1. University of South Bohemia in České Budějovice, Faculty of Agriculture FINANCIAL AND ACTUARIAL MATHEMATICS
INTEREST

2. INTEREST money paid as income on investments or loans
principle – money invested or borrowed on which interest is paid
interest rate – percentual expression of interest
accrued value – sum of principal and interest due

3. TYPES OF INTEREST
simple interest – interest paid once only at the end of the term compound interest – interest which is paid both on the original amount of money invested or borrowed and on any interest which that original amount has collected over a period of time

4. SIMPLE INEREST
interest paid on investments or loans once only at the end of the term
P – principle (present value)
t – time (term)
r – interest rate
A – accrued value (future value)

5. SIMPLE INTEREST
accumulation factor

6. Example: How much money is it necessary to pay back if we borrow CZK 100 for the period of 2 years and a simple interest rate of 8% is charged? How much is the interest?
P = 100, r = 8%  0.08, t = 2.

7. P = 100, r = 8%  0.08, t = 2. I = P  r  t  I = 100  0.08  2 = 16
A = P + I  A = = 116 or
A = P  (1 + r  t )  A = 100  (  2) = = 100  1.16 = 116
It is necessary to repay CZK 116 of which CZK 16 is the interest.

8. How to calculate the time ?
Example: What was the accrued value obtained on 8th July 2001 making the deposit of CZK 1,000 into a bank account on 23rd May 1999 at a 8% simple interest rate?
How to calculate the time ?

9. TYPES OF INTEREST (regarding time calculation)
ordinary
banker’s
exact
30 days
exact number
360 days
365 (366)
1 months =
1 year =

10. TYPES OF INTEREST (regarding time calculation)
ordinary
banker’s
exact
30 days
exact number
360 days
365 (366)
1 months =
1 year =
German method (European Standard)

11. TYPES OF INTEREST (regarding time calculation)
ordinary
banker’s
exact
30 days
exact number
360 days
365 (366)
1 months =
1 year =
French method

12. TYPES OF INTEREST (regarding time calculation)
ordinary
banker’s
exact
30 days
exact number
360 days
365 (366)
1 months =
1 year =
British method

13. TIME CALCULATION
23rd May 1999 – 8th March 2001
ORDINARY INTEREST

14. TIME CALCULATION
23rd May 1999 – 8th March 2001
BANKER’S INTEREST

15. TIME CALCULATION
23rd May 1999 – 8th March 2001
EXACT INTEREST

16. exact interest: A = 1,000  (1 + 0.08  ) = 1,433.42
Example: What was the accrued value obtained on 8th March making the deposit 100 CZK into a bank account on 23rd May 1999 at 8% simple interest rate? A = P  (1 + r  t )
ordinary interest: A = 1,000  (  ) = 1,433.33
banker’s interest: A = 1,000  (  ) = 1,453.33
exact interest: A = 1,000  (  ) = 1,433.42

17. A = P  (1 + r  t )  P = A  (1 + r  t )-1
Example: What was the deposit made into a bank account on 23rd May 1999 at 8% simple ordinary interest rate if the amount of CZK 100 was withdrawn on 8th March 2001?
A = P  (1 + r  t )  P = A  (1 + r  t )-1
principle (present value) determinatiton
discounting

18. BANK (SIMPLE) DISCOUNT
interest on an annual basis deducted in advance on a loan
I = P  r  t  D = A  d  t
discount factor

19. Example: Determine the sum received by the borrower who requests £500 for 60 days if the bank charges 8% simple interest in advance (8% bank discount) on short-term loans.
A = 500, d = 0.08, t =
P = A  (1 – d  t)  500  (1 – 0.08  ) =
The borrower will receive £ and he will pay
back £500 after the period of two months.

20. BILL OF EXCHANGE
a bill of exchange is an unconditional order in writing, addressed by one person to another, signed by the person giving it (drawer), requiring the person (drawee) to whom it is addressed to pay on demand or at a fixed period in the future, a sum of money, to the order of a specified person (payee).

21. BILL OF EXCHANGE / DISCOUNTED BILL
drawer = bill issuer
drawee = bill acceptor
payee = person who is paid
face (nominal) value = value written on bill
of exchange
maturity = date when bill of exchange is due
for payment

22. DISCOUNTING OF BILLS OF EXCHANGE
to discount bills of exchange = to sell bills of exchange for less than the value written on them in order to cash them before their maturity date.

