The Mathematics of Money

366 Chapter 8 Mathematics of Pricing

FORMULA 8.4.1

Straight-Line Depreciation Rate

D
IV ____UL SV

where

D represents the ANNUAL DEPRECIATION AMOUNT

IV represents the INITIAL VALUE OF THE ITEM

SV represents the SALVAGE (RESIDUAL) VALUE

and

UL represents the USEFUL LIFE of the item

FORMULA 8.4.2

Straight-Line Depreciation

DV 
 IV  D(n) when n 
 UL

DV 
 SV when n  UL

where

DV represents the DEPRECIATED VALUE

and

n represents the NUMBER OF YEARS that have passed

If you were comfortable with how we worked Example 8.4.3 without having formal for-

mulas, it is not really necessary to use them explicitly. However, the following example

will illustrate their use.

Example 8.4.4 Rework Example 8.4.3 using Formulas 8.4.1 and 8.4.2.

First we fi nd the annual depreciation amount.

D
___IV UL SV
^

$2,000 ____________ $0

5

(^)
$400
Then, we fi nd the depreciated value after 3 years:
DV
IV  D(n)
$2,000  $400(3)
$2,000  $1,200
$800
Since n
7 and UL
5, and 7  5, we conclude that the value after 7 years is $0.
Comparing Percent to Straight-Line Depreciation
Both of the types of depreciation that we have considered are useful in different situations.
Having seen how each works, it is worthwhile to compare the two methods both math-
ematically and in how they are most commonly used.
As a fi rst step to comparison, let’s reconsider the car from Example 8.4.2, this time with
straight-line depreciation. Suppose we assume a useful life of 10 years, and assume that at
the end of 10 years the car has a residual value of $4,608.25 (the same value we predicted by
using percent depreciation). The depreciation rate would then be ($23,407  $4,608.25)/10

$1,879.88 per year. Showing this in a table similar to the one for this example with 15% declin-
ing balance depreciation (see “Depreciation as a Percent,” above), we get:
Year Starting Price Decrease Ending Price
1 $23,407.00 $1,879.88 $21,527.12
2 $21,527.12 $1,879.88 $19,647.24
3 $19,647.24 $1,879.88 $17,767.36
4 $17,767.36 $1,879.88 $15,887.48
5 $15,887.48 $1,879.88 $14,007.60
6 $14,007.60 $1,879.88 $12,127.72
7 $12,127.72 $1,879.88 $10,247.84
8 $10,247.84 $1,879.88 $8,367.96
9 $8,367.96 $1,879.88 $6,488.08
10 $6,488.08 $1,879.88 $4,608.20