Life Cycle Costing 生命周期成本計算

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Note

22/3/2023: Revised.

6/2/2021: Table of Formulae last two columns merged.

11/3/2020: Interest rate "may be" changed to "must be" when converting to monthly interest rate.

10/3/2020: Conversion from annual interest rate to monthly interest rate added.

28/1/2019: Revised to re-arrange the order and give a simpler summary of the formulae.

7/2/2018: Created.

Time Value of Money

Generally

$1 in possession now can earn interest by saving in a bank.
$1 borrowed now will incur interest until the $1 is repaid.
$1 can buy more things now than in the future because the commodity prices generally increase over time due to inflation.
Therefore, $1 received in the future will be of less value than $1 received now.
Similarly, $1 paid in the future will be less expensive than $1 paid now.
When comparing amounts paid over different times, they should be converted to their equivalent values at the same time for comparison on the same basis.
They may be converted to the present value (present time) or future value (future time) or annuity (annually).

(added, 22/3/2023)

Interest on Loan

Simple interest = Interest calculated only on the original amount (principal) loaned.
Compound interest = Interest calculated on the original amount and accumulated interests.
Total amount to be repaid for $100 principal at interest rate (R%) of 10% per annum:

Loan $100 Now

Repay by End of 1 Year

Repay by End of 2 Years

Repay by End of 3 Years

Repay by End of N Years

Simple interest

100 + 10

= 110

100 + 10 x 2

= 120

100 + 10 x 3

= 130

Loan x (1 + R% x N)

Compound interest

100 x (1 + 10%)

= 110

110 x (1 + 10%)

= 121

121 x (1 + 10%)

= 133

Loan x (1 + R%)^N

Money borrowed now will need repayment in a bigger amount in the future. A longer loan will cost more.

Same Money Paid Later Costs Less Now

On the other hand, if the same amount of loan can be paid later, the cost of the loan will be less.
If a buyer needs to borrow money to pay to buy something, and if he can borrow and pay the same price later, the cost of the loan will be less, and therefore later payment will cost less.
If a buyer does not need to borrow money to pay, and if he can pay the same price later, he can use the spare money now to invest elsewhere to earn money, and therefore later payment will cost less.

(revised, 22/3/2023)

Compounding (PV --> FV)

Converting the present value to the future value is called compounding, because compound interest rates are usually used.

(revised, 22/3/2023)

Discounting (PV <-- FV)

Converting the future value to the present value is called discounting, because the present value is less than the future value of the same amount of money to be paid.

(revised, 22/3/2023)

Factors

Factors are used to simplify all discounting and compounding calculations whereby:
- Result $ = Base $ x Factor per $1
- Future value = Present value x Compounding factor
- Present value = Future value x Discounting factor.

(revised, 22/3/2023)

Definitions

PV = Present value | present worth | principal amount invested now.
FV = Future value | future worth | terminal value | terminal worth.
A = Annuity, a periodic payment.
N = Number of period (year, quarter, month, week or day).
R% = % rate of interest | rate of return | discount rate applicable at the end of a period.
C = 1 + R%.
Due means payable at the beginning of a period.

(revised, 22/3/2023)

Table of Formulae (compound interest)

Factor Description	Formula
FV of 1 after N FV of 1 paid now after N periods (1 payment) Amount of $1 after N periods	C^N
PV of 1 after N PV of 1 paid after N periods (1 payment)	1 / (FV of 1) = 1 / C^N
FVA of 1 after N 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	(C^N - 1) / R%
A of FV of 1 after N Annuity paid at the end* of each period after N periods to give FV of 1 (N payments)*	1 / (FVA of 1 after N) = R% / (C^N - 1)
FVA Due of 1 after N FV of Annuity Due* of 1 paid at the beginning of each period after N periods (N payments)*	(C^N - 1) * C / R%
A Due of FV of 1 after N Annuity Due* paid at the beginning of each period after N periods to give FV of $1 (N payments)*	1 / (FVA Due of 1 after N) = R% / C / (C^N - 1)
PVA of 1 after N 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 PVA of $1 after N years Years Purchase if the period is a year	(1 - 1 / C^N) / R%
A of PV of 1 after N Annuity paid at the end* of each period after N periods to equal PV of $1 (N payments)*	1 / (PVA of 1 after N) = R% / (1 - 1 / C^N)
PVA Due of 1 after N PV of Annuity Due* of $1 paid at the beginning of each period after N payments (N -1 periods)*	(C^N - 1) / C^{N - 1} / R%
A Due of PV of 1 after N Annuity Due* paid at the beginning of each period after N payments to equal PV of $1 (N - 1 periods)*	1 / (PV of A Due of 1 after N) = R% * C^{N - 1} / (C^N - 1)
Converting nominal annual compound interest rate R% to compound interest rate X% for M times regular payments in a year for N times (Verification, after 12 times per year for a year, N = M = 12, (1 + R%)^N/M* becomes (1 + R%), the FV of $1 after 1 year.)*	Used by banks: Interest rate, X% = R%/M. Compounding factor, FV of $1 after N times = (1 + X%)^N = (1 + R%/M)^N Discounting factor, PV of $1 after N times = 1 / (1 + X%)^N = 1 / (1 + R%/M)^N Accurate: Interest rate, X% = (1 + R%)^1/M – 1. Compounding factor, FV of $1 after N times = (1 + X%)^N = (1 + R%)^N/M Discounting factor, PV of $1 after N times = 1 / (1 + X%)^N = 1 / (1 + R%)^N/M

Note: X / Y / Z = X / (Y * Z), e.g., 8/4/2 = 1; 8 / (4 * 2) = 1; but the former is easier for manual calculation or setting formulae in Excel, therefore, the former format is used above.

