As the president of a new start-up organization you know that money is always in

Question

As the president of a new start-up organization you know that money is always in short supply. Therefore, you take every opportunity to save up for anticipated large future purchases. You know that your current production equipment can only produce a limited amount of product each year and when demand increase beyond this point you will know you will need to buy a larger capacity machine. The new machine costs $62,400 to purchase, regardless of the year in which the machine is purchased.

a. If you were to $2150 at the end of each year into a savings account that earns 8% interest per year, when will you be able to purchase the larger capacity machine? Assume that your first deposit will be made at the end of year 1.

b.Based upon your current demand forecasts, you anticipate needing the new machine in 8 years (i.e. the purchase will be made at the end of the 8th year). How much would you have to deposit today, and at the end of each year (year 1 – 8) in order to have enough money saved for the purchase? All deposits will be made into the same account that earns 8% per year.

Explanation / Answer

Cost of new machine = $62,400

a.) Deposit made at end of each year = $2,150

Interest rate = 8%

First payment will be made at end of year 1

In this case, we are depositing same amount every year, therefore, it is annuity

Future value of annuity = [P*[(1+r)n -1]]/r

where p is periodic payment, r is interest rate and n is number of periods

Future value of annuity = 62,400

P = 2150

r = 8%

Now, putting all these value in the equation

62400 = [2150*[(1+0.08)n-1]]/0.08

62400*0.08 = 2150*[(1+0.08)n-1]

4992/2150 = (1+0.08)n-1

2.32+1 = (1+0.08)n

3.32 = 1.08n

Solving this we gets n = 16

So we need to save $2150 for 16 years to get $62,400

b.) In this case, we forecast that we will need new machine in 8 years

Therefore, n = 8

Now, we need to find the amount needs to be deposited each year

We will use same equation as in the previous case just now we know n and we need to find out P.

So, 62400 = [P*[(1+0.08)8-1]]/0.08

We need to solve for P

62400*0.08 = P*[(1+0.08)8-1]

4992 = P*[1.85-1]

4992 = P*0.85

P = $5,866.52

We need to deposit $5,866.52 each year for 8 years to save $62,400.

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