1. Lecture 9 Discounting and Valuation

2. The investment guru Financial Markets and Corporate Strategy, David Hillier Average rate of return on investments is more than 20 percent The purchase of Coca-Cola in 1988 netted him an 800 percent return over twelve years Warren Buffett only invests in firms whose intrinsic value is greater than their cost to purchase

3. Real investments Financial Markets and Corporate Strategy, David Hillier Present value (PV) Cash flows Discount rate Definitions: Present Value: the market price of a portfolio of traded securities that tracks the future cash flows of the proposed project Discount rate: the rate of return that discounts future cash flows to the present.

4. Unlevered Cash Flows Financial Markets and Corporate Strategy, David Hillier Unlevered cash flows Financing cash flows EBIT - earnings before interest and taxes Deriving unlevered cash flows from the accounting cash flow statement Definitions: Unlevered cash flows: cash flows generated directly from the real assets of the project or firm Financing cash flows: associated with (1) issuance or retirement of debt and equity, (2) interest or dividend payments, and (3) any interest-based tax deductions that stem from debt financing.

5. Unlevered Cash Flows - Result 9.1 Financial Markets and Corporate Strategy, David Hillier Unlevered cash flow = operating cash inflow + investing cash inflow (which is usually negative) + debt interest  debt interest tax subsidy Under U.S. GAAP Unlevered cash flow = net cash flow from operating activities + net cash flow from investing activities (which is usually negative) - debt interest tax subsidy Under IFRS

6. Deriving Unlevered Cash Flow from the Income Statement Financial Markets and Corporate Strategy, David Hillier Result 9.2 Unlevered cash flow = Profit before Interest and Taxes + depreciation and amortization  change in working capital  capital expenditures + sales of capital assets  realized capital gains + realized capital losses  Profit before Interest and Taxes  tax rate

7. Exhibit 9.1 Earnings and Cash Flows for Bayer Healthcare AG Financial Markets and Corporate Strategy, David Hillier Year PBIT (a) Depreciatio n (b) PBITDA Increase in Working Capital (d) Pretax Cash Flow (e) = (c) - (d) Taxes (at 40%) (f) = (a)  (0.4) Unlevered Cash Flow (g) = (e) - (f) 1€10,000€100,000€110,000€10,000€100,000€4,000€96,000 210,000100,000110,00010,000100,0004,00096,000 310,000100,000110,00010,000100,0004,00096,000 410,000100,000110,00010,000100,0004,00096,000 510,000100,000110,00010,000100,0004,00096,000 6110,0000 10,000100,00044,00056,000 7110,0000 10,000100,00044,00056,000 8110,0000 10,000100,00044,00056,000

8. Example 9.3: Incremental Cash Flows: Cash Flow Differences in Two Scenarios Financial Markets and Corporate Strategy, David Hillier Flyaway Air is thinking of acquiring a fleet of new fuel-saving jets. The airline will have the following cash flows if it does not acquire the jets: If it does acquire the jets, its cash flows will be What are the incremental cash flows of the project? Answer: The incremental cash flows of the project are given by the difference between the two sets of cash flows: Cash Flows (in € millions) at Date 0123 100140120100 Cash Flows (in € millions) at Date 0123 80180110130 Cash Flows (in € millions) at Date 0123  20 40  10 30

9. Exhibit 9.4 The Value of an Investment over Multiple Periods When Interest (Profit) Is Reinvested Financial Markets and Corporate Strategy, David Hillier The Future Value (Equation 9.2) and Present value (Equation 9.3):

10. Example 9.4 and 9.5 Financial Markets and Corporate Strategy, David Hillier Example 9.4: Computing the Time to Double Your Money How many periods will it take your money to double if the rate of return per period is 4 percent? Answer: Using equation (9.2), find the t that solves which is solved by t = ln (2)/ln(1.04) = 17.673 (periods). Example 9.5: Determining the Yield on a Zero-Coupon Bond Compute the per period yield of a zero-coupon bond with a face value of €100 at date 20 and a current price of €45. Answer: Using the formula presented in equation (9.4): or about 4.07 percent per period.

