## The Time Value of Money

The time value of money concept in financial management is used to compare lump sum cash flows which are received or paid at different times.

The lump sum present and future value formulas can be used to calculate the effect of time and compounding interest rates on the value of the lump sums. They are best looked at by way of example.

### From Present Value to Future Value of a Lump Sum

A lump sum received now and deposited at a compounding interest rate for a number of periods will have a future value.

If you have 100 and deposit it at 5%, after 1 year you would have 100 + 100 x 5% = 105, after 2 years you would have 105 + 105 x 5% = 110.25.

In this example, the 100 is the lump sum received now referred to as the present value, and the 110.25 is the value in 2 years time at an interest rate of 5% and is called the future value.

### From Future Value to Present Value of a Lump Sum

A lump sum received in the future and discounted back at a compounding interest rate (the money you would loose by not being able to invest it now) will have a present value.

If you receive 110.25 in 2 years time, and could have earned 5%, then in 1 years time the value of the lump sum would be 110.25 / 105% = 105. After 2 years the value of the lump sum would be 105 / 105% = 100.

In this example, the 110.25 is the future value of the lump sum, and the 100 is the present value of the lump sum at 5% for 2 years.

## Lump Sum Formulas

The following summarizes for easy reference the formulas for calculating present value of future payments, future value of lump sum, the compounding interest rate, and the number of periods of compounding. An example of using the lump sum formulas is given, together with the corresponding Excel formulas.

The formula to use will depend on which 3 of the 4 variables are already known.

In all present value and future value lump sum formulas the following symbols are used.

- FV means future value
- PV means present value
- i% is the period discount rate
- n is the number of periods
- LN is a natural logarithm
- * means multiply, and ^ means to the power of

The example used below for each of the annuity formulas is based on the following information.

- Future value = FV = 7,335.93
- Present value = PV = 4,622.88
- Period discount rate = i% = 8%
- Number of periods = n = 6

**To Calculate the Future Value of a Lump Sum**

Formula | FV = PV * (1 + i%)^n |

Example | FV = 4622.88 * (1 + 8%)^6 = 7,335.93 |

Excel Future Value Formula | FV = – FV(i%,n,,PV) |

**To Calculate the Present Value of a Lump Sum**

Formula | PV = FV / (1 + i%)^n |

Example | PV = 7,335.93 / (1 + 8%)^6 = 4,622.88 |

Excel Present Value Formula | PV = – PV(i%,n,,FV) |

**The Find the Compounding Discount Rate**

Formula | i = ^{n}√(FV / PV) – 1 |

Example | i = (7335.93 / 4622.88)^(1 / 6)-1 = 8% |

Excel Interest Rate Formula | i = RATE(n,,PV,- FV) |

**To find the Number of Periods**

Formula | N = LN(FV / PV) / LN(1 + i%) |

Example | N = LN(7,335.93/4,622.88)/LN(1+8%) = 6 |

Excel Periods Formula | N = NPER(i%,,PV,- FV) |