If there is one prayer that you should pray/sing every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)
- Samuel Dominic Chukwuemeka
It is the most powerful prayer.
A pure heart, a clean mind, and a clear conscience is necessary for it.
Let everything that has breath praise the LORD! Praise the LORD. - Psalm 150:6
The Joy of a Teacher is the Success of his Students.
- Samuel Chukwuemeka
Evaluation of Projects Calculators
Welcome to Our Site
I greet you this day,
You may use these calculators to check your answers. You are encouraged to solve the questions first,
before checking your answers. Please do not use a comma.
I wrote the codes for these calculators using Javascript, a client-side scripting language.
In addition, I used the JavaScript library, Formula.js for some calculations.
Please use the latest Internet browsers. The calculators should work.
You may need to refresh your browser after each calculation to clear all the results of the calculation.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You
may contact me. Thank you for visiting.
Samuel Dominic Chukwuemeka (SamDom For Peace)
B.Eng., A.A.T, M.Ed., M.S
Symbols and Meanings
- $Per\:\:annum$ OR $Per\:\:year$ OR $Annually$ OR $Yearly$ means for a year (per $1$ year)
- $PV$ = Present Value $(\$)$
- $FV$ = Future Value $(\$)$
- $i$ = Annual Interest Rate Per Period $(\%) \:\:per\:\: period$
- $MARR$ = Minimum Attractive Rate of Return $(\%)$
- $MV$ = Market Value or Salvage Value $\$$
- $APY$ = Annual Percentage Yield or Effective Interest Rate or True Interest Rate $(\%)$
- $t$ = Time $(years)$
- $m$ = Number of Compounding Periods Per Year
- $n$ = Total Number of Compounding Periods $(years)$
- $e$ = Euler Number or Napier's Constant
- $UCF$ = Uniform Cash Flow $(\$)$
- $AR$ = Annual Revenue $(\$)$
- $AE$ = Annual Expenditure $(\$)$
- $ACR$ = Annual Capital Recovery $(\$)$
- $\dfrac{i}{(1 + i)^n - 1}$ = sinking fund factor
- $\dfrac{i(1 + i)^n}{(1 + i)^n - 1}$ = capital recovery factor
- $PW$ = Present Worth Method $(\$)$
- $FW$ = Future Worth Method $(\$)$
- $AW$ = Annual Worth Method $(\$)$
- $IRR$ = Internal Rate of Return Method $(\$)$
- $ERR$ = EXternal Rate of Return Method $(\$)$
- $PP$ = Payback Period Method or Payout Period Method $(\$)$
NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least 5 decimal places (5 or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.
Formulas
It is very important you use these formulas with the meaning of the symbols
In the case of the same formula written in several different ways, use whatever formula is convenient for you.
Basic Formulas
$ (1.)\:\: i = \dfrac{MARR}{m} \\[7ex] (2.)\:\: n = mt \\[5ex] (3.)\:\: APY = MARR \\[5ex] (4.)\:\: For\:\:Compounding\:\:Interest:\:\: use\:\: APY \\[5ex] (5.)\:\: For\:\:Continuous\:\:Compounding\:\:Interest:\:\: use\:\: APY = e^{APY} - 1 \\[5ex] $
| If Compounded: | $m = $ |
|---|---|
| Annually |
$1$ ($1$ time per year) Also means every twelve months |
| Semiannually |
$2$ ($2$ times per year) Also means every six months |
| Quarterly |
$4$ ($4$ times per year) Also means every three months |
| Monthly |
$12$ ($12$ times per year) Also means every month |
| Weekly | $52$ ($52$ times per year) |
| Daily (Ordinary/Banker's Rule) | $360$ ($360$ times per year) |
| Daily (Exact) | $365$ ($365$ times per year) |
Present Worth (PW) Method
$Present\:\:Worth = Present\:\:Values\:\:of\:\:Cash\:\:Inflows - Present\:\:Values\:\:of\:\:Cash\:\:Outflows$Single Cash Flow
$ (1.)\:\: PV = \dfrac{FV}{\left(1 + i\right)^n} \\[7ex] (2.)\:\: PV = \dfrac{FV}{\left(1 + \dfrac{MARR}{m}\right)^{mt}} \\[5ex] $
Uniform Cash Flows (Ordinary Annuities)
$ (1.)\:\: PV = UCF * \left[\dfrac{(1 + i)^n - 1}{i(1 + i)^n}\right] \\[7ex] (2.)\:\: PV = UCF * \left[\dfrac{\left(1 + \dfrac{MARR}{m}\right)^{mt} - 1}{i\left(1 + \dfrac{MARR}{m}\right)^{mt}}\right] \\[10ex] (3.)\:\: PV = m * PMT * \left[\dfrac{1 - \left(1 + \dfrac{MARR}{m}\right)^{-mt}}{MARR}\right] \\[7ex] $
