Chapter 4-6 Time Value of Money Net Present Value Capital Budgeting. Konan Chan Financial Management, Time Value of Money

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1 Chapter 4-6 Time Value of Money Net Present Value Capital Budgeting Konan Chan Financial Management, 2018 Time Value of Money Present values Future values Annuity and Perpetuity APR vs. EAR Five factor framework

2 Present Value & Discounting Value today of a future cash flow Discount rate is used to compute present values of future cash flows C 1 C 2 C T PV Financial Management Konan Chan 3 PV of Multiple Cash Flows Your auto dealer gives you two choices, if your discount rate is 5%, which option do you prefer? pay $15,000 cash now, or make three payments: $8,000 now $4,000 at the end of the following two years Financial Management Konan Chan 4

3 PV of Multiple Cash Flows PV 8, ,000 4, Financial Management Konan Chan 5 Future Value & Compounding Amount to which an investment will grow after earning interest Interest rate is used for compounding C 1 C 2 C T PV FV FV = PV (1+r) T PV = FV / (1+r) T Financial Management Konan Chan 6

4 Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in Was this a good deal? To answer, determine $24 is worth in the year 2016, compounded at 8%. FYI - The value of Manhattan Island is well below this figure. Financial Management Konan Chan 7 Annuities Annuities A series of equal periodical cash flows Ordinary Annuity Each cash flow is at the end of each period Annuity Due Each cash flow is at the beginning of each period Financial Management Konan Chan 8

5 PV of Annuities Tiborn-autos offers payments of $5,000 per year (at the end of each year) for 5 years. If interest rate is 7% per year, what is the cost of the car? Financial Management Konan Chan 9 PV and FV of Annuities Annuity factor Present value of a n-year annuity of $1 PV of ordinary annuity = C * AF PV of annuity due = C * AF * (1+r) FV of ordinary annuity = C * AF * (1+r) n FV of annuity due= C * AF * (1+r) n * (1+r) Financial Management Konan Chan 10

6 Perpetuity Perpetuities A stream of cash payments that never ends Annuity that goes on forever PV of annuity PV of Perpetuity = C / r Financial Management Konan Chan 11 PV of Perpetuity You want to create an endowment to fund a football scholarship, which pays $15,000 per year, forever, how much money must be set aside today if the rate of interest is 5%? PV = 15,000 / 0.05 = 300,000 Financial Management Konan Chan 12

7 Effective Interest Rates Annual Percentage Rate (APR) Annualized interest rate based on simple interest Effective Annual Rate (EAR) Annualized interest rate based on compound interest actual rate interest earned/paid APR = periodic rate * m EAR = (1+APR/m) m 1 m is number of compounding periods per year Financial Management Konan Chan 13 Credit Card EAR A credit card charges 18% APR compounded monthly What is the EAR? APR = 18%, and m=12, so periodic rate = 1.5% EAR = (1+1.5%) 12 1 = 19.56% Financial Management Konan Chan 14

8 Five Factors in TVM Present value: PV Future value: FV Discount rate: r Payment: PMT Number of periods: N Get information of four factors, and find the last one Financial Management Konan Chan 15 Texas Instruments BA-II Plus N = number of periods I/Y = period interest rate (r) PV = present value PMT = payment FV = future value N I/Y PV PMT FV Financial Management Konan Chan 16

9 Time Value of Money N: number of compounding periods I/Y: periodic rate (I/Y = APR/m) PV: present value PMT: periodic payment FV: future value N = m*n (m: number of interests paid per year; n: number of years) It s always to determine one of the five variables If you can t figure out one of the four variables, it must be the case that this one is zero Financial Management Konan Chan 17 Annual &Non-annual Compound Financial Management Konan Chan 18

10 Excel Functions FV(Rate,Nper,Pmt,PV,0/1) PV(Rate,Nper,Pmt,FV,0/1) RATE(Nper,Pmt,PV,FV,0/1) NPER(Rate,Pmt,PV,FV,0/1) PMT(Rate,Nper,PV,FV,0/1) Inside : (RATE,NPER,PMT,PV,FV,0/1) 0/1 Ordinary annuity = 0 (default; no entry needed) Annuity Due = 1 (must be entered) Financial Management Konan Chan 19 Example 1 You have just taken out a 30-year, $120,000 mortgage on your new home. This mortgage is to be repaid in 360 equal end-of-month installments. If each of the monthly installment is $1,500, what is the effective annual interest rate on this mortgage? Financial Management Konan Chan 20

