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finan. Chapter 3: Valuing Firm Output and Pricing Securities How do you assign values to investments and opportunities and how do you compare them? . Valuation Issues 3- 2. Identifying the stream of future benefits

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  1. finan Chapter 3: Valuing Firm Output and Pricing Securities How do you assign values to investments and opportunities and how do you compare them?

  2. Valuation Issues 3-2 • Identifying the stream of future benefits • Valuing them at the owner’s opportunity cost. • Compounding interest to show future amounts at the opportunity cost. • Discounting future sums to present value using opportunity cost. • Discounting a single future period. • Discounting for a period several years removed • Discounting future streams - annuities • Application - valuing corporate bonds: Annuity plus a final payment • Valuing Perpetuities (stock) • Valuing earnings with growth • Valuing Projects • Determining the weighted cost of capital for a firm. • Determining Net Present Value using the cost of capital.

  3. Compoundingat 10%3-3 • Year 1:$1.00 x 1.1 = $1.10; • Year 2$1.10 x 1.1 = $1.21; • Year 3:$1.21 x 1.1 = $1.33; • Year 4:$1.33 x 1.1 = $1.46; • Year 5:$1.46 x 1.1 = $1.61. • Formula: • V = s(1+r) • Where: • V = future value • s = a sum to be received at the end of a period • r = interest rate • Calculator: enter 1+r in memory • Enter sum; “x”, MR, =. Repeat as necessary for additional periods.

  4. Table 3-2 Future Values of $1.00 Invested Today, 3-4Compounded Annually • Interest Rate, r_________________ ____________________ • Number of 2% 4% 6% 8% 10% 12% 14% • Years, n • 1 1.0200 1.0400 1.0600 1.0800 1.1000 1.1200 1.1400 • 2 1.0404 1.0816 1.1236 1.1664 1.2100 1.2544 1.2996 • 3 1.0612 1.1249 1.1910 1.2597 1.3310 1.4049 1.4815 • 4 1.0824 1.1699 1.2625 1.3605 1.4641 1.5735 1.6890 • 5 1.1041 1.2167 1.3382 1.4693 1.6105 1.7623 1.9254 • 6 1.1262 1.2653 1.4185 1.5869 1.7716 1.9738 2.1950 • 7 1.1487 1.3159 1.5036 1.7138 1.9487 2.2107 2.5023 • 8 1.1717 1.3686 1.5938 1.8509 2.1436 2.4760 2.8526 • 9 1.1951 1.4233 1.6895 1.9990 2.3579 2.7731 3.2519 • 10 1.2190 1.4802 1.7908 2.1589 2.5937 3.1058 3.7072 • 15 1.3459 1.8009 2.3966 3.1722 4.1722 4.4736 7.1379 • 20 1.4859 2.1911 3.2071 4.6610 6.7275 9.6463 13.7435 • 25 1.6406 2.6658 4.2919 6.8485 10.8347 17.0001 26.4619 • 30 1.8114 3.2434 5.7435 10.0627 17.4494 29.9599 50.9502 • 35 1.9999 3.9461 7.6861 14.7853 28.1024 52.7996 98.1002 • 40 2.2080 4.8010 10.2857 21.7245 45.2593 93.0510 188.8835 • 50 2.6916 7.1067 18.4202 46.9016 117.3909 289.0022 700.2330

  5. Quick Check Question 3.13-5 • How long will it take for your money to double at 8% • Use rule of 72 72 = 9 years 8 Check Table 3-2: 8% column at 9 years = 1.999

  6. Quick Check Question 3.13-6 • How long will it take for your money to double at 6% • Use rule of 72 72 = 12 years 6 Check Table 3-2: 6% column at 12 years = 2.012

  7. Quick Check Question 3.23- 7 • Franklin invests a penny at 6% in 1750. What is value in 2000? • Relevant Period: 250 years • Interest Rate: 6% • Formula: V = $0.01 (1 + .06)250 • Use Table 3-2 for 50 years @ 6% = $0.18422 • Calculation: V = $0.18422 • V = $0.01 ($0.18422)5 = $21,216. • Alternative Means: • If your money doubles according to the Rule of 72s, at 6% interest, it will double every 12 years. • Thus, in 252 years it would double 21 times (252/12 = 21). • This can be expressed as 1(2)21 = $20,971.

