# Life Cycle Costing 生命周期成本計算

Life Cycle Costing 生命周期成本計算 KCTang Sat, 06/03/2021 - 17:31

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## Note

6 Mar 2021: Table of Formulae last two columns merged.

11 Mar 2020:  Interest rate "may be" changed to "must be" when converting to monthly interest rate.

10 Mar 2020: Conversion from annual interest rate to monthly interest rate added.

28 Jan 2019: Revised to re-arrange the order and give a simpler summary of the formulae.

7 Feb 2018: Created.

## 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

## 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 or Renewal 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)

## Time Value of Money

### Definitions

• PV = Present value / principal amount invested now
• FV = Future value / terminal value
• A = Annuity, a periodic payment
• N = Number of period (usually year)
• R% = % rate of interest / rate of return at the end of a 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

### Table of Formulae (compound interest)

 FV of \$1 paid now after N periods (1 payment) Amount of \$1 after N periods CN PV of \$1 paid after N periods (1 payment) 1 / CN FV of Annuity of \$1 paid at the end of each period after N periods (N payments) FV of \$1 per annum after N years (CN - 1) / R% = D / R% Annuity paid at the end of each period after N periods to give FV of \$1 (N payments) R% / (CN - 1) = R% / D FV of Annuity Due of \$1 paid at the beginning of each period after N periods (N payments) (CN - 1) * C / R% = D * C / R% Annuity Due paid at the beginning of each period after N periods to give FV of \$1 (N payments) R% / C / (CN - 1) = R% / C / D PV of an Annuity of \$1 paid at the end of each period after N periods (N payments) PV of \$1 per annum after N years Years Purchase if the period is a year (CN - 1) / CN  / R%  or (1 / CN - 1) / R% * (-1) for  easier manual calculation = D / CN  / R% Annuity paid at the end of each period after N periods to equal PV of \$1 (N payments) R% / (CN - 1) + R% = R% * D  + R% PV of Annuity Due of \$1 paid at the beginning of each period after N payments (N -1 periods) (CN - 1) / CN-1 / R% = D / CN-1 / R% Annuity Due paid at the beginning of each period after N payments to equal PV of \$1 (N - 1 periods) R% * CN-1 / (CN - 1) = R% * CN-1 / D Converting from annual rate of interest AR% to monthly compound rate of interest MR% MR% = AR%/12 (usual) MR% = ((1 + AR%)1/12) - 1 (accurate)

### Examples Diagrammatically:

• PV-->--FV = a period with compound interest of 10%
• PV--<--FV = discounting
•  = payment of 1 monetary value

FV of 

• -->--1.1-->--1.21-->--1.331

PV of :

• reversing FV of 1, divided by 1.1 each time
• after 1 period: 0.909--<--
• after 2 periods: 0.826--<--0.909--<--
• after 3 periods: 0.751--<--0.826--<--0.909--<--

FV of A of :

• after 1 period: 0-----=1
• after 2 periods: 0-----=1-->--1.1+=2.1
• after 3 periods: 0-----=1-->--1.1+=2.1-->--2.31+=3.31

A of FV of 1:

• A * (FV of A of 1) = FV
• A = FV / (FV of A of 1)
• A of FV of 1 = 1 / (FV of A of 1)
• after 1 period: A of FV of 1 = 1/1 = 1
• after 2 periods: A of FV of 1 = 1/2.1  = 0.476
• after 3 periods: A of FV of 1 = 1/3.31 = 0.302

FV of A Due of :

• after 1 period: -->--1.1
• after 2 periods: -->--1.1+=2.1-->--2.31
• after 3 periods: -->--1.1+=2.1-->--2.31+=3.31-->--3.641

A Due of FV of 1:

• A Due * (FV of A Due of 1) = FV
• A Due = FV / (FV of A Due of 1)
• A Due of FV of 1 = 1 / (FV of A Due of 1)
• after 1 period: A Due of FV of 1 = 1/1.1 = 0.909
• after 2 periods: A Due of FV of 1 = 1/2.31  = 0.433
• after 3 periods: A Due of FV of 1 = 1/3.641 = 0.275

PV of A of  =  FV of A of , all discounted back to now by dividing by 1.10 for each period:

• FV after 1 period: 0-----=1
• now: 0.909--<--1
• FV after 2 periods: 0-----=1-->--1.1+=2.1
• now: 1.736--<--1.909--<--2.1
• FV after 3 periods: 0-----=1-->--1.1+=2.1-->--2.31+=3.31
• now: 2.487--<--2.736--<--3.01--<--3.31

A of PV of 1:

• A * (PV of A of 1) = PV
• A = PV / (PV of A of 1)
• A of PV of 1 = 1 / (PV of A of 1)
• after 1 period: A of PV of 1 = 1/0.909 = 1.10
• after 2 periods: A of PV of 1 = 1/1.736 = 0.576
• after 3 periods: A of PV of 1 = 1/2.487 = 0.402

PV of A Due of  =  FV of A Due of , all discounted back to now by dividing by 1.10 for each period:

• FV after 1 period: -->--1.1
• now: 1--<--1.1
• FV after 2 periods: -->--1.1+=2.1-->--2.31
• now: 1.909--<--2.10--<--2.31
• FV after 3 periods: -->--1.1+=2.1-->--2.31+=3.31-->--3.641
• now: 2.736--<--3.009--<--3.31--<--3.641

