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TIME VALUE OF MONEY

TIME VALUE OF MONEY. Compounding & Discounting Earlier value…later value Implicit rate Implicit time Annuities PVA--Credit card problem FVA--Sinking fund problem Effective rates EAR and APR The “RIPOV FURNITURE” case. Generalization. Formula with factor. Example.

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TIME VALUE OF MONEY

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  1. TIME VALUE OF MONEY • Compounding & Discounting • Earlier value…later value • Implicit rate • Implicit time • Annuities • PVA--Credit card problem • FVA--Sinking fund problem • Effective rates • EAR and APR • The “RIPOV FURNITURE” case

  2. Generalization • Formula • with factor

  3. Example • Assume that we can find an investment that returns 20% per annum with semi-annual compounding. How much should we invest today in order to become a millionaire ten years from now?

  4. Solution • Method 1: “Crunch numbers” • PV=FV/(1+i/m)n x m =1,000,000/(1+.2/2)20 • PV=148,643.6 • Method 2: use the table • PV=FV / FVIF(i/m; n x m)=1000,000/ FVIF(10%;20) • PV= 1,000,000 /6.7275 =148,643.6 • Method 3: Use the calculator • FV=1,000,000; I=10%; n=20; COMPUTE PV

  5. Implicit Rate

  6. Example • Suppose an investment offers to triple your money in 2 years (don’t believe it). What rate are you being offered, if interest rates are compounded twice a year?

  7. Solution

  8. Implicit time

  9. Example • You have $430 today. You need $500. If you earn 1 percent per month, how many months will you wait?

  10. Solution

  11. Ordinary Annuities • Present value of an annuity: Credit card payments, mortgage payments… • PAY-OFF A DEBT • Future value of an annuity: Sinking funds, annuity funds, insurance premium… • INVEST TO ACCUMULATE

  12. PV of an annuity: • Formula: • With factor:

  13. EXAMPLE • Suppose that you have maxed out your credit card at $5,000. The APR on the card is 24%; you make monthly payments and interests are compounded monthly. What payment should you make if you want to reimburse everything in 2 years?

  14. Answer • COMPUTATION: • PVA=PMT x PVIFA(n x m; i/m) • PMT=PVA/PVIFA(n x m; i/m) • =5000/PVIFA(24; 2%)=5000/18.9139=$264.36 • CALCULATOR: • PV=5000; I=2%; n=24; COMPUTE PMT

  15. FV of an annuity • Formula: • with factor:

  16. EXAMPLE • How much should you invest each six months in order to receive $1,000,000 in ten years in an investment that is expected to return 20%?

  17. Answer • COMPUTATION: • FVA=PMT x FVIFA(m x n; i/m) • PMT=FVA/ FVIFA(m x n; i/m) • PMT=1,000,000/FVIFA(20;10%) • PMT=1,000,000/57.275=$17,459.6 • CALCULATOR: • FV=1000000; I=10%; n=20 COMPUTE PMT

  18. APR Vs EAR • EAR is the true rate: it includes the compounding effect • EAR=(1+APR/m)m-1 • EXAMPLE: for the monthly payments on the 24% credit card, the EAR is: • EAR=(1+i/m)m-1=(1+24%/12)12-1 • EAR=1.2682-1=26.82%

  19. EXCEL Functions • InsertFunctionFinancial • PV (i/m , nxm , PMT , FV , Type)Calculate PV or PVA • FV (i/m , nxm , PMT , PV , Type)Calculate FV or FVA • PMT (i/m , nxm , PV , FV , Type)Calculate PMT • Rate (nxm , PMT , PV , FV , Type, Guess)Calculate i/m • NPER (i/m , PMT , PV , FV , Type)Calculate n x m

  20. Capital Budgeting • Real Asset Valuation and Profitability • NPV • IRR • MIRR • Payback • Cross-over rates • Capital Budgeting process • Cash flows that matter • WACC • Sensitivity analysis • Incorporating risk in capital budgeting

  21. Real Asset Valuation • Valuation • Measuring Profitability • The good (NPV) • The bad (IRR) • Cross-over rate • The ugly (Payback) • MIRR

  22. Real Asset Valuation • PV(asset)=PV(future cash flows from asset) • 3 elements: • CF=cash flow • Maturity=n • Interest rate=RAverage cost of moneyCost of capital? • What are the determinants of the firm’s value? • What would the firm’s value be if it had a perpetual cash flow? • Can the firm get value from other factors?

  23. The good: Net Present Value • Formula: • Where I/O is the initial outlay • It measures the $ profitability, taking into account time value of money and risk. It is often referred to as the “extra” $ available to the owners…any comments? • It assumes that cash flows are reinvested at R.