Future Worth (FW) Method
$Future\:\:Worth = Future\:\:Values\:\:of\:\:Cash\:\:Inflows - Future\:\:Values\:\:of\:\:Cash\:\:Outflows$Single Cash Flow
$ (1.)\:\: FV = PV * (1 + i)^{n} \\[5ex] (2.)\:\: FV = PV * \left(1 + \dfrac{MARR}{m}\right)^{mt} \\[5ex] $
Uniform Cash Flows (Ordinary Annuities)
$ (1.)\:\: FV = UCF * \left[\dfrac{(1 + i)^n - 1}{i}\right] \\[7ex] (2.)\:\: FV = UCF * \left[\dfrac{\left(1 + \dfrac{MARR}{m}\right)^n - 1}{\dfrac{MARR}{m}}\right] \\[10ex] (3.)\:\: FV = m * UCF * \left[\dfrac{\left(1 + \dfrac{MARR}{m}\right)^{mt} - 1}{MARR}\right] \\[7ex] $
Annual Worth (AW) Method
$Annual\:\:Worth = Annual\:\:Revenue - Annual\:\:Expenditure - Annual\:\:Capital\:\:Recovery$Uniform Cash Flows (Ordinary Annuities)
$ (1.)\:\:Sinking\:\:Fund:\:\: UCF = FV * \dfrac{i}{(1 + i)^n - 1} \\[7ex] (2.)\:\:Sinking\:\:Fund:\:\: UCF = FV * sinking\:\:fund\:\:factor \\[5ex] (3.)\:\:Sinking\:\:Fund:\:\: UCF = \dfrac{FV * MARR}{m * \left[(1 + i)^{n} - 1\right]} \\[7ex] (4.)\:\:Sinking\:\:Fund:\:\: UCF = \dfrac{FV * MARR}{m * \left[\left(1 + \dfrac{MARR}{m}\right)^{mt} - 1\right]} \\[10ex] (5.)\:\:Amortization:\:\: UCF = PV * \dfrac{i(1 + i)^n}{(1 + i)^n - 1} \\[7ex] (6.)\:\:Amortization:\:\: UCF = PV * capital\:\:recovery\:\:factor \\[5ex] (7.)\:\:Amortization:\:\: UCF = \dfrac{PV * MARR}{m * \left[1 - (1 + i)^{-n}\right]} \\[7ex] (8.)\:\:Amortization:\:\: UCF = \dfrac{PV * MARR}{m * \left[1 - \left(1 + \dfrac{MARR}{m}\right)^{-mt}\right]} \\[10ex] (9.)\:\:Annual\:\:Capital\:\:Recovery:\:\: ACR = Amortization - Sinking\:\:Fund \\[5ex] (10.)\:\:Annual\:\:Capital\:\:Recovery:\:\: ACR = PV * \dfrac{i(1 + i)^n}{(1 + i)^n - 1} - FV * \dfrac{i}{(1 + i)^n - 1} \\[7ex] (11.)\:\:Annual\:\:Capital\:\:Recovery:\:\: ACR = \dfrac{PV * MARR}{m * \left[1 - (1 + i)^{-n}\right]} - \dfrac{FV * MARR}{m * \left[(1 + i)^{n} - 1\right]} \\[7ex] (12.)\:\:Annual\:\:Capital\:\:Recovery:\:\: ACR = \dfrac{PV * MARR}{m * \left[1 - \left(1 + \dfrac{MARR}{m}\right)^{-mt}\right]} - \dfrac{FV * MARR}{m * \left[\left(1 + \dfrac{MARR}{m}\right)^{mt} - 1\right]} \\[10ex] $
Internal Rate of Return (IRR) Method
This is the interest rate at which the Present Worth is zero
Find the interest rate at which the Present Worth is zero.
Do not use the minimum attractive rate of return to calculate the Present Worth in this case.
Equate the Present Worth to zero.
Calculate the interest rate that will make that Present Worth to be zero.
That interest rate is the IRR.
$IRR$ = interest rate when $PW = 0$
External Rate of Return (ERR) Method
This is the rate at which the Future Worth of the cash outflow is equal to the Future Worth of all cash inflows.
$ERR = m\left[\left(\dfrac{FV + MV}{UCF}\right)^{\dfrac{1}{mt}} - 1\right] \\[10ex]$
NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.
Present Worth (PW) Method
- Given: an investment amount/cash outflow, uniform cash inflows, minimum attractive rate of return, market/salvage value
To Determine: if the project is economically justified using the Present Worth Method
(1.) The cash inflows is an ordinary annuity because they occur at the end of each period.
So, we find the Present Value of an Ordinary Annuity
(2.) The market value is also a cash inflow. However, it is the value at the end of the project.
It is a single cash inflow.
It is a future value.
It is the amount.
Hence, we need to find the present value.
Because it is a single cash flow, we use the Compound Interest Formula to solve for the principal.
(3.) The cash outflow is a present value already...it is the investment.
(4.) Using the Present Worth Method:
We add the two present values of the cash inflows and then subtract the present value of the cash outflow.
$Present\:\:Worth = Present\:\:Values\:\:of\:\:Cash\:\:Inflows - Present\:\:Values\:\:of\:\:Cash\:\:Outflows$
(5.) If $\boldsymbol{Present\:\:Worth \ge 0}$, the project is economically justifiable.