A Due of PV of 1:

• A Due * (PV of A Due of 1) = PV
• A Due = PV / (PV of A Due of 1)
• A Due of PV of 1 = 1 / (PV of A Due of 1)
• after 1 period: A Due of PV of 1 = 1/1 = 1
• after 2 periods: A Due of PV of 1 = 1/1.909  = 0.524
• after 3 periods: A Due of PV of 1 = 1/2.736 = 0.365

### ​​​​​​​FV of \$1 paid now at simple interest (1 payment)

• 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 after N periods
```= 1 + N * R% (at simple interest)
```

### FV of \$1 paid now at compound interest (1 payment)

• 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 after N periods
```= (1 + R%)N (at compound interest)
= CN```

### 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 comparison, 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
```

### PV of \$1 paid after N periods at compound interest (1 payment)

• 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 periods
`= 1 / CN`

### FV of an Annuity of \$1 paid at the end of each period after N periods at compound interest (N payments)

• 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                  = (CN - 1) / R%
• Therefore
• FV of A of \$1 after N periods
```= (CN - 1) / R%
```

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

• 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 periods to give FV of \$1
```= 1 / (FV of A of 1 after N)
= R% / (CN - 1)
```

### FV of an Annuity Due of \$1 paid at the beginning of each period after N periods at compound interest (N payments)

• \$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 periods
```= (FV of A of 1 after N) * C
= (CN - 1) * C / 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%
= (CN - 1) * C / R%
```
• Therefore, FV of A Due of 1 after N

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

diagrammatically:

0-----1                                  = payment of 1 at the end of a period

-->--                                     = a period 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, compounding for N periods

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

### Annuity Due (e.g. sinking fund) paid at the beginning of each period after N periods to give FV of \$1 at compound interest (N payments)

• Find A where (FV of A Due after N) = 1
• FV of A Due after N = A * (FV of A Due of 1 after N) = 1
• A Due after N periods to give FV of \$1
```= 1 / (FV of A Due of 1 after N)
= R% / [(CN - 1) * C]
= R% / (CN - 1) / C```

### PV of an Annuity of \$1 paid at the end of each period after N periods at compound interest (N payments)

• Also called: "Present Value of \$1 per annum" or "Years 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 periods
```= (FV of A of 1 after N) * (PV of 1 after N)
= [(CN - 1) / R%] * (1 / CN)
= (CN - 1) / CN / R%
= (1 - 1 / CN) / R%
= (1 / CN - 1) / R% * (-1), for easier manual calculation
= [1 - (PV of 1 after N)] / R%
```

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

• Find A where (PV of A after N) = 1
• PV of A after N =  A * (PV of A of 1 after N) = 1
• A after N periods to equal \$1 now
```= 1 / (PV of A of 1 after N)
= R% * CN / (CN - 1)
= R% * [1 + (CN - 1)] / (CN - 1)
= [R% + R% * (CN - 1)] / (CN - 1)
= R% / (CN - 1) + R%
= (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.

### PV of an Annuity Due of \$1 paid at the beginning of each period after N payments at compound interest (N - 1 periods)

• \$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 periodic payments
```= (FV of A Due of 1 after N) * PV of 1 after N periodic payments
= [(CN - 1) * C / R%] * (1 / CN)
= (CN - 1) * C / CN / R%
= C * (CN - 1) / CN / R%
= C * (PV of A of 1 after N periodic payments)
= (CN - 1) / CN-1 / R%```
• 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)
= (CN - 1 ) / (R% * CN-1)
= C * (CN - 1) / (R% * CN)```

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 Due paid at the beginning of each period after N payments to equal \$1 now at compound interest (N - 1 periods)

• Find A Due where (PV of A Due after N) = 1
• PV of A Due after N = A * (PV of A Due  1 after N) = 1
• A Due after N payments to give \$1 now
```= 1 / (PV of A Due of 1 after N)
= 1 / [(CN - 1) / CN-1 / R%]
= R% * CN-1 / (CN - 1)```

### Equivalent monthly compound interest rate of an annual interest rate

• In the above formulae:
• N = Number of period (usually year)
• R% = % rate of interest / rate of return at the end of a period (usually year)
• The period can be year or month, but the interest rate must be the matching annual interest rate or monthly interest rate.

("may be" changed to "must be", 11 March 2020)

• Very often, when the above formula is used when talking about converting from the annual interest rate to monthly interest rate for monthly payment, e.g. FV of \$1 after N periods = (1 + R%)N
• It would be suggested that
• N = number of years x 12 months/ year
• R% = annual interest rate / 12 month
•  So, if the annual interest rate is 12%, then the monthly interest rate would be 1%.
• However, the above calculation would not take into account the interest earned throughout the months.
• Using the above calculation, the FV of \$1 after 12 months = (1 + 1%)12 = 112.683%, i.e. actual annual interest of 12.683%
• A proper formula should be as follows:
• (1 + annual interest rate AR%) = (1 + monthly interest rate MR%)12
• 1 + AR% = (1 + MR%)12
• MR% = ((1 + AR%)1/12) - 1
• If AR% = 12%, then MR% = ((1 + 12%)1/12) - 1= 0.949%, not 1%.
• Using the above calculation, the FV of \$1 after 12 months = (1 + 0.949%)12 = 112.002%, i.e. actual annual interest of 12.002%.

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