Machine cost a company 1 million and its affective life is estimated to be 20 years. If the scrap is expected to realize 50000 only the sum to be invested every year at 13.25% compounded annually for 20 years to replace the machine which cost 30% more than its present value is:
To calculate the sum to be invested every year to replace the machine over 20 years, we need to consider the present value of the machine and the scrap value at the end.
1. Calculate the present value of the machine: As the machine cost the company 1 million and its effective life is estimated to be 20 years, we can calculate the present value using the formula for compound interest:
PV = Future Value / (1 + r)^n
Where PV is the present value, the Future Value is the cost of the machine (1 million), r is the interest rate (13.25% or 0.1325), and n is the number of years (20).
PV = 1,000,000 / (1 + 0.1325)^20
PV ≈ 143,678.10
The present value of the machine is approximately 143,678.10 dollars.
2. Calculate the cost to replace the machine: Since the cost to replace the machine is 30% more than its present value, we can calculate the replacement cost:
Replacement Cost = Present Value + (Present Value * 30%)
Replacement Cost = 143,678.10 + (143,678.10 * 0.30)
Replacement Cost ≈ 186,398.53
The estimated replacement cost of the machine is approximately 186,398.53 dollars.
3. Calculate the annual investment required: We can use the formula for the future value of an ordinary annuity to find the annual investment required:
FV = PMT * [(1 + r)^n - 1] / r
Where FV is the future value (replacement cost), PMT is the annual investment, r is the interest rate (13.25% or 0.1325), and n is the number of years (20).
186,398.53 = PMT * [(1 + 0.1325)^20 - 1] / 0.1325
Now we can solve this equation for PMT to find the annual investment required:
PMT ≈ 7,014.62
The sum to be invested every year at 13.25% compounded annually for 20 years to replace the machine that cost 30% more than its present value is approximately 7,014.62 dollars.
To find the sum to be invested every year, we need to calculate the present value of the machine and then determine the annual amount required to replace it over 20 years.
First, let's find the present value (PV) of the machine. The machine currently costs 1 million, and its scrap value is expected to be 50,000. The effective life of the machine is 20 years.
Using the formula for present value:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = interest rate
n = number of years
We have:
FV = 1,000,000 - 50,000 = 950,000 (the cost of the machine minus the expected scrap value)
r = 13.25% = 0.1325 (expressed as a decimal)
n = 20 (the number of years)
Plugging in the values:
PV = 950,000 / (1 + 0.1325)^20
Calculating this value gives us:
PV ≈ 208,483.44
Next, we need to determine the cost to replace the machine, which will be 30% more than the present value.
Cost to replace = PV + (30% of PV)
Cost to replace = 208,483.44 + (0.3 * 208,483.44) = 270,077.46 (approx.)
Now, we need to find the annual investment amount required to accumulate the cost to replace the machine in 20 years.
Using the formula for compound interest:
A = P * (1 + r)^n
Where:
A = Amount accumulated
P = Principal (initial investment)
r = interest rate (annual rate)
n = number of years
We have:
A = 270,077.46 (cost to replace the machine)
r = 13.25% = 0.1325 (expressed as a decimal)
n = 20 (number of years)
Plugging in the values:
270,077.46 = P * (1 + 0.1325)^20
Now, we can solve this equation to find the annual investment amount (P).
270,077.46 / (1 + 0.1325)^20 = P
Calculating this value gives us:
P ≈ 7,054.99
Therefore, the sum to be invested every year at 13.25% compounded annually for 20 years to replace the machine is approximately $7,054.99.
The present value of the machine is $1 million.
The scrap value is expected to be $50,000.
The machine's effective life is estimated to be 20 years.
To calculate the sum to be invested every year at 13.25% compounded annually for 20 years, we need to find the future value of the machine's present value:
Future Value = Present Value * (1 + interest rate)^number of years
Future Value = $1,000,000 * (1 + 0.1325)^20
Future Value = $1,000,000 * 2.484905
Future Value = $2,484,905
The replacement machine will cost 30% more than the present value of the old machine:
Replacement Cost = Present Value * (1 + 30%)
Replacement Cost = $1,000,000 * (1 + 0.30)
Replacement Cost = $1,000,000 * 1.30
Replacement Cost = $1,300,000
To find the sum to be invested every year to replace the machine, we subtract the expected scrap value from the replacement cost and divide it by the future value formula:
Sum to be invested = (Replacement Cost - Scrap Value) / Future Value
Sum to be invested = ($1,300,000 - $50,000) / $2,484,905
Sum to be invested = $1,250,000 / $2,484,905
Sum to be invested = 0.502 * 100
Sum to be invested = 50.2%
Therefore, the sum to be invested every year at 13.25% compounded annually for 20 years to replace the machine is 50.2%.