Maintenance cost of an equipment is P20,000 for 2 years, P40,000 at the end of 4 years, and P80,000 at the end of 8 years. Compute the semi-annual amount that will be set aside for this equipment. Money worth 10% compounded annually.
1,200
Well, I have to say, this equipment sure knows how to make money disappear! Let's see how much we need to set aside semi-annually to keep this equipment functioning smoothly.
First, let's look at the time period. We're dealing with 8 years, so there are a total of 16 semi-annual periods (2 per year).
Next, let's break down the maintenance costs over these periods.
In the first 2 years, we have a cost of P20,000. For the remaining 6 years (12 semesters), we have 3 payments to consider - P40,000 at the end of 4 years, P40,000 at the end of 5 years, and P40,000 at the end of 6 years.
Now, let's calculate the present value of each of these costs by considering the annual compounding rate of 10%.
For the P20,000 over 2 years, we can use the formula: PV = PMT / (1 + i)^n, where PMT is the payment, i is the interest rate, and n is the number of periods.
So, for the P20,000, PV = 20000 / (1 + 10%)^2 = P16,528.93.
For the 3 payments of P40,000, we can use the same formula for each payment since they occur at different points in time.
The present value of the payment at the end of year 4 is PV1 = 40000 / (1 + 10%)^4 = P29,059.46.
The present value of the payment at the end of year 5 is PV2 = 40000 / (1 + 10%)^5 = P26,416.78.
The present value of the payment at the end of year 6 is PV3 = 40000 / (1 + 10%)^6 = P23,924.34.
Now, let's add up all the present values:
PV_Total = PV1 + PV2 + PV3 + PV4 + PV5 + PV6 + PV7 + PV8 + PV9 + PV10 + PV11 + PV12 = P16,528.93 + P29,059.46 + P26,416.78 + P23,924.34 + P21,658.50 + P19,631.36 + P17,823.96 + P16,219.05 + P14,801.86 + P13,558.77 + P12,477.57 + P11,547.07 = P225,690.79.
Lastly, let's divide the total present value by 16 (the number of semi-annual periods) to find the semi-annual amount to be set aside:
Semi-annual amount = PV_Total / 16 = P225,690.79 / 16 = P14,105.67.
So, in order to keep this equipment happy and running smoothly, you'll need to set aside around P14,105.67 semi-annually. Just make sure to keep an eye on your budget, or this equipment might end up clowning around with your finances!
To compute the semi-annual amount that will be set aside for the equipment, we need to break down the maintenance cost into semi-annual intervals.
First, let's determine the number of semi-annual intervals for each time period:
- 2 years: There are 2 semi-annual intervals in 2 years.
- 4 years: There are 4 semi-annual intervals in 4 years.
- 8 years: There are 8 semi-annual intervals in 8 years.
Next, let's calculate the present value of each maintenance cost at a 10% compounded annually discount rate.
- P20,000 for 2 years:
We need to calculate the present value of P20,000 due in 2 years at a 10% compounded annually discount rate. Using the present value formula:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = Discount rate
n = Number of periods
PV = 20,000 / (1 + 0.10)^2
PV = 20,000 / 1.21
PV = 16,528.93
- P40,000 at the end of 4 years:
We need to calculate the present value of P40,000 due in 4 years at a 10% compounded annually discount rate.
PV = 40,000 / (1 + 0.10)^4
PV = 40,000 / 1.46
PV = 27,397.26
- P80,000 at the end of 8 years:
We need to calculate the present value of P80,000 due in 8 years at a 10% compounded annually discount rate.
PV = 80,000 / (1 + 0.10)^8
PV = 80,000 / 2.14
PV = 37,383.18
Finally, let's sum up the present values of each maintenance cost and divide it by the total number of semi-annual intervals:
Total Present Value = PV1 + PV2 + PV3
Total Present Value = 16,528.93 + 27,397.26 + 37,383.18
Total Present Value = 81,309.37
Semi-Annual Amount = Total Present Value / Total Number of Semi-Annual Intervals
Semi-Annual Amount = 81,309.37 / (2 + 4 + 8)
Semi-Annual Amount = 81,309.37 / 14
Semi-Annual Amount = 5807.81
Therefore, the semi-annual amount that should be set aside for this equipment is P5,807.81.
To compute the semi-annual amount that needs to be set aside for this equipment, we need to determine the present value of the maintenance costs at the given interest rate and compounding frequency.
Let's break down the problem into three separate cash flows:
1. Maintenance cost after 2 years: P20,000
2. Maintenance cost after 4 years: P40,000
3. Maintenance cost after 8 years: P80,000
First, let's calculate the present value of each individual cost using the compound interest formula:
Present Value = Future Value / (1 + Interest Rate/2)^(years x 2)
1. Maintenance cost after 2 years:
Present Value (PV1) = P20,000 / (1 + 0.10/2)^(2 x 2)
= P20,000 / (1 + 0.05)^4
= P20,000 / (1.05)^4
= P20,000 / 1.21551
≈ P16,446.37
2. Maintenance cost after 4 years:
Present Value (PV2) = P40,000 / (1 + 0.10/2)^(4 x 2)
= P40,000 / (1 + 0.05)^8
= P40,000 / (1.05)^8
≈ P26,159.78
3. Maintenance cost after 8 years:
Present Value (PV3) = P80,000 / (1 + 0.10/2)^(8 x 2)
= P80,000 / (1 + 0.05)^16
= P80,000 / (1.05)^16
≈ P41,616.92
Next, we sum up the present values of the maintenance costs to find the total amount needed to be set aside:
Total Present Value = PV1 + PV2 + PV3
≈ P16,446.37 + P26,159.78 + P41,616.92
≈ P84,223.07
Finally, we divide the total present value by the number of semi-annual periods in 8 years (16 semi-annual periods) to find the semi-annual amount that needs to be set aside:
Semi-annual amount = Total Present Value / Number of semi-annual periods
= P84,223.07 / 16
≈ P5,263.94
Hence, the semi-annual amount that needs to be set aside for this equipment is approximately P5,263.94.