If $\boldsymbol{Present\:\:Worth \lt 0}$, the project is NOT economically justifiable.
Future Worth (FW) Method
- Given: an investment amount/cash outflow,
uniform
cash inflows, minimum attractive rate of return, market/salvage value
To Determine: if the project is economically justified using the Future Worth Method
(1.) The cash inflows is an ordinary annuity because they occur at the end of each period.
So, we find the Future Value of an Ordinary Annuity
(2.) The market value is also a cash inflow.
It is a future value already.
It is a future value cash inflow.
(3.) The cash outflow is a present value.
Hence, we need to find the future value.
Because it is a single cash flow, we use the Compound Interest Formula to solve for the amount.
(4.) Using the Future Worth Method:
We add the two future values of the cash inflows and then subtract the future value of the cash outflow.
$Future\:\:Worth = Future\:\:Values\:\:of\:\:Cash\:\:Inflows - Future\:\:Values\:\:of\:\:Cash\:\:Outflows$
(5.) If $\boldsymbol{Future\:\:Worth \ge 0}$, the project is economically justifiable.
If $\boldsymbol{Future\:\:Worth \lt 0}$, the project is NOT economically justifiable.
Annual Worth (AW) Method
- Given: an investment amount/cash outflow,
uniform
cash inflows, minimum attractive rate of return, market/salvage value
To Determine: if the project is economically justified using the Annual Worth Method
(1.) The Annual Revenues minus the Annual Expenditures is the Uniform Cash Inflow.
(2.) The cash outflow is a present value.
So, we find the Amortization of the cash outflow.
This is the uniform cash flow of the Present Value of Ordinary Annuity.
(3.) The market value is a future value.
So, we find the Sinking Fund of the market value.
This is the uniform cash flow of the Future Value of Ordinary Annuity.
(4.) We find the Annual Capital Recovery.
This is the difference between the Amortization of the cash outflow and Sinking Fund of the market value.
(5.) Using the Annual Worth Method:
We subtract the annual capital recovery from the uniform cash inflow.
$Annual\:\:Worth = Uniform\:\:Cash\:\:Inflow - Annual\:\:Capital\:\:Recovery$
(5.) If $\boldsymbol{Annual\:\:Worth \ge 0}$, the project is economically justifiable.
If $\boldsymbol{Annual\:\:Worth \lt 0}$, the project is NOT economically justifiable.
Internal Rate of Return (IRR) Method
Exact Value (Microsoft Excel value)
- Given: an investment amount/cash outflow,
uniform cash inflows, market/salvage value, minimum attractive rate of return
To Determine: if the project is economically justified using the Internal Rate of Return Method
(1.) Understanding the Present Worth method is required.
Please review that method first.
(2.) The Internal Rate of Return (IRR) is the rate at which the Present Worth is zero
(3.) You can use calculators, spreadsheets, or the Interpolation method to calculate this rate.
As at the time of writing this, there is no specific formula that will give an exact value.
The Interpolation method will usually give an approximate value.
You may use the same values with the two calculators (this one and the one below) and compare the results, so you can see what I mean.
(4.) Using the Internal Rate of Return Method:
If $\boldsymbol{IRR \ge marr}$, the project is economically justifiable.
If $\boldsymbol{IRR \lt marr}$, the project is NOT economically justifiable.
Approximate Value (Interpolation Method)
- Given: an investment amount/cash outflow,
uniform cash inflows, market/salvage value, minimum attractive rate of return
To Determine: if the project is economically justified using the Internal Rate of Return Method (Approximate Value/Interpolation Method)
(1.) Understanding the Present Worth method is required.
Please review that method first.
(2.) The Internal Rate of Return (IRR) is the rate at which the Present Worth is zero
(3.) You can use calculators, spreadsheets, or the Interpolation method to calculate this rate.
As at the time of writing this, there is no specific formula that will give an exact value.
The Interpolation method will usually give an approximate value.
You may use the same values with the two calculators (this one and the one above) and compare the results, so you can see what I mean.
(4.) Using the Internal Rate of Return Method:
If $\boldsymbol{IRR \ge marr}$, the project is economically justifiable.
If $\boldsymbol{IRR \lt marr}$, the project is NOT economically justifiable.
External Rate of Return (ERR) Method
- Given: an investment amount/cash outflow,
uniform
cash inflows, minimum attractive rate of return, market/salvage value
To Determine: if the project is economically justified using the External Rate of Return Method
(1.) Understanding the Future Worth method is required.
Please review that method first.
(2.) The External Rate of Return (IRR) is the rate at which the Future Worth of the cash outflow is equal to the Future Worth of all cash inflows.
(3.) Calculate the Future Worth of all cash inflows using the minimum attractive interest rate as the rate.
(4.) Equate the Future Worth of the cash outflow (unknown rate) to the Future Worth of all cash inflows.
(5.) Determine that rate at which the Future Worth of the cash outflow is equal to the Future Worth of all cash inflows.
(6.) Using the External Rate of Return Method:
If $\boldsymbol{ERR \ge marr}$, the project is economically justifiable.
If $\boldsymbol{ERR \lt marr}$, the project is NOT economically justifiable.