Construction Economics & Finance. Module 3 Lecture-1

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1 Depreciation:- Construction Econoics & Finance Module 3 Lecture- It represents the reduction in arket value of an asset due to age, wear and tear and obsolescence. The physical deterioration of the asset occurs due to wear and tear with passage of tie. Obsolescence occurs to due to availability of new technology or new product in the arket that is superior to the old one and the new one replaces the old even though the old one is still in working condition. The tangible assets for which the depreciation analysis is carried out are construction equipents, buildings, electronic products, vehicles, achinery etc. Depreciation aount for any asset is usually calculated on yearly basis. Depreciation is considered as expenditure in the cash flow of the asset, although there is no physical cash outflow. Depreciation affects the incoe tax to be paid by an individual or a fir as it is considered as an allowable deduction in calculating the taxable incoe. Generally the incoe tax is paid on taxable incoe which is equal to gross incoe less the allowable deductions (expenditures). Depreciation reduces the taxable incoe and hence results in lowering the incoe tax to be paid. Before discussing about different ethods of depreciation, it is necessary to know the coon ters used in depreciation analysis. These ters are initial cost, salvage value, book value and useful life. Initial cost is the total cost of acquiring the asset. Salvage value represents estiated arket value of the asset at the end of its useful life. It is the expected cash inflow that the owner of the asset will receive by disposing it at the end of useful life. Book value is the value of asset recorded on the accounting books of the fir at a given tie period. It is generally calculated at the end of each year. Book value at the end of a given year equals the initial cost less the total depreciation aount till that year. Useful life represents the expected nuber of years the asset is useful in ters of generating revenue. The asset ay still be in working condition after the useful life but it ay not be econoical. Useful life is also known as depreciable life. The asset is depreciated over its useful life. Joint initiative of IITs and IISc Funded by MHRD Page of 34

2 The coonly used depreciation ethods are straight-line depreciation ethod, declining balance ethod, su-of-years-digits ethod and sinking fund ethod. Straight-line (SL) depreciation ethod:- It is the siplest ethod of depreciation. In this ethod it is assued that the book value of an asset will decrease by sae aount every year over the useful life till its salvage value is reached. In other words the book value of the asset decreases at a linear rate with the tie period. The expression for annual depreciation in a given year is presented as follows; D P SV. (3.) n Where D = depreciation aount in year ( =,, 3, 4,., n, i.e. n) P = initial cost of the asset n = useful life or depreciable life (in years) over which the asset is depreciated. In equation (3.), ter d. is the constant annual depreciation rate which is denoted by the n Since the depreciation aount is sae every year, D = D = D 3 = D 4 = = D The book value at the end of st year is equal to initial cost less the depreciation in the st year and is given by; P (3.) D Book value at the end of nd year is equal to book value at the beginning of nd year (i.e. book value at the end of st year) less the depreciation in nd year and is expressed as follows; D. (3.3) As already stated depreciation aount is sae in every year. Now putting the expression of fro equation (3.) in equation (3.3); P D D P D.. (3.4) Siilarly the book value at the end of 3 rd year is equal to book value at the beginning of 3 rd year (i.e. book value at the end of nd year) less the depreciation in 3 rd year and is given by; Joint initiative of IITs and IISc Funded by MHRD Page of 34

3 3 D (3.5) P D D P 3 3D.. (3.6) In the sae anner the generalized expression for book value at the end of any given year can be written as follows; P. (3.7) D Declining balance (DB) depreciation ethod:- It is an accelerated depreciation ethod. In this ethod the annual depreciation is expressed as a fixed percentage of the book value at the beginning of the year and is calculated by ultiplying the book value at the beginning of each year with a fixed percentage. Thus this ethod is also soeties known as fixed percentage ethod of depreciation. The ratio of depreciation aount in a given year to the book value at the beginning of that year is constant for all the years of useful life of the asset. When this ratio is twice the straight-line depreciation rate i.e., the ethod is known as double- n declining balance (DDB) ethod. In other words the depreciation rate is 00% of the straight-line depreciation rate. Double-declining balance (DDB) ethod is the ost coonly used declining balance ethod. Another declining balance ethod that uses depreciation rate equal to 50% of the straight-line depreciation rate i.e..5 is also n used for calculation of depreciation. In declining balance ethods the depreciation during the early years is ore as copared to that in later years of the asset s useful life. In case of declining balance ethod, for calculating annual depreciation aount, the salvage value is not subtracted fro the initial cost. It is iportant to ensure that, the asset is not depreciated below the estiated salvage value. In declining-balance ethod the calculated book value of the asset at the end of useful life does not atch with the salvage value. If the book value of the asset reaches its estiated salvage value before the end of useful life, then the asset is not depreciated further. Representing constant annual depreciation rate in declining balance ethod as d, the expressions for annual depreciation aount and book value are presented as follows; The depreciation in st year is calculated by ultiplying the initial cost (i.e. book value at beginning) with the depreciation rate and is given by; D P d P 0 d d.. (3.8) Joint initiative of IITs and IISc Funded by MHRD Page 3 of 34

