# Life Cycle Costing 生命周期成本計算

Life Cycle Costing 生命周期成本計算 KCTang Thu, 25/12/2014 - 16:33

## Life Cycle Costing 生命周期成本計算

1. Mainly for comparing between different options with different expenditure patterns over a long period of years of use.
主要用來比較在很長的使用年期有不同的支出幅度的不同方案。
2. Considering the whole life cost of a project (or product, service) from birth to death (or before recycling).
考慮某項目(或產品、服務)由生到滅(或再生前)的全部成本。
3. Using discounted cash flow techniques to convert monies spent over different times to the same base.
用貼現金流的方式把不同時間支出的金錢轉換到同一基準上。
4. Impacts on the use of natural resources are now also under the scope of consideration.
對自然資源使用的影響現在亦納入考慮範圍。

## Life Cycle Costs 生命周期成本

1. Capital Costs
資本費用
2. Costs-in-use
使用費用

• Routine Operating, Maintenance, and Repair Costs
日常的使同、保養及維修費用
• Replacement Costs
更換費用
• End of Life Costs (Removal and Disposal Costs less Residual Values (Resale Values or Salvage Values))
終結費用(清拆及處置費用減剩餘價值(轉售價值或廢品價值))
• Non-Monetary Benefits or Costs
非金錢的益處或成本

## Time Value of Money

### Definitions

• PV = Present value / principal amount invested now
• FV = Future value / terminal value
• A = Annuity, a periodic payment
• R% = % rate of interest / rate of return at the end of a period (usually year)
• N = Number of period (usually year)
• C = 1 + R%
• D = CN - 1, which can be understood as the difference between (FV of 1 after N) and 1, or as the total interest earned

### Summary of Formulae - Future Values

• Future Value (FV) of \$1 after N periods = Future Value Factor = Amount of \$1
```FV of 1 after N = 1 + N * R% (at simple interest)
FV of 1 after N = C<sup>N</sup> (at compound interest)```
• Future Value of an Annuity of \$1 after N periods at compound interest = Future Value of \$1 per annum if the period is a year
```FV of A of 1 after N
= D / R%```
• Future Value of an Annuity Due of \$1 after N periods at compound interest
```FV of A Due of 1 after N
= (FV of A of 1 after N) * C, to compound by 1 more period
= C * D / R%```
• Annuity (e.g. sinking fund) invested at the end of each of N periods to give FV of \$1 at compound interest
```A after N to give FV of 1
= 1 / (FV of A of 1 after N)
= R% / D```
• Annuity (e.g. sinking fund) invested at the beginning of each of N periods to give FV of \$1 at compound interest
```A Due after N to give FV of 1
= 1 / (FV of A Due<span style="color:#c0392b"> </span>of 1 after N)
= R% / (C * D)```

### Summary of Formulae - Present Values

• Present Value (PV) of \$1 after N periods at compound interest = Present Value Factor
```PV of 1 after N
= 1 / FV of 1 after N
= 1 / C<sup>N
</sup>```
• Present Value of an Annuity to give \$1 after N periods at compound interest = Present Value of \$1 per annum or Year's Purchase if the period is a year
```PV of A of 1 after N
<span style="color:#c0392b"><></span> 1/ FV of A of 1 after N, like the last formula
but
= FV of A of 1 after N x PV of 1 after N
= (D / R%) * (1 / C<sup>N</sup>)
= D / (R% * C<sup>N</sup>)```
• Present Value of an Annuity Due of \$1 after N payments at compound interest
```PV of A Due of 1 after N
<span style="color:#c0392b"><></span> 1 / FV of A Due of 1 after N
but
= FV of A Due of 1 after N x PV of 1 after N
= (C * D / R%) * (1 / C<sup>N</sup>)
= C * D / (R% * C<sup>N</sup>)
also
= (PV of A of 1 after N) * C, discounted by 1 less period
= C * D / (R% * C<sup>N</sup>)```
• Annuity paid at the end of each of N periods to equal \$1 now at compound interest
```A after N to equal PV of 1
= 1 / (PV of A of 1 after N)
= R% * C<sup>N</sup> / D
= (A after N to give FV of 1) + R%
= Sinking fund at end of each period to repay the loan at the end + interim loan interest R%, where the R% for the sinking fund may be different from the R% for loan interest```
• Annuity due paid at the beginning of each of N periods to equal \$1 now at compound interest
```A Due after N to equal PV of 1
= 1 / (PV of A Due of 1 after N)
= R% * C<sup>N</sup> / (C * D) ```

