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1 Mariusz Próchniak Chair of Economics II Warsaw School of Economics CAPITAL BUDGETING Managerial Economics 1 2 1
2 Future value (FV) r annual interest rate B the amount of money held today Interest is compounded annually (annual capitalisation) The today s sum B will be worth: after one year: after 2 years: after 3 years: after n years: 1 1 FV = B + r 2 FV2 = B 1+ r 1+ r = B 1+ r FV = B + r n ( 1 ) FV = B + r n 3 Future value an example B = $ r = 10% = 0.10 FV 1 = ( ) = = FV 2 2 = = = FV 3 3 = = =
3 Present value (PV) r annual interest rate Annual capitalisation of interest The amount B received one year from now is worth today: PV = B ( 1+ r) The amount B received 2 years from now is worth today: The amount B received 3 years from now is worth today: The amount B received n years from now is worth today: PV = PV = PV = B ( 1+ r) 2 B ( 1+ r) 3 B ( 1+ r) n 5 Present value an example r = 10% = 0.10 The amount $ received one year from now is worth today: PV = = = $ ( + ) The amount $ received two years from now is worth today: PV = = = $ = ( ) ( 1.1) The amount $ received three years from now is worth today: PV = = = $ = ( ) ( 1.1) 6 3
4 Present value and discounting Interest rate used to calculate the present value of cash flows is known as the discount rate, and the process discounting. If cash flows (CF) occur at different points in time, the present value of these future cash flows can be expressed as: PV = T n= 0 1 CF n ( + r) n 7 Net present value (NPV) Net present value (NPV) is equal to discounted cash inflows (operating profits from an investment) minus discounted cash outflows (investment outlays). Net present value indicates the profitability of a given investment project. NPV is calculated according to the formula given on the previous slide, but cash inflows (revenues or profits from investment) have positive sign whereas cash outflows (investment outlays) are included into the summation with a negative sign. 8 4
5 Net present value an example A textile firm is considering building a new facility. Building the plant will require an immediate capital outlay of $ and will take one year. The firm expects that the plant, when in operation, will generate an addition to the firm s operating profit of $ per year for the next 5 years beginning one year from now. The annual interest rate applicable over this time period is 12%. What is the present value associated with building the plant? (Source: Samuelson, Marks, Managerial Economics) 9 Net present value an example Solution: Net present value NPV = NPV = NPV = $ NPV is positive, so the investment should be made. 10 5
6 Net present value an example (cont.) Suppose the firm pays a flat 34% tax rate on its taxable income. The $ annual profit flows are taxable with one exception. The firm is allowed a deduction for the depreciation of its production facility over its lifetime. The firm can depreciate the building on a straightline basis over 5 years. This means the firm can take a deduction of $ ( 1 / 5 of the total capital cost) against its annual income for each of the next 5 years. What is the net present value of this investment project? 11 Net present value an example (cont.) Solution: Net present value (NPV) NPV = NPV = NPV = $ NPV is positive, so the investment should be made. 12 6
7 Annuities Annuity periodic cash flow of fixed amount. 13 Annuity an example During the next 4 years, beginning one year from now, you will be given a fixed amount of 5000 $ annually. The interest rate is 8%. What is the present value of these cash flows? 14 7
8 Annuity an example Solution: This is a fouryear annuity PV = PV = $ 15 Perpetual annuity (perpetuity) A perpetual annuity (a perpetuity) an annuity that goes on forever. The present value of a perpetuity is: PV CF = r CF the constant annual cash flow, r the discount rate. In the above formula it is assumed that constant annual cash flows begin one year from now (and go on forever); i.e. the first cash flow is discounted by a discount factor that corresponds to the first year (division by 1 + r). Derivation of the above formula is easy (see the sum of the infinite geometric sequence). 16 8
9 Perpetual annuity an example You will be given a fixed annual amount of 5000 $ forever, beginning one year from now. The interest rate is 8%. What is the present value of these cash flows? 17 Perpetual annuity an example Solution: This is a perpetual annuity (a perpetuity) PV = K PV = 0.08 PV = $ Possible interpretation: You put a principal of $ in a bank account and leave it there forever. You will withdraw the interest each year till the infinity (8% of $ equals 5000 $). It turns out that the present value of the infinite flow of interests is equal to the principal. 18 9
