**Net present value**

### Table of contents

## Introduction

Present value is simply the current value of an investment or monetary quantity. For example, if you were given 100 DKK today, the present value would equal 100 DKK. However, if you were given 100 DKK on the same date next year, the present value would be __less__ than 100 DKK.

The reasoning is simple: if you were given 100 DKK today, you could invest the money in a bank, and next year it could be worth

100 DKK + interest

Alternatively, you could use the money immediately instead of investing it by purchasing goods worth 100 DKK, e.g. 20 liters of milk. However, if you waited until next year to purchase milk with the same 100 DKK bill, you would no longer be able to buy 20 liters. Prices increase slightly on average across all goods every year, an effect known as *inflation*, corresponding to an increase of 1-2 % annually. A sum of money that can buy 20 liters of milk this year will only be able to buy 19.5 liters of milk next year.

Interest rates set by banks are generally slightly above expected inflation rates, such that money deposited in a bank does not lose purchasing power over time.

## Discounting

Discounting is a practice used to take into account the lesser value of a given amount of money in the future relative to the present. If the discounting rate were set equal to the bank's interest rate, then the value of future income is adjusted downwards to take into account the lost interest that could otherwise have been earned. In other words, 100 DKK earned next year is worth less than 100 DKK earned this year. For example, if the bank's interest rate were set to 3%, then 103 DKK earned next year would be equal to 100 DKK earned this year.

Other factors than interest rates and inflation contribute to the discounting rate, such as personal preference or uncertainty. For example, if you were given the choice between receiving 100 DKK today and maybe receiving 120 DKK next year, you might prefer the 100 DKK today if you wanted to use the money right away. This form of argument is often used when evaluating investment decisions, which depend on giving up a set amount today in order to receive a larger amount in the future - for example, saving up for a pension plan.

Discounting is also commonly used when there is some degree of uncertainty involved in a project proposal. If you were asked to invest in a ferry which would *probably* transport 500 people a day, but *maybe* only 200 people a day, then you might insist that the ferry should be able to pay for itself even if only 200 people a day used it. You would strongly discount potential profits based on the transportation of 500 people a day, since the likelihood of earning such profits would be very low. High discounting rates are used when there is substantial uncertainty about the outcome of a project.

## Practical example

If you wanted to buy a new television, and were unsure whether a plasma screen or an LCD would be more expensive in terms of electricity over time, you could set up the following table:

Table 1. Comparison of televisions |
||
---|---|---|

Screen type |
Annual electricity consumption |
Estimated annual cost |

Inches | KWh | DKK |

LCD 20 | 60-110 | 120-220 |

LCD 32 | 110-260 | 220-520 |

LCD 42 | 160-390 | 320-780 |

Plasma 42 | 260-480 | 520-960 |

Plasma 50 | 360-660 | 720-1320 |

*The data on electricity consumption is taken from www.elsparefonden.dk and is based on average daily tv usage of 4 hours (1 460 hours annually), with the remaining hours on standby. One KWh is expected to cost 2 DKK.*

A 42-inch plasma screen TV has a higher KWh consumption than an LCD screen of the same size, due to its superior contrast and picture quality. In terms of electricity cost, the difference is approximately 180-200 DKK annually. Simply calculated, the difference amounts to an extra 2 000 DKK cost for the plasma screen TV over a ten-year period. However, due to the reasoning explained above, 200 DKK in ten years time is not the same as 200 DKK today.

Using a discount rate of 3 %, the annual minimum electricity costs of a 42-inch plasma screen and LCD screen TV in present value are presented in the table below:

Table 2. Present value of electricity costs over time |
||||
---|---|---|---|---|

Time |
Annual electricity costs |
Present value of electricity costs |
||

Year | Plasma screen | LCD screen | Plasma Screen | LCD screen |

Present | 320 | 520 | 320 | 520 |

2011 | 320 | 520 | 310.7 | 504.9 |

2012 | 320 | 520 | 301.6 | 490.1 |

2013 | 320 | 520 | 292.8 | 475.9 |

... | ... | ... | ... | ... |

2019 | 320 | 520 | 245.3 | 398.5 |

Sum | 3 200 | 5 200 | 2 811.6 | 4 568.8 |

Although annual electricity costs remain constant over the period, the present value falls each year. The longer the time frame, the more the cost is discounted. Note that the present value depends on the year selected. The present value in 2010 of the electricity cost of the LCD screen in 2019 is 245 DKK, but the present value in 2019 of the electricity cost in 2019 is 320 DKK. In other words, although the electricity costs remain the same, the future cost is discounted by 75 DKK relative to its value today. The graph below illustrates this trend for the LCD screen.

## Discounting versus Depreciation

Discounting should not be confused with depreciation. While discounting reflects time preference in valuing money today over money tomorrow, depreciation reflects the decrease in value of an object over time. Using the previous example, a new television is expected to have a lifetime of 20-30 years, after which it will be turned into scrap metal. The value of the TV at the end of its useful lifetime will be substantially lower than its initial cost (if not zero).

The value of a television screen worth 3 999 DKK today, over its lifetime with a constant rate of depreciation and a scrap value equal to zero is shown in the graph below.

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