4 Where D = depreciation aount in st year P = initial cost of the asset as already entioned d = constant annual depreciation rate The book value at the end of st year is equal to initial cost less the depreciation in the st year and is calculated as follows; P D Now putting the expression of D fro equation (3.8) in the above expression; P D P P d P d. (3.9) The depreciation in nd year i.e. D is calculated by ultiplying the book value at the beginning of nd year (i.e. book value at the end of st year) with the depreciation rate d and is given as follows; D d.... (3.0) Now putting the expression of fro equation (3.9) in equation (3.0) results in the following; D d d d P... (3.) Book value at the end of nd year is equal to book value at the beginning of nd year (i.e. book value at the end of st year) less the depreciation in nd year and is expressed as follows; D Now putting the expressions of and D fro equation (3.9) and equation (3.) respectively in above expression results in the following; P d P d d P d d P D d... (3.) Siilarly the depreciation in 3 rd year i.e. D 3 is calculated as follows; D 3 d.... (3.3) Now putting the expression of fro equation (3.) in equation (3.3); D d P d d 3. (3.4) Book value at the end of 3 rd year is calculated as follows; 3 D3 Joint initiative of IITs and IISc Funded by MHRD Page 4 of 34

5 P d P d d P d d P 3 3 D3 d... (3.5) In the sae anner the generalized expression for depreciation in any given year can be written as follows (referring to equations (3.8), (3.) and (3.4)); D - d d P... (3.6) Siilarly the generalized expression for book value at the end of any year is given as follows (referring to equations (3.9), (3.) and (3.5)); P d.. (3.7) The book value at the end of useful life i.e. at the end of n years is given by; n n P d... (3.8) The book value at the end of useful life is theoretically equal to the salvage value of the asset. Thus equating the salvage value (SV) of the asset to its book value ( n ) at the end of useful life results in the following; n n P d SV... (3.9) Thus for calculating the depreciation of an asset using declining balance ethod, equation (3.9) can be used to find out the constant annual depreciation rate, if it is not stated for the asset. Fro equation (3.9), the expression for constant annual depreciation rate d fro known values of initial cost P and salvage value (SV 0) is obtained as follows; SV P SV P d n d n d n SV.... (3.0) P In double-declining balance (DDB) ethod, for calculating annual depreciation aount and book value at the end of different years, the value of constant annual depreciation rate d is replaced by in the above entioned equations. n Joint initiative of IITs and IISc Funded by MHRD Page 5 of 34

6 Exaple - The initial cost of a piece of construction equipent is Rs It has useful life of 0 years. The estiated salvage value of the equipent at the end of useful life is Rs Calculate the annual depreciation and book value of the construction equipent using straight-line ethod and double-declining balance ethod. Solution: Initial cost of the construction equipent = P = Rs Estiated salvage value = SV = Rs Useful life = n = 0 years For straight-line ethod, the depreciation aount for a given year is calculated using equation (3.). D D D D9 D0 Rs The book value at the end of a given year is calculated by subtracting the annual depreciation aount fro previous year s book value. Book value at the end of st year = Rs Rs Rs Book value at the end of nd year = Rs Rs Rs Siilarly the book values at the end of other years have been calculated in the sae anner. The annual depreciation aount and book values at end of years using straightline depreciation ethod are presented in Table 3.. For straight-line ethod, the book value at the end of different years can also be calculated by using equation (3.7). For exaple, the book value at the end of nd year is given by; Rs For double-declining balance ethod, the constant annual depreciation rate d is given by; d n 0 0. The depreciation aount for a given year is calculated using equation (3.6). Depreciation for st year = D Rs Depreciation for nd year = D Rs Joint initiative of IITs and IISc Funded by MHRD Page 6 of 34

7 The book value at the end of a given year is calculated by subtracting the annual depreciation aount fro previous year s book value. Book value at the end of st year = Rs Rs Rs Book value at the end of nd year = Rs Rs Rs Siilarly the annual depreciation and book value at the end of other years have been calculated in the sae anner and are presented in Table 3.. The book value at the end of different years can also be calculated by using equation (3.7). Using this equation, the book value at the end of nd year is given by; Rs Table 3. Depreciation and book value of the construction equipent using straight-line ethod and double-declining balance ethod Year Straight-line ethod Double-declining balance ethod Depreciation (Rs.) Book value (Rs.) Depreciation (Rs.) Book value (Rs.) D D Fro Table 3. it is observed that the book value at the end of useful life i.e. 0 years in double-declining balance ethod is less than salvage value. Thus the equipent will be depreciated fully before the useful life. In other words the book value reaches the estiated salvage value before the end of useful life. As the book value of the equipent is not depreciated further after it has reached the estiated salvage value, the depreciation Joint initiative of IITs and IISc Funded by MHRD Page 7 of 34

8 aount is adjusted accordingly. The book value of the construction equipent at the end of 8 th, 9 th, and 0 th years are Rs , Rs and Rs respectively (fro Table 3., Double-declining balance ethod). Siilarly the depreciation aounts for 9 th and 0 th years are Rs and Rs respectively. Thus to atch the book value with the estiated salvage value, the depreciation in 9 th year will be Rs (Rs Rs ) and that in 0 th year will be zero. This will lead to a book value of Rs (equal to salvage value) at the end of 9 th year and 0 th year. For this case, the switching fro double-declining balance ethod to straight-line ethod can be done to ensure that the book value does not fall below the estiated salvage value and this procedure of switching is presented in Lecture 3 of this odule. Joint initiative of IITs and IISc Funded by MHRD Page 8 of 34