  8. Compounding Quarterly vs. Annually3- 8 • Quarterly Annual • Compounding -10% Compounding -10.38% • Beginning Balance $1,000.00 $1,000.00 • Quarter 1: • Interest: .025 x $1,000 25.00 0.00 • Ending Balance 1,025.00 1,000.00 • Quarter 2: • Interest: .025 x $1,025.00 25.62 0.00 • Ending Balance 1,050.62 1,000.00 • Quarter 3: • Interest: .025 x $1,050.62 26.27 0.00 • Ending Balance 1,076.89 1,000.00 • Quarter 4: • Interest: 0.25 x $1,076.89 26.91 • .1038 x $1,000.000 ________ 103.80 • Ending Balance $1,103.80 $1,103.80

  9. Quick Check Question 3.33-9 • If interest is earned monthly what is your effective annual interest rate? • If interest compounds monthly, you earn 1/12 of 10%. • .10 = .0083, or .83% monthly 12 • Effective annual rate = (1 + .10 )12 - 1 • 12 • Thus: (1.0083)12 -1 = 1.104669 - 1 = $104.67 interest • Effective annual rate of 10.46%.

  10. Basic Discounting to Present Value3- 10 • PV = Sum_____ 1 + interest rate • Where PV = present value, S = sum, and r = interest rate: • PV = 1.00 1 + r • Thus, where the interest rate is 10%: • PV = $1.00 = $0.909 1.10 • To test: what would $0.909 be worth @ 10% in one year? • $0.909 x 1 + r = $0.909 x 1.1 = $0.99999 • To discount to one more year: • PV = $0.909 = $0.826 1.10 • Usually expressed as: PV = S Or more generally PV = S _ (1+r)2 (1+r)n • Where n = number of periods

  11. Table 3-4 Present Values of $1.00 3-11 • Interest Rate, r _________________________________ • Number of 2% 4% 6% 8% 10% 12% 14% Years, n • 1 0.9804 0.9615 0.9434 0.9259 0.9091 0.8929 0.8772 • 2 0.9612 0.9246 0.8900 0.8573 0.8264 0.7972 0.7695 • 3 0.9423 0.8890 0.8396 0.7938 0.7513 0.7118 0.6750 • 4 0.9239 0.8548 0.7921 0.7350 0.6830 0.6355 0.5921 • 5 0.9057 0.8219 0.7473 0.6806 0.6209»0.5674 0.5194 • 6 0.8880 0.7903 0.7050 0.5835 0.5645 0.5066 0.4556 • 7 0.8706 0.7599 0.6651 0.5835 0.5132 0.4523 0.3996 • 8 0.8535 0.7307 0.6274 0.5403 0.4665 0.4039 0.3506 • 9 0.8368 0.7026 0.5919 0.5002 0.4241 0.3606 0.3075 • 10 0.8203 0.6756 0.5584 0.4632 0.3855 0.3220 0.2697 • 15 0.7430 0.5553 0.4173 0.3152 0.2394 0.1827 0.1401 • 20 0.6730 0.4564 0.3118 0.2145 0.1486 0.1037 0.0728 • 30 0.5521 0.3083 0.1741 0.0994 0.0573 0.0334 0.0196 • 35 0.5000 0.2534 0.1301 0.0676 0.0356 0.0189 0.0102 • 40 0.4529 0.2083 0.0972 0.0460 0.0221 0.0107 0.0053 • 50 0.3715 0.1407 0.0543 0.0213 0.0085 0.0035 0.0014

  12. Quick Check Question 3.43-12 • Brother gets a $10,000 bond maturing in 8 years and says he received $10,000. If the rate on 8 year government bonds is currently 7%. What is his bond really worth today?

  13. Quick Check Question 3.43-13 • Discounting a $10,000 savings bond due in 8 years @ 7% • Solution: PV = $10,000 (1 + .07)8 • PV = $10,000 = $5,820.3829 1.7181

  14. Quick Check Question 3.53-14 Law School receives a pledge of $1,000,000.00 bequest from a person with a 20 year life expectancy. How should it report the gift?

  15. Quick Check Question 3.53-15 Two choices $1,000,000 Or PV of $1M paid in 20 years. According to Table 3-4 at 7% value is $258,000 Which way should you report it?

