ppt's-sem-II- business mathematics-ii

1. 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)

2. 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 P
at 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.

3. 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.

4. 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.

5. 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
NR
A P 
(1 )
100
NKR
A P
K
 

6. 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

7. Solution
a. Annually.
Here, P = 1000, R = 8, K = 1 and N = 3
3*1
3
3
(1 )
100
8
1000(1 )
100*1
108
1000( )
100
1000(1.08)
1259.712
1260
NKR
A P
K
 
 





8. b. Semiannually.
Here, P = 1000, R = 8, N = 3 and K = 2.
3*2
6
6
(1 )
100
8
1000(1 )
100*2
208
1000( )
200
1000(1.04)
1000(1.2653)
1265.319
1265
NKR
A P
K
 
 






9. c. Quarterly.
Here, P = 1000, R = 8, N =3 and K = 4.
3*4
12
12
(1 )
100
8
1000(1 )
100*4
408
1000( )
400
1000(1.02)
1000(1.2682)
1268.24
1268
NKR
A P
K
 
 






10. d. Monthly.
Here, P = 1000, R = 8, N = 3 and K = 12.
3*12
36
36
(1 )
100
8
1000(1 )
100*12
1208
1000( )
1200
1000(1.001)
1000(1.2702)
1270.23
1270
NKR
A P
K
 
 






11. e. Daily.
Here, P = 1000, R = 8, N= 3 and K = 365.
3*365
1095
1095
(1 )
100
8
1000(1 )
100*365
36508
1000( )
36500
1000(1.0002)
1000(1.2712)
1271.21
1271
NKR
A P
K
 
 






12. 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.

13. 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.

14. The amount A is given by
Interest = A – P = 4330 – 4000 = 330
1*4
4
(1 )
100
8
4000(1 )
100*4
4000(1 0.02)
4000*1.08243
4329.73 4330
NKR
A P
K
 
 
 

 

15. 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%.

16. 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.

17. 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
a
V
i i
 


18. • 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
1
1
1
nk
ak
V
i i
k
 
 
  
  
  
  
 
1
1 1
(1 )n
a
V i
i i
 
    

19. • If annuity is paid or received ‘k’ times in a
year at the beginning of each period, then the
formula becomes
1
1 1
1
nk
i ak
V
k i i
k
 
 
           
  
  

20. 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.

21. • 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 ) 1n
i
A a
i
  
  
 

22. 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”

23. References
• www.shsu.edu/ldg005/data/mth199/chapter4
• Business Mathematics by G.C. Patel and
A.G.Patel by Atul Prakashan