10 Growing perpetuity An annuity pays an amount of CF after the first year and the payment rises by g% each year thereafter. This is a growing perpetuity. The present value of a growing perpetuity is: 2 3 CF CF 1+ g CF 1+ g CF 1+ g PV = K r 1+ r 1+ r 1+ r or: PV CF = r g Inflation often causes cash flows to grow. 19 Nominal and real cash flows and interest rates There are two equivalent ways to compute present values while properly accounting for inflation: Listing cash flows in nominal terms and discounting by a nominal interest rate. Listing cash flows in real terms and discounting by a real interest rate
11 Internal rate of return (IRR) The internal rate of return (IRR) of an investment is the discount rate at which the project s cash flows have a zero present value
12 MAKING INVESTMENT DECISIONS 1. A singleinvestment investment decision. 2. Mutually exclusive investments. 3. Making investment decisions with constrained resources A singleinvestment investment decision The firm should undertake the project if and only if the project s net present value is positive. In other words: The firm should undertake an investment project if and only if the project s internal rate of return is greater than the discount rate
13 2. Mutually exclusive investments In a choice among a number of mutually exclusive investment alternatives, the manager should choose the one that offers the greatest present value Making investment decisions with constrained resources A firm has 1 million $ to invest and faces the following potential investment projects: Project Initial investment ($) NPV ($) NPV per 1 $ invested A B C D E F G If the firm were not constrained, it would undertake all the programs because NPV > 0. Given constrained resources the firm should choose the programs with the highest NPV per dollar invested first. The firm will choose the projects: D, C, B, and E
14 27 How to determine the discount rate? The discount rate should correspond to the rate of return for projects in a comparable risk class. One of the methods of determining the discount rate: Weighted average cost of capital (WACC) 28 14
15 Weighted average cost of capital (WACC) This method assumes that the considered investment project has the same risk characteristics as does the firm in general. The weighted average cost of capital is the average of the rate of return on the firm s debt and the rate of return on its equity. 29 Weighted average cost of capital (WACC) an example The firm has 40% debt and 60% equity. The firm pays 10% on its debt. The firm returns 19.5% on its stock. WACC = = 15.7% The rate of return on the firm s debt is e.g. the (aftertax) tax) interest rate (the rate of return) of the bonds offered by the firm. The rate of return on equity: we could calculate the internal rate of return of an investment in shares, say, 5 years ago if cashed in today. Unfortunately, this method lacks precision because returns depend not only on firmspecific risks but also on general market conditions
16 The rate of return on equity A more precise measure of the rate of return on equity: rs = rf + rp r s the rate of return on the stock r f the riskfree rate of return r p the risk premium of a firm s stock The riskfree rate of return is typically the current rate of return on shortterm term bonds issued by the federal government. The risk premium can be estimated using the capital asset pricing model (CAPM). 31 Capital asset pricing model (CAPM) r = β r r p m f r p the risk premium of a firm s stock β beta r m r f the market risk premium A firm s risk (and, therefore, its expected rate of return) depends on its correlation with movements in the stock market as a whole. Beta measures the relationship between the individual stock s return and the return of the stock market. E.g. if β = 1, the systematic risk of the stock is the same as the market, meaning that the stock should have the same risk premium as the market
17 The rate of return on equity an example r = r + β r r s f m f In the U.S., over the last 50 years, the annual rate of return on stocks has averaged 12% and the return on riskfree, shortterm term treasury securities has averaged 3.5%. Thus, the risk premium on the stock market (r m r f ) is: 12% 3.5% = 8.5%. The beta on the firm s stock is 1.4. Shortterm term government bonds currently have a return of 7.6%. The firm s rate of return on equity (the firm s cost of equity): r = = = = 19.5% s
18 Appendix to Chapter 19: Present Value Tables 35 Appendix to Chapter 19: Present Value Tables 36 18
19 Thank you for the attention!!! 37 19
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