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 rearrange 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
 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  Costsinuse
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))  NonMonetary Benefits or Costs
非金錢的益處或成本
 Routine Operating, Maintenance, and Repair Costs
Present Values of Life Cycle Costs
 Being sum of:
 Capital Costs
 (spend now) * 1
 Costsinuse
 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)
 Routine Operating, Maintenance, and Repair Costs
 Capital Costs
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 = C^{N}  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)

C^{N} 

1 / C^{N} 

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

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

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

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

(C^{N}  1) / C^{N} / R% or (1 / C^{N } 1) / R% * (1) for easier manual calculation = D / C^{N} / R% 

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

(C^{N}  1) / C^{N1 }/ R% = D / C^{N1 }/ R% 

R% * C^{N1} / (C^{N}  1) = R% * C^{N1} / D 

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
 [1] = payment of 1 monetary value
FV of [1]

[1]>1.1>1.21>1.331
PV of [1]:
 reversing FV of 1, divided by 1.1 each time
 after 1 period: 0.909<[1]
 after 2 periods: 0.826<0.909<[1]
 after 3 periods: 0.751<0.826<0.909<[1]
FV of A of [1]:
 after 1 period: 0[1]=1
 after 2 periods: 0[1]=1>1.1+[1]=2.1
 after 3 periods: 0[1]=1>1.1+[1]=2.1>2.31+[1]=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 [1]:
 after 1 period: [1]>1.1
 after 2 periods: [1]>1.1+[1]=2.1>2.31
 after 3 periods: [1]>1.1+[1]=2.1>2.31+[1]=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 [1] = FV of A of [1], all discounted back to now by dividing by 1.10 for each period:
 FV after 1 period: 0[1]=1
 now: 0.909<1
 FV after 2 periods: 0[1]=1>1.1+[1]=2.1
 now: 1.736<1.909<2.1
 FV after 3 periods: 0[1]=1>1.1+[1]=2.1>2.31+[1]=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 [1] = FV of A Due of [1], all discounted back to now by dividing by 1.10 for each period:
 FV after 1 period: [1]>1.1
 now: 1<1.1
 FV after 2 periods: [1]>1.1+[1]=2.1>2.31
 now: 1.909<2.10<2.31
 FV after 3 periods: [1]>1.1+[1]=2.1>2.31+[1]=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) = C^{N}
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 = C^{N}
 PV of $1 after N periods
= 1 / C^{N}
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 = C^{N}
 FV of 1 invested 1 period later for N1 periods = C^{N1}
 FV of 1 invested 2 periods later for N2 periods = C^{N2}
 FV of 1 invested N1 periods later for 1 period = C
 FV of 1 invested N periods later for 0 period = 1 (no interest)
 Sum = C^{N1} + C^{N2} + .... + C^{2} + 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^{2 }+ C + 1
 Sum after N1 periods = C^{N2} + ... + C^{2} + C + 1
 Sum after N periods = C^{N1} + C^{N2} + ... + C^{2} + C + 1
 Sum = C^{N1} + C^{N2} + .... + C^{2} + C + 1
 Mutiply all on both sides by C
 Sum * C = C^{N } + C^{N1} + C^{N2} + .... + C^{2} + C
 Subtract between the two equations
 Sum * R% = C^{N }  1
 Sum = (C^{N}  1) / R%
 Therefore
 FV of A of $1 after N periods
= (C^{N}  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% / (C^{N}  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 = (C^{N}  1) * C / 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^{N}  1) * C / R%
 Therefore, FV of A Due of 1 after N
also = (FV of A of 1 after N+1)  1
diagrammatically:
01 = payment of 1 at the end of a period
> = a period with interest
01>1>1 = FV of A of 1 after N payments, compounding for N1 periods
01>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% / [(C^{N}  1) * C]
= R% / (C^{N}  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) = [(C^{N}  1) / R%] * (1 / C^{N}) = (C^{N}  1) / C^{N }/ R% = (1  1 / C^{N}) / R% = (1 / C^{N } 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% * C^{N} / (C^{N}  1) = R% * [1 + (C^{N}  1)] / (C^{N}  1) = [R% + R% * (C^{N}  1)] / (C^{N}  1) = R% / (C^{N}  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 = [(C^{N}  1) * C / R%] * (1 / C^{N}) = (C^{N}  1) * C / C^{N }/ R% = C * (C^{N}  1) / C^{N }/ R% = C * (PV of A of 1 after N periodic payments) = (C^{N}  1) / C^{N1 }/ 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 N1) + 1, which is
= (C^{N1}  1) / (R% * C^{N1}) + 1 = (C^{N1}  1 + R% * C^{N1}) / (R% * C^{N1}) = (C^{N1} * (1 + R%)  1 ) / (R% * C^{N1}) = (C^{N1} * C 1 ) / (R% * C^{N1}) = (C^{N } 1 ) / (R% * C^{N1}) = C * (C^{N}  1) / (R% * C^{N})
Therefore, PV of A Due of 1 after N
also = (PV of A of 1 after N1) + 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 N1 payments, discounting for N1 periods
1<1<1<1 = Add 1 to the beginning = PV of A Due of 1 after N payments, discounting for N1 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 / [(C^{N}  1) / C^{N1 }/ R%] = R% * C^{N1} / (C^{N}  1)
Equivalent monthly compound interest rate of an annual interest rate
(added 10 Mar 2020)
 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%.