Chapter 5: Essential Formulae in Project Appraisal A Coverage of the Formulae and Symbols Used to Evaluate Investment Projects
Study objectives • After studying this chapter the reader should be able to: • Apply a discounting rate to cash flows occurring at different points in time to translate them into a common measure of value • Calculate present value, net present value (NPV) and internal rate of return (IRR) from a given cash flow series • Calculate monthly loan (mortgage) repayments, their interest and principal components and the loan balance of a mortgage loan • Understand the financial mathematics involved in the discounted cash flow techniques (such as NPV and IRR) and mortgage loans • Apply the relevant annuity formulae for project appraisal
Internal rate of return • The internal rate of return is an alternative measure for evaluating projects. • It is the calculated rate of return (or discount rate) r at which the NPV will be equal to zero. • In project evaluation this rate has to be equal to or greater than the required rate of return for the project to be acceptable • It is calculated manually by trial and error, or by a special routine in computerized spreadsheets.
Internal Rate of Return: IRR 0 1 2 3 CF0 CF1 CF2 CF3 Cost Inflows IRR is the discount rate that forces PV inflows = cost. This is the same as forcing NPV = 0.
NPV: Enter r, solve for NPV. IRR: Enter NPV = 0, solve for IRR.
What’s Project N’s IRR? 0 1 2 3 IRR = ? -100.00 10 60 80 PV1 PV2 PV3 0 = NPV Enter CFs in CFLO, then press IRR: IRRN = 18.13%.
Internal Rate of Return: IRR • Arriving at the IRR solution will involve a number of iterations, so where there are more than two cash flows, a computer package (such as Excel) or a financial calculator (with an IRR function) is recommended.
Internal Rate of Return: IRR • In the NPV model, the NPV is clearly defined. • In the IRR equation, however, it is difficult to define IRR in its own terms, because it effectively means something like: ‘the rate of return at which all funds, if borrowed at the IRR, could be repaid from the project, without the firm having to make any cash contribution’. • The IRR criterion does not measure the project’s contribution to the firm’s value.
Internal Rate of Return: IRR Example 5.14 • Net cash inflows of a project are estimated as: EOY 1 $3,000; EOY 2 $4,000; and EOY 3 $8,000. Capital outlays for the project will occur during the first and second years and they are estimated as $2,000 and $1,500, respectively. They are assumed to occur at the beginning of each period. The timing notation for the correct discounting of these will be: EOY 0 $2,000; and EOY 1 $1,500. The discount rate is 10% per annum. What is the IRR?
Internal Rate of Return: IRR • IRR = 133.76%
What is the payback period? The number of years required to recover a project’s cost, or how long does it take to get the business’s money back?
Payback for Project N’s(Long: Most CFs in out years) 2.4 0 1 2 3 CFt -100 10 60 100 80 Cumulative -100 -90 -30 0 50 PaybackL = 2 + 30/80 = 2.375 years
Project M’s (Short: CFs come quickly) 1.6 0 1 2 3 CFt -100 70 100 50 20 Cumulative -100 -30 0 20 40 PaybackS = 1 + 30/50 = 1.6 years
Strengths of Payback: 1. Provides an indication of a project’s risk and liquidity. 2. Easy to calculate and understand. Weaknesses of Payback: 1. Ignores the TVM. 2. Ignores CFs occurring after the payback period.
Evaluation of Project Cash Flows. • Cash flows occurring within investment projects are assumed to occur regularly, at the end of each year. • Since they are unlikely to be equal, they will not be annuities. • Annuity calculations apply more to loans and other types of financing. • All future flows are discounted to calculate a Net Present Value, NPV; or an Internal Rate of Return, IRR.
Decision Making With Cash Flow Evaluations • If the Net Present Value is positive, then the project should be accepted. The project will increase the present wealth of the firm by the NPV amount. • If the IRR is greater than the required rate of return, then the project should be accepted. The IRR is a relative measure, and does not measure an increase in the firm’s wealth.
Calculating NPV and IRR With Excel -- Basics. • Ensure that the cash flows are recorded with the correct signs: -$, +$, -$, +$ etc. • Make sure that the cash flows are evenly timed: usually at the end of each year. • Enter the discount rate as a percentage, not as a decimal: e.g. 15.6%, not 0.156. • Check your calculations with a hand held calculator to ensure that the formulae have been correctly set up.
Calculating MIRR and PB With Excel. • Modified Internal Rate of Return – the cash flow cell range is the same as in the IRR, but both the required rate of return, and the re-investment rate, are entered into the formula: MIRR( B6:E6, B13, B14) • Payback – there is no Excel formula . The payback year can be found by inspection of accumulated annual cash flows.
ARR and Other Evaluations With Excel. • Accounting Rate of Return – there is no Excel formula. Average the annual accounting income by using the ‘AVERAGE’ function, and divide by the chosen asset base. • Other financial calculations – use Excel ‘Help’ to find the appropriate function. Read the help information carefully, and apply the function to a known problem before relying on it in a live worksheet.
Calculating Financial Functions With Excel -- Worksheet Errors. Common worksheet errors are: • Cash flow cell range wrongly specified. • Incorrect entry of interest rates. • Wrong NPV, IRR and MIRR formulae. • Incorrect cell referencing. • Mistyped data values. • No worksheet protection.
Calculating Financial Functions With Excel -- Error Control. Methods to reduce errors: • Use Excel audit and tracking tools. • Test the worksheet with known data. • Confirm computations by calculator. • Visually inspect the coding. • Use a team to audit the spreadsheet.
Essential Formulae -- Summary 1.The Time Value of Money is a cornerstone of finance. 2. The amount, direction and timing of cash flows, and relevant interest rates, must be carefully specified. 3. Knowledge of financial formulae is essential for project evaluation.
Essential Formulae -- Summary 4. NPV and IRR are the primary investment evaluation criteria. 5. Most financial functions can be automated within Excel. 6. Spreadsheet errors are common. Error controls should be employed.
Essential Formulae -- Summary 7. To reduce spreadsheet errors: -document all spreadsheets, keep a list of authors and a history of changes, use comments to guide later users and operators. 8. Financial formulae and spreadsheet operation can be demanding. Seek help when in doubt. $ % $ % $ % $ % $ % $ % $ % $ % $ %$ %