  24. NPV Calculation R=10% A B Year CF CF 0 -350 -250 1 50 125 2 100 100 3 150 75 4 200 50

  25. NPV Calculation • For A: NPV(A)=27.4 • For B: NPV(B)=36.78

  26. The Bad: Internal rate of return • IRR is the minimum return (yield) on a real investment so that the present value of the future cash flows is equal to the I/O--It is the (break-even) rate that sets NPV equal to zero. • IRR=Additional cents on the $ invested • It assumes that CFs are reinvested at IRR • It might include several (irrelevant) solutions • It might provide contradictory results with NPV

  27. IRR Calculation R=10% A B A-B Year CF CF CF 0 -350 -250 -100 1 50 125 -75 2 100 100 0 3 150 75 75 4 200 50 150 IRR 12.91% 17.80% 8.1%???

  28. NPV vs. IRR • NPV and IRR will generally give us the same decision • Exceptions • Non-conventional cash flows – cash flow signs change more than once • Mutually exclusive projects • Initial investments are substantially different • Timing of cash flows is substantially different

  29. Another Example – Non-conventional Cash Flows • Suppose an investment will cost $90,000 initially and will generate the following cash flows: • Year 1: 132,000 • Year 2: 100,000 • Year 3: -150,000 • The required return is 15%. • Should we accept or reject the project?

  30. NPV Profile IRR = 10.11% and 42.66%

  31. Summary of Decision Rules • The NPV is positive at a required return of 15%, so you should Accept • If you use the financial calculator, you would get an IRR of 10.11% which would tell you to Reject • You need to recognize that there are non-conventional cash flows and look at the NPV profile

  32. IRR and Mutually Exclusive Projects • Mutually exclusive projects • If you choose one, you can’t choose the other • Example: You can choose to attend graduate school at either Harvard or Stanford, but not both • Intuitively you would use the following decision rules: • NPV – choose the project with the higher NPV • IRR – choose the project with the higher IRR

  33. Example With Mutually Exclusive Projects The required return for both projects is 10%. Which project should you accept and why?

  34. NPV Profiles IRR for A = 19.43% IRR for B = 22.17% Crossover Point = 11.8%

  35. Conflicts Between NPV and IRR • NPV directly measures the increase in value to the firm • Whenever there is a conflict between NPV and another decision rule, you should always use NPV • IRR is unreliable in the following situations • Non-conventional cash flows • Mutually exclusive projects

  36. Summary – Discounted Cash Flow Criteria • Net present value • Difference between market value and cost • Take the project if the NPV is positive • Has no serious problems • Preferred decision criterion • Internal rate of return • Discount rate that makes NPV = 0 • Take the project if the IRR is greater than the required return • Same decision as NPV with conventional cash flows • IRR is unreliable with non-conventional cash flows or mutually exclusive projects • Payback period • Length of time until initial investment is recovered • Take the project if it pays back within some specified period • Doesn’t account for time value of money and there is an arbitrary cutoff period

  37. A Better Method: MIRR • Assume that Cash Flows are reinvested at the opportunity cost rate. • Bring all positive cash flows to the future=FV(Positive cash flows) • Bring all negative cash flows to the present =PV(Negative cash flows) • Then, • FV(Positive cash flows)= PV(Negative cash flows) x FVIF(n, MIRR)

  38. Example: MIRR • For Project A • Do Project B… R=10% A B Year CF CF 0 -350 -250 1 50 125 2 100 100 3 150 75 4 200 50

  39. The Ugly: Payback • Payback: length of time until the sum of an investment’s cash flows equals its cost. Year CF Cumulated CF 1 200 200 2 400 600 3 600 1200 I/O=$1,000 Payback=2 year + 400/600=2 2/3 year • No time value • No risk • Focuses on liquidity; thus, biased against long term projects • What is the most common measure of profitability in corporate America?

  40. Payback Calculation R=10% A B Year CF CF 0 -350 -250 1 50 125 2 100 100 3 150 75 4 200 50 Payback 3.25 years 2.33 years

  41. Capital Budgeting • Capital budgeting • Cash flow • Start form nothing=CFA • Expand or Replace=ΔCFA • Cost of capital

  42. Cash Flows That Matters... • Stand-alone principle: • Cash flow that matters in a new project: Cash flow from assets • Cash flow that matters in a replacement or expansion project: Incremental Cash flow from assets • Also,

  43. Incremental Cash Flow Analysis (case of replacement or expansion Project) Δ revenues + Δ costs (“-” for an increase in costs, “+” for savings in costs) + Δ Depreciation (“+” for an increase in DPR, “-” for a decrease in DPR) + Δ taxes (“-” for an increase in taxes, “+” for savings in taxes) + Δ NWC sp.(“-” for an increase in NWC sp., “+” for a decrease in NWC sp.) + Δ Fixed Assets spending (“-” for an increase in FA sp., “+” for a decrease in FA sp.) --------------------------------------- Incremental (Δ )Cash flow from assets

  44. Costs that matter…or not • Sunk costs (R&D, consulting fee) • Opportunity cost and externalities: cost of using a rented vs. own building space (opportunity cost: you could lease/rent it for a certain amount of dollar) • NWC: it is recovered at the end (2 techniques) • Terminal value (the value at the end…) • Initial outlay • Financing costs • Are they included in “cash flow from assets”? • Would you consider them in evaluating the profitability of a project? Why? How?

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