Question

5) An assembly plant anticipates needing to replace some of its machinery in 4 years, at a current cost of $2 million. The company expects annual inflation of 3%. It also believes it can earn an 8% return on its money, compounded quarterly. How much money would the company have to put into an account now in order to have the anticipated future cost of the machinery available to withdraw at the end of 4 years?

6) (10%) What is the future value of $80,000 invested now for 5 years at a 6% rate under the following compounding options? A. Annual B. Semi-annual 2 NOT FOR DISTRIBUTION OR SHARING OUTSIDE OF THIS COURSE C. Quarterly D. Monthly

7) (10%) For each of the 4 compounding options in question 6, calculate the Effective Annual Rate.

Answer 1

5. Current cost of machine = $ 2 million

Rate of inflation = 3% p.a

Therefore Cost of machine after 4 years = $ 2 million * (1.03)^4 = $ 2.25 million

Applying the formula of compounding interest,

A = P * ( 1 + [r / n] ) ^ (n * t)

where A = final amount

P = initial amount

r = rate of interest

n = no of compounding

t = time period

Here A = final amount = $ 2.25 million ; n = 4 (compounded quarterly)

Thus,

2.25 million = P * ( 1 + [0.08 / 4] ) ^ ( 4 * 4 )

2.25 million = P * ( 1.02 ) ^ 16

P = 2.25 million / 1.02 ^ 16

= $ 1.64 million

Money that the company would have to put into an account now in order to have the anticipated future cost of the machinery available to withdraw at the end of 4 years is $ 1.64 million.

6. Future Value = Cash Flow * ( 1 + [r / n] ) ^ (n * t)

where r = rate of interest

n = no of compounding

t = time period

A. Annual compounding

Future Value = 80000 * ( 1 + 0.06 ) ^ 5

= $ 107058

B. Semi Annual compounding

Future Value = 80000 * ( 1 + [0.06 / 2] ) ^ (5 * 2)

= $ 107513

C. Quarterly compounding

Future Value = 80000 * ( 1 + [0.06 / 4] ) ^ (5 * 4)

= $ 107748

D. Monthly compounding

Future Value = 80000 * ( 1 + [0.06 / 12] ) ^ (5 * 2)

= $ 107908

7. Effective Annual Rate = (1 + [ r / n ] ) ^ n − 1

where r = rate of interest

n = no of compounding

A. Annual compounding

Effective Annual Rate = (1 + [ 0.06 / 1 ] ) ^ 1 - 1 = 6%

B. Semi annual compounding

Effective Annual Rate = (1 + [ 0.06 / 2 ] ) ^ 2 - 1 = 6.09%

C. Quarterly compounding

Effective Annual Rate = (1 + [ 0.06 / 4 ] ) ^ 4 - 1 = 6.14%

D. Monthly compounding

Effective Annual Rate = (1 + [ 0.06 / 12 ] ) ^ 12 - 1 = 6.17%