9 Lecture- Depreciation:- Su-of-years-digits (SOYD) depreciation ethod:- It is also an accelerated depreciation ethod. In this ethod the annual depreciation rate for any year is calculated by dividing the nuber of years left (fro the beginning of that year for which the depreciation is calculated) in the useful life of the asset by the su of years over the useful life. The depreciation rate d for any year is given by; n d... (3.) SOY Where n = useful life of the asset as stated earlier nn SOY = su of years digits over the useful life = Rewriting equation (3.); d n nn.. (3.) The depreciation aount in any year is calculated by ultiplying the depreciation rate for that year with the total depreciation aount (i.e. difference between initial cost P and salvage value SV ) over the useful life. Thus the expression for depreciation aount in any year is represented by; D d P SV.... (3.3) Putting the value of d fro equation (3.) in equation (3.3) results in the following; D n SOY P SV. (3.4) The depreciation in st year i.e. D is obtained by putting equal to in equation (3.4) and is given by; n P SV D P SV n.... (3.5) SOY SOY Joint initiative of IITs and IISc Funded by MHRD Page 9 of 34

10 The book value at the end of st year is equal to initial cost less the depreciation in the st year and is given by; P D Now putting the expression of D fro equation (3.5) in the above expression results in following; P SV P n.... (3.6) SOY The depreciation in nd year i.e. D is given by; n P SV D P SV n.. (3.7) SOY SOY Book value at the end of nd year is equal to book value at the beginning of nd year (i.e. book value at the end of st year) less the depreciation in nd year and is given by; D Now putting the expressions of and D fro equation (3.6) and equation (3.7) respectively in above expression results in the following; P SV P SV D P n n SOY SOY P SV P SV P n n P n. (3.8) SOY SOY Siilarly the expressions for depreciation and book value for 3 rd year are presented below. n 3 P SV D 3 P SV n.. (3.9) SOY SOY P SV P SV 3 D3 P n n SOY SOY P SV P SV 3 P n n P 3n 3. (3.30) SOY SOY The expressions for depreciation and book value for 4 th year are given by; n 4 P SV D 4 P SV n 3 (3.3) SOY SOY Joint initiative of IITs and IISc Funded by MHRD Page 0 of 34

11 P SV P SV 4 3 D4 P 3n 3 n 3 SOY SOY P SV P SV 4 P 3n 3 n 3 P 4n 6.. (3.3) SOY SOY Siilarly the expressions for depreciation and book value for 5 th year are given by; n 5 P SV D 5 P SV n 4... (3.33) SOY SOY P SV P SV 5 4 D5 P 4n 6 n 4 SOY SOY P SV P SV 5 P 4n 6 n 4 P 5n 0 (3.34) SOY SOY The expressions for book value in different years are presented above to find out the generalized expression for book value at end of any given year. Now referring to the expressions of book values,, 3, 4, 5 in above entioned equations, it is observed that the variable ters are n, (n-), (3n-3), (4n-6) and (5n-0) respectively. These variable ters can also be written (n+-), (n+-), (3n+-4), (4n+-7) and (5n+-) respectively. The nubers,, 4, 7, and in these variable ters follow a series and it is observed that value of each ter is equal to value of previous ter plus the difference in the values of current ter and previous ter. On this note, the general expression for value of ter of this series is given by; value of ' ' ter (3.35) Now the generalized expression for book value for any year is given by; P P SV SOY n P SV P n SOY. (3.36) In this ethod also, the annual depreciation during early years is ore as copared to that in later years of the asset s useful life. Joint initiative of IITs and IISc Funded by MHRD Page of 34

12 Sinking fund (SF) depreciation ethod:- In this ethod it is assued that oney is deposited in a sinking fund over the useful life that will enable to replace the asset at the end of its useful life. For this purpose, a fixed aount is set aside every year fro the revenue generated and this fixed su is considered to earn interest at an interest rate copounded annually over the useful life of the asset, so that the total aount accuulated at the end of useful life is equal to the total depreciation aount i.e. initial cost less salvage value of the asset. Thus the annual depreciation in any year has two coponents. The first coponent is the fixed su that is deposited into the sinking fund and the second coponent is the interest earned on the aount accuulated in sinking fund till the beginning of that year. For this purpose, first the unifor depreciation aount (i.e. fixed aount deposited in sinking fund) at the end of each year is calculated by ultiplying the total depreciation aount (i.e. initial cost less salvage value) over the useful life by sinking fund factor. After that the interest earned on the accuulated aount is calculated. The calculations are shown below. The first coponent of depreciation i.e. unifor depreciation aount A at the end of each year is given by; i A. (3.37) P SV A/ F, i, n P SV n i Where i = interest rate per year Depreciation aount for st year is equal to only A as this is the aount (set aside every year fro the revenue generated) to be deposited in sinking fund at the end of st year and hence there is no interest accuulated on this aount. Therefore 0 D A A i.. (3.38) Now book value at the end of st year is given by; 0 P D P A i.. (3.39) Depreciation aount for nd year is equal to unifor aount A to be deposited at the end of nd year plus the interest earned on the aount accuulated till beginning of nd year i.e. on depreciation aount for st year. Thus depreciation for aount for nd year is given by; Joint initiative of IITs and IISc Funded by MHRD Page of 34