  16. Calculating the Discounted Present Value of a Bond3-16 • $1,000 principle amount 10 year bond paying 10% interest a year. What is its current present value? Bond terminology: • Principle is the amount paid by the issuer to the holder at maturity. For bonds without original issue discount this is the same amount paid for the bond at issue. • Interest the payment made each year calculated as a percentage of the principle amount • Bond interest is usually paid semiannually but for simplicity we will assume annual. Interest payment date often called coupon date

  17. Calculating the Discounted Present Value of a Bond3-17 • Year 1: $100 ÷ 1.1000 = $90.91 • Year 2: $100 ÷ 1.2100 = $82.64 • Year 3: $100 ÷ 1.3310 = $75.13 • Year 4: $100 ÷ 1.4641 = $68.30 • Year 5: $100 ÷ 1.6105 = $62.09 • Year 6: $100 ÷ 1.7716 = $56.45 • Year 7: $100 ÷ 1.9487 = $51.32 • Year 8: $100 ÷ 2.1436 = $46.65 • 9 years $100 ÷ 2.3579 = $42.41 • 10 years $100 ÷ 2.5937 = $38.55 • Total of interest payments: 614.45 • Principal $1,000 ÷ 2.5937 385.55 • Discounted Present Value $1,000.00

  18. Table 3-5 Present Value of an Annuity3-18 • Present Value of an Annuity Payable at the End of Each Period for n Periods • Discount Rate, r No. Yrs. 2% 4% 6% 8% 10% 12% 14% • 1 0.9804 0.9615 0.9434 0.9259 0.9091 0.8929 0.8772 • 2 1.9416 1.8861 1.8834 1.7833 1.7355 1.6901 1.6467 • 3 2.8839 2.7751 2.6730 2.5771 2.4869 2.4018 2.3216 • 4 3.8077 3.6299 3.4651 3.3121 3.1699 3.0373 2.9137 • 5 4.7135 4.4518 4.2124 3.9927 3.7908 3.6048 3.4331 • 6 5.6014 5.2421 4.9173 4.6229 4.3553 4.1114 3.8887 • 7 6.4720 6.0021 5.5824 5.2064 4.8684 4.5638 4.2883 • 8 7.3255 6.7327 6.2098 5.7466 5.3349 4.9676 4.6389 • 9 8.1622 7.4353 6.8017 6.2469 5.7590 5.3282 4.9464 • 10 8.9826 8.1109 7.3601 6.7101 6.1446» 5.6502 5.2161 • 15 12.8493 11.1184 9.7122 8.5595 7.6061 6.8109 6.1422 • 20 16.3514 13.5903 11.4699 9.8181 8.5136 7.4694 6.6231 • 30 22.3965 17.2920 13.7648 11.2578 9.4269 8.0552 7.0027 • 35 24.9986 18.6646 14.4982 11.6546 9.6442 8.1755 7.0700 • 40 27.3555 19.7928 15.0463 11.9246 9.7791 8.2438 7.1050 • 50 31.4236 21.4822 15.7619 12.2335 9.9148 8.3045 7.1327

  19. Quick Check Question 3.63-19 • If the market rate on comparable bonds (10 year 10%) drops to 8% what is the present value fo the bond now? The bond pays $100 interest annually & $1,000 at maturity.

  20. Quick Check Question 3.63-20 • The bond pays $100 interest annually & $1,000 at maturity. • 10 annual payments of $100: • Use the annuity table in Table 3-5: • 10 years @ 8% = 6.7101 x $100 = $671.01 • The principal payment is a lump sum after 10 years, discounted at 8% in Table 3-3: • $1,000 x .4632 = $463.10 • Value of the bond in today's market: $1,134.11

  21. Quick Check Question 3.63-21 • As interest rates fall prices (present value) on issued bonds go up • As interest rates rise prices (present value) falls. • Why?

  22. Value of a Perpetuity3-22 • Present Value of a Perpetuity = Payment = P • Discount Rate r PV = 1 = $10.00 .10

  23. Valuing a Perpetuity Intended to be Sold 3-23 • Assume collection of dividend & sale at close of one year: • PV of Dividend: = $1.00 = $1.00 = $0.909 1 + r 1 + .10 • Value of the sale:= $10.00 = $10.00 = $9.090 1 + r 1 + .10 • Total: $9.99999

  24. Valuing a Perpetuity Intended to be Sold 3-24 • Thus if dividends held constant and discount rate constant stock prices would never change. • If dividends held constant price of stocks would be related to changes in doscount rate applied • In the real world both are variables.

