{"id":110,"date":"2024-01-09T08:27:01","date_gmt":"2024-01-09T08:27:01","guid":{"rendered":"https:\/\/openbook.ums.edu.my\/financialmathematicsineconomics\/chapter\/6-1-depreciation-in-accounting\/"},"modified":"2024-09-25T08:37:56","modified_gmt":"2024-09-25T08:37:56","slug":"6-1-depreciation-in-accounting","status":"publish","type":"chapter","link":"https:\/\/openbook.ums.edu.my\/financialmathematicsineconomics\/chapter\/6-1-depreciation-in-accounting\/","title":{"raw":"6.1 Depreciation in Accounting","rendered":"6.1 Depreciation in Accounting"},"content":{"raw":"

Economic life of most assets is limited. After being used for a number of years, it will come to the end of their lives. As the time of retirement, assets may have a small trade-in or scrap value, or even may be worthless.<\/p>\r\n

Computation of depreciation may involve compound interest and ordinary annuity. The computation of the periodic depreciation include allocating the cost of the property as an expense\/ a cost of goods manufactured to the proper business operating periods. The amount of accumulated depreciation is known as accumulated depreciation or allowance for depreciation.<\/p>\r\n

The difference between the original cost of an asset and its total amount in the accumulated depreciation is known as book value. Among reasons for depreciation include time factor, decay, worn out, etc.<\/p>\r\n

In this chapter, we will use the following notation:<\/p>\r\n

C\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 =\u00a0 \u00a0original cost of an asset<\/p>\r\n

T\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 =\u00a0 \u00a0estimated trade-in value or scrap value<\/p>\r\n

C \u2013 T\u00a0 \u00a0=\u00a0 total depreciation charges or expenses<\/p>\r\n

n\u00a0 \u00a0 \u00a0=\u00a0 useful life of the asset estimated in years, service-hours, or product-units.<\/p>\r\n

r\u00a0 \u00a0 =\u00a0 rate of depreciation expense per year, per service-hour, or per product-units.<\/p>\r\n\r\n

6.1.1 Method of Averages<\/h1>\r\n

6.1.1.1 The Straight-Line Method<\/h2>\r\n

The simplest and most popular method of accounting for depreciation is the straight-line method. The depreciation charges are equal for each year. If R is the depreciation expense per year and n is the number of years, then R is the total depreciation charges divide by n, the number of years or can be written as:<\/p>\r\n

[latexpage]<\/p>\r\n

R = $\\frac {C - T}{n}$<\/p>\r\n\r\n

Example 6.1<\/header>\r\n
\r\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Use the straight line method to find the depreciation charges for each year, and construct a depreciation schedule.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

6.1.1.2 Service Hours Method<\/h2>\r\n

The service hours method relates depreciation to the estimated productive capacity of the asset in terms of its hours of useful service. The depreciation rate per service hour r is:<\/p>\r\n

[latexpage]<\/p>\r\n

r = $\\frac {C - T}{n}$<\/p>\r\n

Where n is the number of service hours.<\/p>\r\n

The depreciation charges for a given year is equal to r multiplies the number of service hours of that year.<\/p>\r\n\r\n

Example 6.2<\/header>\r\n
\r\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Assume useful life of the machine is 20,000 service hours and actual number of service hours in each year is as follows.<\/p>\r\n

Construct a depreciation schedule using the service hours method to find the depreciation charges.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
1st year<\/td>\r\n5000 service hours<\/td>\r\n<\/tr>\r\n
2nd year<\/td>\r\n4500 service hours<\/td>\r\n<\/tr>\r\n
3rd year<\/td>\r\n4200 service hours<\/td>\r\n<\/tr>\r\n
4th year<\/td>\r\n3400 service hours<\/td>\r\n<\/tr>\r\n
5th year<\/td>\r\n2900 service hours<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n

6.1.1.3 Product Units Method<\/h2>\r\n

The product units method defined depreciation to the estimated number of units produced by each asset during its useful life. The depreciation rate per unit product r is:<\/p>\r\n

[latexpage]<\/p>\r\n

r = $\\frac {C - T}{n}$<\/p>\r\n

Where n is the number of product units. The depreciation charges for a given period equal to r multiplies the number of units produced in that period.<\/p>\r\n\r\n

