Ford Motor Company was able to reduce by 64% the cost required for installing data acquisition instrumentation on test vehicles by using MTS-developed spinning wheel force transducers. The cost this year (.e. end of year 1) is expected to be $2500.
If the costs are expected to increase by $250 each year for the next 4 years, what is the equivalent annual worth of the costs (years 1- 5) at an interest rate of 7% per year?
The equivalent annual worth is $
To calculate the equivalent annual worth of the costs (years 1-5), we need to determine the present value of the costs over that period.
First, we calculate the present value (PV) of the costs in year 1 using the formula:
PV = CF / (1 + r)^n
Where CF is the cash flow in year 1, r is the interest rate, and n is the number of years.
PV1 = $2500 / (1 + 0.07)^1
PV1 = $2500 / 1.07
PV1 ≈ $2336.45
Next, we calculate the present value of the increasing costs in years 2-5 using the same formula. The cash flow for each year is $250, and the number of years increases by 1 each year:
PV2 = $250 / (1 + 0.07)^2
PV2 ≈ $232.48
PV3 = $250 / (1 + 0.07)^3
PV3 ≈ $217.61
PV4 = $250 / (1 + 0.07)^4
PV4 ≈ $203.90
PV5 = $250 / (1 + 0.07)^5
PV5 ≈ $191.15
Finally, we calculate the equivalent annual worth (EAW) of the costs by summing the present values and dividing by the annuity factor of the cash flows over 5 years. The annuity factor can be calculated using the formula:
annuity factor = (1 - (1 + r)^-n) / r
EAW = (PV1 + PV2 + PV3 + PV4 + PV5) / annuity factor
EAW = ($2336.45 + $232.48 + $217.61 + $203.90 + $191.15) / [(1 - (1 + 0.07)^-5) / 0.07]
EAW ≈ $612.73
Therefore, the equivalent annual worth of the costs (years 1-5) at an interest rate of 7% per year is approximately $612.73.
To calculate the equivalent annual worth of the costs, we can use the formula for the equivalent annual cost (EAC). The EAC represents the equal annual cost over a period of time.
We are given the initial cost in year 1 as $2500, and the costs are expected to increase by $250 each year for the next 4 years. We need to find the equivalent annual worth over the 5-year period at an interest rate of 7% per year.
Let's calculate the equivalent annual worth using the formula:
EAC = (Cost in Year 1 + Cost in Year 2 + Cost in Year 3 + Cost in Year 4 + Cost in Year 5) / [(1 + i) + (1 + i)^2 + (1 + i)^3 + (1 + i)^4 + (1 + i)^5]
where i = interest rate (7% = 0.07)
EAC = (2500 + 2750 + 3000 + 3250 + 3500) / [(1 + 0.07) + (1 + 0.07)^2 + (1 + 0.07)^3 + (1 + 0.07)^4 + (1 + 0.07)^5]
EAC = 15000 / (1.07 + 1.1449 + 1.226 + 1.3138 + 1.4071)
EAC = 15000 / 6.0988
EAC ≈ $2,459.92
Therefore, the equivalent annual worth of the costs over the 5-year period is approximately $2,459.92.
To calculate the equivalent annual worth of the costs, we need to determine the present worth of each individual cost and then find the equivalent uniform annual cost.
First, let's calculate the present worth of the costs for years 1 to 5. We know that the cost for year 1 is $2500, and it is expected to increase by $250 each year for the next 4 years.
Using the formula for calculating present worth (PW), we can calculate the present worth of each year's cost:
PW = Cost / (1 + interest rate)^n
Where:
- Cost: the cost for a specific year
- interest rate: the annual interest rate
- n: the number of years into the future
Let's calculate the present worth for each year:
Year 1: PW = $2500 / (1 + 0.07)^1 = $2500 / (1.07) = $2336.45
Year 2: PW = $2750 / (1 + 0.07)^2 = $2750 / (1.07)^2 = $2327.81
Year 3: PW = $3000 / (1 + 0.07)^3 = $3000 / (1.07)^3 = $2319.24
Year 4: PW = $3250 / (1 + 0.07)^4 = $3250 / (1.07)^4 = $2310.72
Year 5: PW = $3500 / (1 + 0.07)^5 = $3500 / (1.07)^5 = $2302.26
Now, let's find the equivalent uniform annual cost (EUAC) by summing up the present worth values and dividing by the present worth factor:
EUAC = (PW1 + PW2 + PW3 + PW4 + PW5) / Present Worth Factor
Present Worth Factor = [(1 - (1 + interest rate)^(-n)) / interest rate]
Using the values calculated above, let's find the Present Worth Factor:
Present Worth Factor = [(1 - (1 + 0.07)^(-5)) / 0.07] = 3.6349
Finally, let's calculate the Equivalent Annual Worth (EUAC):
EUAC = ($2336.45 + $2327.81 + $2319.24 + $2310.72 + $2302.26) / 3.6349
EUAC ≈ $1279.26
Therefore, the equivalent annual worth of the costs (years 1-5) at an interest rate of 7% per year is approximately $1279.26.