### Future Value (FV) of \$1 at simple interest

• Also called "Future Value Factor" or "Amount of \$1"
• A principal of \$1 invested now earning R% interest per number of period
• After 1 period, FV of 1 = 1 + 1 * R% = 1 + R%
• Interest earned taken away leaving the principal there for the next period
• Base to earn interest for the 2nd period, still = 1
• After 2 periods, FV of 1 = 1 + R% + R% = 1 + 2 * R%
• After N periods, FV of 1 = 1 + N * R%
• FV of 1 = 1 + N * R% (at simple interest)

### Future Value (FV) of \$1 at compound interest

• Also called "Future Value Factor" or "Amount of \$1"
• A principal of \$1 invested now earning R% interest per number of period
• After 1 period, FV of 1 = 1 + 1 * R% = 1 + R%
• Interest earned kept with the principal there for the next period
• New base to earn interest for the 2nd period = 1 + R%
• After 2 periods, FV of 1 = (1 + R%) * (1 + R%)
• After N periods, FV of 1 = (1 + R%)N
• FV of 1 = (1 + R%)N (at compound interest)
• FV of 1 = CN (at compound interest)

### Discounting

• Whatever method, \$1 received in the future will be of less value than \$1 received now.
• When comparing costs spending over a long period of time, they must be brought to the same basis for comparision, using bringing back to the present value.
• The process of calculating the present value is called "discounting".

### Compounding

• If ones want to know the future value of monies incurred over a period of time, the process of calculating the future value is called "compounding".

### Factors

• For all the compounding and discounting calculations, it would be easier if the conversion factor is based on \$1 such that:
• Result \$ = Base \$ * Factor/\$1.

### Future Value of an Annuity of \$1 after N periods at compound interest

• Also called: "Future Value of \$1 per annum" if the period is a year
• \$1 invested at the END of each of N periods earning R% interest per number of period
• One way to calculate the future sum:
• FV of 1 = CN
• FV of 1 invested      1 period later for  N-1 periods = CN-1
• FV of 1 invested     2  periods later for N-2 periods = CN-2
• FV of 1 invested N-1  periods later for     1 period  = C
• FV of 1 invested N     periods later for     0 period   = 1 (no interest)
• Sum                    =                      CN-1 + CN-2 + .... + C2 + C + 1
• Another way to calculate the future sum:
• Sum after 1 period with 1 invested at the end = 1
• with sum earning interest next period and another 1 invested at the end of next period
• Sum after 2 periods     = C + 1
• Sum after 3 periods     = C+ C + 1
• Sum after N-1 periods = CN-2 + ... + C2 + C + 1
• Sum after N periods    = CN-1 + CN-2 + ... + C2 + C + 1
• Sum                   =                      CN-1 + CN-2 + .... + C2 + C + 1
• Mutiply all on both sides by C
• Sum * C = C + CN-1 + CN-2 + .... + C2 + C
• Subtract between the two equations
• Sum * R%         = C - 1
• Sum                  = D / R%
• Therefore
• FV of A of 1 after N

= D / R%

### Future Value of an Annuity Due of \$1 after N periods at compound interest

• \$1 invested at the BEGINNING of each of N periods earning R% interest per number of period.
• This is equivalent to keeping the total money at the end and investing it for 1 more period to earn interest without adding 1 at the end of the 1 more period:
• FV of A Due of 1 after N

= (FV of A of 1 after N) * C

= C * D / R%

• Compare with (FV of A of 1 after N+1) - 1

= (CN+1 - 1) / R% - 1

= (CN+1 - 1 - R%) / R%

= [CN+1 - (1 + R%)] / R%

= (CN+1 - C) / R%

= C * D / R%

• Therefore, FV of A Due of 1 after N

also = (FV of A of 1 after N+1) - 1

diagrammatically:

-->--                                = Compounding with interest

0-----1-->--1-->--1           = FV of A of 1 after N payments, compounding for N-1 periods

0-----1-->--1-->--1-->--1  = FV of A of 1 after N+1 payments, compouding for N periods

1-->--1-->--1-->--0         = Subtract 1 at the end = FV of A Due of 1 after N payments, compouding for N periods

### Annuity (e.g. sinking fund) invested at the end of each of N periods to give FV of \$1 at compound interest

• Find A where (FV of A after N) = 1
• FV of A after N = A * (FV of A of 1 after N) = 1
• A after N to give FV of 1

= 1 / (FV of A of 1 after N)

