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Time Value of Money and NPV

1. The Money Market . Individuals and institutions have different income streams and different intertemporal consumption preferences. Because of this, a market has arisen for money. The price of money is the interest rate.. 2. Security Markets. Individuals and Institutions also have different risk

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Time Value of Money and NPV

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    1. 0 Time Value of Money and NPV We introduce the time value of money This leads to the opportunity cost of capital, Which in turn is used to calculate (net) present value and future value

    2. 1 The Money Market Individuals and institutions have different income streams and different intertemporal consumption preferences. Because of this, a market has arisen for money. The price of money is the interest rate.

    3. 2 Security Markets Individuals and Institutions also have different risk preferences and disagree about the prospects of industries and corporations. This gave rise to security markets, where risky assets are traded.

    4. 3 The Efficient, Competitive Market In a competitive market: Trading is costless. Information is available to all participants There are many traders; no individual can move market prices. In an efficient market, all similar securities promise the same expected return. Otherwise: potential for arbitrage. Financial markets are the most efficient and competitive of all markets.

    5. 4 The Opportunity Cost of Capital I Assume that you as CFO have $100 million available cash. You could… Return the cash to investors, who can trade in financial markets. Buy financial securities such as debt and equity of other companies OR Engage in a “real” investment project.

    6. 5 The Opportunity Cost of Capital II When should you undertake a project? Only if you can achieve a higher return than what you could get for similar financial assets. The rate of return you could obtain from similar financial assets is called the (opportunity) cost of capital.

    7. 6 The Opportunity Cost of Capital III What is a “similar asset”? Similar in Liquidity? Firm size? Firm age and maturity? Yes, all of these, but most important: SIMILAR IN RISKYNESS In the case of debt, the cost of capital equals the interest you pay

    8. 7 Net Present Value We can calculate how much better off in today’s dollar the investment makes us by calculating the Net Present Value:.

    9. 8 Corporate Investment Decision Making In reality, shareholders do not vote on every investment decision faced by a firm and the managers of firms need decision rules to operate by. All shareholders of a firm will be made better off if managers follow the NPV rule—undertake positive NPV projects and reject negative NPV projects.

    10. 9 The “Separation Theorem” The separation theorem in financial markets says that all investors will want to accept or reject the same investment projects by using the NPV rule, regardless of their personal preferences. Logistically, separating investment decision making from the shareholders is a basic requirement of the modern corporation.

    11. 10 Summary Financial markets exist because people want to adjust their consumption over time. They do this by borrowing or lending. An investment should be rejected if a superior alternative exists in the financial markets. If no superior alternative exists in the financial market, an investment has a positive net present value.

    12. 11 In this course… … we will first see how much mileage we get as long as we know the opportunity cost of capital Corporations typically have a good estimate of their cost of capital We will later on talk about how the cost of capital is determined

    13. 12 Let’s go into the details… … and look at some examples of present values and future values We start out simply and then move to multiple periods. You should already be familiar with these – please review the concepts!

    14. 13 The One-Period Case: Future Value In the one-period case, the formula for FV can be written as: FV = C1×(1 + r) Where C1 is cash flow at date 1 and r is the cost of capital.

    15. 14 The One-Period Case: Future Value If you were to invest $10,000 at 5-percent interest for one year, your investment would grow to $10,500 $500 would be interest ($10,000 × .05) $10,000 is the principal repayment ($10,000 × 1) $10,500 is the total due. It can be calculated as: $10,500 = $10,000×(1.05). The total amount due at the end of the investment is called the Future Value (FV).

    16. 15 The One-Period Case: Present Value If you were to be promised $10,000 due in one year when interest rates are at 5-percent, your investment be worth $9,523.81 in today’s dollars.

    17. 16 The One-Period Case: Present Value In the one-period case, the formula for PV can be written as:

    18. 17 The One-Period Case: Net Present Value The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment. Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Cost of Capital is 5%. Should you buy?

    19. 18 The One-Period Case: Net Present Value In the one-period case, the formula for NPV can be written as:

    20. 19 The Multiperiod Case: Future Value The general formula for the future value of an investment over many periods can be written as: FV = C0×(1 + r)T Where C0 is cash flow at date 0, r is the appropriate interest rate, and T is the number of periods over which the cash is invested.

    21. 20 The Multiperiod Case: Future Value Suppose that Jay Ritter invested in the IPO of the Modigliani company. Modigliani pays a current dividend of $1.10, which is expected to grow at 40-percent per year for the next five years. What will the dividend be in five years? FV = C0×(1 + r)T $5.92 = $1.10×(1.40)5

    22. 21 Future Value and Compounding Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40-percent on the original $1.10 dividend: $5.92 > $1.10 + 5×[$1.10×.40] = $3.30 This is due to compounding.

    23. 22 Future Value and Compounding

    24. 23 Present Value and Compounding How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%?

    25. 24 How Long is the Wait? If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?

    26. 25 Assume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child’s education? Answer: 21.15%. What Rate Is Enough?

    27. 26 Compounding Periods Compounding an investment m times a year for T years provides for future value of wealth:

    28. 27 Effective Annual Interest Rates A reasonable question to ask in the above example is what is the effective annual rate of interest on that investment?

    29. 28 Effective Annual Interest Rates (continued) So, investing at 12.36% compounded annually is the same as investing at 12% compounded semiannually.

    30. 29 Continuous Compounding (Advanced) The general formula for the future value of an investment compounded continuously over many periods can be written as: FV = C0×erT Where C0 is cash flow at date 0, r is the stated annual interest rate, T is the number of periods over which the cash is invested, and e is a transcendental number approximately equal to 2.718. ex is a key on your calculator. e is a transcendental number because it transcends the real numbers.e is a transcendental number because it transcends the real numbers.

    31. 30 Simplifications Perpetuity A constant stream of cash flows that lasts forever. Growing perpetuity A stream of cash flows that grows at a constant rate forever. Annuity A stream of constant cash flows that lasts for a fixed number of periods. Growing annuity A stream of cash flows that grows at a constant rate for a fixed number of periods.

    32. 31 A Perpetuity is a… constant stream of cash flows that lasts forever.

    33. 32 Perpetuity: Example What is the value of a century-old British “consol” that promises to pay Ł15 each year forever? The interest rate is 10-percent.

    34. 33 A Growing Perpetuity is a … growing stream of cash flows that lasts forever.

    35. 34 Growing Perpetuity: Example The expected dividend next year is $1.30 and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream?

    36. 35 An Annuity is a … constant stream of cash flows with a fixed maturity.

    37. 36 Annuity: Example If you can afford a $ 400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans?

    38. 37 And finally: A Growing Annuity

    39. 38 Growing Annuity A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by 3-percent each year. What is the present value at retirement if the discount rate is 10-percent?

    40. 39 What Is a Firm Worth? Conceptually, a firm should be worth the present value of the firm’s cash flows. The tricky part is determining the size, timing and risk of those cash flows.

    41. 40 Summary and Conclusions We introduced future value and present value. Interest rates are commonly expressed on an annual basis, but semi-annual, quarterly, monthly and even continuously compounded interest rate arrangements exist. The formula for the net present value of an investment that pays $C for N periods is:

    42. 41 Summary and Conclusions: Four Useful Formulas

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