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Valuation Under Certainty

Valuation Under Certainty. Investors must be concerned with: - Time - Uncertainty First, examine the effects of time for one-period assets. Money has time value. $100 in one year is not as attractive as $100 today.

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Valuation Under Certainty

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  1. Valuation Under Certainty • Investors must be concerned with: - Time - Uncertainty • First, examine the effects of time for one-period assets. • Money has time value. $100 in one year is not as attractive as $100 today. • Rule 1: A dollar today is worth more than a dollar tomorrow, because it can be reinvested to earn more by tomorrow.

  2. Session 1 • Topics to be covered: • Time value of money • Present value, Future value • Interest rates • compounding intervals • Bonds • Arbitrage

  3. Present Value • The value today of money received in the future is called the Present Value • The present value represents the amount of money we would be prepared to pay today for something in the future. • The interest rate, i is the price of credit in financial markets. • Interest rates are also known as discount rates.

  4. Present Value • The Present Value Factor or Discount Factor is the number we multiply by a future cash flow to calculate its present value. • Present Value (PV)=Discount Factor*Future Value(FV) - Discount factor = 1/(1+i) • Example (i=10%) - Discount factor = 1/(1+.10) - The present value of $200 received in 1 year is

  5. Future Value • Alternatively, we may use the interest rate, i, to convert dollars today to their value in the future. • Suppose we borrow $50 today, and must repay this plus 5% interest in one year. Future Value (FV) = Present Value (PV)(1+i) FV = PV(1+i) = 50(1+.05) = $52.50

  6. Bonds • A bond is a promise from the issuer to pay the holder - the principal, or face value, at maturity. - Interest, or coupon payments, at intervals up to maturity. • A $100 face value bond with a coupon rate of 7% pays $7.00 each year in interest, and $100 after a pre-specified length of time, called maturity.

  7. Zero Coupon Bonds • A zero coupon bond has no coupon payments. • The holder only receives the face value of the bond at maturity. • Suppose the interest rate is 10%. A zero coupon bond promises to pay the holder $1 in one year. Its price today is therefore • The discount factor is just the price of a zero coupon bond with a face value of $1.

  8. Net Present Value • The Net Present Value is the present value of the payoffs minus the present value of the costs. • Suppose Treasury Bills yield 10%. • The present value of $110 in one year is • Suppose we could guarantee this payoff by investing in a project that only costs $98 today. • The NPV of this project is

  9. NPV • The formula for calculating the NPV (one-period case) is • Note that C0 is usually negative, a cost or cash outflow. • In the above example, C0 = -98 and C1 = 110.

  10. Rate of Return • The rate of return is the interest rate expected to be earned by an investment. • The rate of return for this project is • We only want to invest in projects that return more than the opportunity cost of capital. • The cost of capital in this case is 10%.

  11. Decision Rules • We know: 1.) This project only costs $98 to guarantee $110 in one year. In “the market”, it costs $100 to buy $110 in one year. 2.) This project returns 12.2%. In the market, our return is only 10%. • This project looks good.

  12. Decision Criteria • We have equivalent decision rules for capital investment (with a ONE-PERIOD investment horizon): - Net Present Value Rule: accept investments that have a positive NPV. - Rate of Return Rule: accept investments that offer a return in excess of their opportunity cost of capital. • These rules are equivalent for one period investments. • These rules are NOT equivalent in more complicated settings.

  13. Example: Market Value • Continue to suppose you can borrow or lend money at 10%. • Assume the price of a one-year zero-coupon bond with a FV of $110 is $98. • The price of this bond is less than its present value. • We may use this example to illustrate the concept of “arbitrage.”

  14. No Arbitrage • Arbitrage is a “free lunch,” a way to make money for sure, with no risk and no net cost. • For example: - Buy something now for a low price and immediately sell it for a higher price. - Buy something now and sell something else such that you have no net cash flows today, but will earn positive net cash flows in the future. • Assets must be priced in financial markets to rule out arbitrage.

