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Chapter 2 The Two Key Concepts in Finance. It’s what we learn after we think we know it all that counts. - Kin Hubbard. Outline. Introduction Time value of money Safe dollars and risky dollars Relationship between risk and return. Introduction.
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It’s what we learn after we think we know it all that counts. - Kin Hubbard
Outline • Introduction • Time value of money • Safe dollars and risky dollars • Relationship between risk and return
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
Time Value of Money • Introduction • Present and future values • Present and future value factors • Compounding • Growing income streams
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
Present and Future Values • Basic time value of money relationships:
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
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
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
Present and Future Value Factors • Single sum factors • How we get present and future value tables • Ordinary annuities and annuities due
Single Sum Factors • Present value interest factor and future value interest factor:
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?
Single Sum Factors (cont’d) Example (cont’d) Solution: Solve for the future value:
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
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)
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
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?
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:
Compounding • Definition • Discrete versus continuous intervals • Nominal versus effective yields
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
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
Discrete Versus Continuous Intervals (cont’d) • Mathematical adjustment for discrete compounding:
Discrete Versus Continuous Intervals (cont’d) • Mathematical equation for continuous compounding:
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?
Discrete Versus Continuous Intervals (cont’d) Example (cont’d) Solution: For quarterly compounding:
Discrete Versus Continuous Intervals (cont’d) Example (cont’d) Solution (cont’d): For continuous compounding:
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
Growing Income Streams • Definition • Growing annuity • Growing perpetuity
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
Growing Annuity • A growing annuity is an annuity in which the cash flows grow at a constant rate g:
Growing Perpetuity • A growing perpetuity is an annuity where the cash flows continue indefinitely:
Safe Dollars and Risky Dollars • Introduction • Choosing among risky alternatives • Defining risk
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
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
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?
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.
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.
Defining Risk • Risk versus uncertainty • Dispersion and chance of loss • Types of risk
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
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
Dispersion and Chance of Loss (cont’d) Investment value Investment A Investment B Time
Dispersion and Chance of Loss (cont’d) • Investments A and B have the same arithmetic mean • Investment B is riskier than Investment A
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
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
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
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
Direct Relationship (cont’d) Expected return Rf 0 Risk
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