Economic analysis: replace machine now or later

[gfd43tg]

Homework Statement

I know it's not directly related to engineering, but its from an engineering economics course!

Homework Equations

The Attempt at a Solution

Hello, I'm trying to solve this problem, and my first thought it to find the present value of each machine
$$ PV = C_{D}t[1-(1+i)^{-n_{t}}]/(in_{t}) $$
Where ##C_{D}## is the depreciated value, ##i## is the interest rate, ##n_{t}## is the depreciation life (tax purposes), and ##t## is the tax rate.
When comparing the old machine and the new machine, I found ##PV_{old} = $516.9##, and ##PV_{new} = $614.4##, so it would mean the new one is presently worth more. I used ##n_{t,old} = 5## and ##n_{t,new}=10##. ##C_{D} = C_{I}-C_{S}## where ##C_{I}## is the initial cost, and ##C_{S}## is the salvage value. However, I am not sure how I should incorporate operating expenses in my calculation, which leads me to believe what I am doing is incorrect.

 

[Chestermiller]

Homework Statement

I know it's not directly related to engineering, but its from an engineering economics course!
View attachment 91560

Homework Equations

The Attempt at a Solution

Hello, I'm trying to solve this problem, and my first thought it to find the present value of each machine
$$ PV = C_{D}t[1-(1+i)^{-n_{t}}]/(in_{t}) $$
Where ##C_{D}## is the depreciated value, ##i## is the interest rate, ##n_{t}## is the depreciation life (tax purposes), and ##t## is the tax rate.
When comparing the old machine and the new machine, I found ##PV_{old} = $516.9##, and ##PV_{new} = $614.4##, so it would mean the new one is presently worth more. I used ##n_{t,old} = 5## and ##n_{t,new}=10##. ##C_{D} = C_{I}-C_{S}## where ##C_{I}## is the initial cost, and ##C_{S}## is the salvage value. However, I am not sure how I should incorporate operating expenses in my calculation, which leads me to believe what I am doing is incorrect.

I wouldn't use an equation to work this out. I would start by determining the cash flows for the two scenarios and then determine the net present value of each scenario.

 

[gfd43tg]

Not sure if I am going about this the right way
Straight-line depreciation
$$ D = (15,000-0)/10 = 1,500 $$
So the salvage value today is
$$ 1,500 = (15,000-C_{s})/5 $$
$$ C_{s} = 7,500 $$

For using the old machine for 5 years and then replacing (no salvage value)

$$ 15,000*1.1^{5} + 2,500(5) = $36,657.65 $$
Then using the new machine for 5 years
$$ 22,000*1.1^{5} + 1,500(5) = $42,931.22 $$
So for the next 10 years you get total cost ##$79,588.87##

Then Just buying the new machine now and using for the next 10 years with subtracting salvage value for the old machine today
$$ 22,000*1.1^{10}+1,500(10) - 7,500 = $64,562.33 $$

So it seems that purchasing the new machine is better.

 

[Chestermiller]

I really don't like what you did here, and I don't think that your results are correct. Here is an assessment of the cash flows for the scenario of waiting the 5 years before purchasing:

End of Year 1:
Operating cost= -2500
Tax refund on operating cost = +1250
Tax refund on depreciation = +750
Net cash flow at end of year 1 = -500

Same for years 2-4.

End of Year 5
Same as years 1-4, except purchase cost of 22000.
Net cash flow at end of year 5 = -22500

End of years 6-15:
Operating cost = -1500
Tax refund on operating cost = +750
Tax Refund on depreciation = +1100
Net cash flow at end of years 6-16 = +350

These case flows can be used to calculate the NPV of scenario 1.

In my judgement, scenario 2 (immediate purchase) is somewhat ambiguous, for the following reasons:
1. Scenario 1 involves having a machine to use for 15 years. In scenario 2, what do you do after 10 years (when the useful life of the new machine has expired)?
2. Can you sell the original machine at time zero to recover its salvage value?

