Find the accumulated value of an investment of $ 20 comma 000 for 5 years at an interest rate of 6.5 % if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly d. compounded continuously.

a. $27,538

b. $27,608

c. $27,662

d. $54,366

Explanation:

Future value of the investment is the sum of principal value and interest value including its compounding effect.

Use following formula of FV to calculate the Value of Investment.

FV = PV x ( 1 + r )^n

As per given data

PV = $20,000

Number of years = 5 years

r = 6.5%

FV = ?

a. compounded semiannually

n = 5 x 2 = 10 compounding periods

r = 6.5% / 2 = 3.25%

FV = $20,000 x ( 1 + 3.25% )^10 = $27,538

b. compounded quarterly

n = 5 x 4 = 20 compounding periods

r = 6.5% / 4 = 1.625%

FV = $20,000 x ( 1 + 1.625% )^20 = $27,608

c. compounded monthly

n = 5 x 12 = 60 compounding periods

r = 6.5% / 12 = 0.542%

FV = $20,000 x ( 1 + 0.542% )^60 = $27,662

d. compounded continuously.

FV   =   Pe^Yr = $20,000 x 2.71828 = $54,366

2.71828 is a constant value for continuous compounding

A) $27,537.89

B) $27,608.40

C) $27,656.90

D) $27,680.61

Explanation:

present value = $20,000

n = 10 , 20 , 60

i = 3.25% , 1.625% , 0.5417%

a. compounded semiannually;

future value = $20,000 x 1.0325¹⁰ = $27,537.89

b. compounded quarterly;

future value = $20,000 x 1.01625²⁰ = $27,608.40

c. compounded monthly

future value = $20,000 x 1.005417⁶⁰ = $27,656.90

d. compounded continuously.

A = Peˣⁿ

where:

A = $20,000 x 2.71828⁰°⁰⁶⁵ˣ⁵ = $20,000 x 2.71828⁰°³²⁵ = $20,000 x 1.384 = $27,680.61