1. THE TIME VALUE OF MONEY Aswath Damodaran

2. 2 Intuition Behind Present Value  There are three reasons why a dollar tomorrow is worth less than a dollar today  Individuals prefer present consumption to future consumption. To induce people to give up present consumption you have to offer them more in the future.  When there is monetary inflation, the value of currency decreases over time. The greater the inflation, the greater the difference in value between a dollar today and a dollar tomorrow.  If there is any uncertainty (risk) associated with the cash flow in the future, the less that cash flow will be valued.  Other things remaining equal, the value of cash flows in future time periods will decrease as  the preference for current consumption increases.  expected inflation increases.  the uncertainty in the cash flow increases.

3. 3 Discounting and Compounding  The mechanism for factoring in these elements is the discount rate. The discount rate is a rate at which present and future cash flows are traded off. It incorporates (1) Preference for current consumption (Greater....Higher Discount Rate) (2) Expected inflation(Higher inflation....Higher Discount Rate) (3) Uncertainty in the future cash flows (Higher Risk....Higher Discount Rate)  A higher discount rate will lead to a lower value for cash flows in the future.  The discount rate is also an opportunity cost, since it captures the returns that an individual would have made on the next best opportunity.  Discounting future cash flows converts them into cash flows in present value dollars. Just a discounting converts future cash flows into present cash flows,  Compounding converts present cash flows into future cash flows.

4. 4 Present Value Principle 1  Cash flows at different points in time cannot be compared and aggregated.  All cash flows have to be brought to the same point in time, before comparisons and aggregations are made.  That point of time can be today (present value) or a point in time in the future (future value).

5. 5 Time lines for cash flows  The best way to visualize cash flows is on a time line, where you list out how much you get and when.  In a time line, today is specified as “time 0” and each year is shown as a period.

6. 6 Cash Flow Types and Discounting Mechanics  There are five types of cash flows -  simple cash flows,  annuities,  growing annuities  perpetuities and  growing perpetuities  Most assets represent combinations of these cash flows. Thus, a conventional bond is a combination of an annuity (coupons) and a simple cash flow (face value at maturity). A stock may be a combination of a growing annuity and a growing perpetuity.

7. 7 I.Simple Cash Flows  A simple cash flow is a single cash flow in a specified future time period. Cash Flow:CF t ______________________________________________ _| Time Period:t  The present value of this cash flow is PV of Simple Cash Flow = CF t / (1+r) t  The future value of a cash flow is FV of Simple Cash Flow = CF 0 (1+ r) t

8. 8 Application: The power of compounding - Stocks, Bonds and Bills  Between 1926 and 2013, stocks on the average made about 9.55% a year, while government bonds on average made about 4.93% a year and T.Bills earned 3.53% a year.  If your holding period is one year, the difference in end-of-period values is small:  Value of $ 100 invested in stocks in one year = $ 109.55  Value of $ 100 invested in bonds in one year = $ 104.93  Value of $100 invested in T.Bills for one year= $103.53

9. 9 Holding Period and Value

10. 10 Concept Check  Most pension plans allow individuals to decide where their pensions funds will be invested - stocks, bonds or money market accounts.  Where would you choose to invest your pension funds? a. Predominantly or all equity b. Predominantly or all bonds and money market accounts c. A Mix of Bonds and Stocks  Will your allocation change as you get older? a. Yes b. No

11. 11 The Frequency of Compounding  The frequency of compounding affects the future and present values of cash flows. The stated interest rate can deviate significantly from the true interest rate –  For instance, a 10% annual interest rate, if there is semiannual compounding, works out to- Effective Interest Rate = 1.05 2 - 1 =.10125 or 10.25% Frequency RatetFormulaEffective Annual Rate Annual10%1r10.00% Semi-Annual10%2(1+r/2) 2 -110.25% Monthly10%12(1+r/12) 12 -110.47% Daily10%365(1+r/365) 365 -110.5156% Continuous10%exp r -110.5171%