= R% / D

### Annuity due (e.g. sinking fund) invested at the beginning of each of N periods to give FV of \$1 at compound interest

• Find A where (FV of A Due after N) = 1
• FV of A Due after N = M * (FV of A Due of 1 after N) = 1
• A Due after N to give FV of 1

= 1 / (FV of A Due of 1 after N)

= R% / (C * D)

### Present Value (PV) of \$1 after N periods at compound interest

• Also called "Present Value Factor"
• PV * (FV of 1) = FV
• PV = FV / (FV of 1)
• PV of 1 in the future = 1 / (FV of 1)
• Since FV of 1 = CN
• PV of 1 after N = 1 / CN

### Present Value of an Annuity of \$1 after N periods at compound interest

• Also called: "Present Value of \$1 per annum" or "Year's Purchase" if the period is a year
• \$1 paid at the END of each of N periods to be discounted back to the present value at R% interest per number of period passed
• Instead of discounting the individual payments back for different periods, the total future value of all payments after N is discounted back once
• PV of A of 1 after N

= (FV of A of 1 after N) * (PV of 1 after N)

= (D / R%) * (1 / CN

= D / (R% * CN)

= (CN - 1) / (R% * CN

= (1 - 1 / CN) / R%

= [1 - (PV of 1 after N)] / R%

### Present Value of an Annuity Dueof \$1 after N payments at compound interest

• \$1 paid at the BEGINNING of each of N periods to be discounted back to the present value at R% interest per number of period passed
• Instead of discounting the individual payments back for different periods, the total future value of all payments after N is discounted back once
• PV of A Due of 1 after N payments

= (FV of A Due of 1 after N) * PV of 1 after N

= (C * D / R%) * (1 / CN

= C * D / (R% * CN), which is easier for doing manual calculations

= D / (R% * CN-1)

• C * D / (R% * CN)

= C * [D / (R% * CN)]

= C * (PV of A of 1 after N)

• This is equivalent to discounting by 1 less period as compared with paying the annuity at the end of each period.
• The discounting factor for 1 more period =  * 1 / C
• The discounting factor for 1 less period = * C
• Compare with (PV of A of 1 after N-1) + 1, which is

= (CN-1 - 1) / (R% * CN-1) + 1

= (CN-1 - 1 + R% * CN-1) / (R% * CN-1)

= (CN-1 * (1 + R%) - 1 ) / (R% * CN-1)

= (CN-1 * C- 1 ) / (R% * CN-1)

= (C- 1 ) / (R% * CN-1)

= D / (R% * CN-1)

• Therefore, PV of A Due of 1 after N

also = (PV of A of 1 after N-1) + 1

diagrammatically:

--<--                                = Discounting with interest

0--<--1--<--1--<--1--<--1 = PV of A of 1 after N payments, discounting for N periods

0--<--1--<--1--<--1           = PV of A of 1 after N-1 payments, discounting for N-1 periods

1--<--1--<--1--<--1           = Add 1 to the beginning = PV of A Due of 1 after N payments, discounting for N-1 periods

### Annuity paid at the end of each of N periods to equal \$1 now at compound interest

• Find A where (PV of A after N) = 1
• PV of A after N = M * (PV of A of 1 after N) = 1
• A after N to equal PV of 1

= 1 / (PV of A of 1 after N)

= R% * CN / D

= R%* (D + 1) / D, because D = CN -1

= R% / D + 1/ D

= (A after N to give FV of 1) + R%

which can be considered as of two portions

= Sinking fund at end of each period to repay the loan at the end + interim loan interest R%, where the R% for the sinking fund may be different from the R% for loan interest.

### Annuity due paid at the beginning of each of N periods to equal \$1 now at compound interest

• Find A Due where (PV of A Due after N) = 1
• PV of A Due after N = M * (PV of A Due of 1 after N) = 1
• A Due after N to give PV of 1

= 1 / (PV of A Due of 1 after N)

= (R% * CN)  / (C * D), which is easier for doing manual calculations

= (R% * CN-1)  / D

## Present Values of Life Cycle Costs

• Being sum of:

• Capital Costs

• (spend now) * 1
• Costs-in-use
• Routine Operating, Maintenance, and Repair Costs

• (average amounts spent every year) * (PV of A of 1 after N years)
• Replacement Costs
• (specific amounts spent after M years) * (PV of 1 after M years)
• End of Life Costs (Removal and Disposal Costs less Residual Values (Resale Values or Salvage Values))
• (specific amounts spent or received after N years) * (PV of 1 after N years)