13 D A D i Now putting D equal to A in the above expression results in; D A Ai A i.. (3.40) Book value at the end of nd year is given by; D Now putting the expressions of and D fro equation (3.39) and equation (3.40) respectively in above expression results in the following; 0 0 i A i P A i i D P A (3.4) Depreciation aount for 3 rd year is equal to unifor aount A to be deposited at the end of 3 rd year plus the interest earned on the aount accuulated till beginning of 3 rd year i.e. on su of depreciation aounts for st year and nd year. Thus depreciation for aount for 3 rd year is given by; D 3 A D D i Putting the expressions of D and D fro equation (3.38) and equation (3.40) respectively in above expression results in the following; D 3 D D i AA A i i A Ai A ii A i A ii A On siplifying the above expression; D 3 i i A i A (3.4) Now book value at the end of 3 rd year is given by; 3 D 3 Putting the expressions of and D 3 fro equation (3.4) and equation (3.4) respectively in above expression results in; i i A i P A i i i 3 D P A... (3.43) Now the generalized expression for depreciation in any given year can be written as follows (referring to equations (3.38), (3.40) and (3.4)); i D A... (3.44) Putting the value of A fro equation (3.37) in equation (3.44) results in the following; Joint initiative of IITs and IISc Funded by MHRD Page 3 of 34

14 i D A i P SV i. (3.45) n i Siilarly the generalized expression for book value at the end of any year is given by (referring to equations (3.39), (3.4) and (3.43)); P A 0 3 i i i i i i.. (3.46) The expression in the square bracket of equation (3.46) is in the for of a geoetric series and its su represents the factor known as unifor series copound aount factor (stated in Module-). Therefore replacing the expression in square bracket of equation (3.46) with unifor series copound aount factor results in; P A F / A, i,.. (3.47) Putting the value of A fro equation (3.37) and expression for unifor series copound aount factor in equation (3.47) results in the following generalized expression for book value at the end of any year ( n). P A F / A, i, P P SV i i n i i i P P SV n.... (3.48) i In this ethod, the depreciation during later years is ore as copared to that in early years of the asset s useful life. Exaple - Using the inforation provided in previous exaple, calculate the annual depreciation and book value of the construction equipent using su-of-years-digits ethod and sinking fund ethod. The interest rate is 8% per year. Solution: P = Rs SV = Rs n = 0 years Joint initiative of IITs and IISc Funded by MHRD Page 4 of 34

15 For su-of-years-digits ethod, the depreciation aount for a given year is calculated using equation (3.4). SOY = su of years digits over the useful life = Depreciation for st year = D Book value at the end of st year: Rs Rs Rs Rs Depreciation for nd year = D Book value at the end of nd year: Rs Rs Rs Rs Siilarly the annual depreciation and book value at the end of other years for the construction equipent have been calculated and are presented in Table 3.. The book value at the end of different years can also be calculated by using equation (3.36). Using this equation, the book value at the end of nd year is calculated as follows; Rs For sinking fund ethod, the depreciation aount for a given year is calculated using equation (3.45). The interest rate per year = i = 8% Depreciation aount for st year: D Book value at the end of st year: 0.08 Rs Rs Rs Rs Depreciation aount for nd year: D Book value at the end of nd year: 0.08 Rs Rs Rs Rs Joint initiative of IITs and IISc Funded by MHRD Page 5 of 34

16 Siilarly the annual depreciation and book value at the end of other years are calculated in the sae anner and are given in Table 3.. The book value at the end of a given year can also be calculated by using equation (3.48). Using this equation, the book value at the end of nd year is given by; Rs Table 3. Depreciation and book value of the construction equipent using su-of-years-digits ethod and sinking fund ethod Year Su-of-years-digits ethod Depreciation (Rs.) Book value (Rs.) Sinking fund ethod Depreciation (Rs.) Book value (Rs.) D D * = * This inor difference is due to effect of decial points in the calculation. The book value at the end of all years over the useful life of the construction equipent obtained fro different depreciation ethods (Exaple and Exaple ) are shown in Fig. 3.. Joint initiative of IITs and IISc Funded by MHRD Page 6 of 34

17 Book value (Rs.) NPTEL Civil Engineering Construction Econoics & Finance Book Value fro different depreciation ethods SL Method DDB Method SOYD Method SF Method Line representing salvage value End of Year Fig. 3. Book value of the construction equipent using different depreciation ethods In the above figure, the line representing the salvage value is also shown. In doubledeclining balance ethod only, the calculated book value at the end of useful life i.e. 0 th year is not sae as the estiated salvage value. Joint initiative of IITs and IISc Funded by MHRD Page 7 of 34

