# Discounted cash flow methods

Rate of return and payback period calculations, described in the previous subsections, provide a good starting point for appraising an investment, but they ignore what is often termed the *time value of money*.

If you were offered the choice of accepting £100 now or in one year’s time, you would almost certainly take the money now to spend and enjoy the proceeds or perhaps invest and earn some interest. In contrast, if the choice were between £100 now and £200 in a year’s time you might well decide that it is better to wait and get the higher sum – assuming that you are confident the offer will still stand in a year and that you don’t have an urgent need for the money now. But if the choice were between £100 now and £110 or maybe £120 in a year’s time, the situation is less clear-cut.

Organisations have to take decisions like this one all the time and one way of making the decision is to look at the interest rate that could be earned on the money. Suppose that the interest rate on a savings account offered by a reliable and trustworthy bank is 3%.

Investing the £100 for one year would give

If the £103 is reinvested for a further year the result would be

You may recognise this processing as compounding. Calculating compound interest has the following general formula

where

*P* = the initial amount invested (the principal)

*r* = the percentage interest rate

*n* = the number of years.

Or in financial language you can say that if the interest rate is 3% the *future value *of £100 in one year is £103 and in two years is approximately £106, and so on.

The converse of this is known as the *present value* (PV) and, rearranging the above formula, the present value of £*P* in *n* years’ time is given by

## Activity 6 Present value

What is the present value of £215 in three years’ time if the interest rate is 5%?

### Answer

Using the above equation for the present value

Table 3 shows the present value of £1 receivable at the end of each year for periods ranging from 0 to 30 years and interest rates ranging from 0 to 10%. If you have access to a spreadsheet package, you might find it interesting to recreate this table yourself.

## Table 3 Present values of £1

Time(years) | Interest rate (%) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

1 | 0.990 | 0.980 | 0.971 | 0.962 | 0.952 | 0.943 | 0.935 | 0.926 | 0.917 | 0.909 |

2 | 0.980 | 0.961 | 0.943 | 0.925 | 0.907 | 0.890 | 0.873 | 0.857 | 0.842 | 0.826 |

3 | 0.971 | 0.942 | 0.915 | 0.889 | 0.864 | 0.840 | 0.816 | 0.794 | 0.772 | 0.751 |

4 | 0.961 | 0.924 | 0.888 | 0.855 | 0.823 | 0.792 | 0.763 | 0.735 | 0.708 | 0.683 |

5 | 0.951 | 0.906 | 0.863 | 0.822 | 0.784 | 0.747 | 0.713 | 0.681 | 0.650 | 0.621 |

6 | 0.942 | 0.888 | 0.837 | 0.790 | 0.746 | 0.705 | 0.666 | 0.630 | 0.596 | 0.564 |

7 | 0.933 | 0.871 | 0.813 | 0.760 | 0.711 | 0.665 | 0.623 | 0.583 | 0.547 | 0.513 |

8 | 0.923 | 0.853 | 0.789 | 0.731 | 0.677 | 0.627 | 0.582 | 0.540 | 0.502 | 0.467 |

9 | 0.914 | 0.837 | 0.766 | 0.703 | 0.645 | 0.592 | 0.544 | 0.500 | 0.460 | 0.424 |

10 | 0.905 | 0.820 | 0.744 | 0.676 | 0.614 | 0.558 | 0.508 | 0.463 | 0.422 | 0.386 |

11 | 0.896 | 0.804 | 0.722 | 0.650 | 0.585 | 0.527 | 0.475 | 0.429 | 0.388 | 0.350 |

12 | 0.887 | 0.788 | 0.701 | 0.625 | 0.557 | 0.497 | 0.444 | 0.397 | 0.356 | 0.319 |

13 | 0.879 | 0.773 | 0.681 | 0.601 | 0.530 | 0.469 | 0.415 | 0.368 | 0.326 | 0.290 |

14 | 0.870 | 0.758 | 0.661 | 0.577 | 0.505 | 0.442 | 0.388 | 0.340 | 0.299 | 0.263 |