Chet

 

[gfd43tg]

Thanks Chet, I didn't understand what the tax credit of 50% meant. I didn't realize you get 50% of your operating cost back. I assumed it meant you pay 22,000 for the machine plus 50% tax, just like buying something at the store. This is more clear, I will give this a try

 

[SteamKing]

I really don't like what you did here, and I don't think that your results are correct. Here is an assessment of the cash flows for the scenario of waiting the 5 years before purchasing:

End of Year 1:
Operating cost= -2500
Tax refund on operating cost = +1250
Tax refund on depreciation = +750
Net cash flow at end of year 1 = -500

Same for years 2-4.

End of Year 5
Same as years 1-4, except purchase cost of 22000.
Net cash flow at end of year 5 = -22500

End of years 6-15:
Operating cost = -1500
Tax refund on operating cost = +750
Tax Refund on depreciation = +1100
Net cash flow at end of years 6-16 = +350

These case flows can be used to calculate the NPV of scenario 1.

In my judgement, scenario 2 (immediate purchase) is somewhat ambiguous, for the following reasons:
1. Scenario 1 involves having a machine to use for 15 years. In scenario 2, what do you do after 10 years (when the useful life of the new machine has expired)?
2. Can you sell the original machine at time zero to recover its salvage value?

Chet

I think something got mixed up here.

According to the OP, Machine 1 has a service life of 10 years. Machine 1 is currently 5 years old, and it will be replaced when it is 10 years old, at the end of its service life. Machine 2, which also has a service life of 10 years, will replace Machine 1. The question asks if it is more economical to replace Machine 1 now by purchasing Machine 2, while Machine 1 still has some salvage value, rather than wait 5 years until Machine 1 is fully depreciated and has no scrap value. In any event, it is not anticipated to keep Machine 1 past 10 years of service life

 

[gfd43tg]

I agree with steamking, I think the old machine has a salvage value of $7500 if sold today

 

[Chestermiller]

Thanks Chet, I didn't understand what the tax credit of 50% meant. I didn't realize you get 50% of your operating cost back. I assumed it meant you pay 22,000 for the machine plus 50% tax, just like buying something at the store. This is more clear, I will give this a try

Yes. When the company fills out its tax return, it gets a deduction for operating expenses (since this reduces their net income) and it also gets a tax deduction on depreciation (which is a bookkeeping entry).

 

[SteamKing]

Thanks Chet, I didn't understand what the tax credit of 50% meant. I didn't realize you get 50% of your operating cost back. I assumed it meant you pay 22,000 for the machine plus 50% tax, just like buying something at the store. This is more clear, I will give this a try

At those tax rates (50%), no one would ever replace any equipment until it vaporized.

 

[Chestermiller]

I agree with steamking, I think the old machine has a salvage value of $7500 if sold today

OK. That settles my first question. My second question still stands. If the new machine is put into operation at time zero, what does the company use from years 11 - 15?

 

[gfd43tg]

OK. That settles my first question. My second question still stands. If the new machine is put into operation at time zero, what does the company use from years 11 - 15?

Agreed, the question doesn't mention it, so it is unknown.

 

[Chestermiller]

At those tax rates (50%), no one would ever replace any equipment until it vaporized.

Actually, a high tax rate is favorable from the standpoint of tax writeoffs for expenses and depreciation. The high tax rate hurts when when it comes to paying taxes on income.

 

[gfd43tg]

I just realized, this thing doesn't give the depreciation of the new machine, so how can I do the tax credit for depreciation for years 6-10? Assume zero salvage value at the end of its operating life as well?

 

[Chestermiller]

Agreed, the question doesn't mention it, so it is unknown.

I would tend to assume that a 2nd new machine is bought at the end of year 10, and that it is sold for salvage value at the end of year 15.

Chet

 

[Chestermiller]

I just realized, this thing doesn't give the depreciation of the new machine, so how can I do the tax credit for depreciation for years 6-10? Assume zero salvage value at the end of its operating life as well?

Sure, use straight line depreciation like the original machine.

 

[SteamKing]

OK. That settles my first question. My second question still stands. If the new machine is put into operation at time zero, what does the company use from years 11 - 15?

I'm not sure what you consider 'time zero'.

