Present Value: Calculations and Interpretation

1. Present Value: Calculations and Interpretation Classes 3 & 4: March 5 and 7 (LA) and March 1 and 6 (OCC)

2. From last classes . . . • What should be the goal of financial managers? • What do we need to know to pursue goal? • How can we assess progress towards that goal? • What is a firm’s market value? Market cap? How do we compute them?

3. Overview: Classes 3 to 6 • Discounted present value: basic tool given projections of cash flows and discount rate • Present value and wealth creation • One and multi-period cash flows • Patterns in cash flows = formulas • Applications to valuation: bonds • Application to valuation: stocks • To be addressed later: projecting cash flows, choosing a discount rate (Class 3 & 4) (Class 5 & 6)

4. Determinants of Value • Cash, Time, Riskdetermine value • Present value analysis deals with the effect of time or timing on value • Cash flow estimation is the subject of the next part of the course (classes 5 to 8) • Risk is incorporated in the discount ratethat we discuss in Part 3 of the course • In discussing present value analysis now, we assume that cash flows and discount rates are given

5. Emphasis on Present Values • Chapter 4 raises a number of topics relevant to the calculation of present values: • Simple versus compound interest • Compounding interval • Continuous compounding • Future values • Calculation of number of periods of cash flows to achieve a given present or future value • We will not emphasize these issues, we concentrate on basic present value calculations

6. Present Value of Cash Flows • Calculation of present values is key technique to assign values • Present value calculations are applications or simplications of two basic formulas: PV of single cash flow = PV of multiple cash flows =

7. Calculation of Present Values

8. Examples / Applications • U. S. Treasury strip prices are examples of market determined discount factors for default-risk free cash flows • The structure of present value tables like those in the text (A.1 and A.2) are very straightforward • Time in discounting in in terms of periods, usually one year, but often shorter intervals • Compounding interval will affect present or future values

9. Present Value Calculations • Present values can be calculated using present value tables and paper, calculators and paper, routines programmed into calculators, and spreadsheets • All correct methods produce the same answers • There is often more than one way to calculate the answers using formulas or individual cash flows but, if correct, they are all mathematically equivalent

10. Example of Three Approaches • Present value of $1000 received at the end of each year for five years discounted at 10% • Three (at least) ways produce same answer: (Using Appendix Table A.1) (Using Appendix Table A.2) (Using Perpetuity formula and Appendix Table A.1 discussed later)

11. Characteristics of Present Value • Present value calculations are non-linear in the discount rate and growth rates, means changes in present values are not proportional to changes in the discount rate • Changes in timing or patterns of growth must always be calculated, relying on intuition is dangerous • Terminology may be confusing: discount rate, discount factor, interest rate, cost of capital, opportunity cost, and yield all can mean the same thing in a calculation

12. Example of Dangers • Change discount rate in previous example to 20% from 10%, PV becomes $2,991, reduced to 78.9% of $3,791 at 10%, not half. • Change times to $1,000 for ten years at 10%, PV becomes $6,146, not double. • Delay first cash flow by one year, PV reduced by about 10%, or if by three years, PV reduced by about 25%, difference between delay of one or three years is not three times greater.

13. Meaning of Present Value and Equality of Present Values • Present Value of $1,000 for five years at 10 percent (Table A.2) • $3,790.80 is equivalent to $1,000 at the end of every year for five years at 10 percent • Future value of $3,790.80 at end of five years is $3,790.80x(1.10)5=$6,105.12 • This is also future value of $1,000 for five years at 10 percent (see Table A.4)

14. Equivalence of Present Valueto Annual Cash Flows

15. Example of Future Value

16. Summary of PV/FV Examples • Present value is the amount that can replicatecash flows if discount rate is the future interest rate • Maximizing present values also maximizes future values if interest rates do not change (in this case, they are equivalent) • Present values and future values of different patterns of cash flows will differ from calculations using constant discount rate if interest-rates vary through time

17. Net Present Value • Net present value (NPV) is the difference between the present value of the future cash flows and the cost of acquiring the cash flows • In most examples, costs are immediate and are not discounted, while cash flows are in the future and must be discounted • More generally, costs and benefits may both be discounted if some costs occur in the future • Net present value is a measure of how much more something is worth than it costs, or a wealth increase, as we discuss and illustrate later

18. Positive Net Present Values • A positive net present value means that future cash flows represent earnings higher than the discount rate • Net present value represents the excess returns (returns above the discount or opportunity rate) represented by the future cash flows • Net present values represent value added relative to the opportunity rate

