Accumulation of money Discounting problem Nominal and effective rates

1. ACCUMULATION OF MONEY
DISCOUNTING PROBLEM
NOMINAL AND EFFECTIVE RATES

3. ACCUMULATION OF MONEY
 P = principal amount of the loan or
investment
 i = periodic interest rate
 n = Time period (term) of the loan or
investment
 F = Maturity value of the loan or
investment
 I = Amount of interest paid or received

4. COMPUTATION OF PERIODIC
RATE AND TIME
•i = Given rate / given
time
•n = number of times per year
this rate is compounded/ given
year

5. FORMULA
 F = P ( 1 + i )n
 P = F
( 1 + i )n
 I = F - P

6. EXAMPLE PROBLEM
 Determine the amount
due at the end of 8.1/2
years if 10,000 is
invested 5%
compounded monthly .

7. • GIVEN :
• P = 10,000
• R = 5% COMPOUNDED MONTHLY
• i = 5%
• 12
• n = ( 12 x 8.1/2 YEARS ) 102

8. • F = P ( 1 + i )n
• F = 10,000 ( 1 + 0.05 )102
• 12
• F = 10,000 ( 1.528240 )
• F = 15,282.40
• I = F – P
• I = 15,282.40 – 10,000
• I = 5,282.40

9. EXERCISE PROBLEM
• Accumulate P50,000 for
5.3/4 years at 8%
compounded quarterly .
Determine the compound
interest earned

10. SOLUTION
• GIVEN :
• P =50,000
• R= 8% compounded quarterly
• i = 8/4 is 2%
• n = 4 ( 5.3/4 years ) is 23

11. • F = P ( 1 + i )n
• SOLUTION:
• F = 50,000 ( 1 + 0.02 )23
• F = 50,000 ( 1.576899 )
• F = 78,844.95
• I = F-P
• I = 78,844.95 – 50,000
• I = 28,844.95 INTEREST
EARNED

12. DISCOUNTING PROBLEM
•P = F ( 1 + i )-n
•I = F - P

13. TAKE NOTE
 In Accumulation formula , the
exponent is POSITIVE n ( +n)
In Discounting formula , the
exponent is NEGATIVE n ( -n)

14. EXAMPLE
• If money is worth 6%
compounded semi-annually
determine the present value
of 5,000 due at the end of 5
years.

15. • GIVEN:
• F = 5,000
• R = 6% compounded semi-annually
• i = 3%
• n = ( 5x2 ) = 10

16. DISCOUTING FORMULA
•P = F ( 1 + i )-n
• P = 5,000 ( 1 + 0.03 )-10
• P = 5,000 ( 0.744093 )
• P = 3,720.47

17. ACCUMULATION FORMULA
P = F
( 1 + i )n
 P = 5,000
( 1+ 0.03 )10
 P = 5,000
1.343916
 P = 3,720.47

18. EXERCISE PROBLEM
• Discount P50,000 pesos for
20 years at 5%
compounded semi-annually
. Determine
present value and discount
earned.

19. • GIVEN :
• F = 50,000
• R = 5% compounded semi-annually
• i = 2.5%
• n = ( 20 x 2 ) 40

20. •P = F ( 1 + i )-n
• P = 50,000 ( 1 + 0.025 )-40
• P = 50,000 ( 0.372430 )
• P = 18,621.50
• I = F- P
• I = 50,000 – 18,621.50
• I = 31,378.50 DISCOUNT EARNED

21. NOMINAL AND EFFECTIVE
RATES
• j = nominal rate
• m = number of times per year this
rate is compounded
• i = j/m
• w = if compounded annually , is
equivalent to given rate ( effective
rate )

22. FORMULA
•w = ( 1 + j )m – 1
m
•1 + w = ( 1+ j )m
m

23. EXAMPLE PROBLEM
•What rate converted
quarterly yields the
effective rate 6 % ?

24. •GIVEN:
•w = 6%
•m = 4
•j = ?

25. • 1 + w = ( 1+ j )m j = 4 ( 1.01467385 – 1 )
m
• 1 + 0.06 = ( 1 + j )4 j = 4 ( 0.01467385 )
4
• √( 1.06 )1 = 1 + j j = 0.0586 or 5.86%
4
• ( 1.06 )1/4 = 1 + j
4
• ( 1.06 )1/4 - 1 = j
4
• 4[( 1.06 )1/4 – 1 ] = j

26. QUIZ
• 1. How much will be the compounded interest of
P1,000 pesos is invested for one year at 6%
compounded monthly?
• 2. Discount P32,000 for 16.1/2 years at 8%
compounded monthly. How much is the
discount?
• 3. How much should you invest today to provide
your only son a brand new car worth P
1,000,000 as a graduation gift 5 years from now
if money is worth 6% compounded semi-annually?