15 | 0.861 | 0.743 | 0.642 | 0.555 | 0.481 | 0.417 | 0.362 | 0.315 | 0.275 | 0.239 |

16 | 0.853 | 0.728 | 0.623 | 0.534 | 0.458 | 0.394 | 0.339 | 0.292 | 0.252 | 0.218 |

17 | 0.844 | 0.714 | 0.605 | 0.513 | 0.436 | 0.371 | 0.317 | 0.270 | 0.231 | 0.198 |

18 | 0.836 | 0.700 | 0.587 | 0.494 | 0.416 | 0.350 | 0.296 | 0.250 | 0.212 | 0.180 |

19 | 0.828 | 0.686 | 0.570 | 0.475 | 0.396 | 0.331 | 0.277 | 0.232 | 0.194 | 0.164 |

20 | 0.820 | 0.673 | 0.554 | 0.456 | 0.377 | 0.312 | 0.258 | 0.215 | 0.178 | 0.149 |

21 | 0.811 | 0.660 | 0.538 | 0.439 | 0.359 | 0.294 | 0.242 | 0.199 | 0.164 | 0.135 |

22 | 0.803 | 0.647 | 0.522 | 0.422 | 0.342 | 0.278 | 0.226 | 0.184 | 0.150 | 0.123 |

23 | 0.795 | 0.634 | 0.507 | 0.406 | 0.326 | 0.262 | 0.211 | 0.170 | 0.138 | 0.112 |

24 | 0.788 | 0.622 | 0.492 | 0.390 | 0.310 | 0.247 | 0.197 | 0.158 | 0.126 | 0.102 |

25 | 0.780 | 0.610 | 0.478 | 0.375 | 0.295 | 0.233 | 0.184 | 0.146 | 0.116 | 0.092 |

26 | 0.772 | 0.598 | 0.464 | 0.361 | 0.281 | 0.220 | 0.172 | 0.135 | 0.106 | 0.084 |

27 | 0.764 | 0.586 | 0.450 | 0.347 | 0.268 | 0.207 | 0.161 | 0.125 | 0.098 | 0.076 |

28 | 0.757 | 0.574 | 0.437 | 0.333 | 0.255 | 0.196 | 0.150 | 0.116 | 0.090 | 0.069 |

29 | 0.749 | 0.563 | 0.424 | 0.321 | 0.243 | 0.185 | 0.141 | 0.107 | 0.082 | 0.063 |

30 | 0.742 | 0.552 | 0.412 | 0.308 | 0.231 | 0.174 | 0.131 | 0.099 | 0.075 | 0.057 |

The principles of discounted cash flow (DCF) allow possible investments to be reviewed by using the table of present values (Table 3), or discount factors as they are also known, to determine the time in a project’s life when payments are made and when income is earned. The term *discount rates* is used rather than interest rates. You can think of the discount rate as being the rate of return or profit that a project will make. In general you can assume that a project will not go ahead if investing in it, which carries some risk, is estimated to give a lower rate of return than investing the money in a secure bank account.

The use of discounted cash flow is best illustrated by means of a worked example, as demonstrated in Activity 7.

## Activity 7 Discounted cash flow

A landfill site operator is considering installing an engine to generate power by burning landfill gas. It is estimated that the capital cost of the engine, generator, control equipment and connections to the grid will be £1000 000. The annual staffing, operating and maintenance costs will be £300 000. The annual income from the sale of power will be £315 000 in the first year and £630 000 in each following year. The capital costs have to be paid at the start of the project and the total project life is expected to be 5 years. If company policy states that a rate of return of 5% must be achieved, is the project worth pursuing?

### Answer

The first stage in the process involves drawing up a cash flow statement. In the example, this is relatively straightforward. However you should appreciate that because I have said that the capital investment takes place at the start of the project, its present and future values are the same. This is indicated by placing this expenditure in year 0 – more complex projects will involve staged payments over more than one year. I have also assumed that the income starts to arrive in year 1.

The cash flow statement is shown in Table 4.