11. Value Additivity and Present Values of Cash Flow Streams Financial Markets and Corporate Strategy, David Hillier Result 9.4 Let C1, C2,..., CT denote cash flows at dates 1, 2,..., T, respectively. The present value of this cash flow stream (9.5) if for all horizons the discount rate is r. Example 9.6: Determining the Present Value of a Cash Flow Stream Compute the present value of the unlevered cash flows of the MRI of Bayer Healthcare AG, computed in Exhibit 9.1. Recall that these cash flows were €96,000 at the end of each of the first five years and €56,000 at the end of years six to eight. Assume that the discount rate is 10 percent per year. Answer: The present value of the unlevered cash flows of the project is

12. Inflation Financial Markets and Corporate Strategy, David Hillier Nominal discount rates Nominal cash flows Inflation-adjusted cash flows Real cash flows Real discount rates Result 9.5 Discounting nominal cash flows at nominal discount rates or inflation-adjusted cash flows at the appropriately computed real interest rates generates the same present value.

13. Annuities and Perpetuities Financial Markets and Corporate Strategy, David Hillier Perpetuities: Example 9.7: The Value of a Perpetuity In 1752, the British government decided to consolidate all of its debt into one perpetuity that paid a 3 ½ percent coupon. The bond, which is known as a consol, still exists today except that its coupon has now changed to 2 ½ percent payable. Assuming that the discount rate on the bond is 5 percent per annum, what is the value of the bond today? Answer: Using Result 9.6, the value is PV = £50 = £2.50/.05 Result 9.6 If r is the discount rate per period, the present value of a perpetuity with payments of C each period commencing at date 1 is C/r.

14. Annuities and Perpetuities - continue Financial Markets and Corporate Strategy, David Hillier Annuities : Example 9.9: Computing Annuity Payments Flavio has just borrowed £100,000 from his rich professor to pay for his doctoral studies. He has promised to make payments each year for the next 30 years to his professor at an interest rate of 10 percent. What are his annual payments? Answer: The present value of the payments has to equal the amount of the loan, £100,000. If r is.10, equation (9.8) indicates that the present value of £1 paid annually for 30 years is £9.427. To obtain a present value equal to £100,000, Flavio must pay at the end of each of the next 30 years. Result 9.7 If r is the discount rate per period, the present value of an annuity with payments commencing at date 1 and ending at date T is

15. Growing Perpetuities and Annuities Financial Markets and Corporate Strategy, David Hillier Growing Perpetuities: Result 9.8 The value of a growing perpetuity with initial payment of C dollars one period from now is Growing Annuities :

16. Example 9.10: Valuing a Share of Equity Financial Markets and Corporate Strategy, David Hillier Agfa-Gevaert N.V. (Agfa) is an imaging technology firm that is listed on NYSE Euronext and is a component of the Bel-20 index, an index of the twenty largest Belgian firms. For the last four years, it paid a gross dividend of €0.50 per year. Assume that next year, Agfa will pay a dividend of €0.50 and that this will grow thereafter by 7 percent per annum forever. If the relevant discount rate is 10 percent, how much should you pay for Agfa equity? Answer: Using equation (9.11), the per share value is The actual price of Agfa-Gevaert N.V. at the end of October 2007 was €15.40.

17. Simple Interest: Time Horizons and Compounding Frequencies Financial Markets and Corporate Strategy, David Hillier Annualized Rates Equivalent Rates Exhibit 9.5 Translating Annualized Interest Rates with Different Compounding Frequencies into Interest Earned per Period Annualized Interest Rate Quotation Basis Interest per PeriodLength of a Period Annually compoundedR1 year Semiannually compoundedr /26 months Quarterly compoundedr /43 months Monthly compoundedr /121 month Weekly compoundedr /521 week Daily compoundedr /3651 Compounded m times a yearr /m1/m years

18. Example 9.11: Finding Equivalent Rates with Different Compounding Frequencies Financial Markets and Corporate Strategy, David Hillier An investment of £1.00 that grows to £1.10 at the end of one year is said to have a return of 10 percent, annually compounded. What are the equivalent semiannually and continuously compounded rates of growth for this investment? Answer: Its semiannually compounded rate is approximately 9.76 percent and its continuously compounded rate is approximately 9.53 percent. These are found respectively by solving the following equations for r and

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