Example 6.3<\/header>\r\n
\r\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Assume useful life of machine is 70,000 product units and number of units produced each year is estimated as follows.<\/p>\r\n

Use the product units method to find the depreciation charges for each year, and construct a depreciation schedule.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
1st year<\/td>\r\n14000 units<\/td>\r\n<\/tr>\r\n
2nd year<\/td>\r\n15000 units<\/td>\r\n<\/tr>\r\n
3rd year<\/td>\r\n16500 units<\/td>\r\n<\/tr>\r\n
4th year<\/td>\r\n17000 units<\/td>\r\n<\/tr>\r\n
5th year<\/td>\r\n7500 units<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n

6.1.2 Reducing Charge Method<\/h1>\r\n

6.1.2.1 Sum of the years-digit method<\/h2>\r\n

The total depreciation is fixed. It is the difference between the original cost and the trade-in or scrap value (C-T). The rate of depreciation is expressed in a changing fraction which becomes smaller each year. The numerator of the fraction is the number of remaining years of life of the asset. Whereas the denominator is the sum of digits that represent the years of life.<\/p>\r\n\r\n

Example 6.4<\/header>\r\n
\r\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Use the sum of the years-digits method to find the depreciation charges for each year and construct the depreciation schedule.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

6.1.2.2 Constant\/ Fixed rate method on diminishing book value<\/h2>\r\n

The depreciation expense for earlier year is higher than the later years. The depreciation expense is equal to fixed\/ constant annual rate multiply the diminishing book value of an asset. There are two ways of computation: i) the declining balance method, ii) the fixed rate method.<\/p>\r\n

For the declining balance method, first we need to find the annual depreciation rate by dividing 100% by the number of years of useful life of the asset. Then, find the maximum annual depreciation rate by multiplying the annual depreciation rate obtained by 2.<\/p>\r\n

For the fixed rate method, the book value at the end of the life of the asset is equal to the scrap or trade-in value. The rate of annual depreciation charges (r) can be written as:<\/p>\r\n[latexpage]\r\n\r\nr = $1 - \\sqrt[n]{\\frac{T}{C}}$\r\n

6.1.3 Compound Interest Method<\/h1>\r\n

6.1.3.1 Annuity method<\/h2>\r\nThe annuity method resembles the method of amortizing a debt. The depreciation charges are equal and include not only a part of the cost of the asset but also the interest on the book value for each operating period. The step by step calculation are as follows:\r\n\r\nStep 1: Find\u00a0 the present value of the total depreciation charges. The present value is, P=T(1+i)-n<\/sup>\r\n\r\nStep 2: The annual depreciation charges are required to be equal. Thus, we can assume that the present value of an annuity (An<\/sub>) with payments consisting of equal depreciation charges as:\r\n\r\n[latexpage]\r\n\r\nC-P = C- T(1+i)-n<\/sup> = An<\/sub> = R $a_{\\bar{n}\\mid{i}}$\r\n\r\nStep 3: R represents the annual depreciation charges, and can be obtained as follows:\r\n\r\nR = $\\frac {A_{n}}{a_{\\bar{n}\\mid{i}}} = \\frac{C \u2013 T(1 + i) ^{-n}}{a_{\\bar{n}\\mid{i}}}$\r\n
Example 6.5<\/header>\r\n
\r\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Use the annuity method to find the depreciation charges for each year and construct a depreciation schedule. Assume that the effective interest rate is 6%.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

6.1.3.2 Sinking fund method<\/h2>\r\n

It is assumed that a sinking fund is established for the purpose of replacing an asset at the end of its useful life. The periodic depreciation charges are exactly the same as the periodic increases including the periodic deposit and interest) in the sinking fund. Hence, the depreciation charges are not equal. But the total of the depreciation charges is equal to the amount in the sinking fund (Sn<\/sub>) at the end of the useful life of the asset. The size of each deposit (R) made in the sinking fund is given by the annuity formula:<\/p>\r\n[latexpage]\r\n\r\n$S_{n}$ = R${s_{\\bar{n}\\mid{i}}}$ or R \u00a0= \u00a0$\\frac {S_{n}}{s_{\\bar{n}\\mid{i}}} = \\frac{C \u2013 T}{s_{\\bar{n}\\mid{i}}}$\r\n\r\n \r\n