  15. Example (cont.) • To arbitrage this opportunity, we • 1.) buy the bond • 2.) borrow $100 for one year. • The cash flows from this strategy today and at the end of one year are: Today One Year Buy the bond -98 +110 Borrow $100 (1 yr) +100 -110 Net cash flow +2 0

  16. Suppose the price of the zero-coupon bond were $102. Our arbitrage strategy would be reversed. - Lend $100 for one year. - Short Sell the zero-coupon bond. The cash flows from this strategy would be Today One Year Sell the bond +102 -110 Lend $100 (1 yr) -100 +110 Net cash flow +2 0 Short Selling

  17. Market Value • As the above example illustrates, the only price for a bond which rules out arbitrage is $100. • $100 is also the present value of the payoff of the bond. • RULE 2: Assets must be priced in the market to rule out arbitrage (i.e., “no arbitrage”) • Therefore, the present value of an asset is its market price.

  18. Compound Interest Vs. Simple Interest • Next we consider assets that last more than one period. • How is multi-period interest paid? • Invest $100 in bonds earning 9% per year for two years: - After one year: $100(1.09) = $109 - Reinvest $109 for the second year: $109(1.09) = 118.81 • We do NOT earn just 9% * 2 = 18% . • We earn “interest on our interest”, or COMPOUND • SIMPLE INTEREST: interest paid only on the initial investment • COMPOUND INTEREST: interest paid on the initial investment and on prior interest.

  19. $100 invested at 10% with no compounding becomes: Example: Simple Interest

  20. Example: Compound Interest • $100 invested at 10% compounded annually becomes:

  21. Compound Interest • A present value $PV invested for n years at an interest rate of i per year grows to a future value • (1 + in)n is the Compound Amount Factor. • Above, the FV of $100 compounded annually at 10% for 3 years is • In principle, the interest rate in may vary with the length of the investment horizon, n. More later . . .

  22. Present Value • We may use the above relation to calculate the present value of an n-period investment, with compound interest: where is the discount factor, or present value factor. • For example, the present value of $100 in 6 years at 10% per year with annual compounding:

  23. Semi-Annual Compounding • So far, we assume cash flows occur at annual intervals. - In Europe, most bonds pay interest annually. - In the U.S., most bonds pay interest semiannually. • A $100 bond pays interest of 10% per year, but payments are semi-annual. - Half of the interest (5% or $50) is paid after 6 months. - Reinvest this $50 for the second 6 months. • By the end of the year, we would have

  24. Example • This return is as if we earned if we had only received our payment at the end of the year. • 10% compounded semiannually is equal to 10.25% compounded annually. - 10% is called the nominal interest rate. - 10.25% is called the effective interest rate.

  25. Example • Suppose you buy $100 of a 7-year Treasury note that pays interest at a nominal rate of 10% per year, compounded semiannually. • Define one period as 6 months: • The interest rate per period is 5%. • There are 14 (6-month) periods until the 7-year maturity. • So, we can use our general formula for future values to compute the value at maturity:

  26. Extending the PV Formula • RECALL: For a project with one cash flow, C1, in one year, • If a project produces one cash flow, C2 after TWO years, then the present value is • If a project produces one cash flow, C1 after one year, and a second cash flow C2 after two years , then

  27. General Present Value • By extension, the present value of an extended stream of cash flows is • This is called the Discounted Cash Flow or Present Value formula: • Similarly, the Net Present Value is given by

  28. Example • Suppose a project will produce $50,000 after 1 year, $10,000 after 2 years, and $210,000 after 4 years. • It costs $200,000 to invest. • We may earn 9% per year (compounded annually) on 1, 2, or 4 year zero-coupon bonds. • The present value of this project is • The NPV of this project is

  29. Net Present Value Rule • In the last example, the PV of payoffs exceeded the PV of the costs, so the project is a good one. • Investment Criterion (The NPV Rule): Accept a project if the NPV is greater than 0. • This criterion is a good general rule for all types of projects. • The NPV Rule can also be used to rank projects; a project with a larger (positive) NPV is better than one with a smaller NPV.

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