Complete Markets
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Definitions
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Event
State of the world
State Contingent Claim (State Claim)
 Payoff Vector
 Market is a payoff vector
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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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Uncertainty
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Market complete?
Interest rate?
Probability of War?
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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.
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Financial Decision Making
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Market prices determine value
 Competitive markets
 One-sided markets
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Time Value of Money
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$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
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Time Value of Money
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Interest rate is the exchange rate across time
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Time Value of Money
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PV, NPV
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Time Value of Money
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NPV + Borrowing and Lending
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Arbitrage
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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
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$1000, 1 year, 5%
What if over-priced or under-priced?
Determine interest rates from bond prices
Other securities
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Separation Principle
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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?
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Portfolio Valuation
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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
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Price of Risk
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$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
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Risk Premium
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Expected return
Risk premium
No-arbitrage pricing of a risky security
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Risk Premiums
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Depends on risk
 Riskier securities command higher risk premium
Risk is relative to the overall market
 Risk premium can be negative
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Risk Premiums
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Risk premium depends on risk:
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rs = rf + (risk premium for investment s)
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Arbitrage and Transaction Costs
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Two types of costs:
 Commissions
 Bid-ask spreads
No arbitrage conditions hold “up to transaction costs”
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