#### Table 4 Cash flow statement (all money values in £000s)

Year | 0 | 1 | 2 | 3 | 4 | 5 | Total |
---|---|---|---|---|---|---|---|

Capital cost | –1000 | –1000 | |||||

Operating cost | –300 | –300 | –300 | –300 | –300 | –1500 | |

Income | 315 | 630 | 630 | 630 | 630 | 2835 | |

Net cash flow | Total –1000 | Total 15 | Total 330 | Total 330 | Total 330 | Total 330 | Total 335 |

The next stage is to adjust the net cash flow in each year by multiplying it by the appropriate discount factor. This is shown in Table 5 for a discount rate of 5% (with the values taken from Table 3).

#### Table 5 Discounted cash flow (all money values in £000s)

Year | 0 | 1 | 2 | 3 | 4 | 5 | Total |
---|---|---|---|---|---|---|---|

Capital cost | –1000 | –1000 | |||||

Operating cost | –300 | –300 | –300 | –300 | –300 | –1500 | |

Income | 315 | 630 | 630 | 630 | 630 | 2835 | |

Net cash flow | –1000 | 15 | 330 | 330 | 330 | 330 | 335 |

Discount factor | 1.00 | 0.952 | 0.907 | 0.864 | 0.823 | 0.784 | |

Discounted cash flow | –1000 | 14.28 | 299.31 | 285.12 | 271.59 | 258.72 | 129.02 |

The final cell in the right-hand column (£129 020) represents the net present value (NPV) of the project. This is positive, so the project has met the target rate of return of 5%.

It is often useful to calculate the rate of return of a project, which is represented by the discount rate that achieves an NPV of zero. Again, this is something you might want to try if you have access to a spreadsheet. If you do so, you will find that the NPVs of 8% and 9% are £25 928 and –£5406 respectively, so the rate of return is somewhere between 8% and 9%. If you use a ‘goal seek’ function on a spreadsheet, you will find that the rate of return is 8.82%.

In principle all DCF calculations can be performed in this way, but it can get long-winded for projects with long lifetimes, such as a reservoir or road. If it is clear that a cash flow will occur over a long period, it is possible to sum the discount factors and multiply this sum by the cash flow.

## Example 1

What is the present value of £300 000 received annually for a period of 5 years if the discount rate is 4%? Use Table 3 to look up the discount factors.

Table 6 shows the present value of £1 received annually for different time periods.

## Table 6 Present value of £1 received annually

Time (years) | Interest rate (%) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

1 | 0.990 | 0.980 | 0.971 | 0.962 | 0.952 | 0.943 | 0.935 | 0.926 | 0.917 | 0.909 |

2 | 1.970 | 1.942 | 1.913 | 1.886 | 1.859 | 1.833 | 1.808 | 1.783 | 1.759 | 1.736 |

3 | 2.941 | 2.884 | 2.829 | 2.775 | 2.723 | 2.673 | 2.624 | 2.577 | 2.531 | 2.487 |

4 | 3.902 | 3.808 | 3.717 | 3.630 | 3.546 | 3.465 | 3.387 | 3.312 | 3.240 | 3.170 |

5 | 4.853 | 4.713 | 4.580 | 4.452 | 4.329 | 4.212 | 4.100 | 3.993 | 3.890 | 3.791 |

6 | 5.795 | 5.601 | 5.417 | 5.242 | 5.076 | 4.917 | 4.767 | 4.623 | 4.486 | 4.355 |

7 | 6.728 | 6.472 | 6.230 | 6.002 | 5.786 | 5.582 | 5.389 | 5.206 | 5.033 | 4.868 |

8 | 7.652 | 7.325 | 7.020 | 6.733 | 6.463 | 6.210 | 5.971 | 5.747 | 5.535 | 5.335 |

9 | 8.566 | 8.162 | 7.786 | 7.435 | 7.108 | 6.802 | 6.515 | 6.247 | 5.995 | 5.759 |

10 | 9.471 | 8.983 | 8.530 | 8.111 | 7.722 | 7.360 | 7.024 | 6.710 | 6.418 | 6.145 |

11 | 10.368 | 9.787 | 9.253 | 8.760 | 8.306 | 7.887 | 7.499 | 7.139 | 6.805 | 6.495 |