(revised, 22/3/2023)

Examples

Note: The above shows that if the interest rate is higher, the future value will be bigger but the present value is smaller.
Diagrammatically:
- PV --> FV = add 10% to compound PV to FV
- PV <-- FV = divide by 1.1 to discount FV to PV
- [1] = add payment of 1 monetary value.
FV of [1]:

[1] --> 1.1

1.1 --> 1.21

1.21 --> 1.331

PV of [1]:

0.751 <-- 0.826

0.826 <-- 0.909

0.909 <-- [1]

FV of A of [1]:

0 --- [1]

1 --> 1.1 + [1] = 2.1

2.1--> 2.31 + [1] = 3.31

A of FV of 1 = 1 / (FV of A of 1):

1 / 1 = 1

1 / 2.1 = 0.476

1 / 3.31 = 0.302

FV of A Due of [1]:

[1] --> 1.1

1.1 + [1] = 2.1 --> 2.31

2.31 + [1] = 3.31 --> 3.641

A Due of FV of 1 = 1 / (FV of A Due of 1):

1 / 1.1 = 0.909

1 / 2.31 = 0.433

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:

0.909 <-- 1

1.736 <-- 1.909 <-- 2.1

2.487 <-- 2.736 <-- 3.009 <-- 3.31

A of PV of 1 = 1 / (PV of A of 1):

1 / 0.909 = 1.1

1 / 1.736 = 0.576

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:

1 <-- 1.1

1.909 <-- 2.10 <-- 2.31

2.736 <-- 3.009 <-- 3.31 <-- 3.641

A Due of PV of 1 = 1 / (PV of A Due of 1):

1 / 1 = 1

1 / 1.909 = 0.524

1 / 2.736 = 0.365

(revised, 22/3/2023)

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

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 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 + .... + 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 = C²+ C + 1
- Sum after N-1 periods = C^N-2 + ... + C² + C + 1
- Sum after N periods = C^N-1 + C^N-2 + ... + C² + C + 1
Sum = C^N-1 + C^N-2 + .... + C² + C + 1
Mutiply all on both sides by C
Sum * C = C^N+ C^N-1 + C^N-2 + .... + C² + 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

= (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)

Converting Nominal Annual Compound Interest Rate to Compound Interest Rate for M Times Regular Payments in a Year for N Times

From the above, FV of 1 after N @ R% interest rate each time = (1 + R%)^N.
- When N = years, R% = annual interest rate.
- When N = times, R% = interest rate each time. Let’s call it X%.
When there are M times regular payments in a year, the usual method used by banks to calculate the interest rate X% each time is:
- X% = R% / M
Accurate method:
Compounding at X% interest rate each time after M times a year for 1 year should be equal to (1 + R%)
- (1 + X%)^M = (1 + R%)
- (1 + X%) = (1 + R%)^1/M
- X% = (1 + R%)^1/M – 1
Test based on R = 10%, M = 12 for 1 year:
- Banks’ method:
  - monthly rate = 10%/12 = 0.000833
  - compounded total after 1 year = (1 + 10%/12)¹² = 1.1047
- Accurate method:
  - monthly rate = (1 + 10%)^1/12 – 1 = 0.007974
  - compounded total = (1 + 0.007974)¹² = 1.10
- Compare with annual compounding only:
  - (1 + 10%) = 1.10
FV of 1 after compounding at M times a year for N times at X% discount rate each time
- = (1 + X%)^N
- = [1 + (1 + R%)^1/M – 1]^N
- = [(1 + R%)^1/M]^N
- = (1 + R%)^N/M
An easier way to remember:
- Let F be the compounding factor each time
- F compounded for M times a year should be equal to (1 + R% annual interest rate)
- F M = (1 + R%)
- F = (1 + R%)^1/M
- F compounded for N times should become:
- F N = (1 + R%)^N/M
PV of 1 after compounding at M times a year for N times at X% discount rate each time
- = 1 / FV
- = 1 / (1 + X%)^N
- = 1 / (1 + R%)^N/M

(revised, 22/3/2023)

Inflation, Deflation and Escalation

Inflation is a general increase in the prices of goods and services in an economy over a period of time.
Deflation is a general decrease in the prices of goods and services in an economy over a period of time.
Escalation is a general increase in the prices of specific group of goods and services over a period of time.
For simplicity, an inflation rate or escalation rate may be positive or negative.
When the inflation rate or escalation rate (E%) is said to be 10% per annum year on year. It means that the price in a year is 10% high than the price a year ago, e.g.:

