumulated amount is the final value of an investment. It is the sum total of principal and
interest earned. Suppose you deposit ` 50,000 in your bank for one year with an interest rate of
5% p.a. you would earn interest of ` 2,500 after one year. (method of computing interest will be
illustrated later). After one year you will get ` 52,500 (principal+ interest), ` 52,500 is accumulated
amount here.
Amount is also known as the balance.
Now we can discuss the method of computing interest. Interest accrues as either simple interest
or compound interest. We will discuss simple interest and compound interest in the following
paragraphs:
4.4.1 Simple Interest
Now we would know what is simple interest and the methodology of computing simple interest
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
4.4
and accumulated amount for an investment (principal) with a simple rate over a period of time.
As you already know the money that you borrow is known as principal and the additional money
that you pay for using somebody else’s money is known as interest. The interest paid for keeping
` 100 for one year is known as the rate percent per annum. Thus if money is borrowed at the rate
of 8% per annum then the interest paid for keeping ` 100 for one year is ` 8. The sum of principal
and interest is known as the Amount.
Clearly the interest you pay is proportionate to the money that you borrow and also to the period
of time for which you keep the money; the more the money and the time, the more the interest.
Interest is also proportionate to the rate of interest agreed upon by the lending and the borrowing
parties. Thus interest varies directly with principal, time and rate.
Simple interest is the interest computed on the principal for the entire period of borrowing. It is
calculated on the outstanding principal balance and not on interest previously earned. It means
no interest is paid on interest earned during the term of loan.
Simple interest can be computed by applying following formulas:
I = Pit
A = P + I
= P + Pit
= P(1 + it)
I = A – P
Here,
A = Accumulated amount (final value of an investment)
P = Principal (initial value of an investment)
i = Annual interest rate in decimal.
I = Amount of Interest
t = Time in years
Let us consider the following examples in order to see how exactly are these quantities
related.
Example 1: How much interest will be earned on ` 2000 at 6% simple interest for 2 years?
Solution: Required interest amount is given by
I = P × i × t
= 2,000 × 
6
100
 × 2
= ` 240
Example 2: Sania deposited ` 50,000 in a bank for two years with the interest rate of 5.5% p.a.
How much interest would she earn?
Solution: Required interest amount is given by
© The Institute of Chartered Accountants of India
Page 5


MATHEMATICS OF FINANCE
4
CHAPTER
After studying this chapter students will be able to understand:
? The concept of interest, related terms and computation thereof;
? Difference between simple and compound interest;
? The concept of annuity;
? The concept of present value and future value;
? Use of present value concept in Leasing, Capital expenditure and Valuation of Bond.
Interest
Annuity Effective rate of
Interest
Simple
Interest
Applications of Annuities Sinking Funds
Leasing Valuation
of Bond
Annuity Types
Capital Expenditure
(Investment
decision)
Future value
of the annuity
regular
Present value
 of annuity
regular
Future value of
annuity due
Present value of
annuity due
Compound
Interest
First Payment or/
Receipt at the end
of the period
First Payment/ Receipt in
the first (beginning) of
the period
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
4.2
People earn money for spending it on housing, food, clothing, education, entertainment etc.
Sometimes extra expenditures have also to be met with. For example there might be a marriage
in the family; one may want to buy house, one may want to set up his or her business, one may
want to buy a car and so on. Some people can manage to put aside some money for such expected
and unexpected expenditures. But most people have to borrow money for such contingencies.
From where they can borrow money?
Money can be borrowed from friends or money lenders or Banks. If you can arrange a loan from
your friend it might be interest free but if you borrow money from lenders or Banks you will
have to pay some charge periodically for using money of money lenders or Banks. This charge is
called interest.
Let us take another view. People earn money for satisfying their various needs as discussed
above. After satisfying those needs some people may have some savings. People may invest
their savings in debentures or lend to other person or simply deposit it into bank. In this way
they can earn interest on their investment.
Most of you are very much aware of the term interest. Interest can be defined as the price paid by
a borrower for the use of a lender’s money.
We will know more about interest and other related terms later.
Now question arises why lenders charge interest for the use of their money. There are a variety of
reasons. We will now discuss those reasons.
1. Time value of money: Time value of money means that the value of a unity of money is
different in different time periods. The sum of money received in future is less valuable than
it is today. In other words the present worth of money received after some time will be less
than money received today. Since money received today has more value rational investors
would prefer current receipts to future receipts. If they postpone their receipts they will
certainly charge some money i.e. interest.
2. Opportunity Cost: The lender has a choice between using his money in different investments.
If he chooses one he forgoes the return from all others. In other words lending incurs an
opportunity cost due to the possible alternative uses of the lent money.
3. Inflation: Most economies generally exhibit inflation. Inflation is a fall in the purchasing
power of money. Due to inflation a given amount of money buys fewer goods in the future
than it will now. The borrower needs to compensate the lender for this.
4. Liquidity Preference: People prefer to have their resources available in a form that can
immediately be converted into cash rather than a form that takes time or money to realize.