18 Lecture-3 Switching between different depreciation ethods:- Switching fro one depreciation ethod to another is done to accelerate the depreciation of book value of the asset and thus to have tax benefits. Switching is generally done when depreciation aount for a given year by the currently used ethod is less than that by the new ethod. The ost coonly used switch is fro double-declining balance (DDB) ethod to straight-line (SL) ethod. In double-declining balance ethod, the book value as calculated by using equation (3.7) never reaches zero. In addition the calculated book value at the end of useful life does not atch with the salvage value. Switching fro double-declining balance ethod to straight-line ethod ensures that the book value does not fall below the estiated salvage value of the asset. In the following exaples the procedure of switching fro double-declining balance ethod to straight-line ethod is illustrated. Exaple -3 The initial cost of an asset is Rs It has useful life of 9 years. The estiated salvage value of the asset at the end of useful life is zero. Calculate the annual depreciation and book value using double-declining balance ethod and find out the year in which the switching is done fro double-declining balance ethod to straight-line ethod. Solution: Initial cost of the asset = P = Rs Useful life = n = 9 years Salvage value = SV = 0 For double-declining balance (DDB) ethod, the constant annual depreciation rate d is given by; d 0. n 9 The annual depreciation and the book value at the end different years are calculated using the respective equations stated earlier and are presented in Table 3.3. Joint initiative of IITs and IISc Funded by MHRD Page 8 of 34

19 Table 3.3 Depreciation and book value fro double-declining balance ethod Year Depreciation (Rs.) Book value (Rs.) Fro the above table it is observed that the book value at the end of useful life is Rs , which is ore than the estiated salvage value i.e. 0. The asset is not copletely depreciated. Thus switching is done fro double-declining balance ethod to straight-line ethod and is shown in Table 3.4. Year Table 3.4 Double-declining balance ethod and switching to straight-line ethod Depreciation aount (Rs.) (DDB ethod) Depreciation aount (Rs.) (SL ethod) Selected depreciation aount (Rs.) Book value (Rs.) ** ** Switching fro DDB ethod to SL ethod Joint initiative of IITs and IISc Funded by MHRD Page 9 of 34

20 In the above table, annual depreciation values fro double-declining balance (DDB) ethod are also presented to copare with those obtained fro straight- line ethod. For straight-line (SL) ethod the depreciation aount for a given year is calculated by dividing the difference of the book value (at the beginning of that year) and salvage value by the nuber of years reaining fro beginning of that year till the end of useful life and is shown below; D SV ( n ) For illustration, the straight-line depreciation in 4 th year ( = 4) is calculated as follows; D (9 4 ) 4 Rs Fro the annual depreciation values by straight-line ethod as shown in 3 rd colun of above table, it is observed that, the annual values are not unifor. This is because when switching is done fro DDB ethod to SL ethod, the larger depreciation aount between the two the ethods for a given year is subtracted fro the previous year s book value to calculate the book value at the end of desired year and this book value is used for calculating the depreciation aount for next year in straight-line ethod. The larger annual depreciation values (i.e. selected depreciation aount) between DDB ethod and SL ethod are provided in 4 th colun of Table 3.4. The book value at the end of a given year presented in 5 th colun of above table is obtained by subtracting selected depreciation aount of that year fro the previous year s book value. Fro the above table it is observed that the annual depreciation aount in DDB ethod is greater than that in SL ethod fro st year to 5 th year and fro 6 th year onwards the annual depreciation is greater in SL ethod than that in DDB ethod. Thus switching fro DDB ethod to SL ethod takes place in 6 th year. With switchover fro DDB ethod to SL ethod the book value at the end of useful life i.e. 9 th year is equal to zero (sae as the estiated salvage value) as copared to Rs without switching to straightline ethod. In addition the total depreciation aount over the useful life of the asset (initial cost inus salvage value) with switchover fro DDB ethod to SL ethod is Rs (i.e. su of values in 4 th colun of Table 3.4) as desired whereas the total Joint initiative of IITs and IISc Funded by MHRD Page 0 of 34

21 depreciation aount without switchover is Rs (su of values in nd colun of Table 3.4). Exaple -4 A piece of construction equipent has initial cost and estiated salvage value of Rs and Rs respectively. The useful life of equipent is 0 years. Find out the year in which the switching fro double-declining balance ethod to straight-line ethod takes place. Solution: Initial cost of the asset = P = Rs , Salvage value = SV = Rs Useful life = n = 0 years The constant annual depreciation rate d for double-declining balance (DDB) ethod is calculated as follows; d n 0 0. The annual depreciation aount and the book value at the end different years using DDB ethod are presented in Table 3.5. Table 3.5 Depreciation and book value using double-declining balance ethod Year Depreciation (Rs.) Book value (Rs.) Joint initiative of IITs and IISc Funded by MHRD Page of 34

22 Fro the above table it is noted that the book value at the end of useful life i.e. 0 th year is Rs.606.7, which is less than the estiated salvage value i.e. Rs Thus the total depreciation aount over the useful life is ore than the desired. Hence switching is done fro double-declining balance ethod to straight-line ethod and is presented in Table 3.6. In Table 3.6, annual depreciation by SL ethod, selected depreciation aount and book values are calculated in the sae anner as that in the previous exaple. The annual depreciation for 0 th year in DDB ethod is Rs and subtracting this aount fro 9 th year book value results in a book value (at the end of 0 th year) less than the estiated salvage value (As seen fro Table 3.5). Further the book value at the end of 9 th year (i.e. Rs ) is greater than salvage value. Thus switching fro DDB ethod to SL ethod takes place in 0 th year. With switching fro DDB ethod to SL ethod, the book value at the end of 0 years is equal to Rs (sae as the estiated salvage value) as copared to Rs without switching to straight-line ethod. Year Table 3.6 Double-declining balance ethod with switching to straight-line ethod Depreciation aount (Rs.) (DDB ethod) Depreciation aount (Rs.) (SL ethod) Selected depreciation aount (Rs.) Book value (Rs.) ** ** Switching fro DDB ethod to SL ethod Further the total depreciation aount (i.e. initial cost less estiated salvage value) with switchover fro DDB ethod to SL ethod is Rs (i.e. su of values in 4 th Joint initiative of IITs and IISc Funded by MHRD Page of 34