6.1.4 Composite Rate Method<\/h1>\r\n

The composite rate method is used for computing\u00a0 the depreciation charges of a group of assets. The composite rate is obtained by dividing the total annual depreciation charges by the total cost\u00a0 of\u00a0 the\u00a0 group\u00a0 of assets. The annual depreciation charges of each asset are obtained by the straight line method. The composite rate may be used to compute depreciation charges\u00a0 during\u00a0 the\u00a0 later\u00a0 years\u00a0 if\u00a0 there\u00a0 is\u00a0 no\u00a0 significant changes in the values and useful lives of assets. The work of computing the depreciation expense of each item can be avoided. \u00a0The composite rate of an asset can be calculated as:<\/p>\r\nThe composite rate = $\\frac{Total \\: annual \\: depreciation \\: charges}{Total \\: cost}$\r\n

The composite life of a group of assets is the average life of the group. In computing the composite life, the annual depreciation charges of the group of assets may be obtained by using: i) the composite rate, ii) the depreciation charges are assumed to be equal for each year.<\/p>\r\nThe composite life = $\\frac{Total \\: depreciation \\: charges}{Total \\: annual \\: depreciation \\: charges}$\r\n

Example 6.6<\/header>\r\n
\r\n

Find the composite rate and composite life of the group of assets in the table below:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Asset<\/td>\r\nOriginal cost<\/td>\r\nScrap value<\/td>\r\nEstimated life<\/td>\r\n<\/tr>\r\n
A<\/td>\r\nRM10000<\/td>\r\n9000<\/td>\r\n10 years<\/td>\r\n<\/tr>\r\n
B<\/td>\r\n5000<\/td>\r\n4800<\/td>\r\n12 years<\/td>\r\n<\/tr>\r\n
C<\/td>\r\n4500<\/td>\r\n4225<\/td>\r\n5 years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n

[h5p id=\"33\"]<\/h2>\r\n

6.1.4.1 The sinking fund method<\/h2>\r\n

If the depreciation charges are not equal for each year, it is not categorised as a fixed composite rate. The annual deposits in the fund for each asset are equal during the life of the asset. The composite life is the time necessary for the total annual deposits at the given interest rate to accumulate to the total depreciation charges of the group of assets.<\/p>\r\n\r\n

\r\n

Example 6.7<\/p>\r\n\r\n<\/header>\r\n

\r\n

Find the composite life of the group of assets in the table below using the sinking fund method. Assume that the effective interest rate is 6%.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Asset<\/td>\r\nOriginal cost<\/td>\r\nScrap value<\/td>\r\nEstimated life<\/td>\r\n<\/tr>\r\n
A<\/td>\r\nRM10000<\/td>\r\n9000<\/td>\r\n10 years<\/td>\r\n<\/tr>\r\n
B<\/td>\r\n5000<\/td>\r\n4800<\/td>\r\n12 years<\/td>\r\n<\/tr>\r\n
C<\/td>\r\n4500<\/td>\r\n4225<\/td>\r\n5 years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[h5p id=\"34\"]\r\n\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

Economic life of most assets is limited. After being used for a number of years, it will come to the end of their lives. As the time of retirement, assets may have a small trade-in or scrap value, or even may be worthless.<\/p>\n

Computation of depreciation may involve compound interest and ordinary annuity. The computation of the periodic depreciation include allocating the cost of the property as an expense\/ a cost of goods manufactured to the proper business operating periods. The amount of accumulated depreciation is known as accumulated depreciation or allowance for depreciation.<\/p>\n

The difference between the original cost of an asset and its total amount in the accumulated depreciation is known as book value. Among reasons for depreciation include time factor, decay, worn out, etc.<\/p>\n

In this chapter, we will use the following notation:<\/p>\n

C\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 =\u00a0 \u00a0original cost of an asset<\/p>\n

T\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 =\u00a0 \u00a0estimated trade-in value or scrap value<\/p>\n

