# Discrete Compounding Formulas

## Compounding formulas for discrete payments

### Single Payment

#### Compound Amount

Converts a single payment (or value) today - to a future value.

F = P [(1 + i)^{n}] (1)

where

F = future value

P = single payment today

i = interest rate per period

n = number of periods

##### Example - Future Value of an Initial Amount Received Today

An amount of *5000* is received today. Calculate the future value of this amount after *7 years* with interest rate *5%*.

The interest rate can be calculated

*i = (5 %) / /100 %)*

* = 0.05*

The future value of the amount can be calculated

*F = (5000) [(1 + 0.05) ^{7}]*

* = 7036 *

##### Future Value - Online Calculator

P - single payment today

i - interest (%)

n - number of periods

Note that interest rate in *%* is used in the calculator - not in the equation.

#### Present Worth (or Value)

Converts a future payment (or value) - to present wort (or value).

P = F [(1 + i)^{-n}] (2)

where

P = present value

F = single future payment

i = discount rate per period

n = number of periods

##### Example - Present Value of a Future Payment

An payment of *5000* is received after 7* years*. Calculate the present worth (or value) of this payment with dicount rate *5%*.

The discount rate can be calculated

*i = (5 %) / /100 %)*

* = 0.05*

The present worth of the future payment can be calculated

*F = (5000) [(1 + 0.05) ^{-7}]*

* = 3553 *

##### Present Value - Online Calculator

F - single future payment

i - discount rate (%)

n - number of periods

Note that discount rate in *%* is used in the calculator - not in the equation.

### Uniform Series

#### Compound Amount - Annuity

Converts a uniform amount (annuity) - to a future value.

F = A [((1 + i)^{n}- 1) / i ] (3)

where

F = future value

A = uniform amount per period

i = interest rate

n = numbers of periods

##### Example - Present Value of Uniforms Payments

An uniform amount of *5000* is paid every year in 7* years*. Calculate the future value of this amount with interest rate *5%*.

The interest rate can be calculated

*i = (5 %) / /100 %)*

* = 0.05*

The future value of the annuity can be calculated

*F = **5000* * [((1 + 0.05) ^{7} - 1 ) / 0.05] *

* = 40710 *

##### Compound Amount - Online Calculator

A - uniform amount per period

i - interest rate (%)

n - number of periods

Note that interest rate ín *%* is used in the calculator - not in the equation.

#### Sinking Fund

Converts a specific future value to uniform amounts (annuities)*.*

A = F [i / ((1 + i)^{n}- 1)] (4)

where

A = uniform amount per period

F = future value

i = interest rate

n = number of periods

##### Example - Uniforms Payments required to reach a Future Value

The future value of a *7 years* annuity is *5000*. Calculate the required annuity to reach this value with interest rate *5%*.

The interest rate can be calculated

*i = (5 %) / /100 %)*

* = 0.05*

The uniform payments (annuity) can be calculated

*A = 5000 [0.05 / ((1 + 0.05) ^{7} - 1)]*

* = 614*

##### Sinking Fund - Online Calculator

F - future value

i - interest rate (%)

n - number of periods

Note that interest rate in *%* is used in the calculator - not in the equation.

#### Present Worth

Converts a uniform amount (annuity) - to a present value*.*

P = A [((1 + i)^{n}- 1) / ( i (1 + i)^{n })] (5)

where

P = present value

A = amount per interest period

i = discount rate

n = discount periods

##### Example - Present Value of Uniform Amounts

The uniform amount (annuity) paid from a *7 years* project is *5000*. Calculate the present value with interest rate *5%*.

The interest rate can be calculated

*i = (5 %) / /100 %)*

* = 0.05*

The present value of the uniform amounts can be calculated

*P = 5000 [((1 + 0.05) ^{7} - 1) / ( 0.05 (1 + 0.05)^{7 })] *

* = 28932*

##### Present Worth or Value - Online Calculator

A - uniform amount per interest period

i - discount rate (%)

n - number of periods

Note that discount rate *%* is used in the calculator - not in the equation.

#### Capital Recovery

Converts a present value - to a uniform amount (annuity).

A = P [(i (1 + i)^{n}) / ((1+i)^{n}- 1)] (6)

where

P = present value

A = amount per interest period

i = interest rate

n = discount periods

##### Capital Recovery - Online Calculator

P - present value

i - interest rate (%)

n - number of periods

Note that interest rate in *%* is used in the calculator - not in the equation.