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?