UNIT-4

MATHEMATICS IN FINANCE

Points to be covered:

• Simple and Compound interest,

• nominal and effective rate of interest,

• concept of present value and amount of a sum,

• Annuity (only for a fixed period of time),

• present value of annuity,

• Sinking funds (with equal payments and equal

time intervals)

Simple Interest (S.I)

• Simple interest is the interest that is computed on the original principal only.

• If I denotes the interest on a principal Pat an interest rate of R per year for T years, then we have

I = P.R.T

• The accumulated amount A, the sum of the principal and interest after t years is given by

A = P + I = P + P.R.T

= P(1 + R.T)

and is a linear function of T.

Compound Interest

• When the interest at the end of a specified period

is added to the principal and the interest for the

next period is calculated on this aggregate

amount, it is called compound interest.

Example:

Rs. 5000 are borrowed for 2 yrs at 12% rate of interest.

• The interest of the first year is:

• I = P.R.T = 5000* 0.12* 1 = Rs. 600.

• Hence the aggregate amount at the end of the first year is

• A = P + I = 5000 + 600 = 5600.

• The interest for the second year is calculated on this amount.

• The interest on Rs. 5600 for the second year is

• I = P.R.T = 5600* 0.12* 1 = Rs. 672.

• Hence the aggregate amount at the end of the second year is

• A = P + I = 5600 + 672 = 6272.

• Hence the amount for the interest for two years

• = Aggregate amount – Principal amount

• = Rs. 6272 – Rs. 5000

• = Rs. 1272.

Formula For Compound Interest

• If the interest is calculated on yearly basis,

• Where A = Amount

P= Principal

R= Rate per interest

N= No. of years.

• If the interest is calculated on half yearly, quarterly or monthly

basis, the formula is

(1 )100

NRA P

(1 )100

NKRA P

K

Example

• Find the accumulated amount after 3 years if

$1000 is invested at 8% per year compounded

a. Annually

b. Semiannually

c. Quarterly

d. Monthly

e. Daily

Solution

a. Annually.

Here, P = 1000, R = 8, K = 1 and N = 3

3*1

3

3

(1 )100

81000(1 )

100*1

1081000( )

100

1000(1.08)

1259.712

1260

NKRA P

K

b. Semiannually.

Here, P = 1000, R = 8, N = 3 and K = 2.

3*2

6

6

(1 )100

81000(1 )

100*2

2081000( )

200

1000(1.04)

1000(1.2653)

1265.319

1265

NKRA P

K

c. Quarterly.

Here, P = 1000, R = 8, N =3 and K = 4.

3*4

12

12

(1 )100

81000(1 )

100*4

4081000( )

400

1000(1.02)

1000(1.2682)

1268.24

1268

NKRA P

K

d. Monthly.

Here, P = 1000, R = 8, N = 3 and K = 12.

3*12

36

36

(1 )100

81000(1 )

100*12

12081000( )

1200

1000(1.001)

1000(1.2702)

1270.23

1270

NKRA P

K

e. Daily.

Here, P = 1000, R = 8, N= 3 and K = 365.

3*365

1095

1095

(1 )100

81000(1 )

100*365

365081000( )

36500

1000(1.0002)

1000(1.2712)

1271.21

1271

NKRA P

K

Effective Rate of Interest

• If a sum of Rs. 100 is invested at R% rate of

interest, compounded yearly, the interest will

be Rs. R for one year.

• But if the interest is compounded half yearly,

quarterly or monthly, the total yearly interest

on Rs. 100 will certainly be more than Rs. R.

• This interest is known as effective rate of

interest.

• R% is known as nominal rate of interest.

EXAMPLE

Rs. 4000 are invested for one year at 8%

compound rate of interest and the interest is

calculated quarterly, what is the effective rate

of interest?

Solution:

Here P= 4000, R = 8, K = 4, N= 1.

Also, R = 8 is known as nominal rate of interest.

The amount A is given by

Interest = A – P = 4330 – 4000 = 330

1*4

4

(1 )100

84000(1 )

100*4

4000(1 0.02)

4000*1.08243

4329.73 4330

NKRA P

K

1 year’s simple interest

I = PR’N / 100

330 = (4000 * R’ * 1)/ 100

R’ = (330 * 100)/ 4000

R’ = 8.25

Effective rate of interest is 8.25%.

ANNUITY

• A fixed amount received or paid in equal installments at equal intervals under a contract is known as annuity.

• For example, sum deposited in cumulative time deposit in a post office, payment of installment of a loan taken etc.

• Generally annuity is calculated on yearly basis.

• But it can be calculated on half yearly, quarterly or monthly basis also.

• The amount of annuity is the sum of all payments with the accumulated interest.

Present Value of Annuity

• The sum at present which is equivalent to the total value of annuity to be paid in future is called the present value of Annuity.

• Formula for present value of annuity, if it is paid on yearly basis at the end of each year is

• Where V = present value of annuity

• a = periodic payment

• n= no. of payment periods

• i = R/100 = annual interest per rupee

1[1 ]

(1 )n

aV

i i

• If annuity is paid or received ‘k’ times in a year

at the end of each period, is

• If annuity is paid or received on yearly basis at

the beginning of each year, then

11

1

nk

akV

i i

k

1

1 1(1 )n

aV i

i i

• If annuity is paid or received ‘k’ times in a

year at the beginning of each period, then the

formula becomes

11 1

1

nk

i akV

k i i

k

Sinking Fund

• A fund created by setting aside a fixed contribution periodically and investing at compound interest to accumulate is known as sinking fund or pay back fund.

• Public companies satisfy their long term capital needs either by issuing shares or debentures or taking long term loans.

• They have to repay the borrowed money at the end of a definite time period.

• Besides funds are required in large amount, to replace old assets at the end of their useful life.

• For this purpose, many companies set aside certain amount out of their profit, at the end of each year.

• The fund thus accumulated is known as sinking fund.

• The sum ‘a’ to be transferred to the sinking fund can be calculated using the following formula for the present value A of annuity.

Where

A = sum required to fulfill certain liabilities

a = the sum to be transferred to the sinking fund every year.

i = annual interest per rupee on the investment of sinking fund = R/100

n = number of years.

(1 ) 1niA a

i

Difference between Annuity and

Sinking Fund

Sr. No. Annuity Sinking Fund

1. In an annuity you put a certain amount of

money each period into an account. The

longer a payment has been in the account

the more interest it earns.

A sinking fund is an account in which you

are withdrawing a certain amount each

period.

2. The classic example of an annuity is a

retirement fund: you might put $350 each

month into your retirement fund and by

the time you retire you have a nice little

nest egg.

For example, after you retire you withdraw

a monthly stipend from your retirement

fund.

3. With an annuity you have to wait till

you’ve made all your payments into it to

know the total value.

You currently (presently) have amassed

(collective) the total value of a sinking

fund.

4. for an annuity we know the “Future

Value.”

For a sinking fund we know the “Present

Value”

References

• www.shsu.edu/ldg005/data/mth199/chapter4

• Business Mathematics by G.C. Patel and

A.G.Patel by Atul Prakashan