19. Seek Simplifying Patterns in Cash Flows for Short-cuts • Can always evaluate individual annual cash flows but this is cumbersome • Simplest pattern is constant cash flow each year -- • First formula to memorize is Cash flow time

20. Useful Present Value Formulas • Perpetuity: • Growing Perpetuity: • Annuity: • Growing Annuity:

21. Simple Patterns in Cash Flows • Perpetuity = Preferred dividend • Growing perpetuity = Approximate cash flows from new products or stock earnings • Annuity = Retirement fund or car or mortgage loan payments • Growing annuity = Approximate cash flows from investment with limited life or lifetime earnings

22. Graphical Representations • Perpetuity: • Growing Perpetuity: Cash Flow 0 Time Cash Flow 0 Time

23. Graphical Representations • Annuity: • Growing Annuity: Cash Flow 0 T Time Cash Flow 0 T Time

24. Sources of Present Values • Present value of $1 perpetuity at 20% is $5 • Present value of $1 annuity for five years at 20% is $2.99 • Therefore, present values of $1 from years six to infinity at 20% is $5 minus $2.99 = $2.01 (less than half of $5) • Present value of perpetuity growing at 10% starting at $1 and at 20% is $10 • Growing over infinite life is valued at $10 minus $5 or $5

25. Graphical Presentation of Four Present Value Formulas E D C Cash Flow A B T time 0

26. Graphical representation of the four important formulas • Areas in graph represent parts of future cash flows - Perpetuity = A+B • Growing Perpetuity = A+B+C+D+E • Annuity = A • Growing Annuity = A+C • You can solve for value added by a piece of cash flows, for example cash flows after T, by subtracting A from A+B

27. Example: $1 growing at 10% Discounted at 20% PV = $ 10.00 E = $ 3.23 D = 1.23 C = $ .54 $ 1 A =$ 2.99 B = $ 2.01 0 5

28. Present Value and Net PV (NPV) • Present values are calculations assuming expected cash flows and required discount rates • Each may differ for different analysts • Knowledge and skill about future cash flows • Assessment of risk and alternative investments • Net present value = Present value - cost • Contrast present value with intrinsic value, market value, under-valued and over-valued

29. Use of Present Value Formulas • Familiarity with PV formulas important • For example, what is future value of constant annual cash flow? Using annuityobtaining (see. p. 840) • Relations between present value formulas are really simple

30. Using PV Formulas to Find Rates • You can solve for r given PV, in simplest case of perpetuity r = C / PV • With a value for g and PV in growth formula, find r also easy and common in stock analysis (we will use later) • With annuities and other formulas you can also solve for r although the equations are non-linear requiring searches

31. Present Value and Wealth • Wealth = Present value of consumption • Wealth = Present value of cash income • DWealth = Change in value of consumption = Change in present value of cash income • DWealth => Increase in utility from consumption • DWealth = Net present value • Net present value > 0 => Wealth increased

32. Present Value and MVA/EVA (I) • Market value added is how much more assets are worth than they cost • MVA is in part the present value of returns above the opportunity rate on investments thus represents management’s ability to find investments better than alternatives • EVA represents the returns above the opportunity rate and is a measure of management’s superior investment strategy

33. Present Value and MVA/EVA (II) • Market values represent present value of expected future cash flows • If market value is above acquisition cost (MVA), management is expect to produce cash flows are above opportunity rate levels • Excess returns (EVA) can be from existing investments and future growth opportunities or growth options

34. Present Value Summary • Present values represent cash amounts that can reproduce a pattern of cash flows in the future given the discount rate • Two equal present values can represent different patterns of future cash flows • Future values and present values are equivalent measures of value given the discount rate • Net present values are measures of the increase in wealth representing increased utility from increases in present and future consumption

35. Present Value Analysis: Review • Objectives • Vocabulary • Problem Assignments • Relation to syllabus and requirements

36. Basic Steps to Valuation in Finance • Estimate cash flows (CASH, TIME) • Easy or hard depending on asset • Look for patterns in cash flows • Choose a discount rate (TIME, RISK) • Risk adjusted • Opportunity cost • Calculate present value and net present value

37. Valuation in Finance • Applies to all investment opportunities, including • investments in fixed plant and equipment • starting a new business • selling a line of business (spin-off) • buying an existing business • values of bonds and stocks • real estate investments • Used by financial managers, stock and bond analysts, real estate investors

38. For Next Classes • Read Chapter 5, 14 and 20 • Do problems as assigned • Download or call or write for annual report, 10K, and proxy statement, and any other disclosures, for the group project firm • Bring Value Line Investment Survey and Standard and Poor’s reports for the company to class • Look for analysts’ reports and press coverage of the group firm