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Present and Future Value Translating cash flows forward and backward through time

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Future Value Money invested earns interest and interest reinvested earns more interest The power of compounding

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Future Value Problems Solve for any variable, given the other three FV: How much will I have in the future? P: How much do I need to invest now? r: What rate of return do I need to earn? T: How long will it take me to reach my goal?

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Present Value Discounting future cash flows at the “opportunity cost” (cost of capital, discount rate, minimum acceptable return) A dollar tomorrow is worth less than a dollar today

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Present Values can be Added Cash flows further out are discounted more Discount factors are like prices (exchange rates)

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Calculating PV of a Stream (Beware) Calculator assumes first CF you give it occurs now (Time 0) Excel assumes first CF you give it occurs one year from now (Time 1)

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Different Compounding Periods m = # of compounding periods in a year APR = actual rate x m (APR is annualized) EAR = the annually compounded rate that gives the same proceeds as APR compounded m times

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Semiannual Compounding m = 2 APR = 10% EAR = 10.25%

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Quarterly Compounding m = 4 APR = 10% EAR = 10.38%

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Monthly Compounding m = 12 APR = 10% EAR = 10.47%

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Daily Compounding m = 365 APR = 10% EAR = 10.516%

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Continuous Compounding m = APR = 10% EAR = 10.517%

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Annuities All cash flows are the same, so we can factor out the constant payment C and calculate the sum of the discount factors

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Special Case: Perpetuity If all the cash flows are the same each period forever, the sum of the discount factors converges to 1/r

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Perpetuity Example Let C = $100 and r =.05 $100 per year forever at 5% is worth:

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Other Perpetuity Examples British Consol Bonds Canadian Pacific 4% Perpetual Bonds Endowments –How much can I withdraw annually without invading principal?

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Growing Perpetuity Suppose the initial payment C grows at a constant rate g per period (where g < r) This growing stream still has a finite present value:

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Growing Perpetuity Example Suppose the initial payment is $100 and that this grows at 3% per year while the discount rate is 5% The value of this growing perpetuity is:

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Other Growing Perpetuity Examples Stock price = present value of growing dividend stream (see Class #7) M&A: How to estimate terminal value –How fast do earnings grow after the end of the analysis period?

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Finite Annuity= Difference Between Two Perpetuities CCCCCCCC012345678CCCCCCCCCCCC012345678CCCC

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Annuity Example What’s the value of a 4-year annuity with annual payments of $40,000 per year (@5%)?

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Oops, Tuition Payments Due at Beginning of Year

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Other Annuity Applications Lottery winnings Lease & loan contracts Home mortgages Retirement savings/ income

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Home Mortgages 30-year fixed rate mortgage: 360 equal monthly payments Most of early payments goes toward interest; principal repayment gradually accelerates At any point: outstanding balance = present value of remaining payments

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More Annuity Problems Saving, Retirement Planning, Evaluating Loans and Investments

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Net Present Value (NPV) Best criterion for corporate investment: Invest if NPV > 0

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NPV with a Single, Initial Investment Outlay I = initial investment outlay C t = project cash flow in period t r = discount rate (shareholders’ opp. cost) T = project termination period

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Implications of NPV > 0 Project benefits exceed cost (in PV terms) Project is worth more than it costs Project market value exceeds book value Project adds shareholder value

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NPV More Generally Treat inflows as +, outflows as – NPV = PV of all cash flows Investment may occur throughout project life

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Internal Rate of Return IRR sets value of benefits = investment IRR sets NPV = 0 IRR is the rate of return company expects on investment I

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NPV > 0 Implies IRR > r If NPV > 0, IRR must exceed r Investing when NPV > 0 implies company expects to earn more than shareholder’ opp. Cost Equivalent: Invest when NPV > 0 or when IRR>I

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6-1 Time Value of Money Future value Present value Annuities Rates of return Amortization.

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