  25. Quick Check Question 3.73-25 • What is the value of a share of preferred stock carrying an $8.00 annual dividend, discounted at 7%, assuming it is neither redeemable by the company ("callable") nor subject to forced redemption by the holder?

  26. Quick Check Question 3.73-26 Discount Rate of 7% PV = $8.00 = $114.29 .07 Discount rate of 8%, PV = $8.00 = $100.00 .08 Discount rate of 10% PV = $8.00 = $80.00 .10

  27. Price – Earnings Multiples 3-27 • P-E multiples based on (1) current market price & (2) last 4 quarters’ net income • Regularly reported in financial press, e.g., Wall St. Journal:

  28. Price – Earnings Multiples 3-28 YTD 52 Week Yld Vol Net Hi Lo Stock (Sym) Div % PE 100s Last 4.2 57.91 42.90 CocaCola KO .80 1.8 27 39470 45.67 (for Friday, Jan. 17, 2003)

  29. Price – Earnings Multiples 3-29 (for Friday, Jan. 17, 2003) Problems with P/E: • Uses historical earnings, not expected earnings & not cash flows • P/E ratio includes both cap rate and growth rate, bundled (described next) • Seems to assume same cap & growth rate will apply to all future periods (i.e. managers will continue to invest in projects that make the same rate of return). • Based on Net Income, not cash flows.

  30. Valuing Perpetuities with Constant Growth3-30 • The Value of a Perpetuity with Constant Growth is: • PV = P r –g • Where g = constant growth rate

  31. Valuing a Perpetuity with Constant Growth 3-31 • Assumptions: • Earnings (A) = $1.00 per year • Capitalization Rate (r) = .10 • Growth Rate (g) = .04 per year • Calculation: PV = $1.00 = $1.00 = $16.67 .10 -.04 .06 • With no growth PV = $1.00 = $10.00 .10

  32. Valuing a Perpetuity with Initial Growth 3-32 • Most companies do not sustain the same growth rate. There is usually a period of high growth until industry maturity when growth levels off and remains nearly constant.

  33. Valuing a Perpetuity with Initial Growth 3-33 • Assume: Initial Earnings $1.00, growing at 4% for 5 years. • Stable earnings thereafter, discounted @ 10% • Year Dividend x Discount Factor = Present Value ( x 1.04) @ 10% (P / 1.1)n • 1 1.00 0.9091 $0.9091 • 2 1.04 0.8264 0.8595 • 3 1.0816 0.7513 0.8126 • 4 1.1249 0.6830 0.7683 • 5 1.17 0.6209 0.7264 • Subtotal: 4.0759 • 6 Perpetuity of $1.17 = $11.70 x 0.5645 = 6.6046 .10 • Total Present Value: $10.6805

  34. Valuing Investments with Different Timing of 3-34Returns Project A: Project B: • End of Year Return End of Year Return • 1 $200,000 1 $100,000 • 2 150,000 2 100,000 • 3 150,000 3 325,000 • Total: $500,000 Total: $525,000 Discounted present values @ 10%, using Table 3-4: • Project A: Project B: • End of Return x NPV End of Return x NPV Year Year • 1 $200,000 x .9091 $181,920 1 $100,000 x .9091 $90,910 • 2 150,000 x .8264 123,960 2 100,000 x .8264 82,640 • 3 150,000 x .7513 112,695 3 325,000 x .7513 244,172 • Totals: $418,575$417,722

  35. Net Present Value Defined3-35 • PV of Funds to be Received —PV of Funds Invested NPV of Project

  36. Quick Check Question 3.83-36 A factory costs $400,000. You calculate that it will produce net cash after operating expenses of $100,000 in year 1, $200,000 in year 2, and $300,000 in year 3, after which it will shut down with zero salvage value. Calculate its Net present Value.

  37. Quick Check Question 3.83-37 Year Pymnt x Discnt Fctr Prsnt @ 10% Value 1 $100,000 .909 $90,900 2 200,000 .8264 165,280 3 300,000 . 7513 225,390 Total $481,570 • Less: Cost of Capital: (400,000) • Net Present Value: $81,570

  38. Summary3-38 1. Compounding of interest or returns. 2. Discounting future payments to present value. 3. Valuing Annuities (and bonds) 4. Valuing perpetuities. • Valuing perpetuities with growth. • Valuing perpetuities with changing growth. 6. Testing the present value of projects.

