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)
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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