11 Example 1 Solution N=360, PV = 120,000, PMT = -1,500, FV = 0 So, I = 1.235% (periodic rate) APR = periodic rate*12 = 14.82% EAR = ( ) 12-1 = Financial Management Konan Chan 21 Example 2 Suppose you borrow $3,000 at 4% and you are going to make annual payments of $ How long before you pay off the loan? I=4%, PV=3,000, PMT= , FV=0 N = 5 years Financial Management Konan Chan 22

12 Investment Rules Net Present Value (NPV) Internal Rate of Return (IRR) Net Present Value (NPV) Present value of all expected cash flows of a project at the cost of capital Cost of capital is the expected rate of return given up by investing in a project Cash flows can be positive or negative in any period NPV = PV of future cash flows initial costs Financial Management Konan Chan 24

13 NPV Rule Managers increase shareholders wealth by taking projects that are worth more than they cost Therefore, managers should accept all projects with positive net present values That is, accept the project if NPV > 0 Financial Management Konan Chan 25 NPV Example You plan to purchase an apartment You will lease it out, and the tenant will pay $12,000 per year for three years At the end of three years you anticipate selling it for $500,000. If the apartment is offered for a sale price of $395,000 now, would you buy the apartment? Cost of capital (i.e., discount rate) is 8% Financial Management Konan Chan 26

14 Net Present Value $512,000 $500,000 -$395,000 $12,000 $12,000 $12,000 Present Value 11,111 10, ,442 $427, Financial Management Konan Chan 27 Internal Rate of Return (IRR) Discount rate at which NPV = 0 Project s expected rate of return IRR rule: accept the project Accept if IRR > cost of capital Financial Management Konan Chan 28

15 IRR Example You pay $395,000 to buy an apartment which will generate $12,000 per year for three years, and sell for $500,000 at the end. What is the IRR on this investment? IRR = 10.99% Financial Management Konan Chan NPV vs. IRR NPV (,000s) IRR NPV> Discount rate (%) NPV<0 NPV profile Financial Management Konan Chan 30

16 Payback Method Measures how long to recover a project s cost. Easy to calculate and a good sense of a project s risk and liquidity. Decision Rule: Accept the project if Payback < pre-specified period Cash Flows Project C 0 C 1 C 2 C 3 Payback NPV@10% A B C , Financial Management Konan Chan 31 Problems of Payback Method Ignores time value of money Ignores cash flows beyond payback period Penalize projects with long lives, or projects with huge initial outlays Ignores the timing of cash flows within the payback period Arbitrary standard for payback period Financial Management Konan Chan 32

17 Survey on CFOs Financial Management Konan Chan 33 Capital Budgeting Incremental cash flows Common mistakes Operating cash flows Depreciation Salvage

18 Incremental Cash Flow Discount incremental cash flows Incremental cash flow = cash flow with project - cash flow without project Ask yourself this question, Will this cash flow occur ONLY if we accept the project? If yes, include it in your analysis If no, do not include it Financial Management Konan Chan 35 Avoid Common Mistakes How to trace all incremental cash flows (avoid some common pitfalls) Forget sunk costs Include opportunity costs Include all side effects Recognize the investment in working capital Consider discounted cash flows, not profits Financial Management Konan Chan 36

19 Sunk Costs & Opportunity Cost Sunk costs: costs that cannot be recovered even if a project is rejected Costs that the firm has to pay anyway Completed marketing studies Previous new product development and testing Opportunity cost is the cash flow which is given up as a result of undertaking a project Lease an idle asset: costs of asset? Financial Management Konan Chan 37 Side Effects Include (good or bad) impact that a new project would have on existing company sales and expenses Erosion: the introduction of new project reduces the cash flows of existing projects HTC plans to make a new type of smart phone, but has to consider lost sales on existing product Financial Management Konan Chan 38

20 Investment in Working Capital Working capital = current assets current liabilities Most new projects require additional short term assets and often current liabilities, such as Additional receivables from increased credit sales. Additional inventory necessary to produce new products. Additional accounts payables and taxes and wages payable. Working capital may change over the life the project. Any needed increase in (net) working capital is an outflow of cash, but these outflows are recovered at the end of the project (AR are collected, INV are Financial sold) Management Konan Chan 39 Discount Cash Flows, Not Profits Discount actual cash flows Using accounting income, rather than cash flow, could lead to erroneous decisions. Example: A project costs $2,000 and is expected to last 2 years, producing cash income of $1,500 and $500 respectively. The cost of the project can be depreciated at $1,000 per year. Given a 10% required return, compare the NPV using accounting income to the NPV using cash flows. Financial Management Konan Chan 40

21 Cash Flow vs. Accounting Income Accept the project given positive NPV, is it right? Financial Management Konan Chan 41 Cash Flow vs. Accounting Income Reject the project due to negative NPV Financial Management Konan Chan 42