12. 12 II. Annuities  An annuity is a constant cash flow that occurs at regular intervals for a fixed period of time. Defining A to be the annuity, the time line looks as follows: AAAA |||| 01234

13. 13 Present Value of an Annuity  The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present, and adding up the present values. Alternatively, there is a short cut that can be used in the calculation [A = Annuity; r = Discount Rate; n = Number of years]

14. 14 Example: PV of an Annuity  The present value of an annuity of $1,000 at the end of each year for the next five years, assuming a discount rate of 10% is -  The notation that will be used in the rest of these lecture notes for the present value of an annuity will be PV(A,r,n).

15. 15 Annuity, given Present Value  The reverse of this problem, is when the present value is known and the annuity is to be estimated - A(PV,r,n).  This, for instance, is the equation you would use to determine your monthly payments on a home mortgage.

16. 16 Computing Monthly Payment on a Mortgage  Suppose you borrow $200,000 to buy a house on a 30-year mortgage with monthly payments. The annual percentage rate on the loan is 8%. The monthly payments on this loan, with the payments occurring at the end of each month, can be calculated using this equation:  Monthly interest rate on loan = APR/ 12 = 0.08/12 = 0.0067

17. 17 Future Value of an Annuity  The future value of an end-of-the-period annuity can also be calculated as follows-  This is the equation you would use to determine how much money you will accumulate at a future point in time if you set aside a constant amount each period.

18. 18 An Example  Thus, the future value of $1,000 at the end of each year for the next five years, at the end of the fifth year is (assuming a 10% discount rate) -  The notation that will be used for the future value of an annuity will be FV(A,r,n).

19. 19 Annuity, given Future Value  if you are given the future value and you are looking for an annuity - A(FV,r,n) in terms of notation -

20. 20 Application : Saving for College Tuition  Assume that you want to send your newborn child to a private college (when he gets to be 18 years old). The tuition costs are $ 16000/year now and that these costs are expected to rise 5% a year for the next 18 years. Assume that you can invest, after taxes, at 8%.  Expected tuition cost/year 18 years from now = 16000*(1.05) 18 = $38,506  PV of four years of tuition costs at $38,506/year = $38,506 * PV(A,8%,4 years)= $127,537  If you need to set aside a lump sum now, the amount you would need to set aside would be -  Amount one needs to set apart now = $127,357/(1.08) 18 = $31,916  If set aside as an annuity each year, starting one year from now -  If set apart as an annuity = $127,537 * A(FV,8%,18 years) = $3,405

21. 21 Application : How much is an MBA worth?  Assume that you were earning $40,000/year before entering program and that tuition costs are $16000/year. Expected salary is $ 54,000/year after graduation. You can invest money at 8%. For simplicity, assume that the first payment of $16,000 has to be made at the start of the program and the second payment one year later.  PV Of Cost Of MBA = $16,000+16,000/1.08 + 40000 * PV(A,8%,2 years) = $102,145  Assume that you will work 30 years after graduation, and that the salary differential ($14000 = $54000-$40000) will continue through this period.  PV of Benefits Before Taxes = $14,000 * PV(A,8%,30 years) = $157,609  This has to be discounted back two years - $157,609/1.08 2 = $135,124  The present value of getting an MBA is = $135,124 - $102,145 = $32,979 1. How much would your salary increment have to be for you to break even on your MBA? 2. Keeping the increment constant, how many years would you have to work to break even?

