unted for, it is useful to make a sketch to represent the various Appendix 2: Financial Calculations Time (years) Figure A2.6 Annual cash flows are discounted in present value methods amounts and when they occur. Such a sketch is shown in Figure A2.6. Income is shown above the line, costs below it. End of year Actual cash flow: Discount factor Discounted cash flow: $ 000 @ 10 % per annum. $ 000 (1000) 1.000 (1000) 0.909 363.6 0.826 413.0 (100) 0.751 (75.1) 0.683 341.5 0.621 372.6 Total 415.6 (Note: The usual condition for the existence of an internal rate of return is that there must be at least one change of sign in the sequence of cash flows.) INTERNAL RATE OF RETURN (IRR) In the IRR study the aim is to find the discount rate which leads to an NPV of zero, that is; NPV of outgoing cash flows just balances NPV of incoming cash flows. The procedure to find IRR is iterative: guess values for IRR; calculate and plot values of NPV; find the value that gives a zero NPV by interpolation. These calculations can be done with computers (e.g. Excel). IRR for a project can be compared with the corporate discount rate i: if IRR is less than i, the project is not profitable; the more IRR exceeds i, the more profitable the project. Example This example shows a trial-and-error method which also uses interpolation. Understanding Changing Telecommunications - Building a Successful Telecom Business The table below shows a series of cash flows corresponding to an initial investment of $1 million, which results in yearly income of $263 800 per annum over a five-year period. End of Actual cash Discount factor Discounted Discount factor Discounted flow ($M) @ 5 % pa cash flow @ 15 % pa cash flow ($M) (1000) (1000) (1000) 263.80 0.952 251.10 0.867 228.70 263.80 0.907 239.30 0.756 199.40 263.80 0.864 227.90 0.658 173.60 263.80 0.823 217.10 0.572 150.90 263.80 0.784 206.80 0.497 131.10 142.20 (116.30) Deduce the IRR by calculating the NPV of the cash flows for two discount rates, one hopefully larger than the IRR, and the other hopefull