The company can either use Machine 1 for another 5 years from the present time and then purchase Machine 2, or it can purchase Machine 2 now and use it for 10 years. I don't think the planning department has considered what happens in the latter event, when Machine 2 is purchased earlier than anticipated.

 

[Chestermiller]

I'm not sure what you consider 'time zero'.

Presumably, the old machine is sold for salvage value at time zero, when the new machine is put into operation. So there is an immediate cash flow at time zero.

 

[gfd43tg]

Okay, I'm not sure if I should only consider the 10 year expenditure and use that as my comparison

Scenario 1 (use old machine for 5 years then use new machine 5 years)
Expenditure: -500(5) - 22000 + 350(5) = -22750

Scenario 2 (buy new machine now and sell old machine)
Expenditure: -22000 + 7500 + 350(10) = -11,000

So you would spend less money in scenario 2. Is that the right way to determine the better option?

 

[Chestermiller]

Okay, I'm not sure if I should only consider the 10 year expenditure and use that as my comparison

Scenario 1 (use old machine for 5 years then use new machine 5 years)
Expenditure: -500(5) - 22000 + 350(5) = -22750

Scenario 2 (buy new machine now and sell old machine)
Expenditure: -22000 + 7500 + 350(10) = -11,000

So you would spend less money in scenario 2. Is that the right way to determine the better option?

No. You need to get the net present value of each scenario by properly discounting the cash flows. Comparing it the way you did it is not adequate. There is a big difference between laying out the 22000 now and paying it 5 years from now. Don't forget that the discount rate is 10%.

 

[gfd43tg]

Should I use the expenditures for all 10 years of operating life to calculate the NPV (first equation should be 350*10)? Admittedly, I don't know how to calculate NPV from these expenditures.

I guess ##NPV = Cash Flow/(1+i)^{n}##

 

[Chestermiller]

Should I use the expenditures for all 10 years of operating life to calculate the NPV (first equation should be 350*10)? Admittedly, I don't know how to calculate NPV from these expenditures

Each cash flow is divided by 1.1ⁿ, where n is the time at the end of the year that the cash flow occurred. This would mean, for example, that, in scenario 1, the 22000 would be divided by 1.1⁵, while in scenario 2, it would be divided by 1.1⁰.

Chet

 

[gfd43tg]

Okay, so here is the formula for NPV

$$ NPV = \sum_{t=0}^{N} \frac {R_{t}}{(1+i)^{t}} $$

Scenario 1:
$$ NPV = \frac {-500}{1.1^{0}} + \frac {-500}{1.1^{1}} + \frac {-500}{1.1^{2}} + \frac {-500}{1.1^{3}} + \frac {-500}{1.1^{4}} + \frac {-22000}{1.1^{4}} + \frac {350}{1.1^{5}} + \frac {350}{1.1^{6}} + \frac {350}{1.1^{7}} + \frac {350}{1.1^{8}}+ \frac {350}{1.1^{9}} = -$16205 $$

 

[Chestermiller]

Okay, so here is the formula for NPV

$$ NPV = \sum_{t=0}^{N} \frac {R_{t}}{(1+i)^{t}} $$

Scenario 1:
$$ NPV = \frac {-500}{1.1^{0}} + \frac {-500}{1.1^{1}} + \frac {-500}{1.1^{2}} + \frac {-500}{1.1^{3}} + \frac {-500}{1.1^{4}} + \frac {-22000}{1.1^{4}} + \frac {350}{1.1^{5}} + \frac {350}{1.1^{6}} + \frac {350}{1.1^{7}} + \frac {350}{1.1^{8}}+ \frac {350}{1.1^{9}} = -$16205 $$

I would index all the n's up one. The cash flows are occurring at the end of the year.

 

[gfd43tg]

Should I go for all 10 years of the operating life of the new machine? Also, if I index all n's up, should it go until N-1?

 

[Chestermiller]

Should I go for all 10 years of the operating life of the new machine?

It's a judgement call. I personally would go 15 years, and, in scenario 2, I would buy a new machine at the end of 10 years, and then sell it for salvage at the end of 15 years. Otherwise, if you go 10 years, then in scenario 1, you need to claim salvage value of machine 2 at the end of 10 years.

