1 Complete Markets

2 Definitions Event State of the world State Contingent Claim (State Claim)  Payoff Vector  Market is a payoff vector Exchange dollars today for state-contingent bundle of dollars tomorrow Markets are complete  If we can arrange a portfolio with any payoff vector

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4 Uncertainty Market complete? Interest rate? Probability of War?

5 example If I know what pure securities pay TOMORROW ($1 in only one state - e.g. "u" or "d") and I know their prices TODAY (p_u and p_d) then I can figure out the price TODAY of any security generating payoffs (cash flows) TOMORROW. In the example you refer to, we work `backwards'. We know the price TODAY (V_1 = 1) of a security that pays (TOMORROW) 1.5 in the "u" state and 0.5 in the "d" state AND we know the price TODAY of the risk-free bond (b = 1) that pays 1 in BOTH states TOMORROW (that's why it is risk-free - it doesn't matter which state prevails) - note that since b = 1 TODAY, the risk-free rate of interest is 0. Knowing these 2 prices allows us to compute the prices of the pure securities TODAY: p_u = 0.5 and p_d = 0.5. Now we can determine the price TODAY of ANY other security in this world - e.g.: a security that pays (TOMORROW) 0.5 in "u" and 0 in "d," must have a price of 0.25 TODAY... TODAY, in this example, simply means some time before the state (here "u" or "d") is revealed at some later time (perhaps only an instant later) - here called TOMORROW.

6 Financial Decision Making Market prices determine value  Competitive markets  One-sided markets

7 Time Value of Money $1 today is worth more than $1 tomorrow Interest rate is the exchange rate across time $1 in your pocket is worth more than $1 promised  Which is worth more than $1 expected  Which is worth more than $1 hoped for Risk-free rates PV NPV NPV + Borrowing or Lending

8 Time Value of Money Interest rate is the exchange rate across time

9 Time Value of Money PV, NPV

10 Time Value of Money NPV + Borrowing and Lending

11 Arbitrage  Certain profit by exploiting different pricing for the same asset Law of one price  An asset has the same price in all exchanges No-arbitrage and security pricing  Bond $1000, 1 year, 5% What if over-priced or under-priced? Determine interest rates from bond prices  Other securities

12 Separation Principle Security transactions in a normal market do not create nor destroy value This allows us to only focus on the NPV of the project  And not worry about the financing choice Example:  Cost today: $10M  Benefit in 1 year: $12M  Risk-free rate: 10%  Ability to issue $5.5M security today  Does the issuance matter?

13 Portfolio Valuation Value additivity  Price of a portfolio is the sum of the prices of individual securities A firm is a portfolio of projects  The value of the firm is the sum of the values of all projects Maximizing NPV for each decision maximizes the value of the firm

14 Price of Risk $1 in your pocket is worth more than $1 promised  Which is worth more than $1 expected  Which is worth more than $1 hoped for

15 Risk Premium Expected return Risk premium No-arbitrage pricing of a risky security

16 Risk Premiums Depends on risk  Riskier securities command higher risk premium Risk is relative to the overall market  Risk premium can be negative

17 Risk Premiums Risk premium depends on risk: r s = r f + (risk premium for investment s)

18 Arbitrage and Transaction Costs Two types of costs:  Commissions  Bid-ask spreads No arbitrage conditions hold “up to transaction costs”