# Life Cycle Costing 生命周期成本計算

## Table of Contents

- Life Cycle Costing 生命周期成本計算
- Life Cycle Costs 生命周期成本
- Time Value of Money
- Definitions
- Summary of Formulae - Future Values
- Summary of Formulae - Present Values
- Examples
- Future Value (FV) of $1 at simple interest
- Future Value (FV) of $1 at compound interest
- Discounting
- Compounding
- Factors
- Future Value of an Annuity of $1 after N periods at compound interest
- Future Value of an Annuity Due of $1 after N periods at compound interest
- Annuity (e.g. sinking fund) invested at the end of each of N periods to give FV of $1 at compound interest
- Annuity due (e.g. sinking fund) invested at the beginning of each of N periods to give FV of $1 at compound interest
- Present Value (PV) of $1 after N periods at compound interest
- Present Value of an Annuity of $1 after N periods at compound interest
- Present Value of an Annuity Due of $1 after N payments at compound interest
- Annuity paid at the end of each of N periods to equal $1 now at compound interest
- Annuity due paid at the beginning of each of N periods to equal $1 now at compound interest

- Present Values of Life Cycle Costs

## Life Cycle Costing 生命周期成本計算

- 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。 - 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。 - 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。 - 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

- Capital Costs

資本費用 - 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

- Routine Operating, Maintenance, and Repair Costs

## 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 = C
^{N}- 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^{N}(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 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^{N }

- 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 <> 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^{N}) = D / (R% * C^{N})

- Present Value of an Annuity Due of $1 after N payments at compound interest

PV of A Due of 1 after N <> 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^{N}) = C * D / (R% * C^{N}) also = (PV of A of 1 after N) * C, discounted by 1 less period = C * D / (R% * C^{N})

- 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^{N}/ 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^{N}/ (C * D)

### Examples

### 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 = C**^{N}(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 = C
^{N} - FV of 1 invested 1 period later for N-1 periods = C
^{N-1} - FV of 1 invested 2 periods later for N-2 periods = C
^{N-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 = C
^{N-1}+ C^{N-2}+ .... + C^{2}+ C + 1

- FV of 1 = C
- 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
^{2 }+ C + 1 - Sum after N-1 periods = C
^{N-2}+ ... + C^{2}+ C + 1 - Sum after N periods = C
^{N-1}+ C^{N-2}+ ... + C^{2}+ C + 1

- Sum = C
^{N-1}+ C^{N-2}+ .... + C^{2}+ C + 1 - Mutiply all on both sides by C
- Sum * C = C
^{N }+ C^{N-1}+ C^{N-2}+ .... + C^{2}+ C - Subtract between the two equations
- Sum * R% = C
^{N }- 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

= (C^{N+1} - 1) / R% - 1

= (C^{N+1} - 1 - R%) / R%

= [C^{N+1} - (1 + R%)] / R%

= (C^{N+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 = C
^{N} **PV of 1 after N = 1 / C**^{N}

### 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 / C^{N})

**= D / (R% * C ^{N})**

= (C^{N} - 1) / (R% * C^{N})

= (1 - 1 / C^{N}) / R%

= [1 - (PV of 1 after N)] / R%

### Present Value of an Annuity Due of $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 / C^{N})

**= C * D / (R% * C ^{N})**, which is easier for doing manual calculations

= D / (R% * C^{N-1})

- C * D / (R% * C
^{N})

= C * [D / (R% * C^{N})]

= 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

= (C^{N-1} - 1) / (R% * C^{N-1}) + 1

= (C^{N-1} - 1 + R% * C^{N-1}) / (R% * C^{N-1})

= (C^{N-1} * (1 + R%) - 1 ) / (R% * C^{N-1})

= (C^{N-1} * C- 1 ) / (R% * C^{N-1})

= (C^{N }- 1 ) / (R% * C^{N-1})

= D / (R% * C^{N-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% * C ^{N} / D**

= R%* (D + 1) / D, because D = C^{N} -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% * C ^{N}) / (C * D)**, which is easier for doing manual calculations

= (R% * C^{N-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)

- Routine Operating, Maintenance, and Repair Costs

- Capital Costs