Depreciation

October 3, 2008

Observation: most things lose value over time…

…though a few things do not…

Things lose value for several reasons:

they wear out through the passing of time..

They wear out through use…

…they become obsolete or out of fashion

We need to keep track of this wearing out for three reasons:

1. To keep track of the company’s value

3. To know when to replace them

5. We can deduct depreciation costs from company taxes.

Note: we need not use the same formula for all three

Two main ways of calculating depreciation:

(though there are others)

Straight – Line Depreciation

Declining balance depreciation

Straight-line depreciation is simple:

If the asset has a life of N years, it loses 1/N of its original value every year

Ntime

Book Value

Ntime

Book Value

S

P

n

Annual Depreciation, D = (P-S)/N

Book value, BV = P - nD

BV

Declining-balance depreciation is slightly harder:

The asset loses d% of its original value every year

Ntime

Book Value

P

ntime

Book Value

P

Book value (BV) = P(1 – d)n

BV

Dn = BVn-1d

Sample Problem:

A machine costs $10,000 when new.

If it depreciates by straight-line depreciation over 10 years, and has a salvage value of $1,000 at the end of the tenthyear, what is its book value after five years and what is itsdepreciation in the sixth year?

If it instead depreciates 10% per year, what is its book value after five years and what is its depreciation in the sixth year?

Replacement Analysis

I have a 2001 Honda CRV. This week I paid $1,400 in repairs. When should I replace it with a new car?

(Based on a true story)

Background: the Bathtub Curve

N (Years before replacement)

EUAC

Capitalised cost

EUAC = P(A/P,i,N)

Background: the Bathtub Curve

Years before replacement

EUAC

Capitalised cost

Operating costs

Background: the Bathtub Curve

Years before replacement

EUAC

Capitalised cost

Operating costs

Total

Background: the Bathtub Curve

Years before replacement

EUAC

Total

n

n is the economic life of the asset

Terminology

The current asset is the defender, its prospective replacementis the challenger.

The economic life of a defender is the period overwhich its EUAC is the least.

Method: calculate and compare the EUAC’s for all challengers, considering all possible dates for replacement.

Simplest Case: Retirement without Replacement

This is the case where the asset generates some income, butis not vital to the continuation of the business.

Example: video game in a corner store

Method: calculate the PW’s of the asset and associatedcash flows for all feasible lifetimes.

If any of these PW’s are positive, keep it; otherwise,get rid of it.

Example:

A supermarket has a mechanical horse that is nearing theend of its life. It won’t survive past the end of Year 2.

It brings in $1,200/year in quarters, but this is expectedto decline by $100/year as the neighbourhood kids growup. Maintenance costs are expected to be $800 this yearand $900 next year.

Right now it has a salvage value of $600. After one year it will only be worth $300, and after that it’s worthless.

If the supermarket’s MARR is 10%, how long should thehorse be kept?

Second Case: Retirement with Identical Replacement

If the technology of an asset changes slowly with time, we may be able to replace it with an identical item in thefuture (e.g., a wheelbarrow).

In this case, we just look at the EUAC of the asset for various lifetimes and find its economic life.

Is the defender younger than its economic life? If so,keep it. If it’s at the economic life, replace it.

If it’s older than its economic life, calculate the cost ofkeeping it for 1, 2, … years and compare with the minimum EUAC

Example:

A fork-life truck costing $10,000 has a maximum service lifeof 6 years. Its salvage value and operating costs are given bythe following table:

Year Salvage ($) Operating Costs ($)

1 7,000 5,000

2 4,500 5,400

3 2,500 6,100

4 1,000 7,100

5 0 8,400

6 0 11,000

Solution:

Year Sn O. C. Capital Recovery Av. Op. Cost. EUAC

1 7,000 5,000 4,000 5,246 9,246

2 4,500 5,400 3,619 5,446 9,065

3 2,500 6,100 3,266 5,734 9,000

4 1,000 7,100 2,939 6,104 9,043

5 0 8,400 2,638 6,548 9,146

6 0 11,000 2,296 7,195 9,491

Capital Recovery = 10,000(A/P,i,N) – Sn(A/F,i,N)

Average Op Cost* = (Σ PW(Op. Costs)) (A/P,i,N)

EUAC = Cap. Recovery + Op. Costs

* Make mid-period assumption

Third Case: One Challenger, Different from Defender

Third Case: One Challenger, Different from Defender

(This analysis assumes both provide the same service)

1. Calculate the economic life of the challenger and its EUAC for that life (`AC(CN*)’)

2. Compare that EUAC with the cost of keeping the defender one more year (`AC(D1)’)

3. If AC(D1) <AC(CN*), do nothing for a year.

4. Otherwise, find AC(Dn), 1 < n < Physical life

5. If AC(Dn) >AC(CN*) for all n, replace

Fourth Case: Several Challengers

This is a simple extension of the third case:

Find the minimum EUAC for each challenger,

Select the challenger with the lowest minimum to go up againstthe defender

Then follow the procedure for the third case.

The General Case

Your company wants to come up with a replacement strategyfor the next N years, knowing that a different replacement may be used every time (`Chao’s problem’).

Difficulty: to determine the economic life of the next-generationchallenger, we need to know what the next-generation-plus-onechallenger will be.

Solution: Dynamic Programming

Dealing with Trade-Ins

The present cost of a defender is its salvage value (since bykeeping it, we’re losing the opportunity to sell it and investthe money).

Suppose we have a defender and two challengers, A and B.The vendor of A offers us a $5,000 trade-in for the defender,while the vendor of B offers us a $4,000 trade-in. The bestoffer we can get on the open market is $2,000.

What do we use as a salvage value when calculating theEUAC of the defender?

Replacement Analysis Pitfalls

1. In estimating the salvage value of the defender, have you taken into account the cost of getting it ready for re-sale? (Cleaning, mending, crating, shipping, etc.)

1. In estimating the purchase price of the challenger, have you considered the cost of installing, de-bugging, re-training and re-routing?

More Replacement Analysis Pitfalls

3. Discount sunk costs!

1. Include all relevant cash-flows – for example, will the challenger produce fewer defectives?

1. Document all assumptions explicitly, so your analysis can be compared with others.