23 colun of Table 3.6) as desired whereas the total depreciation aount without switchover is Rs (su of values in nd colun of Table 3.6). Joint initiative of IITs and IISc Funded by MHRD Page 3 of 34

24 Lecture-4 Inflation: Inflation is defined as an increase in the aount of oney required to purchase or acquire the sae aount of products and services that was acquired without its effect. Inflation results in a reduction in the purchasing power of the onetary unit. In other words, when prices of the products and services increase, we buy less quantity with sae aount of oney i.e. the value of oney is decreased. For exaple, the quantity of ites we purchase today at a cost of Rs.000 is less than that was purchased 5 years ago. This is due to a general change (increase) in the price of the goods and services with passage of tie. On the other hand deflation results in an increase in the purchasing power of the onetary unit with tie and it rarely occurs. Due to effect of deflation, ore can be purchased with sae aount of oney in future tie period than that can be purchased today. The inflation rate (f) is easured as the rate of increase (per tie period) in the aount of oney required to obtain sae aount of products and services. Till now, the interest rate i that was used in the econoic evaluation of a single alternative or between alternatives by different ethods as entioned in earlier lectures was assued to be inflation-free i.e. the effect of inflation on interest rate was excluded. This interest rate i is also known as also real interest rate or inflation-free interest rate. It represents the real gain in oney of the cash flows with tie without the effect of inflation. However if inflation is there in the general arket, then effect of inflation on the interest rate needs to be taken into account for the econoic analysis. The interest rate that includes the effect of price inflation which is occurring in general econoy is known as the arket interest rate (i c ). It takes into account the adjustent for the price inflation in the arket. Market interest rate is also known as inflated interest rate or cobined interest rate as it cobines the effect of both real interest rate and the inflation. In addition to above paraeters, it is also required to define two paraeters naely actual onetary units and constant value onetary units while considering the effect of inflation in the cash flow of the alternatives. The onetary units can be Rupees, Dollars, Euros etc. The actual onetary units are also referred as future or inflated onetary units. The purchasing power of the actual onetary units includes the effect of inflation on the Joint initiative of IITs and IISc Funded by MHRD Page 4 of 34

25 cash flows at the tie it occurs. The constant value onetary units are also called as real or inflation-free onetary units. The constant value onetary units are expressed in ters of the sae purchasing power for the cash flows with reference to a base period. Mostly in engineering econoic studies, the base period is taken as 0 i.e. now. But it can be of any tie period as required. The effect of inflation on cash flows is deonstrated in the following exaple. Exaple -5 The present (today s) cost of an ite is Rs The inflation rate is 5% per year. How does the inflation affect the cost of the ite for the next five years? Solution: The calculations are shown in Table 3.7. Table 3.7 Effect of inflation on future cost of the ite Year Increase in cost of ite due to 5% per year (Rs.) Cost of the ite in inflated onetary units i.e. inflated cost (Rs.) Cost of the ite in inflated onetary units expressed as function of today s cost (i.e. in inflation-free onetary units) and inflation rate, (Rs.) x 0.05 = = x ( ) = x 0.05 = = x ( ) = x 0.05 = = x ( ) 3 = x 0.05 = = x ( ) 4 = x 0.05 = = x ( ) 5 = Fro above table, the relationship between cost in actual onetary units (inflated) in tie period n and cost in constant value onetary units (inflation-free) can be represented as follows; Cost in actual onetary units (inflated) = Cost in constant value onetary units (+f) n. (3.49) Joint initiative of IITs and IISc Funded by MHRD Page 5 of 34

26 Fro equation (3.49), the expression for cost in constant value onetary units (inflationfree) is written as follows Cost in constant Cost in actual onetary units (inflated) value onetary units (inflation - free) f n. (3.50) In the above expressions, f is the inflation rate per year i.e. 5% and the base or reference period is considered as 0. However the above relationship between constant value onetary units and actual onetary units can also be written for any base period b. Cost in constant value onetary units (inflation - free) Cost in actual onetary units (inflated) f nb.. (3.5) In the above relationship, the base period b defines the purchasing power of constant value onetary units. As shown in Table 3.7, the actual cost (inflated) of the ite in years,, 3, 4 and 5 are Rs.000, Rs.050, Rs.35.50, and Rs respectively. However the cost of the ite in inflation-free or constant value rupees in all the years is always Rs.0000 [i.e. 000/(+0.05), 050/(+0.05), etc.] i.e. equal to the cost at the beginning. In this exaple, effect of interest rate i.e. effect of tie value of oney is not considered. Joint initiative of IITs and IISc Funded by MHRD Page 6 of 34