C \u2013 T\u00a0 \u00a0=\u00a0 total depreciation charges or expenses<\/p>\n

n\u00a0 \u00a0 \u00a0=\u00a0 useful life of the asset estimated in years, service-hours, or product-units.<\/p>\n

r\u00a0 \u00a0 =\u00a0 rate of depreciation expense per year, per service-hour, or per product-units.<\/p>\n

6.1.1 Method of Averages<\/h1>\n

6.1.1.1 The Straight-Line Method<\/h2>\n

The simplest and most popular method of accounting for depreciation is the straight-line method. The depreciation charges are equal for each year. If R is the depreciation expense per year and n is the number of years, then R is the total depreciation charges divide by n, the number of years or can be written as:<\/p>\n

\n

R = \"\frac {C - T}{n}\"<\/p>\n

\n
Example 6.1<\/header>\n
\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Use the straight line method to find the depreciation charges for each year, and construct a depreciation schedule.<\/p>\n<\/div>\n<\/div>\n

6.1.1.2 Service Hours Method<\/h2>\n

The service hours method relates depreciation to the estimated productive capacity of the asset in terms of its hours of useful service. The depreciation rate per service hour r is:<\/p>\n

\n

r = \"\frac {C - T}{n}\"<\/p>\n

Where n is the number of service hours.<\/p>\n

The depreciation charges for a given year is equal to r multiplies the number of service hours of that year.<\/p>\n

\n
Example 6.2<\/header>\n
\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Assume useful life of the machine is 20,000 service hours and actual number of service hours in each year is as follows.<\/p>\n

Construct a depreciation schedule using the service hours method to find the depreciation charges.<\/p>\n\n\n\n\n\n\n\n
1st year<\/td>\n5000 service hours<\/td>\n<\/tr>\n
2nd year<\/td>\n4500 service hours<\/td>\n<\/tr>\n
3rd year<\/td>\n4200 service hours<\/td>\n<\/tr>\n
4th year<\/td>\n3400 service hours<\/td>\n<\/tr>\n
5th year<\/td>\n2900 service hours<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

6.1.1.3 Product Units Method<\/h2>\n

The product units method defined depreciation to the estimated number of units produced by each asset during its useful life. The depreciation rate per unit product r is:<\/p>\n

\n

r = \"\frac {C - T}{n}\"<\/p>\n

Where n is the number of product units. The depreciation charges for a given period equal to r multiplies the number of units produced in that period.<\/p>\n

\n
Example 6.3<\/header>\n
\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Assume useful life of machine is 70,000 product units and number of units produced each year is estimated as follows.<\/p>\n

Use the product units method to find the depreciation charges for each year, and construct a depreciation schedule.<\/p>\n\n\n\n\n\n\n\n
1st year<\/td>\n14000 units<\/td>\n<\/tr>\n
2nd year<\/td>\n15000 units<\/td>\n<\/tr>\n
3rd year<\/td>\n16500 units<\/td>\n<\/tr>\n
4th year<\/td>\n17000 units<\/td>\n<\/tr>\n
5th year<\/td>\n7500 units<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

6.1.2 Reducing Charge Method<\/h1>\n

6.1.2.1 Sum of the years-digit method<\/h2>\n

The total depreciation is fixed. It is the difference between the original cost and the trade-in or scrap value (C-T). The rate of depreciation is expressed in a changing fraction which becomes smaller each year. The numerator of the fraction is the number of remaining years of life of the asset. Whereas the denominator is the sum of digits that represent the years of life.<\/p>\n

\n
Example 6.4<\/header>\n
\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Use the sum of the years-digits method to find the depreciation charges for each year and construct the depreciation schedule.<\/p>\n<\/div>\n<\/div>\n

6.1.2.2 Constant\/ Fixed rate method on diminishing book value<\/h2>\n

The depreciation expense for earlier year is higher than the later years. The depreciation expense is equal to fixed\/ constant annual rate multiply the diminishing book value of an asset. There are two ways of computation: i) the declining balance method, ii) the fixed rate method.<\/p>\n

For the declining balance method, first we need to find the annual depreciation rate by dividing 100% by the number of years of useful life of the asset. Then, find the maximum annual depreciation rate by multiplying the annual depreciation rate obtained by 2.<\/p>\n