22. 22 Application: Savings from Refinancing Your Mortgage  Assume that you have a thirty-year mortgage for $200,000 that carries an interest rate of 9.00%. The mortgage was taken three years ago. Since then, assume that interest rates have come down to 7.50%, and that you are thinking of refinancing. The cost of refinancing is expected to be 2.50% of the loan. (This cost includes the points on the loan.) Assume also that you can invest your funds at 6%. Monthly payment based upon 9% mortgage rate (0.75% monthly rate) = $200,000 * A(PV,0.75%,360 months) = $1,609 Monthly payment based upon 7.50% mortgage rate (0.625% monthly rate) = $200,000 * A(PV,0.625%,360 months) = $1,398  Monthly Savings from refinancing = $1,609 - $1,398 = $211

23. 23 Refinancing: The Trade Off  If you plan to remain in this house indefinitely, Present Value of Savings (at 6% annually; 0.5% a month) = $211 * PV(A,0.5%,324 months) = $33,815 The savings will last for 27 years - the remaining life of the existing mortgage. You will need to make payments for three additional years as a consequence of the refinancing - Present Value of Additional Mortgage payments - years 28,29 and 30 = $1,398 * PV(A,0.5%,36 months)/1.06 27 = $9,532  Refinancing Cost = 2.5% of $200,000 = $5,000  Total Refinancing Cost = $9,532 + $5,000 = $14,532  Net Effect = $ 33,815 - $ 14,532 = $ 19,283: Refinance

24. 24 Follow-up Questions 1. How many years would you have to live in this house for you break even on this refinancing? 2. We've ignored taxes in this analysis. How would it impact your decision?

25. 25 Valuing a Straight Bond  You are trying to value a straight bond with a fifteen year maturity and a 10.75% coupon rate. The current interest rate on bonds of this risk level is 8.5%. PV of cash flows on bond = 107.50* PV(A,8.5%,15 years) + 1000/1.085 15 = $ 1186.85  If interest rates rise to 10%, PV of cash flows on bond = 107.50* PV(A,10%,15 years)+ 1000/1.10 15 = $1,057.05 Percentage change in price = -10.94%  If interest rate fall to 7%, PV of cash flows on bond = 107.50* PV(A,7%,15 years)+ 1000/1.07 15 = $1,341.55 Percentage change in price = +13.03%  This asymmetric response to interest rate changes is called convexity.

26. 26 Bond Pricing Proposition 1  The longer the maturity of a bond, the more sensitive it is to changes in interest rates.

27. 27 Bond Pricing Proposition 2  The lower the coupon rate on the bond, the more sensitive it is to changes in interest rates.

28. 28 III. Growing Annuity  A growing annuity is a cash flow growing at a constant rate for a specified period of time. If A is the current cash flow, and g is the expected growth rate, the time line for a growing annuity looks as follows –

29. 29 Present Value of a Growing Annuity  The present value of a growing annuity can be estimated in all cases, but one - where the growth rate is equal to the discount rate, using the following model:  In that specific case, the present value is equal to the nominal sums of the annuities over the period, without the growth effect.

30. 30 The Value of a Gold Mine  Consider the example of a gold mine, where you have the rights to the mine for the next 20 years, over which period you plan to extract 5,000 ounces of gold every year. The price per ounce is $300 currently, but it is expected to increase 3% a year. The appropriate discount rate is 10%. The present value of the gold that will be extracted from this mine can be estimated as follows –

31. 31 PV of Extracted Gold as a Function of Expected Growth Rate

32. 32 IV. Perpetuity  A perpetuity is a constant cash flow at regular intervals forever. The present value of a perpetuity is-  Forever may be a tough concept for human beings to grasp, but it makes the mathematics much simpler.

33. 33 Valuing a Console Bond  A console bond is a bond that has no maturity and pays a fixed coupon. Assume that you have a 6% coupon console bond. The value of this bond, if the interest rate is 9%, is as follows - Value of Console Bond = $60 /.09 = $667

34. 34 V. Growing Perpetuities  A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity is - where  CF 1 is the expected cash flow next year,  g is the constant growth rate and  r is the discount rate.

35. 35 Valuing a Stock with Growing Dividends  In twelve months leading into January 2014, Con Ed paid dividends per share of $2.52.  Its earnings and dividends had grown at 2% a year between 2004 and 2013 and were expected to grow at the same rate in the long run.  The rate of return required by investors on stocks of equivalent risk was 7.50%.  With these inputs, we can value the stock using a perpetual growth model: Value of Stock = $2.52 (1.02)/(0.075  0.02) = $46.73

36. 36 Value and Growth!

