This document discusses accumulation of money, discounting problems, and nominal vs effective interest rates. It provides formulas for accumulation, discounting, and calculating effective rates. Examples are provided to demonstrate accumulating money over time at a given interest rate, discounting a future amount to calculate present value at a given rate, and determining the nominal rate when an effective rate is given and compounded a specific number of times per period.

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?