# Methods of Depreciation: Formulas, Problems, and Solutions

Updated on July 6, 2018 Ray is a Licensed Civil Engineer and specializes in Structural Engineering. He loves to write anything about education.

## What Is Depreciation?

Depreciation means the decrease in the value of physical properties or assets with the passage of time and use. It is the non-cash method of representing the reduction in value of a tangible asset. Specifically, it is an accounting concept that sets an annual deduction considering the factor of time and use on an asset's value. An asset is depreciable if it has a determinable useful life of more than one year in business or something to produce an income.

## Terminologies in Depreciation

There are some terminologies that you need to remember in understanding the different types of depreciation methods.

a. Adjusted Cost Basis is the asset's original cost basis used to compute depreciation deductions adjusted by allowable increases or decreases.

b. First Cost (FC) or Cost Basis is the unadjusted cost basis of an asset. It is the initial cost of acquiring an asset.

c. Book Value (BV) is the original cost basis of the property including any adjustments, less all allowable depreciation deductions.

d. Market Value (MV) is the amount paid to a willing seller by a willing buyer of an asset.

e. Salvage Value (SV) is the estimated value of a property at the end of a property's life.

f. Recovery Period is the number of years of an asset's recovery.

g. Usual Life (n) is the anticipated period of a property's life.

## 1. Straight Line Method of Depreciation

Straight Line Method is the simplest depreciation method. It assumes that a constant amount is depreciated each year over the useful life of the property. The formulas for Straight Line Method are:

• Annual Depreciation = (FC - SV) / n
• Total Depreciation after five years = [ (FC - SV) (5) ] / n
• Book Value = FC - Total Depreciation

### Problem 1: Straight Line Method

A commercial building has a salvage value of Php 1 million after 50 years. Annual depreciation is Php 2 M. Using the Straight Line Method, how many years after should you sell the building for Php 30 M?

Solution

a. Solve for the first cost.

```Annual depreciation = (FC - SV) / n
2 = (FC - 1) / 50
FC = Php 101 million```

b. Solve for the total depreciation after n years.

```Total depreciation = FC - BV
Total depreciation = 101 - 30
Total depreciation = 71 million```

c. Solve for the number of years.

```Total depreciation  = Annual depreciation (n)
71 = 2 (n)
n = 35.5 years
```

### Problem 2: Straight Line Method

The first cost of a machine is Php 1,800,000 with a salvage value of Php 300,000 at the end of its six years of life. Determine the total depreciation after three years using the Straight Line Method of Depreciation.

Solution

a. Solve for the annual depreciation.

```Annual depreciation = (FC - SV) / n
Annual depreciation = (1,800,000 - 300,000) / 6
Annual depreciation = Php 250,000```

b. Solve for the total depreciation after three years.

```Total depreciation = 250,000 (3)
Total depreciation = Php 750,000```

## 2. Depreciation by Declining Balance Method

Declining Balance Method is sometimes called the Constant-Percentage Method or the Matheson formula. The assumption in this depreciation method is that the annual cost of depreciation is the fixed percentage (1 - K) of the Book Value (BV) at the beginning of the year. The formulas for Declining Balance Method of Depreciation are:

• Annual Rate of Depreciation(K): SV = FC (1 - K)n
• Book Value = FC (1 - K)m
• Depreciation at mth year = FC (1 - K)m-1 (K)
• Total Depreciation = FC - SV

### Problem 1: Declining Balance Method

The equipment bought at a price of Php 450,000 has an economic life of 5 years and a salvage value of Php 50, 000. The cost of money is 12% per year. Compute the first year depreciation using Declining Balance Method.

Solution

a. Solve for the annual rate of depreciation.

```SV = FC (1 - K)^n
50,000 = 450,000 (1 - K)^5
K = 0.356```

b. Solve for the depreciation at the end of the first year.

```Depreciation = (K) (FC) (1 - K)^(m-1)
Depreciation = (0.356) (450,000) (1 - 0.356)^0
Depreciation = Php 160,200```

### Problem 2: Declining Balance Method

The first cost of a machine is Php 1,800,000 with a salvage value of Php 400,000 at the end of its life of five years. Determine the depreciation after three years using Constant-Percentage Method.

Solution

a. Solve for (1 - k).

```SV = FC (1 - K)^n
400,000 = 1,800,000 (1 - K)^5
(1 - K) = 0.74```

b. Solve for the book value at the end of the third year.

```BV = FC (1 - K)^m
BV = 1,800,000 (0.74)^3
BV = Php 730,037.21```

c. Solve for the total depreciation after three years.

```Total depreciation = FC - BV
Total depreciation = 1,800,000 - 730,037.21
Total depreciation = Php 1,069,962.79```

## 3. Depreciation by Sum of Years Digit Method (SOYD)

Sum of the Years Digit Method is an accelerated depreciation technique based on the assumption that tangible properties are usually productive when they are new, and their use decreases as they become old. The formulas for the Sum of the Years Digit Method of Depreciation are:

• Sum of years = (n / 2) (n + 1)
• Annual depreciation at 1st year= (FC - SV) (n / Sum of years)
• Annual depreciation at 2nd year = (FC -SV) ((n-1) / Sum of years)
• Book Value = FC - Total depreciation at the end of nth year

### Problem 1: Sum of the Years Digit Method

An equipment costs Php 1,500,000. At the end of its economic life of five years, its salvage value is Php 500,000. Using Sum of the Years Digit Method of Depreciation, what will be its book value for the third year?

Solution

a. Solve for the sum of years.