37. FINANCIAL STATEMENT ANALYSIS

38. 38 Questions we would like answered…

39. 39 Basic Financial Statements  The balance sheet, which summarizes what a firm owns and owes at a point in time.  The income statement, which reports on how much a firm earned in the period of analysis  The statement of cash flows, which reports on cash inflows and outflows to the firm during the period of analysis

40. 40 The Balance Sheet

41. 41 A Financial Balance Sheet

42. 42 The Income Statement

43. 43 Modifications to Income Statement  There are a few expenses that consistently are miscategorized in financial statements.In particular,  Operating leases are considered as operating expenses by accountants but they are really financial expenses  R &D expenses are considered as operating expenses by accountants but they are really capital expenses.  The degree of discretion granted to firms on revenue recognition and extraordinary items is used to manage earnings and provide misleading pictures of profitability.

44. 44 Dealing with Operating Lease Expenses  Debt Value of Operating Leases = PV of Operating Lease Expenses at the pre-tax cost of debt  This now creates an asset - the value of which is equal to the debt value of operating leases. This asset now has to be depreciated over time.  Finally, the operating earnings has to be adjusted to reflect these changes:  Adjusted Operating Earnings = Operating Earnings + Operating Lease Expense - Depreciation on the leased asset  If we assume that depreciation = principal payment on the debt value of operating leases, we can use a short cut: Adjusted Operating Earnings = Operating Earnings + Debt value of Operating leases * Cost of debt

45. 45 Operating Leases at Boeing and The Home Depot in 1998

46. 46 Imputed Interest Expenses on Operating Leases

47. 47 The Effects of Capitalizing Operating Leases  Debt : will increase, leading to an increase in debt ratios used in the cost of capital and levered beta calculation  Operating income: will increase, since operating leases will now be before the imputed interest on the operating lease expense  Net income: will be unaffected since it is after both operating and financial expenses anyway  Return on Capital will generally decrease since the increase in operating income will be proportionately lower than the increase in book capital invested

48. 48 R&D Expenses: Operating or Capital Expenses  Accounting standards require us to consider R&D as an operating expense even though it is designed to generate future growth. It is more logical to treat it as capital expenditures.  To capitalize R&D,  Specify an amortizable life for R&D (2 - 10 years)  Collect past R&D expenses for as long as the amortizable life  Sum up the unamortized R&D over the period. (Thus, if the amortizable life is 5 years, the research asset can be obtained by adding up 1/5th of the R&D expense from five years ago, 2/5th of the R&D expense from four years ago...:

49. 49 Capitalizing R&D Expenses: Boeing

50. 50 Boeing ’ s Corrected Operating Income

51. 51 The Effect of Capitalizing R&D  Operating Income will generally increase, though it depends upon whether R&D is growing or not. If it is flat, there will be no effect since the amortization will offset the R&D added back. The faster R&D is growing the more operating income will be affected.  Net income will increase proportionately, depending again upon how fast R&D is growing  Book value of equity (and capital) will increase by the capitalized Research asset  Capital expenditures will increase by the amount of R&D; Depreciation will increase by the amortization of the research asset; For all firms, the net cap ex will increase by the same amount as the after-tax operating income.

