Question

A piece of equipment has a first cost of $60000, a maximum useful life of 4 years, and a market (salvage) value described by the relation Sk = 48000 – 8400k, where k is the number of years since it was purchased. The AOC series is estimated using AOC = 24000 + 3600k. The interest rate is 9% per year. When should the company replace this asset?

Answer 1

To determine when the company should replace the asset, we have to find the economic service life of the asset which is the number of years at which the equivalent uniform annual worth of costs is the minimum.

The computation of economic service life is shown in the following table.

Capital recovery = - first cost ( A/P, i%, n years) + Salvage value (A/F, i%, n years)

( A/P, i%, n years) = capital recovery factor

(A/F, i%, n years) = sinking fund factor

Capital recovery year 1 = - $ 60,000 ( A/P, 9%, 1 year) + $ 39600 ( A/F, 9%, 1 year) =   - $ 60,000 1.0900 + $ 39600 1.0000

Capital recovery year 1 = - $ 25800

Capital recovery year 2 = - $ 60,000 ( A/P, 9%, 2 year) + $ 31200 ( A/F, 9%, 2 year) =   - $ 60,000   .5685   + $ 31200   .4785

Capital recovery year 2 = - $ 19180.8

Capital recovery year 3 = - $ 60,000 ( A/P, 9%, 3 year) + $ 22800 ( A/F, 9%, 3 year) =   - $ 60,000 0.3951 + $ 22800 0.3051

Capital recovery year 3 = - $ 16749.72

Capital recovery year 4 = - $ 60,000 ( A/P, 9%, 4 year) + $ 14400 ( A/F, 9%, 4 year) =   - $ 60,000 0.3087 + $ 14400 0.2187

Capital recovery year 4 = - $ 15372.72

Annual worth of AOC year 1 = Present worth of AOC year 1 Capital recovery factor

Annual worth of AOC year 1 = - $ 27600 (P/F, 9%, 1 year) ( A/P, 9%, 1 year) = - $ 27600 0.9174 1.0900

Annual worth of AOC year 1 = - $ 27599.06

Annual worth of AOC year 2 = ( present worth of AOC year 1 + present worth of AOC year 2 ) capital recovery factor year 2

Annual worth of AOC year 2 = ( - $ 27600 0.9174 - 31200 0.8417 ) 0.5685

Annual worth of AOC year 2 = - $ 29323.96

Annual worth of AOC year 3 = (  - $ 27600 0.9174 - 31200 0.8417 - 34800 0.7722) 0.3951

Annual worth of AOC year 3 = -$ 30997.11

Annual worth of AOC year 4 = (  - $ 27600 0.9174 - 31200 0.8417 - 34800 0.7722 - 38400 0.7084 ) 0.3087

Annual worth of AOC year 4 = - $ 32616.13  

Year	Salvage value ($)	AOC ( $)	Capital recovery, ($)	Annual worth of AOC, $	Total annual = capital recovery + annual worth of AOC
1	39600	27600	- $ 25800	-$27599.06	- 25800 - 27599.06 = - $ 53399.06
2	31200	31200	-$ 19180.8	- $ 29323.96	-$ 19180.8 - $ 29323.96 = -$ 48504.76
3	22800	34800	-$16749.72	-$ 30997.11	-$ 30997.11 -  $16749.72 = -$ 47746.83
4	14400	38400	-$15372.72	- $ 32616.13	- $ 32616.13 - $15372.72 = - $ 47988.85

From the above table , the economic service life of the asset is 3 years because the total annual worth of the costs is lowest in year 3 compared to other years.

Hence the company should replace the asset in year 3.