23. $ 1, New York, May 11, 1995
Ninety Days after date, I promise to pay to the order of J. D. Green Fifteen hundred and 00/100 dollars.
(signed) J. B. Smith
face value
payee
drawer

24. Example: On July 3, 1995, Mr. Green sold the bill of exchange shown below to a bank at 9% bank discount rate.
$ 1, New York, May 11, 1995
Ninety Days after date, I promise to pay to the order of J. D. Green Fifteen hundred and 00/100 dollars.
(signed) J. B. Smith

25. On July 3, 1995, Mr. Green sold the bill of exchange with a face value of $1,500 to a bank at 9% bank discount rate. How much money was he paid by the bank?
A = 1,500; d = 0.09; t (11/05/ - 03/07) =
P = A · (1 – d · t) = 1,500 · (1 – 0.09 · ) =
1,500 · (1 - 0,013) = 1,480.50
Mr. Green was paid $ 1,

27. INTEREST CALCULATION
English Method
German Method
French Method

28. ENGLISH METHOD 31/12 ----- 3,000 15/04 deposit 1,500 4,500 15/06
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
15/04
deposit
1,500
4,500
15/06
withdrawal
1,000
3,500
01/10
01/01

29. ENGLISH METHOD 31/12 ----- 3,000 105 15/04 deposit 1,500 4,500 60
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
105
15/04
deposit
1,500
4,500 60
15/06
withdrawal
1,000
3,500
01/10 90
360
01/01

30. ENGLISH METHOD 31/12 ----- 3,000 105 3,150 15/04 deposit 1,500 4,500
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
105
3,150
15/04
deposit
1,500
4,500 60
15/06
withdrawal
1,000
3,500
01/10 90
360
01/01

31. ENGLISH METHOD 31/12 ----- 3,000 105 3,150 15/04 deposit 1,500 4,500
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
105
3,150
15/04
deposit
1,500
4,500 60
2,700
15/06
withdrawal
1,000
3,500
3,675
01/10 90
4,050
360
01/01

32. ENGLISH METHOD interest rate : r = 3% 31/12 ----- 3,000 105 3,150
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
105
3,150
15/04
deposit
1,500
4,500 60
2,700
15/06
withdrawal
1,000
3,500
3,675
01/10 90
4,050
360
 13,575
01/01
interest rate : r = 3%

33. ENGLISH METHOD interest rate: r = 3% 31/12 ----- 3,000 105 3,150 15/04
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
105
3,150
15/04
deposit
1,500
4,500 60
2,700
15/06
withdrawal
1,000
3,500
3,675
01/10 90
4,050
360
 13,575
113.13
01/01
4,613.13
interest rate: r = 3%

34. GERMAN METHOD 31/12 ----- 3,000 15/04 deposit 1,500 4,500 15/06
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
15/04
deposit
1,500
4,500
15/06
withdrawal
1,000
3,500
01/10
01/01

35. GERMAN METHOD 31/12 ----- 3,000 360 15/04 deposit 1,500 4,500 255
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
360
15/04
deposit
1,500
4,500
255
15/06
withdrawal
1,000
3,500
195
01/10 90
01/01

36. GERMAN METHOD 31/12 ----- 3,000 360 10,800 15/04 deposit 1,500 4,500
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
360
10,800
15/04
deposit
1,500
4,500
255
15/06
withdrawal
1,000
3,500
195
01/10 90
01/01

37. GERMAN METHOD 31/12 ----- 3,000 360 10,800 15/04 deposit 1,500 4,500
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
360
10,800
15/04
deposit
1,500
4,500
255
3,825
15/06
withdrawal
1,000
3,500
195
-1,950
01/10 90
900
 13,575
01/01

38. GERMAN METHOD interest rate: r = 3% 31/12 ----- 3,000 105 3,150 15/04
date
bank operation
amount
balance
number
of days
interest
31/12
-----
3,000
105
3,150
15/04
deposit
1,500
4,500 60
2,700
15/06
withdrawal
1,000
3,500
3,675
01/10 90
4,050
 13,575
113.13
01/01
4,613.13
interest rate: r = 3%

39. Make all necessary calculations in the given account to find the interest (r = 2%):
date
bank operation
amount
balance
number
of days
interest
31/12
-----
10,000
10/02
withdrawal
3,500
28/08
2,000
03/11
deposit
6,000 ?
01/01

40. Make all necessary calculations in the given account to find the interest (r = 2%):
date
bank operation
amount
balance
number
of days
interest
31/12
-----
10,000
10/02
withdrawal
3,500
6,500
28/08
2,000
4,500
03/11
deposit
6,000
10,500 ?
01/01

41. Make all necessary calculations in the given account to find the interest (r = 2%):
date
bank operation
amount
balance
number
of days
interest
31/12
-----
10,000 40
10/02
withdrawal
3,500
6,500
198
28/08
2,000
4,500 65
03/11
deposit
6,000
10,500 57
 360 ?
01/01

42. Make all necessary calculations in the given account to find the interest (r = 2%):
date
bank operation
amount
balance
number
of days
interest
31/12
-----
10,000 40
4,000
10/02
withdrawal
3,500
6,500
198
12,870
28/08
2,000
4,500 65
2,925
03/11
deposit
6,000
10,500 57
5,985
 360
 25,780 ?
01/01

43. Make all necessary calculations in the given account to find the interest (r = 2%):
date
bank operation
amount
balance
number
of days
interest
31/12
-----
10,000 40
4,000
10/02
withdrawal
3,500
6,500
198
12,870
28/08
2,000
4,500 65
2,925
03/11
deposit
6,000
10,500 57
5,985
 360
 25,780
214.83
01/01
10,714.83