Time Value of Money
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I greet you this day,
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Samuel Dominic Chukwuemeka (SamDom For Peace)
B.Eng., A.A.T, M.Ed., M.S
Overview
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 $(\%)$
- $APR$ = Annual Percentage Rate $(\%)$
- $APY$ = 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 $(\$)$
- $(1 + i)^n$ = single payment compound amount factor
- $(1 + i)^{-n}$ = single payment present worth factor
- $\dfrac{(1 + i)^n - 1}{i}$ = uniform series compound amount factor
- $\dfrac{(1 + i)^n - 1}{i(1 + i)^n}$ = uniform series present worth factor
- $\dfrac{i}{(1 + i)^n - 1}$ = sinking fund factor
- $\dfrac{i(1 + i)^n}{(1 + i)^n - 1}$ = capital recovery factor
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{APY}{m} \\[7ex] (2.)\:\: n = mt \\[5ex] (3.)\:\: For\:\:Compounding\:\:Interest:\:\: effective\:\:interest\:\:rate = APY \\[5ex] (4.)\:\: For\:\:Continuous\:\:Compounding\:\:Interest:\:\: effective\:\:interest\:\:rate = 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) |
Single Cash Flow
$ (1.)\:\: PV = \dfrac{FV}{\left(1 + i\right)^n} \\[7ex] (2.)\:\: PV = FV * \left(1 + i\right)^{-n} = FV * single\:\:payment\:\:present\:\:worth\:\:factor \\[5ex] (3.)\:\: PV = \dfrac{FV}{\left(1 + \dfrac{APY}{m}\right)^{mt}} \\[10ex] (4.)\:\: FV = PV * (1 + i)^{n} = PV * single\:\:payment\:\:compound\:\:amount\:\:factor \\[5ex] (5.)\:\: FV = PV * \left(1 + \dfrac{APY}{m}\right)^{mt} \\[7ex] (6.)\:\: APY = m\left[\left(\dfrac{FV}{PV}\right)^{\dfrac{1}{mt}} - 1\right] \\[10ex] (7.)\:\: APY = m\left(10^{\dfrac{\log\left(\dfrac{FV}{PV}\right)}{mt}} - 1\right) \\[10ex] (8.)\:\: APY = m\left[\left(\dfrac{FV}{PV}\right)^{\dfrac{1}{n}} - 1\right] \\[10ex] (9.)\:\: APY = m\left(10^{\dfrac{\log\left(\dfrac{FV}{PV}\right)}{n}} - 1\right) \\[10ex] (10.)\:\: t = \dfrac{\log\left(\dfrac{FV}{PV}\right)}{m\log\left(1 + \dfrac{APY}{m}\right)} \\[10ex] (11.)\:\: t = \dfrac{\log\left(\dfrac{FV}{PV}\right)}{m\log(1 + i)} \\[7ex] $
Uniform Cash Flows (Ordinary Annuities)
$ (1.)\:\: PV = UCF * \left[\dfrac{(1 + i)^n - 1}{i(1 + i)^n}\right] \\[7ex] (2.)\:\: PV = UCF * uniform\:\:series\:\:present\:\:worth\:\:factor \\[5ex] (3.)\:\: PV = UCF * \left[\dfrac{\left(1 + \dfrac{APY}{m}\right)^{mt} - 1}{i\left(1 + \dfrac{APY}{m}\right)^{mt}}\right] \\[10ex] (4.)\:\: PV = m * UCF * \left[\dfrac{1 - \left(1 + \dfrac{APY}{m}\right)^{-mt}}{APY}\right] \\[10ex] (5.)\:\: FV = UCF * \left[\dfrac{(1 + i)^n - 1}{i}\right] \\[7ex] (6.)\:\: FV = UCF * uniform\:\:series\:\:compound\:\:amount\:\:factor \\[5ex] (7.)\:\: FV = UCF * \left[\dfrac{\left(1 + \dfrac{APY}{m}\right)^n - 1}{\dfrac{APY}{m}}\right] \\[10ex] (8.)\:\: FV = m * UCF * \left[\dfrac{\left(1 + \dfrac{APY}{m}\right)^{mt} - 1}{APY}\right] \\[10ex] (9.)\:\:Sinking\:\:Fund:\:\: UCF = FV * \dfrac{i}{(1 + i)^n - 1} \\[7ex] (10.)\:\:Sinking\:\:Fund:\:\: UCF = FV * sinking\:\:fund\:\:factor \\[5ex] (11.)\:\:Sinking\:\:Fund:\:\: UCF = \dfrac{FV * APY}{m * \left[(1 + i)^{n} - 1\right]} \\[10ex] (12.)\:\:Sinking\:\:Fund:\:\: UCF = \dfrac{FV * APY}{m * \left[\left(1 + \dfrac{APY}{m}\right)^{mt} - 1\right]} \\[10ex] (13.)\:\:Amortization:\:\: UCF = PV * \dfrac{i(1 + i)^n}{(1 + i)^n - 1} \\[7ex] (14.)\:\:Amortization:\:\: UCF = PV * capital\:\:recovery\:\:factor \\[5ex] (15.)\:\:Amortization:\:\: UCF = \dfrac{PV * APY}{m * \left[1 - (1 + i)^{-n}\right]} \\[7ex] (16.)\:\:Amortization:\:\: UCF = \dfrac{PV * APY}{m * \left[1 - \left(1 + \dfrac{APY}{m}\right)^{-mt}\right]} \\[10ex] (17.)\:\: t = \dfrac{\log\left[\dfrac{FV * APY}{m * UCF} + 1\right]}{m * \log\left(1 + \dfrac{APY}{m}\right)} \\[10ex] (18.)\:\: t = -\dfrac{\log\left[1 - \dfrac{PV * APY}{m * UCF}\right]}{m * \log\left(1 + \dfrac{APY}{m}\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.
Calculators
Compound Interest
- Given: present value, number of compounding periods per year, interest rate, time
To Find: future value, interest
- Given: future value, number of compounding periods per year, interest rate, time
To Find: present value, interest
- Given: present value, number of compounding periods per year, future value, time
To Find: interest, interest rate per year
- Given: present value, number of compounding periods per year, interest rate, future value
To Find: interest, time
Ordinary Annuity
- Given: periodic uniform cash flows, interest rate, time
To Find: future value, interest
- Given: periodic uniform cash flows, interest rate, time
To Find: present value, interest
Sinking Fund
- Given: future value, interest rate, time
To Find: periodic uniform cash flows, interest
Amortization
- Given: present value, interest rate, time
To Find: periodic uniform cash flows, interest
- Given: periodic uniform cash flows, interest rate, future value
To Find: time, interest
- Given: periodic uniform cash flows, interest rate, present value
To Find: time, interest
- Given: periodic uniform cash flows, time, future value
To Find: interest rate, interest
- Given: periodic uniform cash flows, time, present value
To Find: interest rate, interest
Deferred Ordinary Annuity
- Given: periodic uniform cash flows, interest rate, time
To Find: future value of deferred ordinary annuity, interest
- Given: periodic uniform cash flows, interest rate, time
To Find: present value of deferred ordinary annuity, interest