Chapter 2 The Two Key Concepts in Finance

1. Chapter 2The Two Key Concepts in Finance

2. It’s what we learn after we think we know it all that counts. - Kin Hubbard

3. Outline • Introduction • Time value of money • Safe dollars and risky dollars • Relationship between risk and return

4. Introduction • The occasional reading of basic material in your chosen field is an excellent philosophical exercise • Do not be tempted to include that you “know it all” • E.g., what is the present value of a growing perpetuity that begins payments in five years

5. Time Value of Money • Introduction • Present and future values • Present and future value factors • Compounding • Growing income streams

6. Introduction • Time has a value • If we owe, we would prefer to pay money later • If we are owed, we would prefer to receive money sooner • The longer the term of a single-payment loan, the higher the amount the borrower must repay

7. Present and Future Values • Basic time value of money relationships:

8. Present and Future Values (cont’d) • A present value is the discounted value of one or more future cash flows • A future value is the compounded value of a present value • The discount factor is the present value of a dollar invested in the future • The compounding factor is the future value of a dollar invested today

9. Present and Future Values (cont’d) • Why is a dollar today worth more than a dollar tomorrow? • The discount factor: • Decreases as time increases • The farther away a cash flow is, the more we discount it • Decreases as interest rates increase • When interest rates are high, a dollar today is worth much more than that same dollar will be in the future

10. Present and Future Values (cont’d) • Situations: • Know the future value and the discount factor • Like solving for the theoretical price of a bond • Know the future value and present value • Like finding the yield to maturity on a bond • Know the present value and the discount rate • Like solving for an account balance in the future

11. Present and Future Value Factors • Single sum factors • How we get present and future value tables • Ordinary annuities and annuities due

12. Single Sum Factors • Present value interest factor and future value interest factor:

13. Single Sum Factors (cont’d) Example You just invested $2,000 in a three-year bank certificate of deposit (CD) with a 9 percent interest rate. How much will you receive at maturity?

14. Single Sum Factors (cont’d) Example (cont’d) Solution: Solve for the future value:

15. How We Get Present and Future Value Tables • Standard time value of money tables present factors for: • Present value of a single sum • Present value of an annuity • Future value of a single sum • Future value of an annuity

16. How We Get Present and Future Value Tables (cont’d) • Relationships: • You can use the present value of a single sum to obtain: • The present value of an annuity factor (a running total of the single sum factors) • The future value of a single sum factor (the inverse of the present value of a single sum factor)

17. Ordinary Annuities and Annuities Due • An annuity is a series of payments at equal time intervals • An ordinary annuity assumes the first payment occurs at the end of the first year • An annuity due assumes the first payment occurs at the beginning of the first year

18. Ordinary Annuities and Annuities Due (cont’d) Example You have just won the lottery! You will receive $1 million in ten installments of $100,000 each. You think you can invest the $1 million at an 8 percent interest rate. What is the present value of the $1 million if the first $100,000 payment occurs one year from today? What is the present value if the first payment occurs today?

19. Ordinary Annuities and Annuities Due (cont’d) Example (cont’d) Solution: These questions treat the cash flows as an ordinary annuity and an annuity due, respectively:

20. Compounding • Definition • Discrete versus continuous intervals • Nominal versus effective yields

21. Definition • Compounding refers to the frequency with which interest is computed and added to the principal balance • The more frequent the compounding, the higher the interest earned

22. Discrete Versus Continuous Intervals • Discrete compounding means we can count the number of compounding periods per year • E.g., once a year, twice a year, quarterly, monthly, or daily • Continuous compounding results when there is an infinite number of compounding periods

23. Discrete Versus Continuous Intervals (cont’d) • Mathematical adjustment for discrete compounding:

24. Discrete Versus Continuous Intervals (cont’d) • Mathematical equation for continuous compounding:

25. Discrete Versus Continuous Intervals (cont’d) Example Your bank pays you 3 percent per year on your savings account. You just deposited $100.00 in your savings account. What is the future value of the $100.00 in one year if interest is compounded quarterly? If interest is compounded continuously?