Now	End of 1 Year	End of 2 Years	End of 3 Years
Present price = 100	100 x 1.1 = 110	100 x 1.1 x 1.1 = 121	100 x 1.1 x 1.1 x 1.1 = 133

The purchasing power of the same amount of money decreases with inflation. Alternatively, it is said to have a devaluation of the money.
One has to cumulate more money to buy the same quantity of goods and services (assuming same quality and still available) in the future.
The future cost to buy = current cost x (1 + E%)^N.
In principle, discounting to the present value should be based on:
- discounted cost = future cost x discounting factor.
The future cost should be the cost taking into account the impact of inflation and escalation, and is called “nominal cost”.
The future cost can only be estimated based on:
- future cost = current cost x escalation factor, where the escalation (specific) is inclusive of inflation (general).
Therefore, discounted cost = current cost x escalation factor x discounting factor.
General inflation is difficult to predict. Since it lowers purchasing power, the discount rate used to calculate the discounting factor has to compensate the decrease in purchasing power. The discount rate is called “nominal discount rate”. Since both the inflation factor and the adjustment to the discount rate apply to all goods and services being considered for comparison, it is considered in order to ignore the general inflation and the corresponding adjustment to the discount rates.
The cost exclusive of the impact of general inflation is called “real cost”. The corresponding discount rate excluding the impact of general inflation is called “real discount rate”.
However, for a specific group of goods and services, their escalation rates can be different from the general inflation rates, it is suggested by some people that their future costs should be based on the differential escalation rates beyond the general inflation rates, while the real discount rates are still used.
Therefore, the above equation can be re-written as:
- discounted cost = nominal cost x nominal discounting factor
- discounted cost = real cost x escalation factor x nominal discounting factor
- discounted cost = real cost x real discounting factor
- discounted cost = current cost x differential escalation factor beyond general inflation factor x real discounting factor
- discounted cost = current cost x (1 + E%)^N / (1 + R%)^N,
  - where E% is the full or differential escalation rate
  - R% is the nominal discount rate or real discount rate respectively.
The above shows that:
- escalation factor x nominal discounting factor = real discounting factor
- (1 + E%)^N / (1 + nominal R%)^N = 1 / (1 + real R%)^N
- (1 + E%) / (1 + nominal R%) = 1 / (1 + real R%)
- (1 + nominal R%) = (1 + E%) * (1 + real R%)
- nominal R% = (1 + E%) * (1 + real R%) - 1
- nominal R% = E% + real R% + E% * real R%, the last product is very small and can be ignored
- i.e., nominal discount rate = escalation rate + real discount rate
- or, real discount rate = nominal discount rate – escalation rate
ISO 15686-5 suggests to use real costs and real discount rates, and assume no differential escalation rates first, but then adjust the discount rates while doing sensitivity analysis to test the impact of the possible differential escalation rates.

(added, 22/3/2023)

Guidance on Discount Rates given by ISO 15686-5 (cited in italics)

The type of discount rate, either real or nominal, shall be clearly distinguished.
NOTE 1 The real discount rate applied to costs (and benefits) that are also measured in real terms assumes that inflation/deflation applies equally to all.
Occasionally, escalation rates may be used as a form of sensitivity analysis where there are grounds to anticipate that the standard rate of inflation does not apply in the case of a specific scenario. Typically, real rates should be used; these exclude the impact of future inflation. Nominal rates may be used by agreement, if that is what is required by the client or justified by the situation.
The discount rate in the private sector should represent the opportunity cost of investing the capital, which can be:
- a) the interest cost of a loan for the investment;
- b) the interest lost on reduction of cash on deposit;
- c) the returns lost on investment elsewhere (e.g. in bonds or equities);
- d) the actual return achieved on capital investment in the business;
- e) the required rate of return of an investor in a new business.
Within the public sector, a discount rate can be determined by the central government (sometimes termed the social discount rate) as a test requirement for their investments, based on an assessment of the long-term opportunity cost to the public sector of selecting one investment rather than another.
NOTE 2 Historically, the real discount rate has reflected the general productivity rate of the producer, sector or field. Generally, productivity has been within 0 % and 2 % over the long term. However, rates as low as these are not universal. Discount rates of between 0 % and 4 % are typically used. A higher rate discourages long-term investments, while a lower rate encourages them.
NOTE 3 Where the discount rate is not a requirement fixed by the client (public or private), it is normal to undertake sensitivity analysis using a range of rates to test the validity of the conclusions if the input conditions change.

(added, 22/3/2023)

Tax

If the discount rates are based on investment interest rates which are subject to tax, the discount rates should subtract the tax charged, e.g., interest rate x (1 – % tax on interest rate).

(added, 22/3/2023)

Sensitivity Analysis

In life cycle costing, since the discount rates are estimates only, it is suggested that discount rates should be adjusted up or down within the possible range to test whether the relative order of the costs of the options would be changed or not. If not, the choice will be more certain. If changed, more careful review should be taken.
The same apply to other data which may likely be varied.

(added, 22/3/2023)