Also, if I index all n's up, should it go until N-1?

No, you go to N, the end of year 10 or year 15.

Chet

 

[gfd43tg]

Turns out all this work may have been moot, here is what the professor said when I asked about the two possibilities

 

[Chestermiller]

Turns out all this work may have been moot, here is what the professor said when I asked about the two possibilities
View attachment 91623

Ah. That makes things a lot easier. All you need to do then is sum some infinite geometric progressions.

 

[gfd43tg]

Does that mean I should just use the equation in the OP

$$ PV = C_{D}t[1-(1+i)^{-n_{t}}]/(in_{t}) $$

Where PV is the present value, ##C_{D}## is the depreciated value, ##i## is the interest rate, and ##n_{t}## is the number of taxable years. The present value for each machine would be

$$ PV = 7500(0.5)(1-1.1^{5})/(0.1*5) = $2843 $$

and

$$ PV = 22000(0.5)(1-1.1^{10})/(0.1*10) = $6759 $$

This doesn't seem to count anything about switching or not...

I found this page for the present value of a perpetuity

http://www.financeformulas.net/Perpetuity.html

and the equation is ##PV = R/i##, so it would be trivial to say ##PV = 1500/0.1 = 15000## and ##PV = 2500/0.1 = 25000##, so something doesn't seem right, even though the prof. said to calculate the present value of a perpetuity.

 

[Chestermiller]

Does that mean I should just use the equation in the OP

$$ PV = C_{D}t[1-(1+i)^{-n_{t}}]/(in_{t}) $$

Where PV is the present value, ##C_{D}## is the depreciated value, ##i## is the interest rate, and ##n_{t}## is the number of taxable years. The present value for each machine would be

$$ PV = 7500(0.5)(1-1.1^{5})/(0.1*5) = $2843 $$

and

$$ PV = 22000(0.5)(1-1.1^{10})/(0.1*10) = $6759 $$

This doesn't seem to count anything about switching or not...

Definitely not. For scenario 1, you have:

$$NPV=∑_{n=1}^5{\frac{-500}{1.1^n}}+∑_{n=6}^∞{\frac{350}{1.1^n}}+∑_{n=1}^∞{\frac{-2200}{1.1^{10n-5}}}$$

 

[gfd43tg]

What does that last term mean?

 

[gfd43tg]

For scenario 2, then the sum would be

$$ \sum_{n=1}^{\infty} \frac {350}{1.1^{n}} + \sum_{n=1}^{\infty} \frac {-22,000}{1.1^{10n-1}} + \sum_{n=1}^{\infty} \frac {7500}{1.1^{10n-1}} $$

 

[Chestermiller]

For scenario 2, then the sum would be

$$ \sum_{n=1}^{\infty} \frac {350}{1.1^{n}} + \sum_{n=1}^{\infty} \frac {-22,000}{1.1^{10n-1}} + \sum_{n=1}^{\infty} \frac {7500}{1.1^{10n-1}} $$

For scenario 2, then the sum would be

$$ \sum_{n=1}^{\infty} \frac {350}{1.1^{n}} + \sum_{n=1}^{\infty} \frac {-22,000}{1.1^{10n-1}} + \sum_{n=1}^{\infty} \frac {7500}{1.1^{10n-1}} $$

My understanding is:
$$ \sum_{n=1}^{\infty} \frac {350}{1.1^{n}} + \sum_{n=0}^{\infty} \frac {-22,000}{1.1^{10n}} +7500 $$

 

[gfd43tg]

I don't understand how you are picking your exponents for the interest rates in the sums. Why 10n, 10n-5??

 

[Chestermiller]

What does that last term mean?

It means that you replace the old machine after 5 years, and then replace each new machine every subsequent 10 years.

 

[Chestermiller]

I don't understand how you are picking your exponents for the interest rates in the sums. Why 10n, 10n-5??

In scenario 2, the 10n (starting with n = 0), means that you replace the machine immediately, and then every subsequent 10 years.

In scenarion 1, the 10n-5 (starting with n = 1) means that you replace the machine after 5 years, and then every subsequent 10 years.