5. Risk Factor: There is always a risk that the borrower will go bankrupt or otherwise default
on the loan. Risk is a determinable factor in fixing rate of interest.
A lender generally charges more interest rate (risk premium) for taking more risk.
© The Institute of Chartered Accountants of India
4.3 MATHEMATICS OF FINANCE
Now we can define interest and some other related terms.
4.3.1 Interest
Interest is the price paid by a borrower for the use of a lender’s money. If you borrow (or lend)
some money from (or to) a person for a particular period you would pay (or receive) more money
than your initial borrowing (or lending). This excess money paid (or received) is called interest.
Suppose you borrow (or lend) ` 50,000 for a year and you pay (or receive) ` 55,000 after one year
the difference between initial borrowing (or lending) ` 50,000 and end payment (or receipts) `
55,000 i.e. ` 5,000 is the amount of interest you paid (or earned).
4.3.2 Principal
Principal is initial value of lending (or borrowing). If you invest your money the value of initial
investment is also called principal. Suppose you borrow (or lend) ` 50,000 from a person for one
year. ` 50,000 in this example is the ‘Principal.’ Take another example suppose you deposit `
20,000 in your bank account for one year. In this example ` 20,000 is the principal.
4.3.3 Rate of Interest
The rate at which the interest is charged for a defined length of time for use of principal generally
on a yearly basis is known to be the rate of interest. Rate of interest is usually as expressed as
percentages. Suppose you invest ` 20,000 in your bank account for one year with the interest rate
of 5% per annum. It means you would earn ` 5 as interest every ` 100 of principal amount in a
year.
Per annum means for a year.
4.3.4 Accumulated amount (or Balance)
Accumulated amount is the final value of an investment. It is the sum total of principal and
interest earned. Suppose you deposit ` 50,000 in your bank for one year with an interest rate of
5% p.a. you would earn interest of ` 2,500 after one year. (method of computing interest will be
illustrated later). After one year you will get ` 52,500 (principal+ interest), ` 52,500 is accumulated
amount here.
Amount is also known as the balance.
Now we can discuss the method of computing interest. Interest accrues as either simple interest
or compound interest. We will discuss simple interest and compound interest in the following
paragraphs:
4.4.1 Simple Interest
Now we would know what is simple interest and the methodology of computing simple interest
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
4.4
and accumulated amount for an investment (principal) with a simple rate over a period of time.
As you already know the money that you borrow is known as principal and the additional money
that you pay for using somebody else’s money is known as interest. The interest paid for keeping
` 100 for one year is known as the rate percent per annum. Thus if money is borrowed at the rate
of 8% per annum then the interest paid for keeping ` 100 for one year is ` 8. The sum of principal
and interest is known as the Amount.
Clearly the interest you pay is proportionate to the money that you borrow and also to the period
of time for which you keep the money; the more the money and the time, the more the interest.
Interest is also proportionate to the rate of interest agreed upon by the lending and the borrowing
parties. Thus interest varies directly with principal, time and rate.
Simple interest is the interest computed on the principal for the entire period of borrowing. It is
calculated on the outstanding principal balance and not on interest previously earned. It means
no interest is paid on interest earned during the term of loan.
Simple interest can be computed by applying following formulas:
I = Pit
A = P + I
= P + Pit
= P(1 + it)
I = A – P
Here,
A = Accumulated amount (final value of an investment)
P = Principal (initial value of an investment)
i = Annual interest rate in decimal.
I = Amount of Interest
t = Time in years
Let us consider the following examples in order to see how exactly are these quantities
related.
Example 1: How much interest will be earned on ` 2000 at 6% simple interest for 2 years?
Solution: Required interest amount is given by
I = P × i × t
= 2,000 × 
6
100
 × 2
= ` 240
Example 2: Sania deposited ` 50,000 in a bank for two years with the interest rate of 5.5% p.a.
How much interest would she earn?
Solution: Required interest amount is given by
© The Institute of Chartered Accountants of India
4.5
I = P × i × t
= ` 50,000 × 
5.5
100
 × 2
= ` 5,500
Example 3: In example 2 what will be the final value of investment?
Solution: Final value of investment is given by
A = P(1 + it)
= ` 50,000 
? ?
? ?
? ?
5.5
1+ ×2
100
= ` 50,000 
? ?
? ?
? ?
11
1+
100
= ` 
50,000×111
100
= ` 55,500
   or
A = P + I
= `
 
(50,000 + 5,500)
= ` 55,500
Example 4: Sachin deposited ` 1,00,000 in his bank for 2 years at simple interest rate of 6%. How
much interest would he earn? How much would be the final value of deposit?
Solution: (a) Required interest amount is given by
I = P × it
= ` 1,00,000 × 
6
100
 × 2
= ` 12,000
(b) Final value of deposit is given by
A = P + I
= ` (1,00,000 + 12,000)
= ` 1,12,000
Example 5: Find the rate of interest if the amount owed after 6 months is ` 1050, borrowed
amount being ` 1000.
Solution: We know A = P + Pit
i.e. 1050 = 1000 + 1000 × i × (6/12)
MATHEMATICS OF FINANCE
© The Institute of Chartered Accountants of India