12 | 11.255 | 10.575 | 9.954 | 9.385 | 8.863 | 8.384 | 7.943 | 7.536 | 7.161 | 6.814 |

13 | 12.134 | 11.348 | 10.635 | 9.986 | 9.394 | 8.853 | 8.358 | 7.904 | 7.487 | 7.103 |

14 | 13.004 | 12.106 | 11.296 | 10.563 | 9.899 | 9.295 | 8.745 | 8.244 | 7.786 | 7.367 |

15 | 13.865 | 12.849 | 11.938 | 11.118 | 10.380 | 9.712 | 9.108 | 8.559 | 8.061 | 7.606 |

16 | 14.718 | 13.578 | 12.561 | 11.652 | 10.838 | 10.106 | 9.447 | 8.851 | 8.313 | 7.824 |

17 | 15.562 | 14.292 | 13.166 | 12.166 | 11.274 | 10.477 | 9.763 | 9.122 | 8.544 | 8.022 |

18 | 16.398 | 14.992 | 13.754 | 12.659 | 11.690 | 10.828 | 10.059 | 9.372 | 8.756 | 8.201 |

19 | 17.226 | 15.678 | 14.324 | 13.134 | 12.085 | 11.158 | 10.336 | 9.604 | 8.950 | 8.365 |

20 | 18.046 | 16.351 | 14.887 | 13.590 | 12.462 | 11.470 | 10.594 | 9.818 | 9.129 | 8.514 |

21 | 18.857 | 17.011 | 15.415 | 14.029 | 12.821 | 11.764 | 10.836 | 10.017 | 9.292 | 8.649 |

22 | 19.660 | 17.658 | 15.937 | 14.451 | 13.163 | 12.042 | 11.061 | 10.201 | 9.442 | 8.772 |

23 | 20.456 | 18.292 | 16.444 | 14.857 | 13.489 | 12.303 | 11.272 | 10.371 | 9.580 | 8.883 |

24 | 21.243 | 18.914 | 16.936 | 15.247 | 13.799 | 12.550 | 11.469 | 10.529 | 9.707 | 8.985 |

25 | 22.023 | 19.523 | 17.413 | 15.622 | 14.094 | 12.783 | 11.654 | 10.675 | 9.823 | 9.077 |

30 | 25.808 | 22.396 | 19.600 | 17.292 | 15.372 | 13.765 | 12.409 | 11.258 | 10.274 | 9.427 |

35 | 29.409 | 24.999 | 21.487 | 18.665 | 16.374 | 14.498 | 12.948 | 11.655 | 10.567 | 9.644 |

40 | 32.835 | 27.355 | 23.115 | 19.793 | 17.159 | 15.046 | 13.332 | 11.925 | 10.757 | 9.779 |

## Example 2

A water supply company is proposing to construct a reservoir on a river at point X at a cost of £5.86 million. No further increment in water storage would be needed for 29 years. It has been suggested that three smaller reservoirs costing £3.25 million, £3.60 million and £4.33 million would be financially and environmentally preferable. However this scheme would require the construction of the second reservoir after 13 years and the third 9 years later. Investigate the financial aspects of the two alternatives for discount rates of 6% and 8%. Assume that the cash to build the first reservoir is required in year 0, the cash for the second is required in year 13, and in year 22 for the third reservoir.

### Answer

The capital costs of the first scheme are all met at the present time, so the present value is £5.86 million for both rates of return.

In the case of the second scheme at a rate of 6% the present value is given by

giving the total as

Therefore, the one-reservoir scheme is the more financially attractive.

At a rate of 8% the PV of the second scheme is given by

giving the total as

At 8% the three-reservoir alternative is preferable, even though the total capital cost is almost twice as great as the single-reservoir scheme. Looked at in another way, if the difference between the cost of the larger and the first-stage reservoir (£5.86m – £3.25m = £2.61m) were to be invested at 8% (say), by the time the second reservoir had to be built the amount available (£6.58m) would be more than sufficient to meet the cost of £3.6m, and the accrued surplus again invested would likewise be more than sufficient to meet the cost of the last reservoir (in fact £5.96m would be available).