Question

Chuck needs to purchase a piece of machinery in 10 years. The item costs $1,800 today but its price inflates 4% per year. He will also have to pay sales tax of 7% on the purchase. To finance the purchase, Chuck deposits $50 into an account at the beginning of each quarter for the next 10 years. The effective annual interest rate is 6.5%. a) Will the balance of the fund be sufficient to purchase the piece of machinery in 10 years? (b) How much extra money will Chuck have to pay or how much extra will he have saved?

Answer 1

First we will calculate the future value of Machinery;-

Future Value = Present Value (1+infiliation rate)^{time period}

= 1800(1+0.04)¹⁰

= $ 2664.44

Since, the purchase of machine attracts sales Tax of 7%.

Final Cost of Machinery = 2664.44(1+0.07)

= $ 2850.95

So, the final price of machinery will be $ 2850.95 in 10 years.

Now, we will calculate future value of quarterly investment made by Chuck using the Future Value of Annuity Due formula:-

FV = P* [(1+r)" – 1] *(1+r)

where, P= periodic payment ;= $50

FV= future Value

r = Periodic Interest rate ;= (.065/12)*3 ;= 0.01625

n = No of period ;= 10 years*4 quarters ;= 40

FV = 50* 1 [(1 + 0.01625) 40 - 11 0.01625 *(1 + 0.01625)

FV = $2831.61

a). So, the balance of the fund is $ 2831.61

Since, Machinery will cost $ 2850.95 in 10 years. Hence, the balance fund is insufficient to purchase machinery in 10 years.

b). Since, the balance fund is less, Chuck has to pay Extra money which is = $2850.95-$2831.61

= $19.34

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