27 Lecture-5 Equivalent worth calculation including the effect of inflation:- For calculations of equivalent worth of cash flows of alternatives with price inflation in action, the interest rate to be adopted (i.e. real interest rate or cobined interest rate) depends on whether the cash flows are expressed in inflated rupees or inflation-free rupees. When the cash flows are calculated in constant value onetary units, inflationfree interest rate (real interest rate) i is used. Siilarly when the cash flows are estiated in actual onetary units, cobined interest rate (inflated interest rate) i c is used. The relationship between present worth and future worth considering the tie value of oney is given by; F P.. (3.5) c i n P = present worth (at base period 0 ) F c = future worth in constant value onetary units (inflation-free) In the above equation, future worth in constant value onetary units (inflation-free) can be represented in ters of future worth in actual onetary units (inflated) by using the relationship stated in equation (3.50). Equation (3.5) can be rewritten by including the inflation rate. P F a f n i n.. (3.53) In equation (3.53), F a is the future worth in actual onetary units (inflated) and f is the inflation rate as stated earlier. Equation (3.53) can be rewritten as follows; P F a P F a i f i f f i n f i n. (3.54)... (3.55) Joint initiative of IITs and IISc Funded by MHRD Page 7 of 34

28 The expression i f f i in the above equation that cobines the effect of real interest rate i and inflation rate f is tered as cobined interest rate or arket interest rate (i c ) as stated earlier. i c i f f i.. (3.56) Equation (3.55) can be rewritten as follows; P F a i n c. (3.57) Using equation (3.57), the future worth in inflated onetary units can be calculated fro known value of present worth P. Equation (3.57) can also be written in functional representation as follows; P F P / F, i n.... (3.58) a c, Siilarly the relationship between annual worth A and present worth P and that between A and future worth F of the cash flows considering the effect of inflation can be obtained. Fro known values of cobined interest rate i c and inflation rate f, the real interest rate i can be obtained fro equation (3.56) and is given as follows; i c i f f i f i f i c f i... (3.59) f The real interest rate i (inflation-free) in equation (3.59) represents the equivalence between cash flows occurring at different periods of tie with sae purchasing power. In the following exaples the effect of inflation on equivalent worth of cash flows is illustrated. Exaple -6 A person has now invested Rs at a arket interest rate of 4% per year for a period of 8 years. The inflation rate is 6% per year. i) What is the future worth of the investent at the end of 8 years? ii) What is the accuulated aount at the end of 8 years in constant value onetary unit i.e. with the sae buying power when the investent is ade? Joint initiative of IITs and IISc Funded by MHRD Page 8 of 34

29 Solution: The arket or cobined interest rate per year (i c ) and inflation rate per year (f) are4% and 6% respectively. i) The future worth i.e. aount of oney accuulated at the end of 8 years will include the effect of inflation and the interest aount accuulated will be calculated using arket or cobined interest rate. FW P F / P, i, c n Putting P, i c, n equal to Rs , 4% and 8 years respectively in the above expression; FW F / P,4%, 8 FW Thus the future worth of the investent at the end of 8 years in actual onetary unit (inflated) is Rs ii) The future worth in constant value onetary unit i.e. with sae buying power when the investent is ade (i.e. now) can be obtained by excluding the effect of inflation fro the inflated future worth. Thus the future worth in constant value onetary unit is calculated by putting the values of tie period n, base period b, inflation rate f and the future worth (inflated) as calculated above in equation (3.5). In this case, the value of base period b is zero. FW in constant value onetary unit (inflation - free) ' FW' in actual onetary unit (inflated) FW in constant value onetary unit (inflation - free) FW in constant value onetary unit (inflation - free) Rs f Thus the future worth of the investent at the end of 8 years in constant value onetary unit (inflation-free) is Rs The future worth of the investent with sae buying power as now i.e. when the investent is ade can also be calculated by using the real interest rate. The real interest rate can be calculated fro arket interest rate and inflation rate using equation (3.59). i c f i f nb Joint initiative of IITs and IISc Funded by MHRD Page 9 of 34

30 i i = 7.547% Now the future worth of the investent with sae buying power as now is calculated as follows; FW P F / P, i, n. F / P, 7.547%, FW The calculated future worth of investent is sae as that calculated earlier. The inor difference between the values is due to the effect of decial points in the calculations. Exaple -7 The cash flow of an alternative consists of payent of Rs per year for 6 years (unifor annual series). The real interest rate is 9% per year. The inflation rate per year is 5%. Find out the present worth of the unifor series when the annual payents are in actual onetary units (inflated). Solution: For deterination of present worth of the unifor annual series payent, which is in actual onetary units (inflated), first the cobined interest rate (i c ) is calculated fro the known values of real interest rate (i) and inflation rate (f) using equation (3.56). i = 9%, f = 5% i c i c i f f i i c = 4.45% Now the present worth of the unifor series payent (in actual onetary units) with unifor annual aount A (Rs.00000) is given by; PW A P / A, i, c n P / A,4.45%, PW PW = Rs.3840 The present worth of the unifor series payent (in actual onetary unit) can also be calculated by converting the unifor annual aount A occurring in different years into constant value onetary units (inflation-free) and then deterining the present worth of the series at real interest rate (i) by using P/F factor. Joint initiative of IITs and IISc Funded by MHRD Page 30 of 34