26. Discrete Versus Continuous Intervals (cont’d) Example (cont’d) Solution: For quarterly compounding:

27. Discrete Versus Continuous Intervals (cont’d) Example (cont’d) Solution (cont’d): For continuous compounding:

28. Nominal Versus Effective Yields • The stated rate of interest is the simple rate or nominal rate • 3.00% in the example • The interest rate that relates present and future values is the effective rate • $3.03/$100 = 3.03% for quarterly compounding • $3.05/$100 = 3.05% for continuous compounding

29. Growing Income Streams • Definition • Growing annuity • Growing perpetuity

30. Definition • A growing stream is one in which each successive cash flow is larger than the previous one • A common problem is one in which the cash flows grow by some fixed percentage

31. Growing Annuity • A growing annuity is an annuity in which the cash flows grow at a constant rate g:

32. Growing Perpetuity • A growing perpetuity is an annuity where the cash flows continue indefinitely:

33. Safe Dollars and Risky Dollars • Introduction • Choosing among risky alternatives • Defining risk

34. Introduction • A safe dollar is worth more than a risky dollar • Investing in the stock market is exchanging bird-in-the-hand safe dollars for a chance at a higher number of dollars in the future

35. Introduction (cont’d) • Most investors are risk averse • People will take a risk only if they expect to be adequately rewarded for taking it • People have different degrees of risk aversion • Some people are more willing to take a chance than others

36. Choosing Among Risky Alternatives Example You have won the right to spin a lottery wheel one time. The wheel contains numbers 1 through 100, and a pointer selects one number when the wheel stops. The payoff alternatives are on the next slide. Which alternative would you choose?

37. Choosing Among Risky Alternatives (cont’d)

38. Choosing Among Risky Alternatives (cont’d) Example (cont’d) Solution: • Most people would think Choice A is “safe.” • Choice B has an opportunity cost of $90 relative to Choice A. • People who get utility from playing a game pick Choice C. • People who cannot tolerate the chance of any loss would avoid Choice D.

39. Choosing Among Risky Alternatives (cont’d) Example (cont’d) Solution (cont’d): • Choice A is like buying shares of a utility stock. • Choice B is like purchasing a stock option. • Choice C is like a convertible bond. • Choice D is like writing out-of-the-money call options.

40. Defining Risk • Risk versus uncertainty • Dispersion and chance of loss • Types of risk

41. Risk Versus Uncertainty • Uncertainty involves a doubtful outcome • What you will get for your birthday • If a particular horse will win at the track • Risk involves the chance of loss • If a particular horse will win at the track if you made a bet

42. Dispersion and Chance of Loss • There are two material factors we use in judging risk: • The average outcome • The scattering of the other possibilities around the average

43. Dispersion and Chance of Loss (cont’d) Investment value Investment A Investment B Time

44. Dispersion and Chance of Loss (cont’d) • Investments A and B have the same arithmetic mean • Investment B is riskier than Investment A

45. Types of Risk • Total risk refers to the overall variability of the returns of financial assets • Undiversifiable risk is risk that must be borne by virtue of being in the market • Arises from systematic factors that affect all securities of a particular type

46. Types of Risk (cont’d) • Diversifiable risk can be removed by proper portfolio diversification • The ups and down of individual securities due to company-specific events will cancel each other out • The only return variability that remains will be due to economic events affecting all stocks

47. Relationship Between Risk and Return • Direct relationship • Concept of utility • Diminishing marginal utility of money • St. Petersburg paradox • Fair bets • The consumption decision • Other considerations

48. Direct Relationship • The more risk someone bears, the higher the expected return • The appropriate discount rate depends on the risk level of the investment • The risk-less rate of interest can be earned without bearing any risk

49. Direct Relationship (cont’d) Expected return Rf 0 Risk

50. Direct Relationship (cont’d) • The expected return is the weighted average of all possible returns • The weights reflect the relative likelihood of each possible return • The risk is undiversifiable risk • A person is not rewarded for bearing risk that could have been diversified away