  39. Equations3-39 • 1 .Compounding of interest or returns. V = P(1+r)n • 2. Discounting future payments to present value. PV = P__ (1+r)n • 3. Valuing Annuities PV = P - P__ r r (1 + r)n • 4. Valuing Perpetuities PV = P r • 5. Valuing Perpetuities with Growth PV = P r-g • 7. Net Present Value = PV (income) – PV (investments)

  40. Determining the Right Discount Rate 3-40 • So far we have assumed a discount or interest rate. Where does it come from? • It has two parts: Risk Free Rate and Compensation for Risk. • Risk Free Rate: Compensation for delaying other uses of the money. Inflation plus a risk free market rate of return. T-Bill 3.7% could be 3% inflation plus 0.7% return • Compensation for risk?

  41. Table 3-6. Returns to Asset Classes, 1926-19973- 41 • Table 3-6 Returns to Asset Classes • Std. Deviation Risk Premium Nominal Real of over Asset Class Return Return Annual Returns T- Bills • Short-term • Treasury Bills 3.8% 0.7% 3.2% 0% • Intermediate-Term • T- Bonds 5.3% 2.2% 5.7% 1.5% • Long-Term • Treasury Bonds 5.2% 2.1 9.2% 1.4% • Corporate Bonds 5.7% 2.6% 8.7% 1.9% • Large-Co. Stocks 11% 7.9% 20.3% 7.2% • Small-Co. Stocks 12.7% 9.6% 33.9% 8.9%

  42. Dow-Jones Average, May 2000 – 2005 3-42

  43. The Coin Flipping Game3-43 • Outcome Probability Weighted Outcome • Original Bet: $1.00 • $0 0.5 $0 • $2.00 0.5 $1.00 • Expected outcome: 1.0 $1.00 • Outcome Probability Weighted Outcome • Original Bet: $25,000 • $0 0.5 $0 • $50,000 0.5 $25,000 • Expected outcome: 1.0 $25,000

  44. Figure 3-144 • Utility • 120 • 100 • 40 • 0 10,000 35,000 60,000 Wealth

  45. Percentage Gains & Losses in Figure 3-1 3- 45 Money: Start: $35,000 • Win +25,000 = $60,000 - a 71% gain • Lose - 25,000 = $10,000 - a 71% loss • Utility: • Start: 100 • Win: +20 = 120 - a 20% gain • Lose -60 = 40 - a 60% loss

  46. Figure 3-23-46 • Utility • 160 • 100 • 40 • 0 10,000 35,000 75,000 Wealth

  47. Percentage Gains & Losses in Figure 3-2 3- 47 Money: Start: $35,000 • Win +40,000 = $75,000 - a 115% gain • Lose - 25,000 = $10,000 - a 71% loss • Utility: • Start: 100 • Win: +60 = 160 - a 60% gain • Lose -60 = 40 - a 60% loss

  48. Figure 3-3 – Outcome Probabilities3- 48 • Firm A Firm B • Probability Probability • 1.0 1.0 • .9 .9 • .8 .8 • .7 .7 • .6 .6 • .5 .5 • .4 .4 • .3 .3 • .2 .2 • .1 .1 • 0 50 100 150 200 250 Firm Value 0 50 100 150 200 250

  49. Expected Values3-49 • Firm A Firm B • Outcome Probability Product Outcome Probability Product • $0 0 0 0 .1 $ 0 • $ 50 .1 $ 5 $ 50 .2 $10 • $100 .8 $80 $100 .4 $40 • $150 .1 $15 $150 .2 $30 • $200 0 0 $200 .1 $20 • Total $100 Total $100

  50. Table 3-73-50 • Firm A • Deviation Deviation Probability times • Outcome x Prob. from Mean Squared Deviation Squared • 0 0 0 0 0 • 50 .1 -50 2,500 250 • 100 .8 0 0 0 • 150 .1 +50 2,500 250 • Variance 500 • Firm B • Deviation Deviation Probability times • Outcome x Prob. from Mean Squared Deviation Squared • 0 .1 -100 10,000 1,000 • 50 .2 -50 2,500 500 • 100 .4 0 0 0 • 150 .2 +50 2,500 500 • 200 .1 +100 10,000 1,000 • Variance 3,000

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