22 Calculating Cash Flow Total cash flow = Operating cash flow (OCF) Changes in working capital Net capital spending Recall all these are incremental cash flows Financial Management Konan Chan 43 How to Compute OCF? Assume no interest expenses Costs mean the expenses excluding depreciation Top-down approach: OCF = Sales Costs Taxes Bottom-up approach: OCF = Net income + Depreciation Tax shield approach: OCF = (Sales Costs) * (1 T) + Depreciation * T T is the corporate tax rate Tax shield = Dep * T, savings of tax due to dep. Financial Management Konan Chan 44

23 Depreciation Depreciation is a non-cash expense. So, it is only relevant because it affects taxes Straight-line depreciation Dep = (Initial cost ending book value) / number of years MACRS Modified Accelerated Cost Recovery System Assets are classified as different class (3-yr, 5-yr, 7-yr) for tax purposes Multiply percentage given in table by the initial cost Depreciate to zero Financial Management Konan Chan 45 After-tax Salvage At the end of the project, We can sell the equipment (or fixed assets) End book value = initial cost all Dep. If ending market (salvage) value is different from the book value, then there is a tax effect After-tax salvage = Sale (Sale Book)*T If sale>book, too much dep. and too few tax before If sale<book, too few dep. and too much tax before Financial Management Konan Chan 46

24 Example: Sun Blocker A utility company plans to build Sun Blocker, which will cost $80 million today (t = 0). The estimated useful life for Sun Blocker is 3 years with no market value at the end of year 3. Without Sun Blocker, the company expects revenues of $200 million and operating costs of $80 million. With Sun Blocker, the company estimates annual cash revenues of $300 million and operating costs of $100 million respectively for the next 3 years. The firm s tax rate is 40%, and cost of capital is 11%. Financial Management Konan Chan 47 Stage 1: Incremental Cash Flow From year 1 to 3 Incremental Revenue = 100 million Incremental Cost = 20 million Tax rate = 40%, discount rate = 11% For year 1 to 3, incremental cash flows include Incremental Revenues100mil Incremental Cost (20mil) Incremental EBT 80mil Tax (40%) 32mil Net income 48mil Financial Management Konan Chan 48

25 Stage 2: Depreciation Based on tax code, Sun-Blocker can be depreciated to zero using 4-year straight-line depreciation With cost = 80 mil, ending book value=0 Annual depreciation = (80-0)/4 = 20 OCF = 48 mil + 20 * 40% = 56 mil, for year 1-3 At year 3 end (since project horizon is 3 years) Book value = 80-20*3 = 20mil After-tax salvage = 0- (0-20)*40% = 8 Financial Management Konan Chan 49 Non-Operating Cash Flow Project-end non-operating cash flow After-tax salvage Recovery of additional working capital NOCF = Sale (Sale Book) * tax rate + Recovery of WC Financial Management Konan Chan 50

26 Stage 3: Working Capital Change With the setup of Sun Blocker, the firm will need an increase in working capital at the beginning of the project, which is $10 million. This increased investment in working capital can be sold at the end of the project s life. Cost = 80 mil, NWC = 10 mil, C0 = -90 mil C1 = C2 = 56 mil C3 = OCF + after-tax salvage + recover of NWC = 56 mil + 8 mil + 10mil = 74mil NPV = 60 mil, IRR = 44.59% Financial Management Konan Chan 51 Comprehensive Problem A $1,000,000 investment is depreciated using a seven-year MACRS class life. It requires $150,000 in additional inventory and will increase accounts payable by $50,000. It will generate $400,000 in revenue and $150,000 in cash expenses annually, and the tax rate is 40%. What is the incremental cash flow in years 0, 1, 7, and 8? Financial Management Konan Chan 52

27 Annual depreciation expense (based on MACRS): Year 1:.1429 x $1million = $142,900 Year 7:.0893 x $1million = $89,300 Year 8:.0446 x $1million = $44,600 Cash flow NWC = $150,000 - $50,000 = $ 100,000 Year 0 = - ($1million +$100,000) = -$1,100,000 Year 1 = $400,000 - $150,000 ($400,000 - $150, ,900) x.4 = $207,160 Year 7 = $400,000 - $150,000 ($400,000 - $150,000-89,300) x.4 = $185,720 Year 8 = $400,000 - $150,000 ($400,000 - $150,000-44,600) x.4 = $267,840 Financial Management Konan Chan 53 Example: Cash Flow Projection Financial Management Konan Chan 54

28 Example: Computing Cash Flow Financial Management Konan Chan 55 Example: Calculate NPV Financial Management Konan Chan 56