31 The values of unifor annual aount A (Rs.00000) occurring in different years in constant value onetary units (inflation-free) are calculated as follows; In year f 0.05 In year f 0.05 In year f 0.05 In year f 0.05 In year f 0.05 In year f Rs Rs Rs Rs.870. Rs Rs Now the present worth of above aounts at real interest rate i.e. 9% per year is calculated as follows; PW PW P/F,i, P/F,i, P/F,i,3 870.P/F,i, P/F,i, P/F,i,6 P/F, 9%, P/F, 9%, P/F, 9%,3 870.P/F, 9%, P/F, 9%, P/F, 9%,6 PW PW = Rs.3840 Thus the present worth of the unifor series payent (in actual onetary unit) is sae as that calculated earlier. Joint initiative of IITs and IISc Funded by MHRD Page 3 of 34

32 Taxes: In the econoic coparison of utually exclusive alternatives, it is possible to select the best alternative without considering the effect of taxes, as all the alternatives are evaluated on the sae basis of not taking into account the effect of taxes on the cash flows. However the inclusion of taxes in the econoic evaluation of alternatives results in iproved estiate of cash flows and the estiated return on the investent. There are different types of taxes which are iposed by central/state governents on the firs. These are incoe taxes (on taxable incoe), property taxes (on value of assets owned), sales taxes (on purchase of goods/services), excise taxes (on sale of specific goods/services), etc. Aong these taxes, incoe tax is the ost iportant one, which is considered in the engineering econoic analysis. Usually property taxes, excise taxes and sales taxes are not directly associated with incoe of a fir. The calculation of incoe taxes is based on taxable incoe of the fir and the incoe tax to be paid by a fir is equal to taxable incoe ultiplied with the applicable incoe tax rate. The taxable incoe is calculated at the end of each financial year. For obtaining the taxable incoe, all allowable expenses (excluding capital investents) and depreciation cost are subtracted fro the gross incoe of the fir. It ay be noted here that depreciation is not the actual cash outflow but it represents the decline in the value of the asset (depreciable) and is considered as an allowable deduction for calculating the taxable incoe. The taxable incoe is also known as net incoe before taxes. The net incoe after taxes is obtained by subtracting the incoe taxes fro taxable incoe (i.e. net incoe before tax). The expression for taxable incoe is shown below. Taxable incoe= gross incoe - all expenses (excluding capital expenditur es)- depreciation The details about revenues, expenses and profit or loss of a fir are presented in Lecture of Module 6. The capital investents are not included in the expenses for calculating the taxable incoe, as this does not affect the incoe of the fir directly; however the ode of financing the capital investent ay have an effect on the taxes. The incoe tax to be paid is calculated as a percentage of the taxable incoe and is given as follows; Incoe tax=taxable incoe incoetax rate Joint initiative of IITs and IISc Funded by MHRD Page 3 of 34

33 The applicable annual tax rate is usually based on range or bracket of taxable incoe and the taxable incoe is charged as per the graduated tax rates wherein higher tax rates are applied to higher taxable incoe. A different arginal tax rate is applied to each of the taxable incoe bracket. Marginal tax rate indicates the rate that is applied on each additional unit of currency (Rupees, Dollars or Euros) of the taxable incoe. Higher arginal tax rate is applied to higher taxable incoe range. The taxable incoe in a certain bracket is charged at the specified arginal tax rate that is fixed for that particular bracket. Soeties depending upon the taxable incoe, a single tax rate (i.e. flat tax rate) is applied to calculate the incoe tax. Capital gain and loss which represent the gain and loss on the disposal or sale of depreciable assets/property are also considered in the incoe tax analysis. The after-tax econoic analysis of an alternative can be carried out by calculating the annual cash flows after taxes and after-tax rate of return. The after-tax rate of return depends on the before-tax rate of return and the tax rate. The annual cash flows after taxes is obtained by subtracting the taxes fro annual cash flows before taxes. The calculation of incoe tax using arginal tax rates for different taxable incoe brackets is presented in the following exaple. Exaple -8 For a given financial year, the gross incoe of a construction contractor is Rs The expenses excluding the capital investent and the depreciation are Rs and Rs respectively. Calculate the taxable incoe and incoe tax for the financial year with the following assued arginal tax rates for different taxable incoe ranges. Taxable incoe range (Rs.) Incoe tax rate (%) Over Joint initiative of IITs and IISc Funded by MHRD Page 33 of 34

34 Solution: For the financial year, the taxable incoe of the construction contractor is calculated using the expression stated earlier. Taxable incoe = gross incoe -expenses (excluding capital expenditures) - depreciation Taxable incoe = Rs Rs Rs = Rs The incoe tax will be equal to the su of incoe taxes calculated over the taxable incoe ranges at the specified tax rates. The construction contractor will pay 0% on first Rs of taxable incoe, 0% on next Rs (i.e. Rs Rs ) of taxable incoe, 30% on next Rs (i.e. Rs Rs ) of taxable incoe and 35% on the reaining Rs (Rs Rs ) of taxable incoe. Incoe tax = = Rs.9785 The taxable incoe and incoe tax for the financial year are Rs and Rs.9785 respectively. For the construction contractor with the above taxable incoe, the average tax rate fore the financial year is calculated as follows; Average tax rate= taxes paid taxable incoe Rs.9785 = Rs = 0.69=.69% Joint initiative of IITs and IISc Funded by MHRD Page 34 of 34

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