Accumulation of Money
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Transcript of Accumulation of Money
ACCUMULATION OF MONEYDISCOUNTING PROBLEM
NOMINAL AND EFFECTIVE RATES
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
COMPUTATION OF PERIODIC RATE AND TIME
• i = Given rate / given time
•n = number of times per year this rate is compounded/ given year
FORMULA
F = P ( 1 + i )n
P = F ( 1 + i )n
I = F - P
EXAMPLE PROBLEM
Determine the amount due at the end of 8.1/2 years if 10,000 is invested 5% compounded monthly .
• GIVEN :
• P = 10,000
• R = 5% COMPOUNDED MONTHLY
• i = 5%• 12
• n = ( 12 x 8.1/2 YEARS ) 102
• 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
EXERCISE PROBLEM
• Accumulate P50,000 for 5.3/4 years at 8% compounded quarterly . Determine the compound interest earned
SOLUTION
• GIVEN :
• P =50,000
• R= 8% compounded quarterly
• i = 8/4 is 2%• n = 4 ( 5.3/4 years ) is 23
• 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
DISCOUNTING PROBLEM
•P = F ( 1 + i )-n
•I = F - P
TAKE NOTE
In Accumulation formula , the exponent is POSITIVE n ( +n)
In Discounting formula , the exponent is NEGATIVE n ( -n)
EXAMPLE
• If money is worth 6% compounded semi-annually determine the present value of 5,000 due at the end of 5 years.
• GIVEN:• F = 5,000• R = 6% compounded semi-
annually • i = 3% • n = ( 5x2 ) = 10
DISCOUTING FORMULA
•P = F ( 1 + i )-n
• P = 5,000 ( 1 + 0.03 )-10
• P = 5,000 ( 0.744093 )• P = 3,720.47
ACCUMULATION FORMULA
P = F ( 1 + i )n
P = 5,000
( 1+ 0.03 )10
P = 5,000 1.343916 P = 3,720.47
EXERCISE PROBLEM
• Discount P50,000 pesos for 20 years at 5% compounded semi-annually . Determine present value and discount earned.
• GIVEN :
• F = 50,000
• R = 5% compounded semi-annually
• i = 2.5% • n = ( 20 x 2 ) 40
•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
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 )
FORMULA
•w = ( 1 + j )m – 1 m•1 + w = ( 1+ j )m
m
EXAMPLE PROBLEM
•What rate converted quarterly yields the effective rate 6 % ?
•GIVEN:
•w = 6%
•m = 4
• j = ?
• 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
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?