Replacement Analysis

When an asset is evaluated, we typically…

• Keep the asset in use
• Overhaul the asset to improve performance
• Remove the asset from use without replacement
• Replace the asset with another

Obsolescence:

• Technology of an asset is surpassed by a newer and/or different technology Depletion:
• Loss of market value due to consumption or exhaustion of the resource Deterioration due to aging:
• Loss of functionality or efficiency due tot he aging process

We can find the equivalent annual capital costs by…

• $EAC=\left(P-S\left(n\right)\right)\left(\frac{A}{P},i,n\right)+S\left(n\right)i$
• This is the equivalent annual cost of replacing equipment every n years as an annuity
• Our $EAC\left(\text{total}\right)=EAC\left(\text{captial}\right)+EAC\left(O\&M\right)$ which occurs at the point where the two EAC’s intersect on a cost vs time graph at $n^{\ast }$
• $EAC(\text{Capital})=P\left(\frac{A}{P},i,n\right)-S\left(\frac{A}{F},i,n\right)$ which simplifies to $EAC\left(\text{Captial}\right)=\left(P-S\right)\left(\frac{A}{P},i,n\right)+Si$
  • For n over the course oof useful life
• $EAC\left(\text{O\&M}\right)$ = converted all operation and maintenance costs to equivalent annual costs over n years (annuities)
• Now using the two equations, you can find total EAC

Replacement Analysis

• The existing asset is the defender
• The new asset is the challenger
• They do not have to be different, they are often the same

Definitions

Some definitions:

• Capital Costs (P): The difference between the price paid and what it can be sold for later (as an EAC)
• O&M Costs: The cost of using the asset to produce goods/services
  • May be in the form of arithmetic or gradient series
  • Convert to equivalent annual cost (EAC)

We Have 3 Situations Here

1. The Defender and Challenger are identical and the need for the asset continuous indefinitely
2. The Defender is different from the Challenger; the challenger repeats indefinitely
3. The Defender and Challenger are different, and successive Challengers are not identical

Situation 1

Assuming…

• The defender + challenger are technologically identical
• Lives of these identical assets are assumed to be short
• Relative prices / interest rates are assumed constant We model the replacement decision as if it were to take place a large number of times

Now, we determine the minimum-cost life of the assets.
The economic life of the asset is the lifetime that minimizes EAC

We want to solve this iteratively where we calculate EAC’s of the capital and O&M for each n until we find the minimum of the EAC(Total) which is then our $n^{\ast }$

Situation 2

Here, we have a decision rule to minimize costs…

1. Determine the economic life of the challenger and its associated EAC
2. Do the same with the defender
3. If EAC(defender) > EAC(challenger), replace now
   1. Otherwise do not replace now

With this method, if you have an asset that is n years old, we start the comparison at that point in time ignoring the costs / benefit cash flows from before that n.

What is P

Note that if we have things like an installation cost, that is NON-REFUNDABLE and is therefore not considered as our P value for our book value (when calculating salvage value), but for other present values, we consider the entire cost (example, for EAC).

Here, we use a 1 year (marginal cost) principle, when the above exists and the yearly operating costs are monotonically increasing, then we obtain marginal cost for one more year so… If the marginal cost of keeping defender for one more year exceeds the EAC of Challenger at its economic life, we replace defender immediately.

• For the one year approach, we keep calculating until, our net EAC stops decreasing. The EACH for the defender is for the year before the increasing year
• For non-monotonic cases, there can be multiple minimums, so for assignments we do this for all ns but on exams, we can assume that the first n is the global minimum
• For the challenger, repeat for all ns and find the n where the EAC of the challenger is greater than that of the defender

Situation 3 (Challenger Does Not Repeat)

We typically expect future challengers to be better than the current challenger, we enumerate all possible combinations of replacement options and evaluate all to make a choice

• The EAC for each combination needs to be computed
• The number of combinations increases quickly
• Very little information is available for future challengers typically

Replacement Analysis Tip

If the Defender and Challenger have the same technology…

• In this case, obtain the economic life and use cyclic replacement?

If they do not have the same technology…

• Separately obtain the remaining economic life with its minimum EAC for the defender and the challenger
• Then, compare the EAC’s, if the defender’s EAC is $\le$ that of the challenger, we make a replacement now, otherwise, we do not replace and continue

Monotonic

Monotonic means “moving in one direction without reversing”, ex. always increasing without a decrease, or always decreasing without ever increasing.