# Simplified Financial Calculations

## Background

The various process technology choices under investigation in the pre-engineering phase each have different capital and operational costs. In order to assess which of the choices is more financially attractive some form of financial analysis is required.

It is not practical or necessary to do a complete financial model of each technology. A simplified financial analysis is sufficient to enable a choice between two or more options.

# Net Present Value

The usual way to evaluate two or more project options is to calculate the Net Present Value (NPV) of the cash flows from the various project options, and then choose the project which returns the highest NPV provided that the NPV>0, projects with a negative NPV should be rejected.

NPV is calculated as follows (please note each equation is shown twice, firstly in graphical form and secondly in html)

NPV = ΣCF_{i} / (1 + r)^{i} i = 1..n

where

- NPV = Nett Present Value
- CF
_{i}= the cash flow in period i - r = the discount rate
- n = the number of periods

The discount rate used is the weighted average cost of capital.

# Simplified Analysis

## Assumptions

It will be assumed that

- The capital spending all happens in the first year
- The project returns a positive cash flow in the second year
- The inflation rate is zero
- Product selling prices remain constant
- Exchange rates remain constant
- Tax can be ignored
- Terminal values are zero

In other words the project cash flows (year 2 to year n) remain constant.

These assumptions are justified in that fluctuations in any of these parameters will affect each option equally, and secondly the fluctuations in these parameters are random and essentially unknowable.

## Method

Let's call the first cash flow CF_{1} and

CF_{1} = -1 × A

The cash flow in year 2, CF_{2} can be set as

CF_{2} = x · A 0 < x < 1

Also

CF_{j} = CF_{2} = x · A j = 3..n

So the NPV calculation can be restated as

NPV = -A/(1+r) + Σ x·A/(1+r)^{i} i = 2..n

or

NPV = -A/(1+r) + x·A Σ 1/(1+r)^{i} i = 2..n

Now if we set NPV = 0 then

A/(1+r) = x·A Σ 1/(1+r)^{i} i = 2..n

or

1/x = (1+r) Σ 1/(1+r)^{i} i = 2..n

or

1/x = Σ 1/(1+r)^{i} i = 1..n-1

Now 1/x is the payback period of the investment. The calculation of this payback is easily done in a spreadsheet. If, for example, your investment criterion is that a project must return a positive NPV calculated over 10 years at a 14% discount rate, when the above formula is applied with r=14% and n=10 then 1/x is 4.946 years

i | (1+r)^{-i} |
---|---|

1 | 0.877 |

2 | 0.769 |

3 | 0.675 |

4 | 0.592 |

5 | 0.519 |

6 | 0.456 |

7 | 0.400 |

8 | 0.351 |

9 | 0.308 |

sum | 4.946 |

In this example, if a project option has a payback of less than 5 years it is a viable project.

# Application of the Method

One of the options must be chosen as the base case (usually the current process configuration). The difference in costs (both capital and operational) between the base case (on the one hand) and the other choices (on the other) can be calculated

If the ratio of extra capital spent to extra income earned is less than 1/x the project option is viable and the project option which has the lowest payback time is the best option.