For the fixed rate method, the book value at the end of the life of the asset is equal to the scrap or trade-in value. The rate of annual depreciation charges (r) can be written as:<\/p>\n

r = \"1 - \sqrt[n]{\frac{T}{C}}\"<\/p>\n

6.1.3 Compound Interest Method<\/h1>\n

6.1.3.1 Annuity method<\/h2>\n

The annuity method resembles the method of amortizing a debt. The depreciation charges are equal and include not only a part of the cost of the asset but also the interest on the book value for each operating period. The step by step calculation are as follows:<\/p>\n

Step 1: Find\u00a0 the present value of the total depreciation charges. The present value is, P=T(1+i)-n<\/sup><\/p>\n

Step 2: The annual depreciation charges are required to be equal. Thus, we can assume that the present value of an annuity (An<\/sub>) with payments consisting of equal depreciation charges as:<\/p>\n

C-P = C- T(1+i)-n<\/sup> = An<\/sub> = R \"a_{\bar{n}\mid{i}}\"<\/p>\n

Step 3: R represents the annual depreciation charges, and can be obtained as follows:<\/p>\n

R = \"\frac {A_{n}}{a_{\bar{n}\mid{i}}} = \frac{C - T(1 + i) ^{-n}}{a_{\bar{n}\mid{i}}}\"<\/p>\n

\n
Example 6.5<\/header>\n
\n

A machine was purchased for RM1,100 has an estimated useful life of 5 years and a trade-in value of RM120. Use the annuity method to find the depreciation charges for each year and construct a depreciation schedule. Assume that the effective interest rate is 6%.<\/p>\n<\/div>\n<\/div>\n

6.1.3.2 Sinking fund method<\/h2>\n

It is assumed that a sinking fund is established for the purpose of replacing an asset at the end of its useful life. The periodic depreciation charges are exactly the same as the periodic increases including the periodic deposit and interest) in the sinking fund. Hence, the depreciation charges are not equal. But the total of the depreciation charges is equal to the amount in the sinking fund (Sn<\/sub>) at the end of the useful life of the asset. The size of each deposit (R) made in the sinking fund is given by the annuity formula:<\/p>\n

\"S_{n}\" = R\"{s_{\bar{n}\mid{i}}}\" or R \u00a0= \u00a0\"\frac {S_{n}}{s_{\bar{n}\mid{i}}} = \frac{C - T}{s_{\bar{n}\mid{i}}}\"<\/p>\n

 <\/p>\n

6.1.4 Composite Rate Method<\/h1>\n

The composite rate method is used for computing\u00a0 the depreciation charges of a group of assets. The composite rate is obtained by dividing the total annual depreciation charges by the total cost\u00a0 of\u00a0 the\u00a0 group\u00a0 of assets. The annual depreciation charges of each asset are obtained by the straight line method. The composite rate may be used to compute depreciation charges\u00a0 during\u00a0 the\u00a0 later\u00a0 years\u00a0 if\u00a0 there\u00a0 is\u00a0 no\u00a0 significant changes in the values and useful lives of assets. The work of computing the depreciation expense of each item can be avoided. \u00a0The composite rate of an asset can be calculated as:<\/p>\n

The composite rate = \"\frac{Total \: annual \: depreciation \: charges}{Total \: cost}\"<\/p>\n

The composite life of a group of assets is the average life of the group. In computing the composite life, the annual depreciation charges of the group of assets may be obtained by using: i) the composite rate, ii) the depreciation charges are assumed to be equal for each year.<\/p>\n

The composite life = \"\frac{Total \: depreciation \: charges}{Total \: annual \: depreciation \: charges}\"<\/p>\n

\n
Example 6.6<\/header>\n
\n

Find the composite rate and composite life of the group of assets in the table below:<\/p>\n\n\n\n\n\n\n
Asset<\/td>\nOriginal cost<\/td>\nScrap value<\/td>\nEstimated life<\/td>\n<\/tr>\n
A<\/td>\nRM10000<\/td>\n9000<\/td>\n10 years<\/td>\n<\/tr>\n
B<\/td>\n5000<\/td>\n4800<\/td>\n12 years<\/td>\n<\/tr>\n
C<\/td>\n4500<\/td>\n4225<\/td>\n5 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

\n
\n