52. 52 The Statement of Cash Flows

53. 53 The Financial perspective on cash flows  In financial analysis, we are much more concerned about  Cash flows to the firm or operating cash flows, which are before cash flows to debt and equity)  Cash flows to equity, which are after cash flows to debt but prior to cash flows to equity

54. FUNDAMENTALS OF VALUATION

55. 55 Discounted Cashflow Valuation: Basis for Approach  where,  n = Life of the asset  CF t = Cashflow in period t  r = Discount rate reflecting the riskiness of the estimated cashflows

56. 56 Two Measures of Cash Flows  Cash flows to Equity: Thesea are the cash flows generated by the asset after all expenses and taxes, and also after payments due on the debt. This cash flow, which is after debt payments, operating expenses and taxes, is called the cash flow to equity investors.  Cash flow to Firm: There is also a broader definition of cash flow that we can use, where we look at not just the equity investor in the asset, but at the total cash flows generated by the asset for both the equity investor and the lender. This cash flow, which is before debt payments but after operating expenses and taxes, is called the cash flow to the firm

57. 57 Two Measures of Discount Rates  Cost of Equity: This is the rate of return required by equity investors on an investment. It will incorporate a premium for equity risk -the greater the risk, the greater the premium.  Cost of capital: This is a composite cost of all of the capital invested in an asset or business. It will be a weighted average of the cost of equity and the after- tax cost of borrowing.

58. 58 Equity Valuation

59. 59 Valuing Equity in a Finite Life Asset  Assume that you are trying to value the Home Depot ’ s equity investment in a new store.  Assume that the cash flows from the store after debt payments and reinvestment needs are expected will be $ 850,000 a year, growing at 5% a year for the next 12 years.  In addition, assume that the salvage value of the store, after repaying remaining debt will be $ 1 million.  Finally, assume that the cost of equity is 9.78%.

60. 60 Firm Valuation

61. 61 Valuing a Finite-Life Asset  Consider the Home Depot's investment in a proposed store. The store is assumed to have a finite life of 12 years and is expected to have cash flows before debt payments and after reinvestment needs of $ 1 million, growing at 5% a year for the next 12 years.  The store is also expected to have a value of $ 2.5 million at the end of the 12 th year (called the salvage value).  The Home Depot's cost of capital is 9.51%.

62. 62 Expected Cash Flows and present value

63. 63 Valuation with Infinite Life

64. 64 Valuing the Home Depot ’ s Equity  Assume that we expect the free cash flows to equity at th Home Depot to grow for the next 10 years at rates much higher than the growth rate for the economy. To estimate the free cash flows to equity for the next 10 years, we make the following assumptions:  The net income of $1,614 million will grow 15% a year each year for the next 10 years.  The firm will reinvest 75% of the net income back into new investments each year, and its net debt issued each year will be 10% of the reinvestment.  To estimate the terminal price, we assume that net income will grow 6% a year forever after year 10. Since lower growth will require less reinvestment, we will assume that the reinvestment rate after year 10 will be 40% of net income; net debt issued will remain 10% of reinvestment.

65. 65 Estimating cash flows to equity: The Home Depot

66. 66 Terminal Value and Value of Equity today  FCFE 11 = Net Income 11 – Reinvestment 11 – Net Debt Paid (Issued) 11 = $6,530 (1.06) – $6,530 (1.06) (0.40) – (-277) = $ 4,430 million  Terminal Price 10 = FCFE 11 /(k e – g) = $ 4,430 / (.0978 -.06) = $117,186 million  The value per share today can be computed as the sum of the present values of the free cash flows to equity during the next 10 years and the present value of the terminal value at the end of the 10 th year. Value of the Stock today = $ 6,833 million + $ 117,186/(1.0978) 10 = $52,927 million

67. 67 Valuing Boeing as a firm  Assume that you are valuing Boeing as a firm, and that Boeing has cash flows before debt payments but after reinvestment needs and taxes of $ 850 million in the current year.  Assume that these cash flows will grow at 15% a year for the next 5 years and at 5% thereafter.  Boeing has a cost of capital of 9.17%.

68. 68 Expected Cash Flows and Firm Value  Terminal Value = $ 1710 (1.05)/(.0917-.05) = $ 43,049 million YearCash FlowTerminal ValuePresent Value 1$978$895 2$1,124$943 3$1,293$994 4$1,487$1,047 5$1,710$43,049$28,864 Value of Boeing as a firm =$32,743