```Sum of years = (n / 2) (n + 1)
Sum of years = (5 / 2) (5 + 1)
Sum of years = 15 years```

b. Solve for the total depreciation up to the third year.

```Total depreciation = (FC - SV) (5 + 4 + 3) /15
Total depreciation = (1,500,000 - 500,000) (12) / 15
Total depreciation = Php 800,000```

c. Solve for the book value in the third year.

```Book Value = FC - Total depreciation
Book Value  = 1,500,000 - 800,000
Book Value = Php 700,000```

### Problem 2: Sum of the Years Digit Method

A machine costs Php 2,000,000. It has a salvage value of Php 500,000 at the end of its economic life. Using the Sum of the Years Digit Method, the book value at the end of two years is Php 800,000. What is the machine's economic life in years?

Solution

a. Solve for the total depreciation of the machine.

```BV = FC - Total depreciation
800,000 = 2,000,000 - Total depreciation
Total depreciation = Php 1,200,000```

b. Solve for the total depreciation after two years. Compute the machine's economic life in years.

```Sum of years = (n / 2) (n + 1)
Total depreciation = (n + (n + 1)) (FC - SV) / [(n / 2) (n + 1)]
1,200,000 = 2(2n - 1) (2,000,000 - 500,000) / (n (1 + n))
(2n - 1) / (n^2 + n) = 0.4
(2n - 1) = 0.4n^2 + 0.4n
0.4n^2 + 0.4n - 2n + 1 = 0
0.4n^2 - 1.6n + 1 = 0
n = 3.22
n = 4 years
```

## 4. Depreciation by Sinking Fund Method

Sinking Fund Method is a depreciation method wherein funds will accumulate for replacement purposes. The formulas for Sinking Fund Method of Depreciation are:

• Annual depreciation (A) = [ (FC -V) (i) ] / [ (1 + i)^(n) -1 ]
• Total depreciation after x years = A [(1 + i)^x - 1] / i
• Book Value = FC -Total depreciation

### Problem 1: Sinking Fund Method

A machine costs Php 300,000 with a salvage value of Php 50,000 at the end of its life of 10 years. If money is worth 6% annually, use Sinking Fund Method and determine the depreciation at the 6th year.

Solution

a. Solve for the annual depreciation.

```Annual depreciation (A) = [ (FC -SV) (i) ] / [ (1 + i)^(n) -1 ]
A = [ (300,000 - 50,000) (0.06) ] / [ (1 + 0.06)^10 -1 ]
A = Php 18966.98956```

b. Solve for the depreciation in the 6th year.

```Total depreciation after x years = A [(1 + i)^x  - 1] / i
Total depreciation = (18966.98956) [(1 + 0.06)^6 - 1] / 0.06
Total depreciation = Php 132,300.7939```

## 5. Depreciation Using Working Hours Method

Working Hours Method also called as Service Output Method is a depreciation method that results in the cost basis allocated equally over the expected number of units produced during the period of tangible properties. The formula for Working Hours Method of Depreciation is:

• Depreciation per hour = (FC - SV) / Total number of hours

### Problem 1: Working Hours Method

A machine costs Php 400,000 with a salvage value of Php 200,000. Life of it is six years. In the first year, 4000 hours. In the second year, 6000 hours and 8000 hours on the third year. The expected flow of the machine is 38000 hours in six years. What is the depreciation at the end of the second year?

Solution

a. Solve for the depreciation per hour.

```Depreciation per hour = (FC - SV) / Total number of hours
Depreciation per hour = (400,000 - 20,000) / 38000
Depreciation per hour = Php 10```

b. Solve for the depreciation at the end of 2nd year.

```Depreciation = 10 (6000)
Depreciation = Php 60,000```

## 6. Depreciation Using Constant Unit Method

Constant Unit Method is the same with Working Hours Method in the structure of the formula. The formula for Constant Unit Method of Depreciation is:

• Depreciation per unit = (FC - SV) / Total number of units

### Problem 1: Constant Unit Method

A coin machine costing Php 200,000 has a salvage value of Php 20,000 at the end of its economic life of five years. Determine the annual reserve for depreciation for the third year only. The schedule of production per year is as follows:

Year
Number of Coins
1
100,000
2
80,000
3
60,000
4
40,000
5
20,000
Constant Unit Method

Solution

a. Solve for the total number of coins.

```Total number of coins = 100,000 + 80,000 + 60,000 + 40,000 +20,000
Total number of coins = 300,000```

b. Solve for the depreciation per unit.

```Depreciation per unit = (FC - SV) / Total number of coins
Depreciation per unit = (200,000 - 20,000) / 300,000
Depreciation per unit = 0.60```

c. Solve for the depreciation reserve for the third year.

```Depreciation = 0.66 (60,000)
Depreciation = Php 36,000```

See results

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• Sangram Surve

2 months ago

Exam purpose

• sileshi

10 months ago

please release the most point that is required to manage Fixed asset

• AUTHOR

Ray

18 months ago from Philippines

Thanks, Sir Eric. I considered writing this into parts but I just decided to compress it into one article. Thanks for stopping by. Have a great day!

• Eric Dierker

18 months ago from Spring Valley, CA. U.S.A.