If there is one prayer that you should

- Samuel Dominic Chukwuemeka
**pray/sing** every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)

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

- Samuel Chukwuemeka
**Joy** of a **Teacher** is the **Success** of his **Students.**

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

- $Per\:\:annum$ 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 $(\$)$

For all financial calculations, do not round until the final answer.

Do not round intermediate calculations. If it is too long, write it to

Round your final answer to $2$ decimal places.

Make sure you include your unit.

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.

$ (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) |

$ (1.)\:\: PV = \dfrac{FV}{\left(1 + i\right)^n} \\[7ex] (2.)\:\: PV = \dfrac{FV}{\left(1 + \dfrac{MARR}{m}\right)^{mt}} \\[5ex] $

$ (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] $

$ (1.)\:\: FV = PV * (1 + i)^{n} \\[5ex] (2.)\:\: FV = PV * \left(1 + \dfrac{MARR}{m}\right)^{mt} \\[5ex] $

$ (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] $

$ (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] $

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 $PW = 0$

This is the **rate** at which the **Future Worth of the cash outflow** is equal to the **Future Worth of all cash inflows.**

$ERR = m\left[\left(\dfrac{FV + MV}{UCF}\right)^{\dfrac{1}{mt}} - 1\right] \\[10ex]$

For all financial calculations, do not round until the final answer.

Do not round intermediate calculations. If it is too long, write it to

Round your final answer to $2$ decimal places.

Make sure you include your unit.

__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

(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

(3.) The cash outflow is a present value already...it is the investment.

(4.) Using the

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

If $\boldsymbol{Present\:\:Worth \lt 0}$, the

__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

(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

(4.) Using the

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

If $\boldsymbol{Future\:\:Worth \lt 0}$, the

__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

(2.) The cash outflow is a present value.

So, we find the

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

This is the uniform cash flow of the Future Value of Ordinary Annuity.

(4.) We find the

This is the difference between the

(5.) Using the

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

If $\boldsymbol{Annual\:\:Worth \lt 0}$, the

**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

(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

If $\boldsymbol{IRR \ge marr}$, the

If $\boldsymbol{IRR \lt marr}$, the

**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

(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

If $\boldsymbol{IRR \ge marr}$, the

If $\boldsymbol{IRR \lt marr}$, the

__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

(3.) Calculate the

(4.) Equate the

(5.) Determine that

(6.) Using the

If $\boldsymbol{ERR \ge marr}$, the

If $\boldsymbol{ERR \lt marr}$, the