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# Лекц 2

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Present Value of Money

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• e is a transcendental number because it transcends the real numbers.
• ### Лекц 2

1. 1. Present Value of Money
2. 2. 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).
3. 3. The One-Period Case: Future Value  In the one-period case, the formula for FV can be written as: FV = C0×(1 + r) Where C0 is cash flow today (time zero) and r is the appropriate interest rate.
4. 4. The One-Period Case: Present Value  If you were to be promised \$10,000 due in one year when interest rates are at 5percent, your investment would be worth \$9,523.81 in today’s dollars. \$10,000 \$9,523.81 = 1.05 The amount that you would need to set aside today to to able to meet the promised payment of \$10,000 in one year is called the Present Value (PV) of \$10,000. Note that \$10,000 = \$9,523.81×(1.05).
5. 5. The One-Period Case: Present Value  In the one-period case, the formula for PV can be written as: C1 PV = 1+ r Where C1 is cash flow at date 1 and r is the appropriate interest rate.
6. 6. 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. Your interest rate is 5%. Should you buy? \$10,000 NPV = −\$9,500 + 1.05 NPV = −\$9,500 + \$9,523.81 NPV = \$23.81 Yes!
7. 7. The One-Period Case: Net Present Value In the one-period case, the formula for NPV can be written as: NPV = –Cost + PV If we had not undertaken the positive NPV project considered on the last slide, and instead invested our \$9,500 elsewhere at 5-percent, our FV would be less than the \$10,000 the investment promised and we would be unambiguously worse off in FV terms as well: \$9,500×(1.05) = \$9,975 < \$10,000.
8. 8. 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.
9. 9. The Multiperiod Case: Future Value  Suppose that you invested in the initial public offering of a company. The company 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
10. 10. 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 40percent on the original \$1.10 dividend: \$5.92 > \$1.10 + 5×[\$1.10×.40] = \$3.30 This is due to compounding.
11. 11. Future Value and Compounding \$1.10 × (1.40) 5 \$1.10 × (1.40) 4 \$1.10 × (1.40)3 \$1.10 × (1.40) 2 \$1.10 × (1.40) \$1.10 \$1.54 \$2.16 \$3.02 \$4.23 \$5.92 0 1 2 3 4 5
12. 12. 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%? PV 0 \$20,000 1 \$9,943.53 = 2 \$20,000 (1.15) 5 3 4 5
13. 13. The Multi-Period Case: Net Present Value N C1 C2 CN Ct NPV = C0 + + ++ = C0 + ∑ 2 N (1 + r ) (1 + r ) (1 + r ) (1 + r ) t t =1 Ct is the net cash inflow at time t
14. 14. 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? FV = C0 × (1 + r )T \$10,000 = \$5,000 × (1.10)T \$10,000 (1.10) = =2 \$5,000 T ln(1.10)T = ln 2 T ln(1.10) = ln 2 ln 2 0.6931 T= = = 7.27 years ln(1.10) 0.0953
15. 15. What Rate Is Enough? Assume that you will need \$50,000 in 12 years. You have \$5,000 to invest today. What rate of interest must you earn on your investment to get \$50,000 in 12 years? FV = C0 × (1 + r )T \$50,000 (1 + r ) = = 10 \$5,000 12 \$50,000 = \$5,000 × (1 + r )12 (1 + r ) = 101 12 r = 101 12 − 1 = 1.2115 − 1 = .2115 About 21.15%.
16. 16. Compounding Periods Compounding an investment m times a year for T years provides for future value of m×T wealth: r   FV = C0 × 1 +   m For example, if you invest \$50 for 3 years at 12% compounded semiannually, your investment will grow to  .12  FV = \$50 × 1 +  2   2×3 = \$50 × (1.06) 6 = \$70.93
17. 17. 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?  .12  FV = \$50 × 1 +  2   2×3 = \$50 × (1.06) = \$70.93 6 The Effective Annual Interest Rate (EAR) is the annual rate, compounded annually, that would give us the same end-of-investment wealth after 3 years: \$50 × (1 + EAR) 3 = \$70.93
18. 18. Effective Annual Interest Rates (continued) FV = \$50 × (1 + EAR) = \$70.93 \$70.93 3 (1 + EAR) = \$50 3 13  \$70.93  EAR =   − 1 = .1236  \$50  So, investing at 12.36% compounded annually is the same as investing at 12% compounded semiannually.
19. 19. Effective Annual Interest Rates (continued)  Find the Effective Annual Rate (EAR) of an 18% annual interest rate loan that is compounded monthly.  What we have is a loan with a monthly interest rate of 1½ percent.  This is equivalent to a loan with an annual interest rate of 19.56 percent r  1 +   m n×m 12  .18  12 = 1 +  = (1.015) = 1.19561817  12 
20. 20. Continuous Compounding  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 years over which the cash is invested, and e is a transcendental number approximately equal to 2.718